Geophysical Field Theory and Method: Gravitational, Electric, and Magnetic Fields

Geophysical FieldTheory and MethodPart A

This is Volume 49, Part A in theINTERNATIONAL GEOPHYSICS SERIESA series of monographs and textbooksEdited by RENATA pMOWSKA and JAMES R. HOLTON

A complete list of the books in this series appears at the end of this volume.

Geophysical FieldTheory and MethodPart A

Gravitational, Electric, and Magnetic Fields

Alexander A. Kaufman.DEPARTMENT OF GEOPHYSICS

COLORADO SCHOOL OF MINES

GOLDEN, COLORADO

ACADEMIC PRESS, INC.Harcourt Brace Jovanovich, Publishers

San Diego New York BostonLondon Sydney Tokyo Toronto

Front cover photograph: Apollo 16 Earth view. Courtesy of © NASA.

This book is printed on acid-free paper. e

Copyright © 1992 by ACADEMIC PRESS, INC.All Rights Reserved.No part of this publication may be reproduced or transmitted in any form or by anymeans, electronic or mechanical, including photocopy, recording, or any informationstorage and retrieval system, without permission in writing from the publisher.

Academic Press, Inc.1250 Sixth Avenue, San Diego, California 92101-4311

United Kingdom Edition published byAcademic Press Limited24-28 Oval Road, London NWI 7DX

Library of Congress Cataloging-in-Publication Data

Kaufman, Alexander A., dateGeophysical field theory and methods I Alexander A. Kaufman.

p. em. - (International geophysics series; v.49A)Includes bibliographical references.Contents: v. I. Gravitational, electric, and magnetic fieldsISBN 0-12-402041-0 (vol. I). - ISBN 0-12-402042-9 (vol. 2).-

ISBN 0-12-402043-7 (vol. 3).I. Field theory (Physics) 2. Magnetic fields. 3. Electric

fields. 4. Gravitational fields. 5. Prospecting-Geophysicalmethods. I. Title. II. Series.QC173.7.K38 1992550'.1'53014--dc20

PRINTED IN TIlE UNITEDSTATESOF AMERICA

92 93 94 95 96 97 BC 9 8 7 6 5 4 3 2 I

91-48245CIP

To my wife Irina

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Contents

~a ~Acknowledgments xiList of Symbols xiii

Chapter I Fields and Their Generators1.1 Scalars and Vectors, Systems of Coordinates 11.2 The Solid Angle 121.3 Fields 22104 Scalar Field and Gradient 231.5 Geometric Model of a Field 361.6 Flux, Divergence, Gauss' Theorem 401.7 Voltage, Circulation, Curl, Stokes' Theorem 521.8 Two Types of Fields and Their Generators: Field Equations 661.9 •Harmonic Fields 811.10 Source Fields 1001.11 Vortex Fields 123

References 136

Chapter II The Gravitational Field11.1 Newton's Law of Attraction and the Gravitational Field 139II.2 Determination of the Gravitational Field 157II.3 System of Equations of the Gravitational Field and Upward Continuation 178

References 199

Chapter III Electric FieldsIII.1 Coulomb's Law 200III.2 System of Equations for the Time-Invariant Electric Field and Potential 213III.3 The Electric Field in the Presence of Dielectrics 238lIlA Electric Current, Conductivity, and Ohm's Law 251

vii

Vlll Contents

I1I.5 Electric Charges in a Conducting Medium 265I1I.6 Resistance 274IlL7 The Extraneous Field and Its Electromotive Force 286IlL8 The Work of Coulomb and Extraneous Forces, Joule's Law 299Ill.9 Determination of the Electric Field in a Conducting Medium 304Ill.lO Behavior of the Electric Field in a Conducting Medium 326

References 396

Chapter IV Magnetic FieldsIV.! Interaction of Currents, Biot-Savart's Law, the Magnetic Field 398IV.2 The Vector Potential of the Magnetic Field 405IV.3 The System of Equations of the Magnetic Field B Caused by Conduction

Currents 425IVA Determination of the Magnetic Field B Caused by Conduction Currents 432IV.5 Behavior of the Magnetic Field Caused by Conduction Currents 444IV.6 Magnetization and Molecular Currents: The Field H and Its Relation to the

Magnetic Field B 481IV.7 Systems of Equations for the Magnetic Field B and the Field H 493IV.8 Behavior of the Magnetic Field Caused by Currents in the Earth 511

References 565

Index 567International Geophysics Series 579

Preface

In this monograph I describe the theory of fields as applied to gravita-tional, electrical, and magnetic exploration methods. The next volumeswill be devoted to the theory of fields applied to electromagnetic, seismic,nuclear, and geothermal methods.

Geophysical methods are applied in a wide variety of areas. They areused for oil and mineral prospecting, for solving groundwater and engi-neering problems, and in logging. And, of course, geophysics plays afundamental role in studies of the earth's deep layers.

In every geophysical method it is useful to distinguish several elements,such as theory of the method, principles and methods of measuring thefield, systems of survey parameters, data processing, and solving theinverse problem and performing geological interpretation. .

All of these elements together form a geophysical method and everyone of them is of great practical importance. The theory of a specificmethod, however, has a large influence on the main features of otherelements. In fact, the basis of all geophysical methods are physical laws.The choice of distances between observation points along profiles, as wellas the distance between profiles and survey parameters, is usually madebased on an understanding of field behavior. Regardless of the method,we always measure a signal which consists of several parts. One of theseparts contains useful information about certain structures of the earth,such as layers and confined bodies. Other parts are man-made noise andgeologic noise and they have to be reduced as much as possible. Inseparating the useful signal from the noise, which is the main goal of dataprocessing, knowledge of field behavior as a function of coordinates,frequency, and time is extremely important. Finally, the solution of theinverse problem is in essence based on a comparison between the usefulsignal and the results of field modeling.

ix

x Preface

Sometimes, and very briefly, I discuss aspects of measurement, noisereduction, and interpretation, but it is done only as an illustration of fieldbehavior. All elements of a geohysical method, except its theory, are farbeyond the scope of this monograph.

In describing the theory of gravitational, electric, and magnetic fields, Iuse the same approach in each case and discuss only those features thatare relevant to geophysical exploration. This approach includes addressinga series of questions, which I discuss in the following order:

1. Physical principles of the method2. Physical laws which govern field behavior and their areas of applica-

tion3. Influence of a medium on the field and the distribution of field

generators4. Formulation of conditions when physical laws cannot be used di-

rectly for field calculations5. Systems of field equations and their necessity when some of the field

generators are unknown6. Formulation of boundary-value problems and their importance in

determining the field7. Auxiliary fields and their role in the field theory8. Approximate methods of field calculation9. Study of the field behavior in various media corresponding to the

most typical conditions where geophysical methods are used, including:(a) Formulation of boundary-value problems and their solutions(b) Analysis of the distribution of field generators(c) Relationship between the field and parameters of the medium

The theory of these fields is the main subject of the last three chapters.In the first chapter, by contrast, I consider general features of fields,regardless of their nature. This chapter lays down the basis for under-standing physical principles and methods of calculating fields used ingeophysics. Of course, the central concept of this material is the relation-ship between a field and its generators. I hope this book will be useful forgeophysicists working in exploration and global geophysics, as well as forphysicists and electronics engineers.

Acknowledgments

During two semesters, Maureen Pretty, a student of geophysics at theColorado School of Mines, carefully read this book and made manygrammatical corrections. Due to her exceptional efforts, I am able topresent a significantly improved version of this book.

I also wish to thank Dr. L. Tabarovski of Western Atlas for reading thismonograph. Because of his attention to this book, several errors andambiguous expressions were removed.

I have been aided greatly in the preparation of this book by mycolleague Dr. Richard Hansen who spent a great deal of time reading themanuscript not only for scientific content but also for English usage. Thediscussions with him were most instructive and enjoyable, and I wish togratefully acknowledge his generous contributions. I also would like toexpress my thanks to Dr. Norman HarthiIl and Professor Michael Brodskyfor very useful discussions.

I express to all of them my deep gratitude for their considerablecontributions. If the book contains any inaccuracies, however, it is myresponsibility only.

I wish also to express my thanks to Dorothy Nogues who typed themanuscript.

xi


List of 'Symbols

a major semiaxis of spheroidb minor semiaxis of spheroidA magnetic vector potential defined by B = curl AB magnetic field .C velocityD dielectric displacement vector D = lEE or declinatione charge

es surface chargeE vector electric field, volts/meter

En electric field component normal to surfaceEo primary electric field

Eext extraneous forcei5' electromotive force

i5'c contact electromotive forceF attraction force

Fa centripetal forceg':', g~ terminal points of vector lines

g gravitational fieldgN normal gravitational fieldG b geometric factor of boreholeG, geometric factor of formationG Green function

hi' h 2' h 3 metric coefficientshi.q, p) harmonic function

H auxiliary functionj, i current density

jm' i m volume and surface density of molecular currents,respectively

I current or inclination

xiii

XlV List of Symbols

fo(x), fl(x) Bessel functions of first kind of argument x and oforder 0 Or 1 as indicated

lo(), K o( ), ll()' K I ( ) modified Bessel functions, order 0,1 of the first andsecond kinds, respectively

K I 2 contrast coefficientsK f , K d , K, coefficients, describing self potential

L path of integration or depolarization factorLm edge line of normal surface

L qp distance between points q and pd t I'd t 2' d t 3 displacements along coordinate lines

Lop radius vectord t m vector line element

M vector or magnetic dipole momentM k components of Mm separation constant or massn unit vectorn parameter of transmission lineP weight or polarization vectorp observation point

Po' PI Legendre functions of first kindq pointQ heat

Qo' QI Legendre functions of second kindr , cp, z cylindrical coordinatesR, £J, tp spherical coordinates

R resistanceR; grounding resistance

S surface or conductances ratio of conductivitiest time

T scalar field or transversal resistanceU potential of source field

u ", U - mobility of positive and negative charges, respec-tively

V voltagew+, w- velocity of positive and negative charges, respec-

tivelyW energyZ vertical component of magnetic field of the earth

x, y, z coordinates of Cartesian systema polarizabilityf3 dielectric susceptibility'Y gravitational constant, conductivity

List of Symbols XV

E dielectric permittivityEO constant€r relative permittivity8 volume density

8[,8 b volume density of free and bounded charges, respec-tivelysurface densityfree and bounded density of surface chargeslinear densityfluxresistivity

P« apparent resistivity/-Lo constant

/-L magnetic permeabilityw solid angle


Chapter I Fields and Their Generators

1.1 Scalars and Vectors, Systems of CoordinatesScalar and Vector, Position of an Observation PointScalar and Vector Components of Vector M(p)Dot and Cross Products of Vectors and Some of Their CombinationsDifferentiation of Combinations of Scalar and Vector FunctionsScalar and Vector Components of the Vector Near a Surface and a LineOriented Lines and Oriented Surfaces, System of Curvilinear Coordinates

1.2 The Solid Angle1.3 Fields1.4 Scalar Field and Gradient1.5 Geometric Model of a Field1.6 Flux, Divergence, Gauss' Theorem1.7 Voltage, Circulation, Curl, Stokes' Theorem1.8 Two Types of Fields and Their Generators: Field Equations1.9 Harmonic Fields

I.10 Source FieldsI.11 Vortex Fields

References

1.1 Scalars and Vectors, Systems of Coordinates

In this section we will describe some elements of algebra with scalar andvector functions that are used most often in this monograph. However, it isproper to notice that in some cases deeper insight into the theory ofgeophysical methods requires the use of such concepts as tensors.

Scalar and Vector, Position of an Observation Point

In general one will assume that both scalar T and the vector Marefunctions of a position of point p within a volume V; that is, this pointpresents itself as an argument of these functions.

T=T(p) and M=M(p) (1.1)

2

a

I Fields and Their Generators

b

o

d

Fig. I.1 (a) Radius vector; (b) coordinate displacements; (c) projection of a vector on a line;and (d) vector components.

At every point p the scalar value is defined by its magnitude ITI and sign,while the vector value is characterized by its magnitude M(p) and direc-tion.

(1.2)

Here M(p) is the magnitude of the vector M, but i m is the unit vector,directed along M. By definition,

(1.3)

Usually, the point p, where the behavior of these functions is studied, iscalled an observation point; to define its position one can use either theradius-vector Lop or three coordinates of the point: Xl' X Z, x3 •

Of course, both approaches require a choice of the origin at some pointo of known position. Correspondingly, the radius-vector Lop is written as

(104)

Here Lop is the distance between the origin 0 and the observation pointp, and i is the unit vector directed along the radius (Fig I.1a).

1.1 Scalars and Vectors, Systems of Coordinates 3

Thus, either the radius-vector or three coordinates of the point canserve as arguments of functions T(p) and M(p).

and

or

or

(1.5)

Furthermore, let us use only the right curvilinear systems of coordinatesformed by three mutually orthogonal families of coordinate lines([ , f 2 , f 3 ' the direction of which is defined by unit vectors i[, i2 , i 3 ,

respectively (Fig. l.lb). To determine the position of the observation pointits coordinates Xl' X 2' and x 3 are measured or calculated along corre-sponding lines.

Scalar and Vector Components of Vector M(p)

Let us introduce scalar and vector components of vector M along somedirection ( in the following way (Fig. I.1c).

and (1.6)

Here i I is the unit vector along line f and (M, I I) is the angle betweenvectors M and i I .

Notice that the scalar component M I is positive if the angle (M,i /) isacute, but it is negative when the angle becomes obtuse. Very often thevector M is described with the help of its vector and scalar componentsalong the coordinate lines ([ , t 2 ' and f 3 .

(1.7)

or

and

k=(l,2,3) (1.8)

Here (M, i k ) is the angle between the vector M and the unit vector i kdefining a direction of the corresponding coordinate line.

Taking into account orthogonality of coordinate lines we obtain for themagnitude of the vector and its direction (Fig. I.1d),

M = ...;M{ + Mt + Mf ,M k

cos(M,id = M k=(l,2,3) (1.9)

If M is the unit vector i I characterizing a direction of the line f, then in

4 I Fields and Their Generators

accordance with Eqs, 0.7), and (I,8) we have

i t = i 1 cos(i t , i 1) + i 2 cos(i r , i 2) + i 3 cos( it, i 3) (1.10)

That is, the vector it is expressed through the direction cosines cosOr , ik ) .

Dot and Cross Products of Vectors and Someof Their Combinations

The dot product of two vectors

and

is

(1.11)

Here (a, b) is the angle between these vectors.Thus, the dot product is a scalar equal to the sum of products of

corresponding components of vectors, and its sign is defined by the anglebetween these vectors. In particular, if they are perpendicular to eachother, the dot product equals zero. Suppose that one of these vectors is aunit vector; for instance, b = it. Then we have

(1.12)

where at is the projection of the vector a on the line t. In other words, tofind the scalar projection of the vector on some direction, we can form thedot product of the vector and the unit vector along this direction.

If both vectors have the same direction, costa, b) == 1, the dot product isreduced to that of their magnitudes.

As follows from Eq. (1.11), for the dot product of unit vectors of anorthogonal system of coordinates we have

(Ll3)

and

Consider the next operation of vectors. The cross product a X b ofvectors a and b is the vector perpendicular to both of them, and itsmagnitude equals the area of the parallelogram formed by these vectors.

[a x b] = ab sin(a, b) (1.14)

a


b

M

dc

T

Fig. 1.2 (a) Cross product; (b) cross product with unit normal n; (c) tangential and normalcomponents of a vector near a surface; and (d) tangential and perpendicular components of avector near a line.

and

where the vertical lines indicate a determinant.From the latter it follows that

aXb= -bXa

(1.15)

(1.16)

The direction of the cross product c is defined from the condition thatvectors a, b, and c form the right-hand system as is shown in Fig. 1.2a.

In accordance with Eq. 0.14) the cross product of two parallel vectors isequal to zero, but it reaches a maximum when they are perpendicular to


each other. For instance for unit vectors of the orthogonal system we have

(1.17)

and

Suppose that b = n is a unit vector, then

c = a X n = a sin(a, n)c o (1.18)

Here Co is the unit vector located in the plane perpendicular to the vec-tor n.

Thus the cross product of any vector a and the unit vector n forms anew vector c, which is located at the plane perpendicular to n and whosemagnitude lei equals the scalar projection of the vector a into this plane(Fig. I.2b).

Two more useful operations with vectors are described. The mixed ordot-cross product of three vectors a, b, and c is a scalar equal to thevolume of the parallelepiped formed by these vectors.

a . (b X c) = b . (c X a) = c . (a X b)

(1.19)

and

a . (b X c) = - b . (a X c) = - a . (c X b)

The double cross product of the vectors a, b, and c,

aX(bXc)

(1.20)

is more complicated, but it is possible to present it as a difference of twovectors.

a X (b X c) = (a· c)b - (a' b)c (1.21 )

This equality is very useful in simplifying algebraic transformations, and itis often applied in this book.

From the definition of the cross product it follows that

aX(bXc)= -(bXc)Xa (1.22)


Differentiation of Combinations of Scalarand Vector Functions

In those cases when vector functions are continuous, the known rules ofdifferentiation of scalar functions can be applied. For example,

d da db-(a+b)=-+-dx dx dx

d da dcp-cpa = cp- + a-dx dx dx

d da db-(a'b)=-'b+a'-dx dx dx

d da db-(aXb)=-· Xb+aX-dx dx dx

(1.23)

Here cp is a scalar function, but x is an argument of these functions and inparticular it can be a coordinate of an observation point.

Similar relations can be written for more complicated combinations ofvector and scalar functions. Let us make one more comment about thederivative of a vector. In general, both the magnitude and the direction ofthe vector are functions of coordinates of an observation point. Then, inaccordance with Eq. (1.2) the derivative of the vector M(p) with respect toany argument x is

dM dM dim--=-i +M--dx dx m dx

(1.24)

In particular, for the derivative from the vector component along coordi-nate lines we have

(1.25)

Here i k is the unit vector along line t k' but x is one of the coordinates ofan observation point.

In a curvilinear system of coordinates the direction of a unit vector isusually a function of the position of an observation point. Therefore, thesecond term of Eq. (1.25) is not equal to zero.

8 J Fields and Their Generators

Scalar and Vector Components of the VectorNear a Surface and a Line

In studying the behavior of vector functions near some surface S, it isoften useful to present them as a sum of normal and tangential compo-nents (Fig. I.2c).

and

M; = M . 0 = M cos(M, 0)

MT

= M· T =Mcos(M,T)

(1.26)

(1.27)

Here 0 is the normal to the surface, 101 = 1, and T is the unit vectorcharacterizing the direction of the tangential component M

T•

As follows from Eq, (Ll8), the vector n X M is tangential to the surfaceS and its magnitude is equal to M

T•

MT=lnXMI

Forming again the cross product with n we obtain another presentation ofthe tangential component through the normal n and vector M.

(1.28)

(1.29)

It is clear that the angle between vector M and the normal n is defined as

MT

tana= -M n

Finally, for any tangential component along some direction t (Fig. I.2c),we have

and M, = t· M = M cos(M, t) = MT

' t (1.30)

Here t is the unit vector located at the plane tangential to the surface atpoint p.

The behavior of a vector M near some line t can also be described withthe help of tangential and normal components with respect to this line.

(1.31)

Here M r is the component tangential to the line t at point p, and M, islocated in the plane perpendicular to this line (Fig. I.2d). For thesecomponents we have

M=(M'ir)i r (1.32)

1,1 Scalars and Vectors, Systems of Coordinates 9

and

M s =Msin(M,i f), M s = (if X M) X if

Here if is the unit vector along line t, but the vector if X M is located inthe plane perpendicular to this line.

Oriented Lines and Oriented Surfaces, Systemof Curvilinear Coordinates

First we introduce the concept of an oriented elementary displace-ment dt.

dl'= dti f= d.t; + dt; + dtj

= dtl i] + dtz i z + dt3 i 3 (1.33)

Here dt is the magnitude of the vector d/, which equals the length of thissegment, but dtk and d~ = dtk ik are the scalar and vector componentsof the vector dl' along coordinate lines.

Correspondingly, an orientation of a line t in a space is defined by achoice of its positive direction, that is, by the vector d/.

An oriented element surface dS can be expressed as

dS = dS n = dS j + dS z+ dS 3

= dS] i] + dS z i z + dS3 i 3 (1.34)

Here dS, the magnitude of the vector dS I , equals the area; n is the unitvector normal to this surface; and

dSk = dS cos( dS, id, (1.35)

are the scalar and vector components of dS k ' which is perpendicular tocoordinate lines t k .

The orientation of the surface is defined by the orientation of itsnormal n. We will distinguish the front and back sides of the surface andassume that the normal n is directed from the back to the front side.

To characterize a mutual orientation of vectors we will use in this bookonly right-handed systems, which can be illustrated in the following way.Suppose that an observation point changes its position along some path tin the positive direction dl' (Fig. I.3a). Then this vector forms a right-handed system with any direction s, if an observer mentally placed at theend of s sees a movement of the observation point counterclockwise. Forexample, one will consider a surface S with the normal n and bounded bycontour t (Fig. r.n». Then, in accordance with the right-hand rule, thedirection dl' should be chosen in such a way that indicates a rotationaround vector n counterclockwise. In general three vectors a, b, and c


Fig. 1.3 Mutual orientation of lines and surfaces.

form a right-handed system if their directions are defined by the right-handrule, as is shown in Fig. I.3a. In particular in a right-hand system ofcoordinates, unit vectors are related to each other in accordance with Eq.(1.17).

Having defined the concept of the right-hand rule, let us briefly outlinethe main features of a curvilinear orthogonal system of coordinates. Aswas pointed out, three mutually perpendicular coordinates lines t 1 , t 2 '

and t 3 pass through every point and form in a space three families oflines. Along every line only one coordinate varies while two others remainconstant. For instance, along line t 1 coordinates x 2 and x 3 do notchange. At the same time, a position of a point can be characterized bythree families of coordinate surfaces 51' 52' and 53' which are orientedin such a way that the coordinate line t k is perpendicular at every pointto the corresponding surface 5k . At every coordinate surface only onecoordinate does not change. These three families of surfaces, as well asthose of lines, are perpendicular to each other. As can be seen from Fig.I.1b, elements of coordinate surfaces d5k , bounded by coordinate lines,are defined by vectors.

Respectively, an elementary volume surrounded by coordinate surfaces is

(1.37)

Next we will introduce metric coefficients that relate a length of theelementary segment of the coordinate line dtk with a change of thecorresponding coordinate dx i; that is,

(1.38)

Here hI' h 2 , and h3 are metric coefficients of the coordinate system, andthey are usually functions of coordinates of an observation point. As a


rule, analytical expressions for metric coefficients are derived from ananalysis of the geometry of the coordinate lines.

Let us consider three examples corresponding to the simplest systemsof coordinates.

Cartesian System

All coordinate lines present straight lines, while coordinate surfaces areplanes.

h, = h 2 = h 3 = 1

dt; = dx, dt2 = dy, dt3 = dz

dS, =dydz, dS 2=dxdz, dS3=dxdy

dV=dxdydz

Cylindrical System

(1.39)

(lAO)

Coordinate lines t, and t 3 are straight lines and t 2 is a circle. Thecoordinate surface r = constant is a surface of the cylinder, cp = constant isa half plane, but z = constant is a horizontal plane.

h, = 1, h 2 = r, and h 3 = 1

dt, = dr, dt2 = rde , dt3 = dz

dS,=rdcpdz, dS 2 = drdz, dS 3=rdrdcp

dV = r dr dtp dz

Spherical System

The coordinate line t, is a straight line, and lines t 2 and t 3 are a halfcircle and a circle, respectively. The coordinate surface R = constant is aspherical surface, 0 = constant is a cone, but cp = constant is a half plane.

h, = 1, h 2 = R, h 3 = R sin 0

dt, = dR, dt2 = RdO, dt3 = R sin 0 dsp

dS, = R 2 sin 0 dO dip,

dS 3 =RdRdO, and

dS 2 = R sin 0 dR d sp

dV = R 2 sin 0 dRdO dip

(1.41)

12 [ Fields and Their Generators

Fig. [.4 Examples of solid angles.

1.2 The Solid Angle

In this section we will describe the concept of a solid angle, which is veryuseful in deriving field equations and which also allows us, in some cases,to simplify calculations of the field. Consider a point p and a closedcontour Y that has an arbitrary shape (Fig. 1.4). By drawing straight linesfrom point p through every point of the contour 2' we obtain the conewith its apex at point p and the conic surface Sc' Examples of cones withvarious shapes are shown in Fig. 1.4. All possible lines on a conic surfacecan be separated into two groups, the direction and nondirection lines.These groups differ in that every straight line originating at the apex of thecone passes through every point of a direction line; but not necessarilythrough the second type of closed line.

Every cone divides a space into two parts-the internal part D, and theexternal part De (Fig. I.5a). To characterize a cone, let us evaluate a ratiobetween these parts. That procedure perhaps can be done by differentmethods. For example, it seems natural to consider the volumes of D, andDe; but these are infinitely large and so we will apply another approach.We begin by drawing a spherical surface with an origin at the cone apexhaving radius R (Fig. I.5a). Then the cone divides this surface into twoparts, Sj and Se' which correspond to D, and De' It is obvious that thesurface Sj can be, in principle, used to characterize the internal part D j ,

confined by the cone. However, there is some ambiguity related to the factthat Sj also depends on its radius R, which can be arbitrarily chosen.Indeed, the spherical surface S as well as its parts Sj and Se areproportional to the square of the radius. For this reason, to evaluate the

1.2 The Solid Angle 13

c dS

a', qi· .· .· .· .· .~ "..

~ .p

dS'=dS cos a

b

p

d

p

Fig. I.S Definition of a solid angle.

internal part of the cone D j , the ratio

(1.42)

is used. The function w(p) is called the solid angle, and it is characteristicof the cone. Imagine that an observer is placed at the apex p, and theconic surface is not transparent. Then it is natural to treat coi p) as a visualangle under which the surface Sj is seen from point p. This approach willbe developed here in detail, and it may serve as an explanation of the factthat usually in figures showing a cone, parameter w(p) is indicated nearthe apex.

Let us illustrate Eq. (1.42) by several examples.

1. Sj = 0, that is, the conic surface becomes a strip. Thus D, = 0 and,correspondingly, w(p) = O.

2. In the opposite case when the internal part D, occupies a wholespace, we have


and therefore the solid angle corresponding to the whole space is

w( p) = 4'lT

These two examples show that the solid angle varies as

0:5: w(p) S 4'lT

3. In the case when a conic surface becomes a plane, Sj = 2'lTRz, andcorrespondingly

w(p)=2'lTFinally,

4. If the conic surface confines a quarter of the space, S, = 'lTR z, andthe solid angle

w(p)='lT

Thus, we have described the solid angle from two different but relatedpoints of view, namely,

1. The solid angle is a measure of the internal part of the spaceconfined by the cone.

2. The solid angle is a visual angle under which a part of the sphericalsurface is seen from the apex.

The latter is more important for our purposes, and for this reason let usgeneralize and develop this point of view in detail. First, consider anelementary surface dS at the point q and an observation point p (Fig.I.5b). Then from the point p we will draw straight lines through everypoint of the contour surrounding dS and, correspondingly, obtain the conewith the solid angle dioi i»). To calculate this angle, let us project thesurface dS into the spherical surface with radius L p q • Here L p q is thedistance from point p to the elementary surface dS . As can be seen fromFig. I.5c, the projection dS* is

dS* = dS cos(Lpq , n)

Here n is the unit vector perpendicular to dS.Therefore, in accordance with Eq. (1.42), the solid angle equals

dS* dS cos(dS, Lp q )dw(p) = -2-=z

L p q L p q

or

Here dS = dS n.

dS· L p qdw(p) = 3

L p q

(1.43)

1.2 The Solid Angle IS

Unlike Eq, (1.42), the solid angle dw(p) is expressed here through thesurface dS, which is, in general, a nonspherical one, and it can havepositive as wen as negative values. As follows from Eq. (1.43) the solidangle is positive when the back side of the surface dS is seen from anobservation point p, and it is negative in the opposite case. In particular,when both dS and the point p are located at the same plane, the conetransforms into a strip and the solid angle equals zero. In accordance withEq. (1.43) one can say that the solid angle dw is subtended by the surfacedS, being viewed from an observation point p. It is clear that all surfacesdS inside the cone and bounded by direction lines are characterized by thesame magnitude of the solid angle.

Now let us generalize this result for an arbitrary surface S (Fig. LSd).Having mentally divided this surface into many elementary surfaces andthen performing summation, we obtain for the solid angle w(p), sub-tended by surface S as viewed from point p, the following expression:

fdS· L p q

w(p) = L 3S pq

(1.44)

It is clear that a corresponding cone is formed by drawing straight linesfrom point p to all points of the boundary line of the surface S. Thismeans that any surface confined by the cone and bounded by the samedirection line is characterized by the same magnitude of the solid angle.As concerns the sign of scalar w(p), it depends on a position of the apex pwith respect to the back and front sides of the surface. In other words, themagnitude of the solid angle, subtended by any surface S with the sameboundary line, is the same. Assuming that both the normal n to thesurface S and the unit vector ~o directed along the boundary line 2' forma right-handed system, one can say that both the magnitude and the signof the solid angle are defined by the boundary line of the surface S.Therefore the solid angle viewed from point p, w(p), is the same for allsurfaces having an identical boundary line (Fig. LSd). Notice also that thesolid angle for surfaces with different boundary lines will be the same,provided that these lines are located on the same conic surface.

Now making use of Eq. 0.44) we will describe some useful features ofthe solid angle.

1. Suppose that surface S is spherical and its radius equals the distancebetween point p and the surface. Then


Fig. 1.6 Examples of solid angle behavior.

and since L pq is constant we have

1 Sw(p) = -z-fdS = -z-c: S L p q

that coincides with Eq, (1.42).2. Suppose S is an arbitrary closed surface, and the point p is located

somewhere inside volume V, surrounded by this surface (Fig. 1.6a). Alsoassume that the normal n is directed outside the volume. Inasmuch as aspherical surface with its center at point p is characterized by the solidangle equal to 47T, one can say that the solid angle, subtended by anyclosed surface as viewed from point p, located inside the volume, is equalto 47T (Fig. I.6a). If the normal n has an opposite direction, the solid angleis equal to - 47T.

3. Suppose that point p is located outside some arbitrary but closedsurface S. By drawing straight lines from the point p tangent to thesurface S, we form a cone, and the direction line .:J: divides the surface Sinto two parts, Sl and Sz (Fig. 1.6b). At all points of surface Sz functioncostn, L p q ) is positive, while at points of surface Sl it is negative. Taking


into account that both surfaces are bounded by the same line 2' one canconclude that the solid angles subtended by these surfaces have the samemagnitude, but opposite signs. For this reason, the solid angle subtendedby a closed surface, when an observation point is located outside thevolume V, is equal to zero, regardless of the position of the point. Thisvery useful result is often applied in the theory of fields. Thus, the last twoexamples allow us to write

{41T

w(p) = 0p inside volume Vp outside volume V

4. We will find the solid angle subtended by an infinitely extendedplane surface S. Inasmuch as the conic surface becomes a plane parallel tothe surface S, we conclude that the solid angle is either equal to 21T or-21T.

z<oz>O (1.45)

(1.46)

It is essential that at every part of the space the solid angle does notdepend on the position of the point p.

5. We will study the change of the solid angle subtended by a planesurface, having finite dimensions and located at plane z = 0 (Fig. I.6c).

At distances much greater than surface dimensions, the distance be-tween the point p and any point q of the surface is practically the sameand, correspondingly, Eq. (1.44) is greatly simplified.

1 1 L p q • kS Sw(p)z- L .dS= 0 =-

L3 S pqo L3 Z2pqo pq

where k is the unit vector along with axis z, and qo is any point on thesurface S. Thus, far away from the surface the solid angle coincides withthat of the elementary surface, and it decreases at a rate inverselyproportional to the square of the distance.

In approaching the surface, due to a decrease of a distance L q p , thesolid angle increases and near the surface it tends to either 21T or -21T.In fact, when point p is very close to the surface S, the conic surfacealmost transforms into a plane, and correspondingly,

w(p) ~ ±21T

As is seen from Fig. I.6d, with a decrease of the surface dimensions theselimiting values of the solid angle are practically achieved closer to thesurface.


Fig. 1.7 Solid angle behavior.

Comparison with the previous example shows that near a plane surfaceof finite dimensions the solid angle coincides with that for the infiniteplane.

Until now we have discussed the behavior of the solid angle along a lineof observation that intersects the surface. If a profile of observation pointsdoes not intersect the surface S, the solid angle behaves in a differentmanner (Fig. I,7a). In approaching the plane z, as z < 0, it increases, thenreaches a maximum at some distance from surface S, and then it tends tozero. At all points of the plane z = 0 and outside the surface S, the solidangle is equal to zero. Also, it is clear that the solid angle is an antisym-metric function.

w(z) = -we -z)

6. Let us study one special case when the plane surface S is a disk withradius a, and the observation point is located at the axis z passing throughits center (Fig. I,7b). It is clear that the solid angle w(z) subtended by thedisk can be determined by calculating an area of the spherical surfacebounded by the edge line of the disk. To solve this problem we will find an


area of the elementary strip with radius r and width Rde (Fig. I.7b). HereRand e are spherical coordinates, but

r = R sin eAs is seen from this figure,

dS = 27TrRde

or

dS = 27TR 2 sin e de

The angle e varies from zero to a; here

aa = sin- I -

R

Therefore, performing an integration we obtain

S = 27TR2['sin e de = 27TR 2(1 - cos a)o

(1.47)

Correspondingly, the solid angle subtended by the disk with radius a asviewed from the axis z is

W(Z)=27T(I-COSa)=27T(I-,; z )Z2 + a 2

(I.48)

This equation is used often in this book.7. Suppose an arbitrary surface S is bounded by two contours ..2"1 and

..2"2 (Fig. I.7c). Then the conic surface consists of two parts: the internaland external parts, Si and Se' Correspondingly the solid angle can bepresented as a difference of two solid angles formed by every conicsurface.

(1.49)

8. Now consider one feature of a solid angle near a surface of anarbitrary shape (Fig. I.7d). With this purpose let us present the wholesurface S as a sum of two surfaces: one of them is elementary surface dS,with its center at point a and the other is the rest of the surface S.

S =dS +S* (1.50)

Inasmuch as dS is very small it can be considered a plane elementarysurface.

Correspondingly, the solid angle W subtended by the surface S can beconsidered as a sum of two angles.

(1.51 )


Here w I and w* are the solid angles subtended by the surfaces dS andS*, respectively.

As was shown above, the solid angle WI is a discontinuous function nearthe point a, and

wi(p) = 27T, as p~a

Here wi and wi are values of the solid angle from the front and backsides of the surfaces dS, correspondingly. At the same time the solid anglew*(p) is a continuous function at vicinity point a.

Therefore, for the total angle w near the point a we have

(1.52)

and

as p ~a

Hence, the difference of solid angles near the surface is

as p ~a (1.53)

Earlier this result was derived for a plane surface, but Eq, (1.53) showsthat it is valid for any surface. In particular, if there is some hole withinthe surface, then S = S *; the solid angle changes as a continuous functionat the vicinity of this hole.

Now we will describe the calculation of a solid angle subtended by anarbitrary surface, as viewed from an observation point p, It is clear that allspherical surfaces confined by the conic surface and having the center atpoint p are characterized by the same solid angle. On the other hand, asfollows from Eq. 0.42), the solid angle w(p) equals the area of thespherical surface Sj having the unit radius

if R = 1 (1.54)

Thus, the problem of calculation of the solid angle w(p) subtended by anarbitrary surface S is reduced to determination of the corresponding areaof the spherical surface; it is described in detail in spherical trigonometry.

First, we will choose a set of points m 1 , m 2 , m 3 , ••• , mn of the edgeline 2' surrounding the surface S and connect these points by straightlines (Fig. 1.8). Correspondingly, the edge line 2' of an arbitrary shape isrepresented by a polygon, and the coordinates of its corners define theconic surface with the apex p. Thus, instead of the cone with the directionline 2', we obtained a new cone, formed by straight lines drawn from theapex p to every point of polygon sides. Certainly such replacement leads


Fig. 1.8 Illustration of a solid angle calculation.

to some error in calculating the solid angle, but the error becomes smallerwith an increase in the number of polygon sides.

Next, we will present the polygon as a system of triangles that in turnform a system of cones. Correspondingly, our task consists of calculationof an area of the spherical surface with unit radius w/p) for every triangle(Fig. 1.8), and

N

w(p) = L wi(p)i~ 1

(1.55)

where N is the number of triangles.Suppose the corners of some triangles are (Xi' f3i,"'Ii and their position

with respect to the point P is characterized by vectors Ai' Bi , and Ci ,

respectively. Rays drawn from point p to every point of a triangle sideform the spherical triangle on the spherical surface that is bounded bythree great-circle arcs, The area of this triangle is found by Huiler's rule

1/2Wi { f; f; - «, r, - hi t i - Ci }

tan - = tan - tan -- tan -- tan --4 2 2 2 2

Here a.; hi' c i are lengths of sides of the spherical triangle, and

(1.56)


Inasmuch as the spherical surface has unit radius, the length of every sideis equal to the angle 0 of the corresponding corner, and making use of thedot product we have

BI·C.-I I

a j = Oil = cos IB;IIC;I

A··C·-I I I

b, = 0;2 = cos IA;IIC;I

A··B·-I I I

Ci = 0;3 = cos IA;IIB;I

(1.57)

By calculating the solid angle UJ;Cp) for every triangle and performingsummation, we define the solid angle subtended by an arbitrary surface.

1.3 Fields

We will begin by defining a field N as a function of a point p in space;that is,

N=N(p) (1.58)

In other words, coordinates of an observation point p, where the field isconsidered, present themselves as an argument of the function «». Wewill consider here only scalar and vector fields formed by scalar and vectorquantities T(p) and M(p), respectively. In general, it is assumed that thefield is a single-valued function. Also the field will be mainly studied in thevicinity of regular points where it behaves as a continuous function.However, we will pay some attention to singular points, lines, and surfaces,where the field behaves as a discontinuous function.

As is known, continuity of a function T in the vicinity of a point pmeans that any displacement of the observation point p within an in-finitesimally small area, results in an infinitesimal small change of the fieldT. If an infinitesimally small displacement ~ t of the point p along someline t leads to either a finite or an infinitely large change of the field ~T,

the ratio ~T /~ t tends to infinity. Correspondingly, a discontinuity of thefield along the line t is observed. Consider two points located on eitherside of a surface S at an infinitesimally small distance from each other.

The surface S is a surface of discontinuity of the field T if thedifference ~T corresponding to these points has either a finite or infinitelylarge value. In other words, the difference of the field T at two merging

1.4 Scalar Field and Gradient 23

points located on opposite sides of S does not tend to zero. At the sametime, in the direction tangential to such a surface, the field T can be acontinuous function. Similar considerations are applied to points of a linewhere the field T has a singularity. Inasmuch as the vector field M isalways described by scalar components M,(p), MzCp), and Mip), itsdiscontinuity can be studied by considering discontinuity of the scalarfields. Often we will deal with fields that do not change within somevolume V. Such fields are called uniform ones, and in the case of a vectorfield M, this means that both the magnitude and the direction of Mareindependent of the position of an observation point.

Now we will investigate a change of the field due to small displacementsof an observation point p in different directions t with help of spatialderivatives (gradient, divergence, curl, Laplacian, etc.), which are definedby analogy with the derivatives of function y(x) with respect to anargument x.

In reality we consider a field within one volume V, surrounded by aclosed surface S[V]. This can be arbitrarily large. Having chosen anarbitrary point 0 as an origin, one can study the field behavior only atpoints located at finite distances from the origin. However, it is convenientto consider the volume V as infinitely large and, correspondingly, thesurface S[V] becomes the infinitely remote surface ~. Points locatedoutside V are considered infinitely remote points from the origin O.Usually one will represent this surface k as a spherical surface with thecenter at point 0, having an area equal 47TR 2, where R is its radius, whichtends to infinity.

In general, we will study both constant fields that do not depend ontime and alternating fields that vary with time and location.

1.4 Scalar Field and Gradient

(1.59)as f:1t -t 0

Now consider the behavior of a scalar field Tt p) in the vicinity of anobservation point p. With this purpose let us choose some direction t andstudy the change of the field along this line (Fig. 1.9a). This change ischaracterized by the derivative of T in this direction; that is,

st f:1Tat = lim sr:

Here f:1T is a change of function T.

while f:1t is the distance between these two points.

24 1 Fields and Their Generators

a

T(p)

P P P21

T

c

qgradLqp

dJ Pgrad Lqp

d z

x

Fig. 1.9 (a) Change of T along a line; (b) gradient as a derivative; (c) gradient of thedistance L q p ; and (d) gradient as a flux.

As follows from Eq. (1.59) the derivative aT/at is a measure of the rateof change of the field T along line t, and it equals liT normalized by thecorresponding interval lit. It is natural to expect that in general changinga direction of line 1', passing through point p, the derivative aT/at alsovaries; that is, there are an infinite number of derivatives of the scalar fieldat vicinity and observation point.

Now let us attempt to express all these derivatives through only onefunction, which is directly related to a scalar field behavior. To accomplishthis task, let us take into account that coordinates of every point Xl' X 2 ,

and x 3 vary when distance I' changes. This relationship of the field Twith coordinates of the point and the distance I' can be illustrated as


Making use of the chain rule for a derivative, we have

st et aX I et ax z et aX3-=--+--+--at ax] at axz at aX3 atwhere

(1.60)

aX l 1 atl

---;;e= hi ai'ax z 1 atzat = h z ai'

since within small intervals along coordinate lines t I , t z ' t 3 the metriccoefficients do not change. Correspondingly, Eq. (1.60) can be rewritten as

er 1 et atl 1 er atz 1 st at3-=---+---+---at hI aX I at hz ax z at h 3 aX 3 ator

er sr atl «r at'z er at3-=--+--+--at att at atz at at3 at

As follows from Eq. (I.l2),

(1.61)

(1.62)

It is clear that the right-hand side of Eq. (1.62) can be represented as thedot product of two vectors.

Here

aTat = it· grad T (1.63)

it = cos( 1', I't )i l + cos( 1', I'z)iz + cos( 1', 1(3)i3

is the unit vector characterizing the direction of the line t, along whichthe derivative is considered.

The vector

(1.64)

or

(1.65)


is called the gradient of the scalar field, and in accordance with Eq, (1.63)any directional derivative of the scalar field aTjat is expressed throughthe gradient of T. Also from this equation it follows that grad T shows thedirection along which a maximal increase of the field is observed, but itsmagnitude equals the maximal derivative aTjat in the vicinity of anobservation point (Fig. L9b). This means that the gradient characterizesthe field behavior only, and correspondingly it is independent of any otherfactors-in particular, of the system of coordinates. At the same time Eq.0,63) vividly demonstrates the practical meaning of the gradient since itshows that instead of taking a derivative aTjat along any line t, it issufficient and simpler to project grad T along this direction. To emphasizethis fact, let us rewrite Eq. (1.63) as

aTat = grad, T

that is, the derivative of a scalar field in any direction t is the projectionof the gradient along this direction.

For illustration we will present grad T in the simplest systems ofcoordinates.

1. The Cartesian system

et et etgradT= -i + -j+-kax ay az

2. The cylindrical system

er et ergrad T= -i + --i +-iar r racp 'P az Z

3. The spherical system

et et 1 etgradT= -i + --i + -----i

aR R R ao IJ R sin 0 acp 'P

Often it is convenient to express grad T as

grad T= VT

(1.66)

(1.67)

Here V is an operator having different expressions in various systems ofcoordinates; for example, in the Cartesian systems we have

e a eV=i-+j-+k-

ax iJy iJz(1.68)

(1.69)


The gradient, as a vector, has in general all three components; but ifcoordinate lines are chosen in such a way that one of them, for instance,t 3 coincides with the direction of grad T, then we have

1 aTgrad T= grad , T= - -i 3h 3 aX 3

Now we derive an expression for the gradient of a scalar field T(cp),where cp is a function of observation point p.

T= T(cp) = T{cp(p)}

In this case, one can write

(1.70)

aT aT acp-=--aX k acp aXk

Then in accord with Eq, 0.64) we have

k = 1,2,3

aTgrad T = - grad cp

acp(1.71)

(1.72)

Until now it has been assumed that a field T is studied within somevolume V, where T is a function of all three coordinates. If we restrictourselves to consideration of the field on a surface S, it is appropriate tointroduce a corresponding gradient, grad" T, as

er ergrad" T = -tl + -tzat 1 at z

where t l and t z are unit vectors tangential to the surfaces S and perpen-dicular to each other. At the same time, the derivative aTjat along anydirection t, tangential to the surface, is defined in the following way:

aT- = i . grad" Tat t

Of course, an analysis of the field behavior in a volume can be accom-plished with help of the two-dimensional gradient, if also the derivativeaTjan in the direction perpendicular to the surface is considered.

Next consider grad T in the vicinity of a point where a field has asingularity. If the field T near some point p in direction t has adiscontinuity, then aTjat ~ 00, and correspondingly grad T becomesmeaningless. For instance, if the field T has different values on both sidesof a surface S, the difference Tz - T1 characterizes its change through


such a surface, and it is natural to introduce a surface analogy of thegradient as

(I. 73)

Here n is the unit vector directed from back to front sides of the surface,and T, and Tz are the values of the field on these sides, respectively.

Suppose that at some point p, grad T = O. Then in the vicinity of thispoint the derivative aTlat = 0 in any direction; that is, the field does notchange near this point. Therefore, at such as extremal point, the directionof grad T is not defined.

If grad T = 0 within a volume V, the field does not vary in V; that is, Tis a constant. Also, it is obvious that the vector M = grad T defines thefield T to within a constant in the same manner that the derivative dy /dxallows us to define that function y(x).

Now let us consider one very interesting and important feature ofgrad T, and with this purpose we will form the full differential of asingle-valued function TCp). We can write

aT aT aTdT= -dx j + -dxz + -dx3ax] axz aX3

It is clear that the right-hand side of this equation is a dot product.

aTdT = d/· grad T = dtgrade T = at dt (1.74)

Suppose that 2' is an arbitrary path between points a and b. Thenintegrating we have

jbgrad T. d/ = jbdT = T( b) - T( a)a a

(1.75)

TCb) and TCa) are values of the field T at terminal points of the path. Itmeans that the integral of grad T is independent of the path of integra-tion, but it is defined by the values of T at the terminal points. Inparticular, for a closed path we have

¢grad T· d/= 0

This equation is of great importance in the theory of many fields.

(1.76)


We will illustrate the concept of gradient with the help of two examples.

1. First, consider a function describing a distance between two points pand q (Fig. 1.9c).

In general, this function depends on the position of both points, but wesuppose that the point q is fixed while the coordinates of the point p canchange. Then one can imagine an infinite number of displacements dt thatresult in a change of function T. As is seen from Fig. 1.9c, the maximalincrease of distance L qp takes place when dt is directed along a lineconnecting points q and p. In this case a change of the function !J.Tcoincides with a displacement !J.t, and correspondingly,

Inasmuch as the gradient characterizes the maximum rate of change ofa function,

(1.77)

(1.78)

or

since

L qp =LqpL~p

Here L~p is the unit vector directed along line L qp from point q to pointp, and the index "p" indicates that derivatives are taken with respect tothe coordinates of the point p.

In the opposite case, when the point p is fixed, we have

q L pq L qpgradL =-=--qp L pq L qp

Here L pq is the vector with the same magnitude L qp, but directed frompoint p to point q.

Comparing Eqs, (1.77), (1.78) we obtain

p q

grad L qp = - grad L qp (1.79)

We have derived Eqs, (1.77), (1.78) from a geometrical point of view, as


well as from a definition of gradient. Now we will obtain the same resultfrom Eq. (1.64). Taking into account that in a Cartesian system

/ 2 )2 2T=Lqp=V(xp-Xq) +(Yp-Yq +(Zp-Zq)

we have

and therefore

er zp -Zq

azp -:

p L qpgrad T= --,

i;but

q Lp q

grad T=-i;

2. Next consider field T= I/L q p • Making use of Eq, (1.71) and lettingcp = L qp we have

(1.80)

and

q 1 1 q L q pgrad - = - -2- grad L q p = --3-

-: L qp u:Equations (1.78)-0.80) are used often in this monograph.

Until now we have presented the gradient of T through derivatives withrespect to coordinates of an observation point. Now let us express thegradient with help of an integral and with this purpose introduce theCartesian system of coordinates x, Y, z. Then consider an elementaryvolume bounded by coordinate elementary surfaces dS j , dS 2 , and dS 3 ,

and a quantity T dS. It is a vector equal to the product of the scalar T andthe vector dS,

TdS = TdSn (1.81)

and it is called the vector flux of T through the surface dS.Next we will determine this flux through a closed surface surrounding

the volume dV (Fig. 1.9d). First, consider the flux through both sides dS j ,

which are parallel and located at the distance df from each other. It isassumed that the areas dS j are very small and that the field T does notvary over them and other elementary surfaces. Correspondingly, the flux


through a pair of elementary surfaces dS l is

(1.82)

Inasmuch as

and

(1.83)

we have for the flux

Taking into account the distance df equals dx and that it is small, thisdifference can be replaced by the first derivative times dx and we obtain

aT aT{T(p2) - T(Pl)} dy dz i = -dxdydz i = - dViax ax

In a similar manner for the flux through two other pairs of surfaces wehave

aT aT{T(p4) - T( P3)} dxdz j = -dxdydz j = -dVjay ay

and

er «t{T( P6) - T( Ps)} dy dx k = -dxdydz k = -dV kaz az

Performing a summation of these three equalities we have

¢ T dS = grad T ~vS[IlV]

Here S[~V] is the closed surface surrounding an elementary volume ~V.

Thus we have obtained three forms of equations for the gradient,namely,

1. At usual points

2. On the surface of discontinuity


3. The integral presentation

1gradT= -f TdS

.6.V S[W]

or in the limit

1grad T = lim-A:. T dS

.6.V~v->o

As was shown above, the two dimensional gradient is

aT aTgradsT= -i+-j

ax ay

and its integral presentation almost directly follows from Eq, (1.84).

1grad" T = .6.5 t Tv dt

(1.84)

(1.85)

Here .6.5 is an elementary area surrounded by the contour Sf, and v is theunit vector perpendicular to the path Sf and directed outside the area dS.

Fig. 1.10 (a) Illustration of the two-dimensional gradient; (b) integral presentation of thegradient; and (c) geometric interpretation of the gradient.


In fact, as is seen from Fig. 1.1Oa, the integral along the closed path .2' is

¢Tv dt = T( pz) dy i - T( PI) dy i + T( P4) dx j - T( P3) dx j

aT aT= --dydx i + -dxdy j = grad" TdS

ay ay

or

1grad" T = lim t:.S ¢Tv dt (1.86)

Equation (1.84) has allowed us to express the gradient through thesurface integral, provided that a volume is sufficiently small so that thegradient is constant within it. The same comment applies to Eq. (1.86).

As an example of applications of Eq. (1.84), let us derive an equationthat establishes a relation between values of the scalar field T at pointslocated at arbitrary distances from each other. Consider a volume of anysize and shape, and mentally divide it into many elementary volumes. Inaccordance with Eq. (1.84), for every elementary volume t:.V; we can write

AV;gradT=~TdS,s,

i = 1,2, ... , N (1.87)

Here S, is the surface surrounding the volume t:.V;.Performing summation of Eq, (1.87), written for every such volume, we

have

N N

2: t:. V; grad T = 2: ~ T dSi=1 i~l S

(1.88)

Taking into account that integration over every elementary surface S, isperformed twice, in each case with dS having opposite direction (Fig.1.10b), the right-hand side of Eq. (1.88) is replaced by only an integralsurrounding volume V, and therefore in the limit we have

1grad TdV= ~TdSv s

By analogy, for the two dimensional case,

!gradS TdS = ~ Tv deS Sf

(1.89)

(1.90)


It is proper to notice that both these equations are often used in thetheory of geophysical methods. First of all they allow, in many cases,drastic simplification in the calculation of fields, replacing either a volumeintegral by a surface one or the surface integral by a linear one. At thesame time, Eqs. (1.89), (1.90) relate values of the field inside a volume(surface) to its values at the boundary surface (line), and this fact explainstheir important role in the solution of inverse problems of geophysics.

To describe a scalar field T, often a geometrical approach is applied,which is based on the use of level surfaces St. At every point of such asurface field has a constant value (Fig. i.ioe;

T=C on St

Inasmuch as single-valued fields are considered, level surfaces are definedeverywhere except singularities and extremal points; they are closed anddo not intersect each other.' The geometry of this family of level surfacesallows one to visualize a scalar field, and with this aim they are drawn insuch a way that difference aT, corresponding to two neighbor surfaces, isthe same and sufficiently small. Also the normal n of these surfaces showsa direction along which the field increases. Usually a part of the spaceconfined by two neighbor level surfaces is called a level layer. It is clearthat the surface St , in turn, defines a distribution of lines orthogonal tolevel surfaces. The length of a segment of such a line, corresponding to thelevel layer, represents its thickness.

Consider an elementary level layer with small thickness an, which can,in general, change from point to point. As is seen from Fig. 1.1Oc the smalldistance at along an arbitrary line t between surfaces of such layer isrelated to its thickness an,

Here i ( is the unit vector along line t.The change of the field T along this line is

aT aT anaT = -at= -----

t at at cos(it,n)

while along an we have

(1.91 )

(1.92)

(1.93)

Inasmuch as for the level layer a change of the field aT does not depend


on a direction of line t,

t1T( = t1Tn = C = constant

and therefore

aT C-=-at sr

aT aT9 = - cos(i(,n)ac an

(1.94 )

From the last equation it follows that derivative aT jat along anydirection t is defined by the derivative aTjan along the normal n and theangle between these directions. In particular, on the level surface the fieldT does not change and, correspondingly, the derivative in the directiontangent to this surface equals zero (cos 900 = 0).

Comparing Eqs. 0.65) and (1.94) we see that the magnitude of grad T isequal to the derivative of T along the normal n, and its direction coincideswith that of this normal.

and

aTgradT= -0

an

aTIgradTI= -

an

(1.95)

(1.96)

Let us note that Eq. 1.95 can be considered a definition of grad T.Thus, to describe a scalar field it is sufficient to know the direction of

the normal n to the level surface, and the derivative of the field aTjan inthis direction.

Lines perpendicular to level surfaces are often called gradient lines,since vector M = grad T is tangential to them.

In accordance with Eq. 0.94)

aT Can t1n

That is, the magnitude of grad T on level surface is inversely proportionalto the thickness of the level layer.

It is very simple to derive an equation for gradient lines. In fact, takinginto account that an oriented element of this line, d/, and grad Tare


parallel to each other, we have

cos( grad T, d.l) = 1

or

a, dt2 sr,-aT-I-at-

l= et1M2 = aTlat

3

(1.97)

(1.98)

Here dtl ' dt2 ' and dt3 are components of vector d.l along the coordi-nate lines t I , t 2 ' and t 3 , respectively.

The latter can be rewritten as

In particular, in the Cartesian system of coordinates,

ax ay az--- = --- = ---etlax etlay etlaz

(1.99)

(1.100)

If the field is studied on the plane surface, its behavior can be character-ized with the help of level lines,

T = constant

which are equivalent to level surface, as well as by a family of gradientlines indicating a direction of grad" T.

1.5 Geometric Model of a Field

In all geophysical methods we mainly deal with vector fields caused byvarious types of generators, such as masses, electric charges, currents,stresses, etc. For example, gravitational, magnetic, electric, electromag-netic fields, as well as the velocity of seismic waves are vector fields.

In this section we will develop a geometric model of a field and withthis purpose in mind we introduce two concepts, namely, vector lines andnormal surfaces. These will allow us to establish fundamental relationsbetween fields and their generators practically without any application ofmathematics, and in essence this is the main reason for developing thisapproach. As soon as these equation are derived, a geometric model of afield will not usually be used.

Earlier we introduced oriented lines and oriented surfaces. It is essen-tial to distinguish positive and negative passages of oriented lines througha surface S (Fig. Ll l a), If a line t goes from the back to the front

1.5 Geometric Model of a Field 37

Fig. 1.11 Geometric field models.

side-that is, its direction coincides with that of the normal n of thesuface-a positive passage takes place. When the line t goes in theopposite direction, we observe a negative passage. Correspondingly, indetermining the number of oriented lines intersecting an oriented surface,we will use this rule and take into account the sign. A similar approachwill be applied in calculating the number of surfaces intersecting anoriented line, as is shown in Fig. 1.1lb. It is amazing that so simple anapproach will permit us to derive fundamental equations of the fieldregardless of its nature.

The first geometric model of the field is based on the concept of vectorlines. Let us consider a field M( p). Then a vector line t m of this field isdefined from the condition: the angle (M, dl"m) between its element dz?"and field M equals zero; that is, vector M is tangential to this line.

(1.101)

or also


Here i m is the unit vector characterizing the direction of field M, and dl'mis the oriented element of the vector line at the same point. Since vectorM and dl'm are parallel to each other, the equation of the vector line is

(1.102)

For instance, in a Cartesian system we have

(1.103)

In a cylindrical system,

dr rdep dz-=--=-M r M<{! u, (1.104)

and in a spherical system,

(1.105)M<{!

R sin fJ de:dR RdfJ-=--=----

In general, there are two types of vector lines, open and closed. It isobvious that open vector lines have terminal points, which we call initial,q~, and final, qr:! , points. An example of a vector line is shown in Fig.I.1lc. Notice that terminal points can be located at infinity, that is, faraway from observation points where a field is studied. While vector linespresent themselves as a geometric model of a field, terminal pointscharacterize field generators. Because of this we will pay special attentionto determining the number of these points as well as their location. Vectorlines f m can illustrate not only a direction of field M, but also itsmagnitude. To realize this, suppose that they are drawn with density equalto a M, Here a is an arbitrarily chosen constant. It is clear that thenumber of vector lines piercing an elementary surface dS with its center atpoint p and transverse to the field M equals aM(p) dS. The system ofvector lines drawn through every point of some nonvector line f forms avector surface. If the line f is closed, then such a vector surface confinessome part of the space, which is called a vector tube. If the vector field Mcharacterizes a movement-for instance, a motion of a liquid or electriccharges-then vector lines and vector tubes are called current lines andcurrent tubes. In those cases when a force field is considered, they arecalled force lines and force tubes, respectively. In general, the crosssection of a vector tube changes, and the total number of vector lines

1.5 Geometric Model of a Field 39

piercing the cross section is proportional to the product

M(p) dS( p)

where dS( p) is the cross-section area.It follows that a family of vector lines defines a family of surfaces sm,

which are orthogonal to the lines and are called normal surfaces. Thesecan also be used to describe a vector field. The normal n to such a surfaceis directed along the field, and correspondingly at every point of thenormal surface the following condition is met:

cos(M,dS m) =1, dsm=dSmi m (1.106)

Here d.S'" is an elementary area of the normal surface.These surfaces can be either open or closed. The open surface sm has

an edge line Lm, which is always closed since it confines the normalsurfaces; but sometimes these lines can be located at infinity. Also we willassume that edge lines Lm are directed in such a way that they form aright-handed system with the vector lines I'm (Fig. I.11d). By analogy withvector lines we can make use of normal surfaces to characterize both thedirection and the magnitude of the vector field M. With this purpose inmind, imagine that normal surfaces are drawn with a density equal to 13M.Here 13 is some constant that is also arbitrarily chosen. In this case thenumber of normal surfaces intersecting any elementary segment of avector line dt" equals 13Mdt'", The part of space bounded by two normalsurfaces is usually called the normal layer, and the length of a vector linebetween them represents its thickness. Our main attention will be to anelementary layer with a small thickness A, which is in general a function ofthe point p located on an average surface of the normal layer (Fig. Ll ld).

We have mentioned several times a concept of generators of fields,which present themselves as a physical cause of these fields. Some exam-ples of such generators and fields are given below.

Generators

masses

charges

currents

charges, currentsrate of change ofmagnetic andelectric fieldswith time

stresses, strains

Fields

gravitational

electrical

magnetic

electromagnetic

elastic waves


Fig. 1.12 (a) Flux through an elementary surface; (b) illustration of flux; (c) flux through anelementary vector tube; and, (d) the surface analogy of divergence.

1.6 Flux, Divergence, Gauss' Theorem

In this and the next sections we will make use of vector lines and normalsurfaces and derive fundamental relations between a field and its genera-tors. First of all, let us remember that the number of vector lines piercingthe elementary area tiS": of a normal surface, perpendicular to vectorlines, equals

aMdS m

Now we will introduce the notion of a flux of the field M as the integral

4J = fM' dS5

Here S is an arbitrary surface, and M is the vector field.The product

M . dS = M dS cos(M, dS)

(1.107)

(1.108)

is a flux through an elementary surface dS, arbitrarily oriented withrespect to the field (Fig. I.l2a).

1.6 Flux, Divergence, Gauss' Theorem 41

From a mathematical point of view, the flux ¢ is a sum of elementaryfluxes through different parts of the surface S, and they can be eitherpositive or negative, or zero. However, it is much more important todemonstrate that the flux ¢ can characterize some of the essentialfeatures of the field behavior. Consider an element of the normal surfacedS m . The flux through such an element is

(1.109)

since the angle between vector M and dS m equals zero.At the same time the number of vector lines dN piercing the area

dS m is

dN I = aMdS m (I.11O)

Comparing Eqs. (1.109), (1.110) we see that the flux and the number ofvector lines are related by

1d¢ = -dN

a(1.111)

Next let us examine the case when M is not normal to dS. Then, inaccordance with Eq. (1.108) the flux through the surface dS is

d¢ = M· dS =MdS cos(M,dS) (I.112)

As is seen from Fig. 1.12a the number of vector lines piercing the surfacedS and its projection dS m on the normal surface is the same; that is,

dN I = a M d.S'" = «MdS cos(M, dS) = o (M . dS) (I.113)

Here we take into account the fact that if a direction of the normal nandthe field M form an angle exceeding 900

, then the number of vector lines isnegative since they go through the surface dS from the front to back side.

Integrating, we obtain a relation between the flux and the amount ofvector lines through any surface S.

1¢ = f M . dS = - N 1

S a(1.114)

Thus, the flux ¢, as a pure mathematical concept, is expressed through thenumber of vector lines, which is much easier to visualize. In general, onepart of the vector lines go from the back to the front side of the surface,giving a positive contribution, while others go in the opposite direction,defining a negative number of vector lines. Also, some of these lines canbe tangential to the surface and correspondingly do not make any contri-bution.


Now let us consider a closed surface S of arbitrary shape and askourselves the following. What does the flux through a closed surface show?It turns out that the answer to this question is very simple. First assumethat there is one vector line em, only, which passes through a closedsurface S, intersecting it once from the front to the back side and then inanother place from the back to the front side. Correspondingly, the totalamount of passages NI of this vector line through the closed surfaceequals zero. Therefore, in accordance with Eq. (1.114) the flux in this caseis also equal to zero.

Generalizing this result, we can say that if vector lines do not haveterminal points inside the volume surrounded by the closed surface S, theflux of the field through this surface equals zero. Now suppose a vectorline is started somewhere inside the volume V. Then it intersects thesurface S only once from the back to front side, and correspondingly theamount of passages N I is equal to one; but the flux c/J equals l/a, Eq,0.114). In the opposite case when the final point of a vector line is locatedin volume V, the flux equals -1/a. It is obvious that in a general case ofan arbitrary number of vector lines, the flux through a closed surface withan accuracy of the constant equals the total amount of terminal pointsinside the volume surrounded by this surface.

~1 a;

M· dS = -(q~-q~) =-s a a

(1.115)

Here q';; and q~ are the amounts of initial and final points of vector linesinside volume V, respectively; but

(1.116)

It is proper to emphasize that Eq. (1.115) is one of two fundamentalrelations of the field theory. Certainly it is an amazing fact that regardlessof the nature of the field (electric, gravitational, magnetic, seismic, etc.)the integral

c/J=~M'dSs

over any closed surface characterizes the number of terminal points ofvector lines inside volume V. Later we will show that terminal points ofvector lines are a geometric model of one type of field generators, calledsources, and respectively the flux through a closed surface plays a role of"litmus paper," determining the total amount of sources in a spaceconfined by this surface. At the same time, this flux is independent of thedistribution of terminal points within the volume V. In other words, the


flux through any closed surface does not change if the position of terminalpoints of vector lines arbitrarily varies within the volume, provided thatthe total amount of these points remains the same. For this reason it isnatural to make the next step and introduce a new tool that permits us tostudy the distribution of terminal points in detail.

With this purpose in mind, let us consider a relatively small volumewhere terminal points are distributed uniformly. The flux through thesurface S surrounding this elementary volume LiV defines the number ofterminal points within it. Dividing this amount by the volume, we intro-duce a new notion, which characterizes the density of these points; that is,

¢M'dS

LiV(1.117)

Thus, the flux through S, surrounding a small volume LiV and normalizedby this volume, equals within a constant of proportionality l/a the densityof terminal points of vector lines within this volume. This, in essence,represents the density of sources of the field.

The importance of this notion is hard to overestimate since, if we wouldlike to determine the behavior of a field, it is natural to have informationabout the distribution of its sources. This ratio of the flux and thecorresponding elementary volume is called the divergence.

¢M'dSdivM=---

LiV(1.118)

This analysis shows that both the flux ¢M' dS through the surface,surrounding a relatively small volume, and the divergence divM have thesame meaning, since they characterize the number of terminal pointsinside the same volume. There is only one difference between them;namely, the divergence, unlike the flux, determines the density of thesepoints, and respectively they have different dimensions.

As follows from Eq. (1.118), to find the number of terminal points ofvector lines when divM is known; it is necessary to form the product

divM LiV

Usually the divergence of the field M is written as

(1.119)

¢sM'dSdivM = lim LiV as LiV ~ 0 (1.120)


Now let us make several comments:

1. The term divM is a scalar, since a density of distribution of terminalpoints is characterized by its magnitude and a sign only. In the nextsection we will study a geometric model of another type of field generatorsthat require a vector to describe their distribution.

2. In accordance with Eq. (1.117), divM can be rewritten as

1 qmdivM=-lim-

a AV'as AV~ 0 (1.121)

that demonstrates a direct connection between the divergence and thedensity of terminal points.

3. To calculate divM as follows from Eq. (1.120), we have to divide asurface S by several elements so that within every element the magnitudeand the direction of vector M do not change. Then the dot productM . dS = M dS cos(M, dS) is defined and, after summation and normaliza-tion, the divergence is determined. Since this procedure consists of severalrelatively complicated steps, let us replace them by one operation that ismuch simpler.

With this purpose in mind we will introduce a curvilinear and orthogo-nal system (x I , x 2 , x 3) and consider an elementary volume formed bycoordinate lines t\ ' t 2 ' and t 3 with the center at point p (Fig. 1.12b).Let us note that since the divergence characterizes a density of terminalpoints of the field, its value does not depend on a choice of coordinatesystem or the shape of the small volume AV, as long as these points aredistributed uniformly within the volume.

We will proceed from Eq. (1.118). It is obvious that the total flux of thefield through the surface S (Figure 1.12b), consists of six fluxes. First of all,consider flux through two elementary surfaces perpendicular to the coordi-nate line t I . This part of the total flux is

(1.122)

Here dS/PI) and dSj(pz) are elementary surfaces with centers at pointsPI and Pz, respectively.

Taking into account that within every elementary surface vector M doesnot change, integrals in Eq. (1.122) can be replaced by


Inasmuch as

and

but

we have

M(pz) . dSI(pz) + M(PI) . dSI(PI)

=M1(pz) dS1(pz) -M1(PI) dS1(Pl) (1.123)

The distance dfl between points P I and pz is small and correspondinglyone can assume that the function M I dS changes linearly within thisinterval. This allows us to replace the difference at the right-hand side ofEq. (1.123) by

M1(pz) dS1(pz) -M1(PI) dS1(Pl)

a= -f {M1(p) dS1(p)} dt;

a I

where p is the point at the middle of the volume.In the same manner, for two other pairs of flux we have

Mz( P4) dSz( P4) - M z( P3) dS z( P3)

a= -f {Mz(p) dSz(p)}dfza z

and

M3(P6) dS3(P6) -M3(ps) dS3(ps)

a= -f {M3( p ) dS3( p )}.u,

a 3

(1.124)

(1.125)

(1.126)

Thus, for the total flux through the surface S, surrounding the volume ilVwe obtain

(1.127)


whence

(1.128)

Taking into account Eqs. (1.38) we finally obtain

Therefore our problem is solved since we have been able to replace theintegration in Eq. (1.118) by a differentiation that is usually much easier toperform. For illustration let us write Eq. 0.129) in three systems ofcoordinates.

1. Cartesian system

2. Cylindrical system

. 1 [arMr aM<p aMz ]dlvM=- --+-- +r--

r ar a'P az

3. Spherical system

(1.130)

(1.131)

It is clear that Eqs. 0.118) and (1.129) have the same meaning becausethey both characterize the density of terminal points in a very smallvolume with the center at point p.

4. The simplest expression for the divergence is, of course, in a Carte-sian system. However, it may be a source of confusion since it is a greattemptation to consider divM as a sum of derivatives of field componentsMx ' My, and Mz ' with respect to corresponding coordinates, and inter-

1,6 Flux, Divergence, Gauss' Theorem 47

pret divM as a measure of a change of the field, but it is not true.' Forinstance, if in the vicinity of some point vector lines do not have terminalpoints, divM equals zero regardless of how the field M changes near thispoint.

To emphasize the real meaning of divergence we will consider asegment of the elementary vector tube (Fig. 1.12c). The surface surround-ing this volume consists of the vector surface of the tube Stand two crosssections dS I and dSz . Taking into account that the field is tangential tothe vector surface and

We have for the flux through the closed surface

or

Therefore,

since

1 adivM = cr(p) ae{M(p)cr(p)} (1.133)

b.V = dea ( p)

Here de is the length of the tube segment. In accordance with Eq, (1.133)divM can be treated as the rate of a change of the flux (but not the field)in the direction of the field normalized by the cross-section area of anelementary vector tube.

5. In deriving Eq. 0.128) we assumed that all derivatives of the fieldwith respect to coordinates exist in the vicinity of the observation point. Inother words, unlike Eq. (1.118), this expression for the divergence is validat the usual points only.

Now we will consider the special case when one of the field componentsis a discontinuous function and, correspondingly, Eq. (1.128) cannot beapplied. Similar behavior of a field is often observed at interfaces betweenmedia with different physical properties. To determine the distribution ofterminal points of vector lines on such a surface we will consider itselement dS and surround it by a cylindrical surface Sc' as is shown inFig. 1.12d. The total flux through this surface can be presented as a sum of


three flux corresponding to the lateral surface St , as well as dSlpI) anddS/pz)·

Inasmuch as dSz = dS n, dS I = -dS D, where n is the unit vector directedfrom the back to the front side, we can rewrite Eq. (1.134) as

(1.135)

Here M2)(Pz) and M~I)(PI) are the normal components of the field at thefront and back sides of the surface, respectively.

With a decrease of the cylinder height, the integral value at theright-hand side of Eq. (1.135) tends to zero, provided that the field hasfinite values, but points PI and Pz approach point P on the surface.Correspondingly, instead of Eq. 0.135) we obtain

The next step is obvious. To characterize the distribution of terminalpoints on the surface S, it is natural to divide both parts of the lastequation by dS. Then we have

¢M·dS--- = M(Z) - M(1)

dS n n

or

¢M'dS 1 «;=--

dS adS

Here qrn is the number of terminal points located within an elementaryarea dS.

Correspondingly, one will introduce the surface divergence

¢M ·dSDivM=lim ~S

= 2. ~ = M(Z) - M(1)a ~S n n

as ~S ~ 0 (1.136)

In accordance with Eq. (1.136), a difference of normal components ofthe field from both sides of the surface defines the density of terminal


points. In particular, in the vicinity of those points of the surface wherethe normal component M; is a continuous function, terminal points areabsent. Let us remember that a discontinuity can manifest itself as achange of either a magnitude or a sign of the vector field, or both of them.

Thus, we have introduced three equations characterizing a distributionof vector lines.

1¢M .dS = -;;qrn

¢sM'dSdivM = lim ----

LiV

Div M = M(2) - M(l)n n on S

(1.137)

Now, making use of the concept of terminal points of vector lines, let usderive one of the most important relations of field theory. Suppose that anarbitrary volume V, surrounded by the surface S, is mentally divided intomany elementary volumes LiV. Then we will calculate the total number ofterminal points inside the volume. First of all, by definition it is equal tothe surface integral.

On the other hand, the number of terminal points inside any elementaryvolume dV is

divMdV

and therefore the total number of these points inside the volume V is

f. divM dVv

Thus, both the surface and volume integrals characterize the same numberof terminal points of vector lines, and we obtain Gauss' theorem.

¢.M' dS = f. divM dVs v

(1.138)

This equality plays a fundamental role in the theory and interpretationof geophysical methods. It results from the fact that Eq, (1.138) establishesa connection between the field on the surface S and its values inside thevolume V. For this reason Gauss' formula is often used to derive equationsthat allow us to find a field somewhere inside the volume, as its values on


Fig. 1.13 (a) Gauss' theorem applied to a two-dimensional model; (b) the two-dimensionalanalogy of divergence; (c) voltage; and, (d) voltage and edge lines.

the surface S are known either due to calculations or measurements. It isalso proper to notice that it is because of the application of vector lines wehave been able to derive Gauss' formula in such an extremely simple way.

To conclude this section we will obtain an expression for the divergencein the two-dimensional case as well as Gauss' formula, assuming that thefield does not vary along some straight line 2' (Fig. I.13a) With thispurpose in mind we will consider an elementary volume IlV, surroundedby the surface S, which consists of two elementary plane surfaces perpen-dicular to the line 2' and located at a distance Ilh from each other andthe lateral surface Sf. Inasmuch as the field is tangential to two firstsurfaces, the flux through them equals zero and, correspondingly, for thetotal flux we have

~M'dS= f M·dSS Sf

(1.139)

Then, taking into account the fact that the field does not change along line2', but

dS = 11 de'Sh ; dV=dSllh


we obtain for the two-dimensional divergence

as ~S ~ 0

Is M· dS ¢tM' vdedivsM = t = .:....:.... _

~V ~S

¢tMv dt

~S(I.140)

Here v is the unit vector perpendicular to the closed line t, and M; is thefield component in this direction.

Applying the same approach as in the three-dimensional case, div" Mcan be expressed in terms of derivatives as

(1.141)

It is clear that this equation directly follows from Eq. (1.129) if we leth3 = 1 and JM3 / Jx 3 = 0, and it is valid at the usual points only.

In those cases when there is a line where the normal component of thefield is a discontinuous function, one can characterize a distribution ofterminal points along such a line in the following way:

A:.tM· v dtdiv" M = lim 'j' = M(Z) - M(l)

~t v v(1.142)

Here M:Z) and M:l) are components of the field perpendicular to the linefrom both its sides (Fig. I. 13b).

Thus, we have derived three equations describing a distribution ofterminal points in the two-dimensional case, as is summarized below.

1~M'v dt= -qrn'f/ Q'

¢t M' v dtdiv" M = lim .:....:----

~S '

div" M = M(Z) - M(l)v v

~S~O(1.143)

Now we will consider some plane surface S, bounded by a closedcontour t. Then, mentally dividing this surface into many elements dSand applying Eq. (1.140) for every such elementary surface ~Si we have

div" M ~Si = ¢,M . v dtt j

(1.144)


Here t; is the contour surrounding the surface AS;. Then, performingsummation of this equality for every elementary surface and taking intoaccount that integration along any internal contour t; is performed twicewith opposite directions of v, we obtain Gauss' formula for the two-dimensional case.

f divSMdS=~M .vdt=~Mvdts t t

(1.145)

In conclusion let us note that often it is convenient to present div M as

divM= V·M

1.7 Voltage, Circulation, Curl, Stokes' Theorem

In the previous section we introduced the flux of the field through a closedsurface, as well as divergence, to characterize the distribution of terminalpoints of vector lines. Here we will study in detail a second type of toolsused in field theory, circulation and curl. First we will consider only thosefields that can be described with the help of normal surfaces. These resultswill then be generalized for any vector field.

Consider normal surfaces sm and assume that

(a) They are drawn with a density proportional to the field M.(b) The field M and the direction of the edge line L'" of a normal

surface (Fig. 1.14a) form the right system (Fig. t.n.nTo begin we will introduce a concept of the voltage as the integral

(1.146)

along some path t with terminal points a and b.Let us make two comments.

1. Usually this notion is applied to electric and electromagnetic fields,but we will use it for all vector fields.

2. As will be shown later, the voltage generally depends on the path ofintegration as well as the position of the terminal points a and b.However, there is one special type of field that plays a very important rolein the theory of some geophysical methods for which the voltage is pathindependent.

Now let us attempt to evaluate the voltage by making use of normalsurfaces. First, consider an element of the vector line, dl'm. By definition it

1.7 Voltage, Circulatlon, Curl, Stokes' Theorem 53

Fig. 1.14 (a) Voltage and edge lines; (b) illustration of curl; (c) illustration of curl; and,(d) surface analogy of curl.

is perpendicular to the normal surface sm, and correspondingly its direc-tion coincides with normal n. The voltage along this segment is

dV=Mdr (1.147)

since the angle between M and dl'm equals zero.It is clear that the number of normal surfaces dNz intersecting this

element dt'" is

Thus, the voltage and number of normal surfaces are related by

1dV= -dNzf3

or

dNM . dl'm = M dr = _z

f3

(1.148)

(1.149)


Now determine the voltage along an arbitrary oriented element dl'(Fig. 1.13c). Inasmuch as the projection of dl' on the vector line passingthrough point p is intersected by the same number of normal surfaces asthe element d/, we have

dNz = 13Mdt'" = 13Mdteos(M, dl' )

= 13M· dl'= 13 dV (1.150)

Again the voltage characterizes the number of normal surfaces thatpass through the element d/. As Eq. (1.150) shows, the orientation of dl'has a strong influence on the voltage. For instance, if dl'is a segment ofthe vector line dl'ID, then there is a maximum number of passages ofnormal surfaces, and correspondingly the voltage also reaches its maximalvalue. If the element dl' is tangential to the normal surface, it is notintersected by normal surfaces and therefore the voltage vanishes. Finally,if elements dl' and dr form an angle exceeding 900

, the number ofpassages of normal surfaces through dl' becomes negative, as does thevoltage. In particular, if the elements dl' and dl'm have opposite direc-tions, the voltage is negative, but it has a maximal magnitude. Thus, achange of the orientation of the element dl' at vicinity point p leads to achange of the voltage.

Now making use of Eq. (1.150) and applying the principle of superposi-tion, we obtain for the voltage V along an arbitrary contour t

fb 1

V= M· dl'= -Nzat {3

(1.151)

Here N z is the amount of normal surfaces of the field intersecting thepath t , and either their positive or negative contribution is defined by amutual orientation of normal surfaces and the element d/.

As follows from Eq. (US!), a change of the position of the path t doesnot change the voltage if the number of passages of normal surfacesthrough these paths remains the same.

Having established a relation between the voltage and the number ofthe normal surfaces, let us introduce the concept of the circulation as thevoltage along an arbitrary closed path.

V=¢.M'dl'r

By analogy with the flux through a closed surface, we will raise thefollowing question. Does the circulation characterize some essential fea-tures of the field behavior? The answer is yes and it can be obtained with

1.7 Voltage, Circulation, Curl, Stokes' Theorem 55

amazing simplicity. In fact, suppose that one normal surface sm passesthrough a closed contour t, and its edge line Lm is located outside thesurface S, surrounded by the contour t (Fig. LlSd), In this case thenormal surface intersects the contour t twice, so that at one time itsnormal n forms an angle with the contour segment dl'; which is less than90°, while the second time the angle between them exceeds 90°. Corre-spondingly, the number of intersections by this normal surface equals zero,and therefore in accordance with Eq, (1.151) the circulation is also equalto zero. Generalizing this result, one can say that if all normal surfaces donot have edge lines L" inside the contour t, the circulation ¢(M' dl'equals zero.

Next we will assume that the edge line of the normal surface intersectsthe surface S bounded by contour t. This means that this path isintersected by the normal surface only once; that is, the number ofpassages of this surface through the contour t is equal to either + 1 or-1 (Fig. 1.14a). As was pointed out above, the sign of the passage isdefined by the mutual orientation of the edge line Lm and the path dt:Thus, in this case the circulation is

1~M' dl'= +-'Y -{3

Now integrating we can conclude that in general, as there is an arbitrarynumber of normal surfaces, the circulation is

1~M'd/=-Nz( f3

(1.152)

Here N 2 is the total number of edge lines L m intersecting a surface S,bounded by the contour of integration t.

Let us make several comments that describe this equation.

1. Nz is an algebraic sum of positive and negative terms. Those edgelines that form a right-handed system with the vector d/ give a positivecontribution, while the others define negative terms of this sum.

2. Any surface S bounded by the path t is intersected by the samenumber of edge lines Lm, since they are closed.

3. A change of a position of the contour t does not affect thecirculation if this path does not intersect the new edge lines or a sum ofnew intersections equal to zero. Similar behavior is observed in the case ofthe flux through a closed surface. Performing a deformation of the surfacesurrounding the volume, the flux will not change if the surface does not

56 I Field~ and Their Generators

intersect terminal points or the sum of the number of terminal pointsequals zero.

4. Equation (I.152) establishes a very important fact, namely, that thevoltage of this field along a closed path (circulation) characterizes thenumber of edge lines of normal surfaces that intersect any area boundedby this path.

It is clear that the circulation presents itself as an analogy to the fluxthrough a closed surface, since the latter describes the amount of terminalpoints within the volume surrounded by the surface. Moreover, as will beshown in the next section, edge lines and terminal points play a role ofgeometric models of two types of field generators: vortices and sources.This is the main reason why both functions

¢,M'dl't

and

can be considered the most fundamental concepts in the theory of fields.In accordance with Eq. 0.152) the circulation along a contour t is definedby the total amount of edge lines Lm passing through any surface boundedby this contour, but its does not provide any information about theirdistribution. Because of this, let us introduce a new tool that allows us tocharacterize both a density and a direction of these lines. With thispurpose, unlike the case of terminal points, it is natural to use a vector.

First, consider a small contour I' surrounding the plane surface dS(Fig. I.14b). As always, the directions of dl' and dS obey the right-handrule. It is also assumed that within dS edge lines are distributed uniformlyand all of them have the same direction. As is seen from Fig. I.14b, thenumber of edge lines intersecting the surface dS depends on its orienta-tion. For instance, if the path t is located at the plane of edge lines, theydo not intersect dS, and correspondingly the circulation equals zero. Letus choose a contour t around the point p, which is located in the planeperpendicular to the edge lines. Then it is obvious that a maximal numberof these lines intersect the area dS, and the direction of the normal n(dS = dS n) is tangential to the edge lines. Inasmuch as the number ofedge lines N2 is

their density is defined as

N 3 ¢M'd/dS = f3 dS (U53)

1.7 Voltage, Circnlation, Curl, Stokes' Theorem 57

To characterize also the direction of the edge lines we multiply bothparts of this equation by unit vector L'8, which is directed along theselines, and then obtain

N jM'dl'_2LID = a. LIDdS 0 ~ dS 0

The vector given by Eq. 0.154) is called curl M,

~M' dl'curlM = LID

dS 0

or

(1.154)

(1.155)

as liS -) 0~M' dl'

curlM = lim liS L'8

and it describes the density as well as the direction of edge lines in thevicinity of the source point p. For example, if curl M is negative, the edgelines have a direction opposite to that of the normal n.

5. In deriving Eq. (1.155) we have to choose a small closed contour, forinstance a circle, around point p to determine the circulation. However, itdoes not, of course, mean that the geometry of the field is such that itsvector lines should also be closed around this point.

6. As in the case of divergence, it is appropriate to replace thecumbersome procedure of integration on the right-hand side of Eq, (1.155)by differentiation. With this purpose, let us introduce the vector B, whosecomponent in any direction is characterized by unit vector p, defined as

1B = -~M'dl'

p dS 'Jfp

(1.156)

Here dS p is the elementary surface with its center at the point p locatedin the plane perpendicular to p, t is the contour surrounding this surface,and its direction and that of vector B form a right-handed system. Inparticular, along coordinate lines we have

(1.157)

Here dS k is the element of the coordinate surface perpendicular to theline t k , and t [dSk] is the contour bounding this area. Multiplying eachcomponent Bk by the corresponding unit vector and adding all three


products, we obtain for the vector B

It is clear that along this vector its component has the maximal value,which is equal to the magnitude of B. Therefore, by rotating the surfacedS p around point p the right-hand side of Eq. 0.156) Bp reaches themaximal value, equal to IBI, as the direction of p coincides with that of B.At the same time the circulation ¢M . dl'in this equation is defined by theamount of edge lines U" piercing the area dS p ' and it is a maximum whenp is tangential to these lines. In other words, in such a case the surface dS p

is located at the plane perpendicular to the edge lines in the vicinity of thepoint p. Correspondingly, the vector B can be written as

¢M'dl'B = L'g

dS

and comparison with Eq. (U55) shows that it coincides with curl M,

curlM =B (1.158)

but its components are defined from Eq. (1.157).Such presentation of curl M is essential since it gives us an expression

for its components along coordinate lines. Moreover, Eqs. 0.157), (U58)are introduced without the use of normal surfaces and edge lines, andcorrespondingly these equations of curl M are applied for any kind offields. In fact, curl M is usually defined by Eq, (U57).

7. Now we are ready to describe this vector using derivatives. First,consider a contour t, surrounding the elementary surface dS I with itsnormal directed along the coordinate line t I and consisting of two pairsof segments parallel to the coordinate lines t z and t3' respectively (Fig.I.I4c). Taking into account the fact that the field M does not change alongevery segment, we have for the circulation

¢M . dl'= M( PI) . dl'( PI) + M( pz) . dl'( pz)

+ M( P3) . dl'( P3) + M( P4) . dl'( P4)

or

¢M' dl'=Mz(Pl) dtz(pt) +M3(pz) dt3(pz)

- Mz( P3) dtz( P3) - M3(P4) dt3(P4) (1.159)


Inasmuch as distances between opposite sides of the contour are small,the corresponding differences of terms in Eq, 0.159) can be expressedthrough first derivatives. Then we have

or

a(h M ) a(h M )A;M'dl'= 3 3 dx dx _ z z dx dx'Y ax Z 3 ax Z 3

Z 3

In accordance with Eq. (U57) the projection of curl M on the coordi-nate line t j is

(1.160)

(1.161)

since

In the same manner for the two other components we have

1 [ahzMz ahjMj]curl 3 M = -- - ---

hjh Z ax] ax z

Thus we have described curl M with the help of six derivatives in thevicinity of the point p, located at the center of the elementary volumeformed by coordinate lines.

Also, we present curl M in a more compact form as

hji[ hziz h3i31 a a a

curlM =ax] ax z aX3

(1.162)h]h Zh3

hjM] hzMz h3M3

As in the case of gradient and divergence, it is often useful to apply thenabla operator, and then we have

curlM = VX M


It is obvious that Eqs. (1.160)-(1.162) can be used only at usual pointswhere derivatives of the field exist.

To illustrate these results, let us present curl M in the three simplestsystems of coordinates.

1. Cartesian system

JMz JM ycurl M = -- - --

x Jy Jz

JMx JMzcurl M= -- ---

Y Jz Jx

JMy JMxcurl M = -- - --

z Jx Jy

2. Cylindrical system

curl M= ~[JMz _ JrMcp]r r Jr.p Jz

[JM, JMz]

curl M= -- ---cp Jz Jr

1 [JrMcp JMr ]curl M = - -- - --z r Jr Jr.p

3. Spherical system

1 [J JMe]curl , M = --.- - sin (}Mcp ---

R sin e J() Jr.p

1 [JMR JRMcp]curl, M = --- -- - sin ()--R sin () Jr.p JR

curl M = ~[JRMe _ JM R ]cp R JR J(}

(1.163)

(1.164)

(1.165)

8. As was mentioned above, the vector curl M is usually arbitrarilyoriented with respect to the field M, and correspondingly normal surfacescannot be, in general, used to study a field behavior. However, there is onetype of field when at every point the angle between these vectors equals


90°, It occurs if the field M can be presented as

M=',O(p)gradU(p) (1.166)

Here ',O(p) is a scalar function that can be differentiated.To prove that the vector M, defined by Eq, (1.166), and curl Mare

perpendicular to each other, it is sufficient to show that

M· curlM = aWith this purpose, we will make use of Eq. 1.20:

a'(bXc) = -b'(axc) = -a'(cXb)

as well as two relations

curl grad U = aand

curl( ea) = ',0curl a + grad ',0 X a

Letting a = grad U we have

curl M = curl ( ',0grad U)

= ',0curl grad U + (grad ',0 X grad U)

Then taking into account Eqs. (1.166) and 0.168) we obtain

1curl M = grad ',0 X grad U = - grad ',0 X M

',0

whence

(1.167)

(1.168)

(1.169)

M grad ',0M'curlM= - '(grad',OXM) = --- '(MXM) =0

',0 ',0

Thus, the curl M is perpendicular to the field M if Eq. 0.166) is valid.Such fields are called quasi-potential fields, and correspondingly thereexists a family of normal surfaces perpendicular to the vector lines. Ingeneral, this is not true and because of this, curl M is defined by Eq.(1.156). At the same time it is proper to note that any field can always bepresented as a sum of two fields.

Here M 1 is a quasi-potential field, while M 2 is described by the curl ofsome other vector.


Taking into account the fact that our main goal is to illustrate as muchas possible the main concepts, such as curl, we will continue to use normalsurfaces and edge lines with the understanding that the area of theirapplication is limited.

9. It is obvious that Eqs, (1.156) and 0.162) have the same meaning. Inparticular, for a quasi-potential field they characterize the density and thedirection of edge lines in the vicinity of the point p. At the same timeEq. 0.162) cannot be used near points where any component of the field isa discontinuous function, since the corresponding derivative does not existin this case. As was pointed out earlier, a singularity in the behavior of afield is often observed at interfaces of media with different parameters. Tocharacterize a distribution of edge lines located at this type of surface S ina simpler manner than Eq. (1.156), we will introduce a system of coordi-nates where lines t 1 and t 2 are tangential to the surface S; but t 3

coincides with normal n, as directed as always from the back to the frontside (Fig. 1.14d).

First, consider an elementary contour, located on the coordinate sur-face Sl around the point p. It is clear that the circulation along thiscontour is

Here M(2) and M(1) are values of the field from the front and back sides ofthe surface, respectively.

Inasmuch as

but segments of the contour along the normal n tend to zero, we obtain

(1.170)

In accordance with Eq. (1.151) this relation characterizes the number ofedge lines intersecting an element of the coordinate line t 2. Correspond-


ingly, the density of their distribution is proportional to

¢M'dl'--- = - (M(2) - M(1»)de 2 2

2

(1.171)

In a similar manner the density of edge lines passing through theelement dtl of coordinate line t 1 is proportional to

¢M'dl'--- = M(2) _ M(l)

dt 1 II

(1.172)

Correspondingly, the vector that describes a density of edge lines and theirdirection on the surface S can be presented as

Curl M = (- MO) + M(1»)i + (M(2) - M(1»)i2 2 1 I 1 2

and it is called the surface analogy of curl M.The latter can also be written in the form

(1.173)

(1.174)

Here M(2) and M(1) are the fields at either side of the surface.This result follows from the definition of the cross product a X b, if a is

the unit vector,

i l i 2 n

aXb= 0 0 1bx by bz

In accordance with Eq. (1.174) the distribution of edge lines at the surfaceS is characterized by the difference of the tangential components of thefield at either side of the surface. In particular, near those points wherethis component is a continuous function, edge lines are absent.

Now let us generalize this result for an arbitrary field. With thispurpose in mind, let us rewrite Eq. (1.156) as

1B = curl M = --A:.M . dl'

n n dS'tj

Here n is the unit vector perpendicular to dS (dS = dS n),Inasmuch as

(1.175)

B; = B' n or curl, M = n : curlM

(1.176)

64 I Fields and. Their Generators

we have

dS n . curl M = dS . curl M = WM . dl'

The left-hand side of Eq. 0.176) represents the flux of the field B = curl Mthrough the elementary surface dS; that is, it characterizes the amount ofvector lines of the field B = curl M passing through dS. At the same time,in accordance with this equation, the circulation along the contour t alsodescribes the number of vector lines of the field B. In other words, thecirculation along the contour t surrounding the elementary surface dS isequal to the flux of the field curl M through this surface. It is obvious thatEq. 0.176) is valid regardless of the orientation of the surface dS withrespect to curl M.

Applying this result to the surface analogy of curl M, one can say that ingeneral Eq. 0.174) characterizes the density and direction of vector linesof the field curl M on a surface S.

Thus, We have derived three equations that characterize either thedistribution of edge lines of normal surfaces of the quasi-potential field Mor, in a general case, vector lines of the field B = curlM.

These equations are

1~M'dl'=-N'f1 . {32

¢M'dl'curl M = dS to

(1.177)

Here to is the unit vector tangential to the vector lines B. Let usremember the following.

In the first expression, t is an arbitrary contour and the integraldefines the number of vector lines of the field B = curl M intersecting anysurface bounded by the contour. At the same time, this circulation doesnot provide any information about the distribution of lines within suchsurfaces. In particular, the circulation can be zero even in the presence ofvector lines of the field B. This happens when there are equal amounts ofpositive and negative passages of these lines through the surface.

In the second expression t is a small contour surrounding the elementdS, located in the plane perpendicular to the unit vector to' Correspond-ingly, curl M characterizes the density and direction of its vector lines.


Finally, the third expression describes a distribution of these lines on asurface S. As in the previous section it is natural to raise the question:Why do we pay so much attention to the distribution of vector lines of thefield curl M? It turns out that the answer is very simple, and in the nextsection we will show that these lines allow us to visualize the distributionof the second type of field generators called vortices.

11. Now we will prove the second theorem that plays a very importantrole in the theory of fields. First we will start from the quasi-potential fieldand consider the dot product.

curlM'dS

Here dS is an elementary surface, arbitrarily oriented with respect tocurlM. In accordance with Eq. 0.155) this product equals the number ofedge lines passing through the element dS. Then, applying the principle ofsuperposition, the total number of edge lines intersecting an arbitrarysurface S is proportional to

f curlM· dSs

(1.178)

(1.179)

(1.180)

On the other hand, this number of edge lines is proportional to thecirculation of the field along the contour surrounding this surface.

¢.M'de't

In both cases the coefficient of proportionality f3 is the same. Therefore,we obtain

f curlM' dS =~M' dl'S I

Here it is essential that the direction of the normal to the surface Sandthat of the displacement along the contour de' are related to each other bythe right-hand rule. Thus, we have proved Stokes' theorem for quasi-potential fields, which establishes the relation between values of the fieldat points of the surface and the contour surrounding it.

Next we will show that Stokes' theorem remains valid for an arbitraryfield M. With this purpose in mind let us imagine that any surface S isrepresented as a system of elementary surfaces dS i , surrounded by thecontour t i (Fig. 1.14d). In accordance with Eq. (1.176) we have

dS; . curl M = ¢.M . dl'Ii

(1.181)


Then, performing a summation of this equality, written for every elemen-tary area dS], and taking into account that integration along everyinternal contour is done twice but in opposite directions, we obtainagain Eq. (1.181),

f curl M.: dS =~M' dl's t

(1.182)

which is valid for any vector field M. This is Stokes' theorem.Undoubtedly both Gauss' and Stokes' theorem represent a foundation

of the theory of fields applied in geophysics, and many important relationswill be derived making use of these theorems.

In this connection it is proper to notice that both theorems are valid atthose points where field M does not have singularities. In particular, ifsuch points are located at a surface and the field has finite values, thefunctions div M and curl M are replaced by their surface analogies, respec-tively.

1.8 Two Types of Fields and Their Generators: Field Equations

In the last two sections we introduced div M and curl M as operators thatallow one to characterize the distribution of terminal points of vector linesand edge lines of normal surfaces of the quasi-potential field M, respec-tively.

Now it will be demonstrated that these operators playa fundamentalrole in the theory of vector fields. With this purpose in mind, let us firstconsider the quasi-potential field M.

M = 'P( P)grad U( p)

Here 'P(p) and U( p) are continuous scalar functions.Then, in accordance with results derived earlier we can write

(1.183)

divM= Q DivM=Qs

(1.184)

curlM = W CurlM = Ws1M' dl'= w*t

Here Q*,Q, Qs and W*' W,Ws are scalar and vector functions thatdescribe the distribution of terminal points and edge lines. We will treatEqs. 0.184) from two completely different points of view. First, supposethat the field M is known everywhere, and we would like to find places

1.8 Two Types of Fields and Their Generators: Field Equations 67

where, for some reason, vector lines and normal surfaces have terminalpoints and edge lines, respectively. At any point p, where the firstderivative of the field exists, it can be done by calculating divM andcurlM,

Q(p) = divM and W(p) = curlM (1.185)

while at interfaces, as one of the field components is discontinuous, wehave

and (I.186)

Of course, instead of Eqs. (I.185), 0.186) one can use the integral form

and (I.187)

Indeed, by choosing a sufficiently small volume surrounded by a closedsurface, and also a properly oriented small contour, one can again deter-mine a distribution of terminal points and edge lines.

Thus, in accordance with Eqs. (I.185)-(I.187) we can consider div Mand curl M as well as the flux and the circulation as indicators of specialplaces where terminal points and edge lines are located, and they charac-terize their distribution. This is the first approach of interpretation ofEqs. (I.184), which in essence gave the motivation to introduce divergenceand curl. However, we will now treat these equations in a completelydifferent way.

Suppose that terminal points of vector lines and edge lines of normalsurfaces are given, that is, functions

and

are known. Then, at the usual points where derivatives of M exist, we willtreat relations

curlM =W and divM= Q (I.188)

as equations with respect to the unknown field M.At points where the first derivatives do not exist we have instead of

Eqs. 0.188) their surface analogies.

CurlM = Ws and DivM = Qs (1.189)

Also, in accordance with Eqs. 0.184) the third form of the equations with


respect to unknown field M is

and (1.190)

Let us note that the left and right-hand sides of Eqs. (1.188), 0.189)describe the behavior of the field M and functions Q and W in the vicinityof the same point, while Eqs. 0.190) relate to each other all thesefunctions M, Q*, and W*, at different points. It is also essential that theseequations do not contain derivatives and, correspondingly, they can beapplied everywhere.

In considering the first approach, we have seen that determination ofthe density of terminal points as well as that of edge lines and theirorientation requires a differentiation of the field M, and as soon as thefield is known, it is a straightforward problem that can be performedwithout any difficulties. A completely different situation holds when wewould like to find the field M, assuming that functions Wand Q areknown.

Let us start from Eqs. (1.188) and raise three questions concerningthese equations, namely,

1. How do we solve these equations?2. What is the meaning of the right-hand side of Eqs. (1.188), 0.189)?3. Do these systems define the field M uniquely?

The first question presents itself mainly as a subject of mathematicalphysics, and it is not studied in this monograph. However, in some of thesechapters several examples illustrating a solution to these equations havebeen given. Because of this we will restrict ourselves to their briefdescription only. In accordance with Eqs, (1.188) we have four differentialequations with partial derivatives of the first order. In fact, in a curvilinearorthogonal system of coordinates this system is written as

(I.191)


and

Suppose that there is also a surface of singularity where the field Meanbe a discontinuous function. Then from Eqs. (1.189) we have

where all notations are given in the previous two sections.Again we have equations, but linear ones, relating unknown values of

the tangential and normal components of the field from both sides of thesurface S with given functions Ws(p) and Q/p).

Let us examine the second question. First of all, we assume thateverywhere in a space terminal points and edge lines are absent; that is,

and

curlM = 0

CurlM = 0

divM = 0

DivM= 0

(1.193)

(I.194)

Both pairs of equations are rather complicated, but it turns out that it issurprisingly easy to find the field M satisfying these systems.

In fact, inasmuch as

curlM = 0 and CurlM = 0 (I.195)

open normal surfaces having edge lines are absent. Also, the field does nothave closed vector lines. If there were even one closed vector line, thecirculation ¢fM . d/ along this line could not be equal to zero since boththe field M and the displacement dl' have the same direction. Then bydefinition, curl M and Curl M would differ from zero, too.

From equations,

div M = 0 and Div M = 0 (I.196)

it directly follows that open vector lines, having by definition terminalpoints, are absent. Also we have to conclude that closed normal surfacesdo not exist.

If there were such surfaces, the flux through them would differ fromzero since both vectors M and the normal n to the surface have the samedirection. This result means that somewhere inside the volume surroundedby such a normal surface, there are terminal points of vector lines; that, ofcourse, contradicts Eqs. 0.196).


Now we present results of this analysis in the form of a table.

Equations

curlM = 0CurlM = 0

no closedvector lines

no opennormal surfaces

divM=ODivM=O

no openvector lines

no closednormal surfaces

It is obvious that the same conclusion can be derived from the integralform of Eqs. (1.193), (1.194).

~M' dl'= 0f

and ~M' dS =0s

(1.197)

Inasmuch as one can imagine only two types of vector lines and normalsurfaces, namely open and closed ones, the table vividly shows that thefield M, satisfying the systems 0.193), (1.194), must be everywhere equal tozero. In other words, to have a field M that is different from zero, thereshould be places in a space where either terminal points of vector lines oredge lines of normal surfaces or both of them are located; that is, withouttheir presence the field cannot exist. Taking into account that only fieldgenerators such as charges, currents, stresses, etc., can create the field, wehave to conclude that these generators of the field are located at thoseplaces where terminal points of vector lines and edge lines of normalsurfaces are present. In essence, we can say that these points and lineshave been introduced to visualize a distribution of field generators. Corre-spondingly, an interpretation of the system of Eqs, (1.193), 0.194) becomesobvious. Inasmuch as the field M satisfying this system does not havegenerators, it is natural that it equals zero.

M=O (1.198)

Let us make one more step and classify field generators into two types,called sources and vortices.

I Field generators I

/ ~I sources I Ivortices I

Sources are a type of field generators whose distribution can be describedby terminal points of vector lines. Classical examples of sources aregravitational masses and electric charges.


Vortices are the second type of generators, and in the case of aquasi-potential field they can be described with the help of edge lines ofnormal surfaces located inside of some toroidal tubes. Unlike the first typeof generators, vortices are characterized by both a magnitude and adirection. Electric current and a change of the magnetic field with time arewell-known examples of vortices.

It is natural to expect that different types of generators create fields ofvery different behavior, and for this reason it is proper to consider themseparately. Depending on the type of generators, we will call fields eithersource or vortex fields, respectively.

I Field type I

Source field Vortex field

General case dueto both the source

and vortex generators

Now it is time to return to Eqs. (1.188), (1.190). Taking into account.their importance, let us write them as one system again.

( a) ~M' dl'= W* ~M' dS =Q*t s

(b) curlM=W divM = Q(I.199)

CuriM -w DivM = Qs

On the right-hand side the functions W *, W, Ws and Q *, Q, Qs describegenerators of the field, and on the left-hand side we have the field M alongwith its derivatives caused by these generators. What could be a morenatural relation than that between the cause and its effect? And this fact isthe main reason why Eqs. (1.199) are called systems of field equations inthe integral (a) and differential (b) forms.

At the beginning of this section we assumed that the field M is aquasi-potential field; that is,

M(p) =cp(p)gradU(p) (1.200)

and correspondingly the system of Eqs. (1.199) was derived making use ofterminal points of vector lines as well as edge lines of normal surfaces.

72 1 Fields and Their Generators

Before we demonstrate that this system remains valid for any type offield, let us investigate a distribution of generators of the quasi-potentialfield in terms of the function, 'P(p) and vector P, where

P = gradU

Taking into account that

curl M = curl 'PP = 'P curl P - P X grad 'P

= -P X grad 'P

since

curl P = curl grad U == 0

and

(1.201)

(1.202)

(1.203)

divM = div 'PP = 'PdivP + p. grad 'P

within a constant scale factor, the generators of the field M at usual pointsare

w = - P X grad 'P

Q = 'P div P + P . grad 'P

Analysis of Eqs. (1.202), (1.203) allows us to make several comments,such as

1. The field P, expressed through the gradient of function U, is theclassical example of a source field and is often called the potential field.

2. Vortices of the field M are located only in the vicinity of pointswhere vector P and grad 'P are not parallel to each other.

3. Sources of the quasi-potential field consist, in general, of two parts.One of them appears near those points where sources of the potentialfield P exist if 'P(p) =F 0, while the second part of sources arises in placeswhere vectors P and grad 'P are not perpendicular to each other. Ofcourse, these two types of sources can appear at the same as well as atdifferent points. In particular, they can cancel each other if

'P div P = - P . grad 'P

so that at some points the total density of sources vanishes.

Now we will consider surface distributions of sources and vortices of thequasi-potential field.

Inasmuch as


we have

CurlM =" X H( 'Pz - 'PIHPz + PI) + ('PZ+ 'PI)(PZ- PI)}

where 'PZ' Pz , and 'PI' PI are values of the scalar and vector functions atthe front and back sides of the surface, respectively.

Taking into account Eq. (1.201) we can show that the tangential compo-nent of the potential field P is a continuous function; that is,

Therefore, the surface vortices are

Ws =" X t( 'Pz- 'Pl)(PZ+ PI) =('Pz - 'PI)" X pay

orWs = - pay X Grad 'P

Here

is the average value of the potential field P at the surface.By analogy we have for the surface sources

Div M = M(Z) - M(1) = m p(2) - m p(1)n n "1""'2 n ..,..1 n

or

Qs = p av• Grad 'P + 'Pav DivP

Here

(1.204)

(1.205)

av 'PI + 'Pz'P =

2

It is obvious that Eqs. (1.204), (1.205) are very similar to the correspondingEqs. (1.203), and as in the case of volume distribution of field generators, itis appropriate to make the following comments:

1. Surface vortices arise at the vicinity of points where a discontinuityof function 'P(p) takes place and also where the tangential component ofthe potential field P is not equal to zero.

2. In general, there are two types of surface sources. One type arisesexactly in places where sources of the potential field are located if'P I * - 'Pz· At the same time the second type of source appears nearpoints of discontinuity of function 'P(p), provided that the normal compo-nent of the field P does not vanish.


Let us illustrate the quasi-potential field by one example. As followsfrom Ohm's law the current density j(p) and the electric field E(p) arerelated in the following way:

j(p) = y(p)E(p)

where y( p) is the conductivity of the medium and, in general, changesfrom point to point. Inasmuch as the electric field of constant charges is apotential field, we have

j(p) = ~y(p)gradU

That is, the current density field is a quasi-potential field caused bysources as well as vortices.

As was pointed out above in deriving Eqs. (I.199), both vector lines andnormal surfaces have been used, and correspondingly its application wasrestricted to the quasi-potential fields only. Now we will generalize thisresult and show that it is natural to treat Eqs. (1.199) as a system of fieldequations valid for any field. With this purpose we will completely discardthe concept of normal surfaces and make use of vector lines. Our goal is todemonstrate that for any kind of field M, the right-hand side ofEqs. (I.199) still consists of field generators.

Applying the same approach as in the case of the quasi-potential field,we will assume that everywhere, including usual points and surfaces wherethere may be singularities, the field M satisfies the following equations:

From equations

curlM = 0

CurlM = 0

divM = 0

DivM=O(1.206)

curl M = 0 and Curl M = 0

as well as Stokes' theorem, it follows that the circulation of M along anypath must be equal to zero.

Inasmuch as at every point of the vector line both vectors M and dl' havethe same direction, that is,

M· dl'> 0

we have to conclude that the field M does not have closed lines.At the same time, in accordance with equations

divM = 0 and DivM =0

vector lines of this field cannot be open since they have no terminal points.


Thus, we have proved that the field M satisfying Eqs, (1.205) has neitherclosed nor open vector lines, and this simply means that the field M iseverywhere equal to zero. This analysis leads us again to the conclusionthat if at every point space functions W,Ws and Q,Qs are equal to zero,then the field M also vanishes everywhere; for this reason it is natural tosuppose that these functions, as in the case of quasi-potential fields,characterize a distribution of sources and vortices.

Correspondingly, we will assume that for an arbitrary field we have thesame system of equations as in the case of the quasi-potential field; that is,in the differential form

curlM=W

CurlM = Ws

and the integral form is

divM=Q

DivM= Qs(1.207)

and (1.208)

Here M is an arbitrary field, but W,Ws , W* and Q I , Qs, Q * are itsvortices and sources, respectively.

Here it is appropriate to make several comments.

1. The system of field equations, as in the case of the quasi-potentialfield, is a "bridge" between the field and its generators.

2. Each field applied in geophysics has a different physical nature, andcorrespondingly they are governed by different physical laws, such asNewton's, Hooke's, Biot-Savart's, Faraday's, etc. At the same time, thefield equations are always derived from physical laws, as indicated below.

I' I System ofPhysical laws ~ fi ld tiie equa IOns

3. Usually functions E, E, and Q1 and Qs differ from a correspondingdensity of field generators by some constant that is independent of theposition of the observation point.

4. Fields are often studied in the presence of some medium. Due tothis fact some part of generators cannot, in principal, be known beforedetermination of the field. Correspondingly, most of the physical laws suchas Coulomb's, Hooke's, Faraday's, etc., which establish a connectionbetween a field and its generators, become useless for field calculation,and the system of field equations turns out to be the only means of solvingthe problem. At the same time, considering carefully this system, weusually observe the "closed circle problem" since the right-hand side of


Eqs. (1.207), (1.208) is not known. In fact, to find the field M we have toknow the distribution of its generators, but some part of them cannot, inprinciple, be known before the field calculation.

To overcome this problem a new field is usually introduced that isrelated to the field M and its generators, with the help of correspondingparameters of a medium. Respectively, instead of the system of twoequations, we often obtain a system of four equations with respect to twovector fields. The classical example of such a system is Maxwell's equa-tions.

Now we are ready to discuss the third question concerning the unique-ness of a solution of Eqs. (1.207). First of all, from a physical point of viewit is almost obvious that if a distribution of field generators is everywhereknown, the field is uniquely defined by them. Let us illustrate this fact byone example. Suppose there is a certain distribution of chairs and tables insome room that creates a gravitational field, and outside the room massesare absent. In other words, these chairs and table are the sale generatorsof the field which, in accordance with Newton's law of attraction, stronglydepends on a distance between the observation point p and the masses.

If we vary the position of even one chair slightly, but the others remainat rest, then certainly a change of the field will be observed. Of course, atsome points located closer to the moved chair, a strong change of the fieldtakes place while far away from this mass there can be a very smallchange, even unmeasurable by modern sensors. However, it is essentialthat as soon as there is a new distribution of generators, a new field arises,and correspondingly it is proper to expect that the system of Eqs. (1.207)uniquely defines the field. Let us derive the same result making use of adifferent approach. Suppose that there are two different fields, Mj(p) andMip), and that both of them satisfy the system of Eqs. 0.207) with thesame right-hand side; that is,

curl M, = W(p) divM, = Q(p)

CurlM l =Ws(p) DivM j = Qs(p)

and (1.209)

curlM2 = W( p) divM, = Q(p)

Curl M, = Ws( p) DivMz = Qs( p)

Now we will form the difference between these two fields.

which is also a vector field.


Taking into account that curl and divergence are linear operators, wehave

(1.210)curl M, = 0 div M, = 0

curl M, = 0 DivM3 = 0

Inasmuch as Eqs. (1.210) describe the field M 3 everywhere, we can applythe results derived above and conclude that this field equals zero. In otherwords, two fields M] and M 2 , caused by the same distribution of genera-tors, coincide with each other. Therefore, we have proved again that thesystem of Eqs, (1.207) uniquely defines the field provided that its genera-tors are known. everywhere.

Our next step is to study the problem of uniqueness in a morecomplicated case, in which field generators are given only in a certain partof the space. To investigate this problem, let us first suppose that genera-tors are absent in the volume V where the field is studied, but they can belocated outside, creating the field M everywhere including observationpoints of the volume V. Then, taking into account that the functions W,WS ' Q, and Qs are equal to zero within volume V, the system ofEqs. (1.207) becomes homogeneous, and we have

curlM = 0

curlM = 0

divM = 0

divM = 0(1.211)

(1.212a)

(1.212b)divM = 0

DivM = 0

divM=Q

DivM=QsCurlM=Ws

(b) Outside the volume

curlM = 0

CurlM = 0

Thus, the system of equations describing the field behavior within somevolume V is homogeneous if its generators are located outside. It isobvious that by changing the distribution of generators, the field M causedby them also varies, but the system of equations still remains uniformwithin this volume. Correspondingly, we can say that Eqs. (1.211) have aninfinite number of solutions and every one of them can be interpreted as afield whose generators are distributed somewhere outside the volume V.

Next we will assume that field generators are known inside the volumeV, while they are absent outside of it. To some extent, this case is theopposite of the previous one. Then the system of field equations is

(a) Inside volume V

curlM=W

Here W, WS ' Q, and o; are given functions.


It is easy to show that this system uniquely defines the field. In fact,suppose there are two fields, Mt(p) and M 2( p ), satisfying Eqs. (1.212).This means that both of the fields are caused by the same generatorslocated only inside the volume V. Then it is clear that the difference ofthese fields

satisfies everywhere the homogeneous system

and therefore

curl M, = 0

Curl M, = 0

divM, = 0

DivM 3 = 0

or

In other words, the system of Eqs. (1.212) uniquely defines the field M.That is hardly surprising since all of its generators are known.

Now we are ready to discuss the uniqueness of a solution of the system(1.207) provided that the field generators are known only within thevolume V where the field is considered. Unlike the previous case, therecan also be generators of the field outside the volume that are notspecified, and for this reason it is natural to expect that Eqs. (1.207) do notuniquely define the field M. Indeed, suppose that two fields Mt(p) andMip) are solutions of this system. Then their difference

(1.213)

is a solution of the homogeneous system within the volume V.

curl M, = 0

Curl M; = 0

divM, = 0

DivM3 = 0

As was shown above, it means that in general the field M3 differs fromzero since its generators can be located somewhere outside.

Thus, we have demonstrated that the system of field equations definedwithin some volume V has an infinite number of solutions that differ fromeach other by functions describing fields due to generators located outsidethe volume V. Let us represent the solution of Eqs. (1.207) as a sum.

(1.214)

where Mi(p) and Me(p) are fields caused by generators inside and outsidethe volume, respectively. As was shown, the field Mi(p) is uniquelydefined, inasmuch as its generators-that is, functions W, Ws' Q, and Qs


-are known. However, the field Me(p), regardless of distribution of itsgenerators, satisfies a homogeneous system of field equations; this factexplains the nonuniqueness of the solution of Eqs. (1.207). Because thissystem, defined in some volume, is not sufficient by itself to determine thefield, it is natural to raise the following question: What has to be known inaddition to the system of field equations in order to determine the field Muniquely? In this section we will not describe this question in detail, butinstead, just outline this problem.

With this purpose in mind, first we will consider a very simple casewhen all generators of the field are located within the volume V while thefield M due to these generators is studied outside of it. Inasmuch as in thisexternal part generators are absent, the system of field equations ishomogeneous,

curlM = aCurlM = 0

divM = 0

DivM =0(1.215)

regardless of the position of generators within the volume V, and corre-spondingly it has an infinite number of solutions.

If we do not know the distribution of field generators inside the volumeV but are trying to find the field outside, we have to have some informa-tion about the field that can replace the absence of knowledge about itsgenerators. And it is almost obvious that the surface S surrounding thevolume V is the most natural location for such information.

Then the system of Eqs, (1.215) and certain information about the fieldbehavior on the surface S may provide a unique solution. Later we willdemonstrate that in fact the so-called theorem of uniqueness can beformulated in terms of several types of such information on the surface S.Having accepted this fact, we can say that all information providing theuniqueness of the solution of Eqs. 1.215 consists of two parts, namely,

(a) Outside the volume V

curlM = 0

CurlM = 0

divM = 0

DivM = 0

and(b) Information about the field behavior on the surface S, usually the

behavior of some component of the field M, which can be represented as

(1.216)

Here M, is either the tangential or normal component of the field and CPtis the given function.


Therefore, determination of the field includes two steps.

1. Solution of Eqs. 0.215). As was shown above, an infinite number ofvector functions satisfy this system.

2. Choice among these solutions of a field M such that its correspond-ing component becomes equal to the given function 'Pip) on the bound-ary surface S.

Taking into account the importance of this last step, the whole processof determination of the field is called a solution of a boundary-valueproblem.

Here it is also appropriate to make several comments.

(a) The same analysis can be applied to the case in which the genera-tors are located outside the volume V, but the field is considered insideof V.

(b) If a solution of Eqs. (1.215) is considered inside the volume V,surrounded by several surfaces, boundary conditions of either type (1.216)or others have to be defined at every surface.

(c) Usually these boundary conditions are derived from an analysis of aphysical nature of the problem, and as a rule they characterize a fieldbehavior near the given generators that are known and at infinity.

(d) In general, regardless of the type of boundary condition, they donot uniquely define a distribution of generators outside of the volume V.

Now let us generalize on the previous case and assume that generatorscan be distributed everywhere, but that the field M is sought within thevolume V, surrounded by the surface S. Then the solution of the valueboundary problem, as before, consists of two steps, namely,

(a) Solution of the system of field Eq. (1.207) within the volume V,and(b) Selection of such solutions of this system that satisfy the boundary

conditions on the surface S.

In conclusion to this section let us consider one more subject related tothe presentation of the field caused by both types of generators. First, wewill represent the field M within some volume V as a sum of three fields,

(1.217)

where M 1 and M 2 are fields caused by sources and vortices distributedwithin the volume V, respectively, while M3 is the field due to generatorslocated outside this volume. Then, by definition they satisfy the following

1.9 Harmonic Fields 81

systems of field equations:

curl M, = 0

Curl M, = 0

curl M, =W

CurIM2 =Ws

and

cUrIM 3 = 0

Curl.M, = 0

div M, = Q

DivM1 = Qs

divM, = 0

DivM2=O

div M, = 0

DivM 3 = 0

(1.218)

(1.219)

(1.220)

In the next sections we will study some general features of these fields,and it is natural to begin this analysis from the simplest field M3 , which ofcourse is a partial case of either field M, or field M 2 •

Also it is useful to note that the field M can be presented as

M=Mj+Mi +M3

where Mj is the quasi-potential field

Mj =cpgrad U

(1.221)

(1.222)

generators of which are located within the volume V; Mi is a field causedby vortices only, located also within this volume; but M 3 is a field due togenerators distributed somewhere outside of it. Then, we have

curlMj = wt divMj = Q

Curl M] = Wl~ DivMj = Qs

curlMi = W2* divMi = 0

Cur.lMi = W~ DivMi = 0

curl M, = 0 divM, = 0

Curl M, = 0 DivM 3 = 0

where W = W)* + W2* , Ws = W)~ + W~, and Q, Qs ,W,Ws are generatorsof the field Mwithin the volume V.

1.9 Harmonic Fields

We will study a field M in some volume V assuming that the generators,sources, and vortices of the field are located outside of the surface S thatsurrounds this volume. Then, as was shown before, the field satisfies the


homogeneous system of equations

curlM = 0

CurlM = 0

or

~M' dl'=Ot

divM = 0

DivM=O

~M'dS=Os

(1.223)

In the next two sections we will consider fields in the presence ofsources and vortices, but it is obvious that results derived from Eqs. (1.223)can be directly applied to such fields, provided that they are studied nearpoints where generators are absent.

Now proceeding from the system (1.223) let us describe in detail thebehavior of a field M.

1. First, it is clear that inside the volume V vector lines and normalsurfaces of the field M do not have terminal points and edge lines,respectively.

2. Inasmuch as field generators are absent within the volume V, thefield M does not have singularities; that is, an points are usual ones. Inparticular, if we imagine some surface within this volume, then near itspoints the field behaves as a continuous function and correspondingly,

(1.224)

Here M(2) M(2) and M(1) M(1) are the tangential and normal compo-t' nt' n

nents of the field at the front and back sides of the surface, respectively.3. As follows from the system 0.223) the circulation of the field M

equals zero within the volume V, regardless of the contour

~M'dl'=Ot

(1.225)

It is essential that this equality is based solely on the fact that vortices areabsent in this part of space, and correspondingly it remains valid also forsource fields. We will consider an arbitrary closed contour t inside V(Fig. Li5a). Then the voltage along a closed path t can be represented asthe sum of two voltages,

(1.226)


Fig. US (a) Circulation; (b) presentation of Laplace's equation through finite differences;(c) illustration of Green's formula; and (d) illustration of Green's formula.

where

t'=t'1+t'2

Changing the direction of the path t'2 to the opposite one-that is,t'; -we obtain

or

(1.227)

where t'1 and tt are two arbitrary paths from point a to point b. Inother words; the voltage between the two points a and b is path indepen-dent and is defined only by the position of these points.

To calculate the voltage we have to divide the path of integration into acertain number of elements so that within every element dl' both themagnitude and the direction of field M do not change. Then, taking thedot product and performing summation, we obtain a value for the voltage.If the position of the path is varied, we have to use other values of the


field M for determining the voltage along a new path with a differentlength as well as different directions of its elements. For this reason, it isnatural to expect that in general the voltage between two points dependsessentially on the path along which it is calculated. It is a great surprisethen that equality (1.227) shows independence of the voltage of the path ofintegration for fields described by Eqs. (1.223). For instance, it turns outthat the voltage of the gravitational field between two points located ata distance of one meter remains the same whether it is calculated along astraight line with a length of one meter or along an arbitrary path having alength of thousands of kilometers. Let us note that the equality (1.227) canbe derived from the fact that the field M is the simplest case of aquasi-potential field, as any path between two points located in the volumeV is intersected by the same number of normal surfaces.

Now we will demonstrate that the absence of vortices within the volumeV also permits us to express the field M through a scalar field U, whichfacilitates the study of the vector field. With this purpose in mind, we willwrite the first equation of the system 0.223),

in the form

curlM=O (1.228)

curl, M = _1_ { ah 2 M2 _ ah! M1 } = 0h!h 2 ax! aX2

Then it is easy to see that a solution of Eq. (1.228) can be written as

M = -gradU

or (1.229)

For instance, we have


The scalar function Ui.p) is usually called the potential of the field M, andasa rule it is related to the generators in a simpler way than the fielditself.

Here it is appropriate to make several comments, namely,

(a) The field M can be described with the help of an infinite number ofpotentials V, which differ from each other by any constant C.

(1.230)

where V\(p) and Vip) are an arbitrary pair of potentials. This equalityfollows from the fact that grad C == O. This ambiguity in the determinationof the potential V shows that it does not have any physical meaning, but itis an auxiliary function allowing us to simplify the analysis of the vectorfield M.

(b) The quasi-potential field was introduced as

M=q>(p)gradV

and correspondingly we can say that a field satisfying Eqs. (1.223) repre-sents the simplest case in which q>(p) = ± l.

(c) In general the potential V, as well as the field M, can be a functionof the observation point coordinates and of the time of measurement t.

(d) As soon as the potential V(p) is known, Eq. (1.229) permits us tofind the field M in the volume V by the simple operation of differentiationwith respect to coordinates of the point.

(e) It is clear that solution of Eqs. (1.228) can be represented as eitherM = grad V or M = - grad V, and both of these relations are used in thismonograph.

Now we will show that the introduction of the potential V essentiallysimplifies the calculation of the voltage. By definition we have

fb M . d/ = - fb grad V .d/a a

It is also clear that the dot product of vectors

and

(1.231)


equals

au au augradU'dt"= -dx l + -dx2 + -dx3aX l aX2 aX3

or

grad U . dt" = dU

where dU is the differential of the potential U.Substituting Eq. (1.232) into Eq. 0.231) we obtain

t M ' dt"= - t dU= U(a) - U(b)a a

(1.232)

(1.233)

Thus, to calculate the voltage it is sufficient to take the difference ofpotentials at the points a and b. This, of course, is much simpler than anintegration of the field M. Let us also note that Eq. (1.233) can be used todetermine the potential U at any point of the volume V, provided that thefield M is known as well as the value of the potential at some given point.

Having introduced the potential U(p), it is natural to derive equationsdescribing its behavior. From the first field equation

curlM = 0

we have established that

M = -gradU

Then substituting this relation into the second field equation

divM = 0we obtain

divgrad U = 0

This process of derivation is shown below.

~I curl M = 0 I @J div M = 0 Ij j

IM = ±grad U !f--------->.! div grad U = 0 I

(1.234)

Therefore, the system of two field equations in differential form, describ-ing the field behavior at usual points, is replaced by one equation.

divgrad U = 0 (1.235)


Making use of the representation of div M and grad U in an orthogonalsystem of coordinates and denoting the operation

div grad as V2 = .6we obtain

(1.236)or

a (h zh3au) a (h'h3au) a (h'hz au )ax! T ax! + aX2 -,;; aX2 + aX3 -,;; aX3 = 0

We have derived one of the most important equations in the theory offields applied to geophysical methods; it is called the Laplace equation. Inparticular, in a Cartesian system

a2u a2u a2uVZU= - + - + - =0 (1.237)ax 2 ay2 az 2

In a cylindrical system

a au 1 a2u a2u

V 2U= -r- +--+r- =0ar ar r acp2 JZ2 (1.238)

(1.239)

In a spherical system

a ( au) a au- R2sinO- + - sin 0-JR aR ee ee

a ( 1 au)+ acp sin 0 acp = 0

Thus, instead of two partial differential equations of first order contain-ing three unknowns, M!, M2 , and M3 ,

curiM = 0, divM = 0we have obtained one differential equation of second order with partialderivatives of one unknown, U.

Next, we will consider the replacement of equations

CuriM = 0 and DivM = 0

by corresponding relations with respect to the potential U; this can bedone in the following way. As is well known, the equation M = - grad Umeans that the component of the field M in any direction t can bepresented as

auM ---

t - at (1.240)


Therefore, surface analogies of field equations

Mt(2)=MP)

can be written in the form

and

au(2) nr»at=at'

au(2) au (1)

--=--an an (1.241)

(1.242)

Here U(2) and u» are values of the potential U at each side of the surfacelocated inside the volume V. It is easy to see that from continuity of thepotential U follows continuity of tangential components of the field, andcorrespondingly Eqs, 0.240 can be slightly simplified.

mr» au(2)--=--

an an

Let us note that continuity of the potential at some surface does not meancontinuity of the normal derivative aujan, since its calculation requiresthe use of potential values at points not located at the surface.

Thus, we have obtained the system for the potential U, which isequivalent to the system (1.223):

(1.243)

on SU(1)= U(2),

and

eir» au(2)

an anAt the same time, taking into account that within the volume V generatorsare absent, we can essentially simplify both systems (1.223) and (1.243). Infact, all points inside the volume are usual ones and correspondingly thesystem of field equations and the potential are

curlM = 0, divM= 0

and (1.244)

LlU=O

Before we formulate the value boundary problem, let us consider theLaplace equation in detail.

In accordance with Eq. (1.237) we have

a2u a2u a2u

LlU= - + - + - =0ax 2 ay2 az 2

1.9 Hannonic Fields 89

That is, in the vicinity of every point of the volume V the sum of thesecond derivatives along the coordinate lines x, y, z equals zero. At thispoint it is natural to raise the following question. Does this fact reflectsome special features of the potential behavior?

To answer this question we will take an arbitrary point p within thevolume V and imagine an elementary cube around this point with sidesalong coordinate lines. The length of each side is 2h (Fig. USb). As is wellknown, the derivatives of the function at any usual point p can bereplaced by finite differences of this function, taken at points located nearthe point p. For instance, for the first and second derivatives along someline t we have

and

a2u

=_1 [aU(t+ dt) _ aU(t_ ~t)lat2 ~t at 2 at 2

First, applying these relations for derivatives with respect to coordinate xwe obtain

_au_(x_,y_,z_) = _1 [u(x + _LlX y z) _u(x _ _LlX. y z)]ax Llx 2 ' , 2 ' ,

and (1.245)

a2u(x,y,z)

=_I_[aU(x+ Llx y z)- au(x_ Llx y z)]. ax 2 ~x ax 2 " ax 2 ' ,

Here x, y, z are the coordinates of the cube center, that is, those of thepoint p.

Inasmuch as

au ( ~x ) 1ax x+T'Y'z = dX[U(X+LlX,y,Z)-U(x,y,z)]

au(x- Ll2X

,y,z) 1

-"'------'- = -[U(x, y, z) - U(x - ~x, y, z)]ax Llx


we have

aZu(x,y,z) 1--"""'z-- = -_z [U(x+ £\x,y,z)

ax (£\x)

+U(x-£\x,y,z)-2U(x,y,z)] (1.246)

where Ui;x + £\x, y, z) and Ui:x - £\x, y, z) are values of potential U onopposite faces of the cube perpendicular to axis x.

By analogy for the second derivatives with respect to coordinates y andz we have

aZu(x,y,z) 1_.....:....--z-....:.... = --Z [U(x, y + Ay, z) + U(x, y - Ay, z)

ay (Ay)

-2U(x,y,z)]

aZu(x,y,z) 1-""':""--2-":"" = --Z [U(x,y,z+£\z) + U(x,y,z-Az)

az (£\z) .

-2U(x,y,z)]

(1.247)

Taking into account that Ax = Ay = Az = h and substituting Eqs.(1.246), (1.247) into the expression for the Laplacian

a2u a2u aZu£\U=-+-+-ax 2 ay2 az 2

we obtain

V2U = -;[ t U;-6U(P)]h ;=1

or

(1.248)

Here U; is a value of the potential on the ith face of the cube, while o:»is its value at the center of the cube. It is clear that the term

1 6

-LV6 i=1 I

1.9 Harmonic .Flelds 91

is the average value of the potential at the point p,

1 6Uav(p) = - E Vi

6i~1

Correspondingly, the Laplacian can be rewritten as

(1.249)

Thus, the Laplacian of a scalar function is a measure of the differencebetween the average value of the function U'" and its value U at the samepoint.

For example, if the average value exceeds the value of the function, theLaplacian is positive. Now, making use of Eq, 0.237) we obtain thesimplest form of the Laplace equation.

Uav(p) - U(p) = 0 (1.250)

Therefore, we can say that if the function U satisfies the Laplace equation,then it possesses a remarkably interesting feature, namely that its averagevalue calculated around some point p is exactly equal to the value of thefunction at this point. Only a certain class of functions has this feature,and such functions are called harmonic.

Correspondingly, we can conclude that the potential U describing afield M

M= -gradU

is a harmonic function within the volume V, provided that the fieldgenerators are located outside of V. Let us also note that in accordancewith eq. (1.249) the Laplacian can be considered a measure that shows towhat extent some function differs from a harmonic.

In accordance with Eq. 0.237) the sum of second derivatives alongthree mutually perpendicular directions equals zero if the function U isharmonic. At the same time we know that in the one-dimensional casethere is also a class of functions y(x) for which the second derivative isequal to zero; that is,

(1.251)

and these functions are linear.From this comparison of second derivative behavior, it is natural to

assume that harmonic functions are an analogy to linear functions for two-and three-dimensional space and, correspondingly, have similar features.


Let us describe some of these that are, in fact, inherent for harmonicfunctions.

1. It is clear that if values of a linear function are known at terminalpoints of some interval, it can be calculated at every point inside of thisinterval. Consequently, if a harmonic function is known at every point ofthe boundary surfaces surrounding the volume, it can be determined atany point within this volume.

2. If a linear function has equal values at terminal points of theinterval, then it has the same value inside of the interval; that is, the linearfunction is constant. By analogy, if a harmonic function has some constantvalue at all points of the boundary surface, then it has the same value atany point within the volume. This is a very important feature of harmonicfunctions and has many applications. In particular, it allows us to give amathematical proof of the effect of electrostatic screening.

3. A linear function reaches its maximum and a minimum at theterminal points of the interval. The same behavior is observed for har-monic functions, which cannot have their extrema inside of the volume.Otherwise, the average value of the function at some point will not beequal to its value at this point and, correspondingly, the Laplacian woulddiffer from zero.

In conclusion of this comparison, it is appropriate to note that if linearfunctions are the simplest functions in the one-dimensional case, thenharmonic functions are the simplest ones in the two- and three-dimen-sional cases.

In the future we will call a field M satisfying the uniform system of fieldequations (1.223) a harmonic field, since its potential U is a harmonicfunction.

In Section 1.8 we formulated the boundary-value problem and empha-sized the importance of information about the behavior of the field at theboundary surface S surrounding the volume V. Now we will attempt tofind boundary conditions on S such that they uniquely define the harmonicfield M. With this purpose in mind, we will proceed from Gauss' theorem,which is a natural "bridge" between values of the field inside of thevolume and those at the boundary surface.

(1.252)

Here n is the unit vector perpendicular to the surface S and directedoutward, and X n is the normal component of an arbitrary vector X, whichis continuous function within the volume V.


To simplify our derivations we shall make use of the potential V, whichsatisfies Laplace's equation

v 2 V = 0 (1.253)

It is obvious that this equation has an infinite number of solutions thatcan, in particular, correspond to different distributions of generatorsoutside the volume. Let us choose any pair of them, V1(p) and Vip), andform their difference.

(1.254)

(1.257)

(1.256)

(1.258)

(1.259)

Inasmuch as Laplace's equation is linear, the function V/p) is also asolution. To derive boundary conditions, let us introduce some vectorfunction X(p), which has the form

X = V3 grad V3 = V3VV3 (1.255)

Substituting Eq. (1.255) into 0.252) we obtain

J. V(V3VV3 ) dV = ~ V3 grad, V3 dSv s

where grad, U3 is the component of the gradient along the normal n, and

aV3grad U =-n 3 an

It is proper to notice here that the boundary surface S surrounding thevolume can consist of several surfaces.

As is weI! known, the operator V is a differential operator and corre-spondingly we have

V(V3 VV3) = U3 V2V3 + VV3 ' VV3 = (VV3) 2

since

V2 V3 = 0

Taking into account both Eqs. (1.257), 0.258) we can rewrite Eq. (1.256) as

J. 2 ~ aV3(VU3 ) dV= U3 - dS

v s an

This equality, which in essence is Gauss' theorem, will aI!ow us toformulate the most important boundary conditions; but first let us maketwo comments.

(a) The integrand of the volume integral in Eq. (1.259) is nonnegative.(b) In equality (1.259), which relates the values of the function inside of

the volume V to its values on the boundary surface S, V3 is the differenceof two arbitrary solutions of Laplace's equation.

94 I Fields and Tbeir Generators

Now we are prepared to formulate boundary conditions for the poten-tial of the harmonic field M, which uniquely define it inside of the vol-ume V.

With this purpose in mind, suppose that the surface integral on theright-hand side of Eq. 0.259) equals zero. Then

f (VU3)2dV= 0v

and taking into account that its integrand cannot be negative, we have toconclude that at every point of the volume

grad U3 = 0 (1.260)

This means that the derivative of function U3 in any direction t is zero.

aU3

at = 0

Substituting this equation into Eq, (1.254) we discover that if the surfaceintegral in Eq. (1.259) vanishes, then the derivatives of solutions ofLaplace's equation in any direction t are equal to each other

In other words, these solutions can differ by a constant only; that is,

(1.261)

where C is a constant that is the same for all points of the volume V,including the surface. Note that C can be zero.

Now we will define conditions under which the surface integral

au,(.. U3 -

3dS (1.262)

~ anvanishes, and correspondingly Eq. 0.261) becomes valid. At least threesuch conditions are described below.

Case 1

Suppose that the potential Ut.p) is known on the boundary surface; that is,

on S (1.263)

and we are looking for a solution of Laplace's equation that satisfies thecondition (1.263). Let us assume there are two different solutions to this


equation inside of the volume; Uj(p) and U2(p), which coincide on theboundary surface.

on S

Then their difference U3 on this surface becomes equal to zero.

on S

and consequently, the surface integral in Eq. (1.262) vanishes.Therefore, in accordance with Eq. (1.261), solutions of Laplace's equa-

tion satisfying the condition (1.263) can differ from each other by aconstant only. However, this constant is known, and it equals zero since onthe boundary surface all solutions should coincide. In other words, wehave proved that two equations,

and

on S

(1.264)

or

uniquely define the potential U as well as the field M, since

M= -gradU

Equations (1.264) are the Dirichlet boundary value problem.It is proper to notice that in accordance with Eq. (1.263) we can say that

along any direction t tangential to the boundary surface, the componentof the field M 1 is also known, since M 1 = -au/Bt. Consequently theboundary value problem can be written in terms of the field M as

curlM = 0 divM = 0

and (1.265)

acpM = -- on S

I at

This case vividly illustrates the importance of the boundary condition

acpjM=--

t atIndeed, Laplace's equation or the system of field equations have aninfinite number of solutions corresponding to different distributions of


generators located outside of the volume. Certainly we can mentallypicture unlimited variants of the generator distribution and expect aninfinite number of different fields within the volume V. In other words,Laplace's equation as well as Eqs, (1.223) provide relatively limited infor-mation about the field; namely, they tell us only that generators are absentinside the volume. In contrast, the boundary condition 0.263) is vitalinasmuch as it was proved that only one harmonic field satisfies thiscondition inside V.

Here it is appropriate to make two comments.

(a) Taking into account that the boundary condition (1.263) uniquelydefines the field, it is natural to expect that there is an equation thatallows us to find the potential U at every point of the volume if it is knownon the boundary surface. As was pointed out above, this fact demonstratesan analogy between linear and harmonic functions.

(b) We have proved that Eqs. (1.264) uniquely define the potentialU(p). However, it is obvious that uniqueness of the field determination isachieved even if the potential is defined only to within a constant, sinceM= -VU.

( 1.266)on S

Now we will consider one example illustrating efficiency of the use ofDirichlet's problem.

Suppose that the potential «» is constant on the boundary surfaceand, correspondingly, the value-boundary problem is formulated as

I1U=O

U(p) = C

As we have proved, there is only one harmonic function, U(p), thatsatisfies Eqs. 0.266). It turns out that it is very simple to find this function.In fact, let us assume that the potential within the volume is also constant.

U(p) = C in V (1.267)

It is easy to see that this assumption is correct. First of all Laplace'sequation is a sum of second derivatives that is equal to zero, and thereforethe constant function U(p) = C is a solution. At the same time, theboundary condition is automatically satisfied since we have chosen asolution that coincides with the value of the potential on the boundarysurface. Inasmuch as both equations of (1.266) are satisfied, our assump-tion is valid, and the potential U(p) within the volume is also constant,which is equal to that on the boundary surface.

It is essential that due to the uniqueness of the solution of Dirichlet'sproblem, we can say that there is no other solution satisfying Eqs, (1.266).Here we see again an analogy between linear and harmonic functions,


since both of them do not vary within a corresponding interval if they havethe same values at terminal points. As follows from the equation M =- grad U, the field M vanishes within the volume where the potential isconstant.

We can imagine different applications of this example, and in particu-lar, it shows that the principle of electrostatic screening can be proved bysolving Dirichlet's problem.

Case 2

Now let us assume that two arbitrary harmonic functions within thevolume V, U](p) and Uip), have the same normal derivative on thesurface S; that is,

au] et),-=-=ip (p)an an z

on S (1.268)

where ipip) is a known function.From this equality it instantly follows that the normal derivative of a

difference of these solutions vanishes on the boundary surface.

on S

Therefore, the surface integral in Eq. (1.259), as in the previous case,equals zero and correspondingly inside of the volume we have

This means that any solutions of Laplace's equation-for instance,U](p) and Uip)-can differ from each other at every point of the volumeV by a constant only if their normal derivatives coincide on the boundarysurface S.

Thus, this boundary value problem, which also uniquely defines thefield, can be written as

and

on S

(1.269)


or

and

curlM = 0 divM =0

(1.270)

on S

and is called Neumann's problem.Unlike the previous case, Eqs. 0.269) define the potential only to within

a constant, but of course the field is determined uniquely.

Case 3

We suppose that the boundary surface S is an equipotential; that is,

U(p) = C on S (1.271)

(1.272)

In addition, it is assumed that the following integral:

au~ an dS = 'P3( p)

is known. Here S in the boundary surface.Now we will show that two harmonic functions U\(p) and Uip),

satisfying Eqs. 0.271-1.272), can differ from each other by a constant only.Consider again the surface integral in Eq. 0.259).

auA:. U3 -

3 dS'J1, an

Inasmuch as the boundary surface is an equipotential surface for bothpotentials U\ and U2 , their difference, U3(p) = U\(p) - Uip), is alsoconstant on this surface, and consequently we can write

¢, aU3 ~ aU3U3-dS= U3 -dS

s an san

Then, taking into account Eq. 0.272), we have

~ aU3 pau3 {~au\ pauz }U3- dS = U3 - dS = U3 - dS - - dS = 0s an an s an an

and in accordance with Eq. (1.260)

\lU3 =0


or

U2( p ) = U1( p ) + cThus, boundary conditions (1.271), (1.272) define the potential within thevolume V up to some constant. Correspondingly, the third value-boundaryproblem can be presented as

and (1.273)

where S is an equipotential surface, or

curlM = 0,

and

divM=O

(1.274)

where rp3(P) is the known function and Mn is the normal component ofthe field that coincides with the magnitude of the field, since on theequipotential surface S the tangential component vanishes.

There are many cases when harmonic fields can be found by solving thethird boundary value problem. For instance, the determination of theelectric field outside conductors, provided that the total charge on everyconductor surface is known, reduces to a solution of this problem.

In conclusion, let us summarize the main results derived in this section.

1. Three types of boundary conditions have been determined, and theyuniquely define a harmonic field within a volume V.

2. As was pointed out above, the volume can be surrounded by severalsurfaces, and at every point on them on of these conditions has to begiven.

3. The procedure of determination of these conditions, based on theuse of Gauss' theorem, is called the theorem of uniqueness.

4. In general, boundary surfaces can have an arbitrary shape as well aslocation; and here it may be appropriate to distinguish three cases. In thefirst one, the boundary surfaces surrounding generators are located in thevicinity of the generators; and in essence, boundary conditions on such


surfaces replace information about the generator parameters. In thesecond case, the boundary surface is located far away from all generators;and for this reason, it is natural to assume that field M is very small atpoints of this surface. Moreover, from an analysis of physical principles ofa specific problem, it is usually possible to understand the behavior of thefield at great distances from generators, which shows in what manner thefield decreases. This information is a boundary condition at infinity. Andfinally, one more case deserves our attention. Often the boundary surfaceis chosen in such a way that it corresponds to the surface where measure-ments of the field are performed; this case arises in solving inverseproblems.

5. It is obvious that if the field M satisfies Eqs. (1.223) everywhere, itequals zero.

M=O

This result is obvious since this field does not have generators. Corre-spondingly, in terms of its potential we can say that if a function isharmonic everywhere, it is equal to zero, and again we observe an analogywith linear functions.

1.10 Source Fields

In this section we will study a field M within a volume V, where onlysources are located, while outside of the volume, both types of fieldgenerators can be distributed. In general, the field M inside V presents asum of two fields, namely,

1. The field caused by sources located inside the volume, and2. The harmonic field of external generators.

The main attention will be paid here to the first part of the total field,which it is natural to call the source field.

In accordance with Eqs. (1.199) the system of equations of the sourcefield is

~M. dl"= 0 fcM. dS = Qt S

or

curlM = 0 divM=Q (1.275)

CurlM = 0 DivM=Qs

1.10 Source Fields 101

It is clear that all three forms of the first equation of the system indicatethe absence of vortices within the volume V, and correspondingly we canexpect some common features between the source and harmonic fields.

Now, making use of the system (1.275), let us describe the most generalproperties of the source field within the volume V.

1. Vector lines of the field M, unlike those of a harmonic field, haveterminal points where sources are located, while edge lines of normalsurfaces are absent.

2. Near surface sources, the field M can have singularities. In fact, fromthe equation

DivM = Qs

it follows that the normal component of the field is a discontinuousfunction

(1.276)

and this discontinuity is directly proportional to the density of surfacesources. In particular, if near some point of this surface sources areabsent, the component M; is a continuous function.

3. Because of absence of vortices in the volume, tangential componentsof the field are always continuous functions regardless of the distributionof surface sources.

CurlM = 0 or (1.277)

4. In accordance with Eqs. 0.275) the circulation of the field M equalszero for any closed path located within the volume V. For instance, thiscontour can be partly located inside sources as well as outside of them.

5. As follows from the first equation of the system in the differentialform curl M = 0, the field M can be represented as

M = -gradU

or

auM---

t - at' (1.278)

where U is the potential of the source field.Thus, a source field, in the same manner as a harmonic field, can be

described with the help of a scalar function. Such similarity between these


fields is not surprising since in both cases vortices are absent within V. Letus make two obvious comments.

(a) The potential U is defined from Eq. (1.278) to within some constant.(b) Equation (1.278) is valid only at usual points where the field does

not have singularities.6. Taking into account the fact that the circulation

the voltage of the source field path is independent and, as in the case of aharmonic field, it can be expressed through a difference of potentials.

tM' dt'= U(a) - U(b)II

(1.279)

7. By analogy with the harmonic field we will derive an equation for thepotential of the source field. Substituting Eq. 0.278) into the secondequation of the system, divM = Q, we obtain

ordivgradU= -Q (1.280)

(1.281)

\12U= -Q

This equation describes the potential behavior at usual points where itsfirst and second derivatives exist; it is called Poisson's equation. It isobvious that outside the sources, Poisson's equation reduces into Laplace'sequation.

\12U=0

Therefore, we can distinguish two areas within the volume V: The firstarea does not contain sources and the potential U, satisfying Laplace'sequation, is a harmonic function; in the second area, occupied by sources,the potential is a solution of Poisson's equation.

In accordance with Eq, 0.248) we can represent Poisson's equation as

6h2 [uav(p) - U(p)] = -Q

Thus, if the right-hand side of Eq. (1.281) is positive, the average value ofthe potential around some point p exceeds its value at this point. Forinstance, such behavior is observed for the potential of the electric fieldnear a negative charge. At the same time, around a gravitational mass aswell as around a positive electric charge, the potential of both gravity andelectric fields at some point p is greater than its average value near thispoint.


8. Poisson's equation is one of the most fundamental equations of fieldtheory. Here it is appropriate to notice that source fields are often used ingeophysics. Gravitational and electric fields, as well as fields of compres-sional waves, are typical examples of source fields.

9. In the next chapter it will be shown that the potential of thegravitational field can be represented as

fo(q) dV

U(p) = yv L q p

and at the same time it satisfies outside and inside the masses Laplace'sand Poisson's equations, respectively.

and

Generalizing this result we can say that for any source field, a solutionof the equation

has the form

1 QdVU(p)=-f-

47T L qp

and describes the potential inside and outside the sources.Of course, there are an infinite number of solutions of Poisson's

equations that differ from each other by a potential for a harmonic field.10. Poisson's equation is equivalent to the field equations

curIM = 0 divM = Q

written for usual points of the volume. To derive surface analogies ofPoisson's equation we will proceed again from Eq. 0.278). ThenEqs. 0.276-1.277) remain valid if the tangential and normal derivatives ofthe potential U(p) at any surface, located inside the volume V, satisfy thefollowing conditions:

nr» aU(2)

at at andau(2) nr»---=-Qan an S

As was shown in the previous section, the first condition can be replacedby a simpler one,

UO) = U(2)

since continuity of the potential at the surface results in continuity of thetangential derivative. Thus, the behavior of the potential on the surface


located inside of the volume is described by equations

U(l) = U(2) andau(2) su»---=-Qan an s

(1.282)

11. Taking into account Eqs. (1.280), (1.282) the system of field equa-tions can be replaced by an equivalent system with respect to the potentialinside of the volume,

ir» = U(2) andau(2) eu»---=-Qan an s

(1.283)

Comparison with the case of harmonic fields shows that this system ismore complicated.

First, instead of a uniform Laplace equation, we have Poisson's equa-tion with the right-hand side characterizing the distribution of sourceswithin the volume V. Second, in the presence of surface sources thenormal derivative of the potential becomes a discontinuous function.

From this consideration, it is natural to assume that, as in the case ofthe harmonic field, both systems (1.275) and (1.283) do not uniquely definethe source field.

Let us discuss this problem in terms of the field, and afterward we willformulate boundary-value problems with the help of the potential. At thebeginning of this section we have represented the total field M inside ofthe volume as a sum of two fields:

(1.284)

Here M j is the field caused by the sources Q and Qs within the volumeonly, while Me is a harmonic field whose generators are located outsidethe volume.

By definition, the field M j isa solution of the system (1.275).

curl M, =0

Curl M, = 0

divM j =Q

DivMj=Qs(1.285)

Since the given distribution of sources described by the functions Q andQs generates only one field, we can say that the field M, is uniquely by thissystem. However, Eqs. (1.275) have other solutions; to demonstrate this, letus refer to the field Me. First of all, it represents a harmonic field in the


volume V, which satisfies the homogeneous system

curl Me = 0

Curl Me = 0

div M, = 0

Div M, = 0(1.286)

Moreover, this field is a continuous function within the volume V, becauseits generators are located outside V and in essence surface analogies offield equations can be omitted. Now, performing a summation of corre-sponding equations of both systems (1.285), (1.286) and taking into accountthat all equations are linear, we see that the total field M is also a solutionof Eqs. (1.285).

In the previous section we demonstrated that an infinite number ofharmonic fields satisfy the homogeneous system (1.286), corresponding todifferent distributions of generators outside the volume V. Therefore, thesystem of field equations (1.275) or 0.286) has also an infinite number ofsolutions that differ from each other by harmonic field Me' In other words,the system 1.283 is not sufficient to define the field uniquely, and we haveto formulate boundary conditions on the surface 5 surrounding thevolume V. Taking into account that these conditions should uniquelydefine a harmonic field Me in the volume V, it is natural to make use ofthe results derived in the previous section and correspondingly to formu-late three boundary-values problems. They are

1. Dirichlet's problem

U O) = U(2)au(2) nr»----= -Qan an S

Here 50 is a surface surrounding the volume V, while 5 j is a surfacelocated inside of this volume.

2. Neumann's problem

au(2) ur»---=-Qan an S


3. The third problem

U(I) = U(2)

and

U( So) = constant

aU(2) etr»----= -Qan an S

Perhaps it is appropriate to make several comments here.Functions fPl(P), fPip), and fP/p) describe the behavior of the poten-

tial of the total field caused by both the internal and external generators.We can formulate the boundary-value problem with respect to the

external field Me' defining corresponding functions fP~(p), fP2(P), andfP3( p), providing that in the third problem the boundary surface is anequipotential surface for both fields. This is related to the fact that thepotential as well as its derivatives caused by a known distribution ofsources, located inside the volume V, can be found without solving theboundary problem everywhere including the boundary surface So' Suchconsiderations show, in fact, that the theorem of uniqueness for the sourcefield can be reduced to that for the harmonic field studied in the previoussection.

In principle, the field due to known sources distributed inside thevolume can be determined by making use of physical laws such asCoulomb's or Newton's, without solving the boundary-value problem.However, determination of the field with the help of the system of fieldequations also requires boundary conditions that take into account thefield caused by external generators.

In essence, the theorem of uniqueness formulates all the steps thathave to be undertaken to find the field. These steps are

1. A solution of Poisson's equation.2. A selection among these solutions of such functions that satisfy

boundary conditions.3. A choice among this last group of a function that also satisfies the

conditions at interfaces.

At the same time it is proper to notice that in modern numericalmethods of solution of boundary value problems based on replacement of


differential equations by finite differences, all of these stages are per-formed simultaneously.

In accordance with the theorem of uniqueness, the field inside thevolume is defined by the distribution of the volume and surface sourcesand boundary conditions, and correspondingly it is natural to derive anequation establishing this link.

With this purpose in mind, we will proceed from Gauss' theorem,

! div X dV = ~ X . dSv So

(1.287)

assuming that the vector function X(q) and its first derivatives exist in thevolume V. To simplify the derivations we will express the vector X with thehelp of two scalar functions cp(q) and I/!(q,p) in the following way:

X = cp(q)VI/!(q,p) -1/!(q,p)Vcp(q) (1.288)

Here cp and I/! are continuous functions with continuous first and secondderivatives, p is an observation point where the potential is determined,and q is an arbitrary point.

Substituting Eq, (1.288) into Eq. (1.287) and taking into account that

8I/! 8cpX . dS = X n dS = cp an -I/! 8n

and

divX = Vcp . VI/! + cpv 21/! - Vcp . VI/! -I/!V 2cp

= cpv 21/J -I/!V 2cp

we obtain

(1.289)

(1.290)

The latter is called the second Green's formula and in essence it repre-sents Gauss' theorem when the vector X is given by Eq. 0.288). Inparticular, letting I/! = constant we obtain the first Green's formula.

! V 2cp dV = rf.. 8cp dSv 'f'so an

Our goal is to express the potential of the field U( p) within the volumeV in terms of both the sources inside the volume and values of thepotential and its derivatives on the boundary surface So' We will considerseveral ways of solving this task; with this purpose in mind suppose that

(1.291 )


the function cp(q) describes the potential U(p) of the source field. Then,taking into account Eq. (1.280) we can rewrite Eq, (1.289) as

f U(q)V 2GdV+ fG(q,p)Q(q)dVv v

{eo au}=rl- U- - G- dS

~ an ano

Here G(q, p) = rjJ(q, p) is called the Green's function.It turns out that we can obtain different expressions for the potential

U(p) by choosing various functions G(q, p). To illustrate this fact, let usconsider several cases.

Case 1

Suppose that the function G(q, p) is

1G(q,p) = L

qp

(1.292)

where L qp is the distance between points q and p. As was shown in theprevious section, this Green's function satisfies Laplace's equation every-where except at the point p; that is,

1V 2

- = 0 if q =1= p (1.293)i:

Inasmuch as the second Green's formula has been derived assuming thatsingularities of the functions U and G are absent within volume V, wecannot directly use this function G in Eq. (1.291).

To avoid this obstacle let us surround the point p by a small sphericalsurface S * and apply Eq. (1.291) to the volume restricted by both surfaces,So and S*, as shown in Fig. USc. Also at the beginning we assumed thatthe potential and its first derivatives are continuous functions; that is,surface sources are absent inside the volume. Then, taking into accountEq. (1.293) we have instead of Eq. (1.291);

Q(q) ( all su;q) If-dV=~ U(q)------- dSv L q p So an L qp L q p an

(a 1 1 au( q) I

+~ U(q)--- - --- dS (1.294)s, an L qp L qp an


We will consider the behavior of the last integral as the radius of thespherical surface r tends to zero.

Inasmuch as both the potential and its derivatives are continuousfunctions we have

U{q) ~ U{p) andaU{q) aU{p)--~---an an

That is, functions U(q) and aU(q)jan on the surface S* approach theirvalues at the observation point, respectively.

Also from Fig. U5c it follows that the normal and the radius on thissurface are opposite to each other, and therefore for points on this surfacewe have

1 1G=-=-

L qp rand

iJ 1 a 1 1

as r ~ 0 (1.295)

Then, applying the mean value theorem, the surface integral around theobservation point can be represented as

(a 1 1 au(q) )

~ U(q)------ dSs, an L q p L qp an

I 1 aU(q)= - ( U(q) dS - _A:. --dS

,2 is. r '!'s. an

Inasmuch as

au r- = - -' gradU = [grad Ul cos san r

we have

1 aU(q) 1- _A:. -- dS = -I grad U( p) Ir'!'s. an r

A:. cos 8r 2 sin BdBx~ d~

s; r

Therefore, in the limit this integral equals 47TU(p),

(a 1 1 au(q»)

~ U(q)--- - --- dS=47TU(p)s; an L q p L q p an

This is the most important result in our derivations since it permits us to


obtain an expression for the potential in an explicit form. In fact, substitut-ing Eq. (1.295) into Eq. (1.294) we have

1 Q(q) 1 aU(q) 1U(p)=-f-dV+-~ ---dS

47T v L q p 47T So an L q p

1 a 1--~ U(q)--dS

47T So an L q p

(1.296)

That is, the potential U(p) at any point of the volume V can be calculatedif we know the distribution of sources in the volume as well as values ofthe potential and its normal derivative on the boundary surface So'

In particular, if this surface is located at infinity, it is natural to assumethat

and therefore

andaU(q)--- -.70

an

1 Q(q)U(p) = -f --dV

47T v L q p

(1.297)

In other words, if the sources are known everywhere the potential isdefined by Eq. 0.297). This presentation is often called the fundamentalsolution of Poisson's equation.

We will consider another example in which sources are absent in thevolume V. Then, in accordance with Eq. 0.296) we have

1 a 1--~ U(q)--dS

47T So an L q p

(1.298)

Thus, the potential at any point of the volume where sources are absentis defined by both values of the potential and its normal derivative on theboundary surface; and they, in effect, replace information about thedistribution of generators outside the volume.


At the same time, from the boundary conditions of Dirichlet's andNeumann's problems it follows that the field can be found inside thevolume if either the potential U(q) or its normal derivative aU(q) jan isknown. To demonstrate this fact we will consider cases where new Green'sfunctions are chosen.

Case 2

Let us introduce a new Green's function that satisfies the followingconditions:

1.

2.

3.

V2G=O if q =1= P

1G(q,p) ~L if q~p (1.299)

qp

aG(q,p)=0 Soon

an

Comparison with the previous case shows that the surface integral aroundthe point p still equals 47TU(p), because in both cases the Green'sfunctions have the same singularity. Then, taking into account that on theboundary surface So

aG-=0on

instead of Eq, 0.296) we have

1 1 aU(q)U(p) = -4 fQ(q)G(q,p) dV+ -4 rf.. G(q,p)--dS (1.300)

7T v 7T'1';;0 on

Therefore, we have expressed the potential at any point of the volumeV in terms of the distribution of sources and the normal derivative of thepotential on the surface So; that of course corresponds to Neumann'sboundary value problem. Inasmuch as

auM =--

n an


we can rewrite Eq. (1.300) as

1U(p) = 47TfvQ(q)G(q,p)dV

1--f Mn(q)G(q,p) dS47T So

In particular, if sources are absent we obtain

1U(p)=- 47T~Mn(q)G(q,p)dS

o

(1.301)

(1.302)

That is, in this case the potential is defined only by the normal componentof the field given at the boundary surface So'

As follows from Eqs. (1.299), determination of function G(q, p) is asolution of Neumann's boundary-value problem. To emphasize this, wewill write down Green's function as a sum.

1G(q,p) = L +h(q,p)

qp

(1.303)

Here h(q, p) is a harmonic function everywhere within the volume V, andin accordance with Eqs. (1.299) the boundary problem with respect tohi.q, p) is formulated as

where

V 2h( q , p) = 0

aanh(q,p) =CP2(q),

(1.304)

(1.305)

In other words, determination of the potential u» by Eq. (1.301)implies a solution of Neumann's problem with respect to the harmonicfield h(q, p) and is usually a sufficiently difficult task. Its complexitystrongly depends on the shape of the boundary surface. However, thereare cases when it is very simple to find hi q, p) as well as the Green'sfunction G(q, p). For instance, suppose that the boundary surface Soconsists of a plane Sp and a hemispherical surface with infinitely largeradius Ssph' Note that the field tends to zero at points of the surface Ssph

and the Green's functions also decays at infinity. Here PI is a point


located outside the volume V and is a mirror reflection of the point P withrespect to the plane.

Correspondingly, the distances L qp and L qp 1 are

(1.306)

and

since

and Z =-zPI P

(1.307)

Inasmuch as the point Pi is located outside the volume V, as Z> 0, thefunction h(q, Pi) is harmonic and therefore it is a solution of Laplace'sequation; that is,

At the boundary plane Z q = 0 we have

a 1

and since Zq = 0

At the same time, for the function 'PzCq) we obtain

a 1 Zq-Zp zp'P2(q) = - - - = 3 = - -3-

aZq i; u: L q p

Comparing the last two equations, we see that function hi.q, p) satisfiesthe Neumann's boundary problem given by Eqs. (1.304) and, correspond-ingly, we can calculate the potential of the source field by making use ofeither Eq. (1.301) or Eq. (1.302), if the Green's function is

1 1G(q,p)=-+-

i: L q p I

Certainly it is a very simple function, which allows us to find the potentialin one of the half spaces when the normal component of the field is knownon the boundary plane; it is often used in gravimetry and magnetometry to


calculate the field above the earth's surface (upward continuation). It isappropriate to notice that in this case integration in Eqs. (1.301), 0.302) isperformed over the plane Sp only.

Case 3

We can modify Eq, (1.296) in a different way; with this purpose in mind,let us choose a Green's function such that it satisfies the followingconditions:

1. V2G =0 if q*p

12. G(q,p) ~- as q~p (1.308)

i;

3. G(q,p)=O on So

Then, the integral containing the normal derivative of the potential on theboundary surface vanishes and instead of Eq. (1.296) we obtain

1U(p) = -lQ(q)G(q,p) dV

47T v

1 aG--~ U(q)-dS

41T So an

And in particular, as sources are absent in the volume,

1 aGU(p) =-~ U(q)-dS

41T So an

(1.309)

(1.310)

Determination of the Green's function, as in the previous case, requiresa solution of the boundary value problem, which also can be formulated interms of the harmonic field h.

1G(q,p) = L +h(q,p)

qp


Then, we have the following Dirichlet's problem:

1h(q,p) = --c;

(1.311)

Of course, in general it is a very complicated problem, but it isdrastically simplified for some special cases. For instance, in the case ofthe plane boundary surface Sp, z = 0, and a hemispherical surface withinfinitely large radius, the harmonic function is

Here PI is the mirror reflection of the point p with respect to the planeSo' Then, on this plane we have

G(q,p) = 0

while the normal derivative of the function G has the form

Correspondingly, Eq, (I,31O) can be written as

It is easy to see that the function

can be expressed as

if z = 0

(1.313)


where dw is the solid angle suspended by the surface dS as viewed fromthe point p.

Thus, we can present Eq. (1.313) in the form

1U(p)=-f U(q)dw

21T s,(1.314)

Now we will consider one more approach to choosing Green's func-tions.

Case 4

Suppose that the new Green's function satisfies two conditions, namely,

(a) It has a singularity near the observation point p, which is the sameas earlier.

as q ---) p (1.315)

and(b) Unlike the previous cases, the function G satisfies the same equa-

tion as the potential U does.

(1.316)

In other words, we do not assume any more than that the Green's functionis harmonic.

Again we will proceed from Eq. (1.289) and taking into account that

and

the volume integral in this equation vanishes. Then as before, the surfaceintegral consists of two parts.

{eo au},f, U--G- dS

'fJs an ano

{sc au}

+,f, U--G- dS=O'fJs * an an


Inasmuch as the Green's function has the same singularity as in theprevious cases, we obtain

1 {au aG}U(p) = _A.. G- - u- dS47T ~o an an (1.317)

Therefore, we have expressed the potential at any point of the volume interms of its values and those of its normal derivative on the boundarysurface So' This result is valid regardless of presence or absence of sourcesof the field in the volume V.

Let us assume that the Green's function has the following form:

feu)G(q,p)=-

-;

where u depends on the distance L qp and

(1.318)

feu) -s i

Then, substituting Eq. 0.318) into Eq. (1.317) we have

1 auU(p) = _A.. G-ds

47T ~o an

1 f'(u) et.- -¢U(q) -----!!£dS

47T L qp an

1 feu) et:+ -~ U( q) -2- ---!!£ dS

47T So L qp an

where !'(u) = df(u)jan.

(1.319)

We have considered several types of Green's functions, correspondingto different formulas, allowing us to calculate the potential inside thevolume. With this in mind it is appropriate to make several commentsillustrating various aspects of this subject.

1. A Green's function can be treated as a potential caused by a certaintype of sources. For instance G(q, p) = Ij47TL qp is the potential at theobservation point p due to either the elementary mass or a charge withunit magnitude, located at point q.


At the same time, the Green's function

(1.320)

where

describes the potential of a displacement in a medium as the longitudinalwave propagates with velocity C, caused by an elementary source locatedat the point q.

Here it is appropriate to notice that, substituting Eq, (1.320) intoEq. 0.319), we obtain Kirchoff's formula, which plays an important role inthe theory of seismology.

2. In deriving formulas for the potential it was assumed that the field isa continuous function everywhere within the volume V.

Now let us consider the case in which sources are distributed on somesurface Sj where the normal component of the field is a discontinuousfunction. Then, surrounding this surface by the surface So; and applyingGreen's formula, Eq. (1.289), to the volume confined by surfaces So andSo;, we obtain an additional integral over Soi' In the limit, as the surfaceSoi approaches Sj, this integration is reduced to that over the back andfront sides of the surface Sj (Fig. 1.15d). Taking into account that thedirection of the normal to Soj coincides with the normal of S, on the backside and opposite on the front side, we have

{aif/l) aifP) acp(l) acp(2) }

+~ cp(1) __ - cp(2) __ -l/J(1)-- + l/J(2)_- dS (1.321)Sj an an an an

where cp(1), 1/1(1) and cp(2), 1/1(2) are the values of both functions on the backand front sides of the surface Sj, respectively.

Letting cp = U and l/J = G and assuming that both the potential andGreen's function are continuous functions on the surface Si, the last


integral in Eq. (1.321) can be simplified, and we have

{

au(2) aU(1) }tf..G --- dS't;;; an an

= -¢,GQsdSSj

(1.322)

where G = G(2) = G(1), but M~2) and M~I) are the normal components ofthe field at either side of the surface Sj, respectively.

Thus, in the presence of surface sources we have to add the term givenby Eq. (1.322) to the right-hand side of all expressions for the potentialderived above. In particular, if the Green's function is harmonic andbecomes singular as l/Lq p at the point p, we obtain

1 1U(p) = -4 f.Q(q)G(q,p) dV+ -f. Qs(q)G(q,p) dS

rr v 4rr ~

1 aU(q) 1 8G(q,p)+ -4 tf.. --"G(q,p) dS + -4 tf.. U(q) dS (1.323)

rr't;;o an rr't;;o an

This equation, as well as similar ones, can create the impression thatthe solution of a boundary-value problem always consists of an integration.However, in general, this is not true, and it is related to the fact that someterms on the right-hand side of Eq. 0.323) remain unknown until the fieldis calculated. Let us consider this question in more detail.

First, let us briefly discuss terms containing the potential and its normalderivative-that is, boundary conditions. As was already pointed out, thevolume V, where the field is studied, is usually confined by a surface nearprimary sources and one at infinity. Correspondingly, knowing from physi-cal consideration the distribution of primary sources and the behavior ofthe field at infinity, we can assume that the integrals over the surface So inEq. (1.323) are known. This conclusion is also valid for cases when a partof the boundary surface is not located at infinity, but contacts a mediumsuch that it is easy to formulate boundary conditions. For instance, thenormal component of current density and the force due to elastic wavesequals zero at a boundary with a free space.

Next, we will consider the integrals in Eq. (1.323) that contain termsdescribing sources. In general, these sources depend on the field M andthe properties of the medium. For example, sources arise in the vicinity of


points where some parameters of a medium vary along the field. Also theycan appear at interfaces of media with different parameters. In some casesit is appropriate to represent the source as a sum.

and (1.324)

where Qo characterizes sources that can be specified before the field iscalculated, while Q 1 depends on the field. At the same time, let us pointout that the total source remains a function of the field.

In other words, we are faced with a problem that can be characterizedas "the closed circle." Indeed, to find the field we have to know thedistribution of sources, but they in turn depend on the field, as isillustrated below:

This means that in principle total sources cannot be specified if the field isnot determined, and therefore Eq. (1.323) as well as similar ones areuseless in calculating the potential by integration of its right-hand side. Ofcourse, there are some exceptions; for instance, in the case of the gravita-tional field for geophysical problems, masses can always be defined.

Now let us extend this analysis to the system of field equations

curlM = 0

CurlM=O

divM=Q

DivM=Qs

(1.325)

which, in general, contains two unknown functions, the field and itssources. From the point of view of calculation this means that theright-hand side of the system is unknown and therefore in this form it isnot suitable for solution of the boundary-value problems. However, takinginto account that sources depend on their field, we can modify the systemin such a way that a new one would either contain an unknown field onlyor two equations with both unknowns, which allow us to eliminate each ofthem.

A similar problem of "the closed circle" arises in considering thepotential. In particular for usual points, we have Poisson's equation

with unknown right-hand side,and correspondingly this also requires somemodifications.


Inasmuch as these changes for both the system of field equations andPoisson's equation depend essentially on the physical nature of the field(electric, elastic waves), let us illustrate this procedure by considering twoexamples.

Example 1 The Electric Field in Dielectrics

In this case, the system of field equations is

curlE = 0

CurlE = 0(1.326)

where 80' 8b and uo ' u b are free and bounded charges, but

are total volume and surface charges.Assuming that due to the electric field there is a displacement of

positive charges with respect to negative ones-that is, polarization of themedium takes place-and that this is directly proportional to the electricfield, we can show that bounded charges are related with the field by

8b = -divaE, (Tb = -DivaE (1.327)

where a is the parameter characterizing the polarization ability of themedium and is called the polarizability.

Then, substituting Eqs. (1.327) into Eqs. (1.326) we obtain

curlE = 0

CurlE = 0(1.328)

where E = 1 + a is the dielectric constant of the medium.Thus, we have obtained a new system for the field with a known

right-hand side, and it becomes possible to solve this system because twoassumptions have been made, namely,

(a) In the presence of the field, polarization of the medium occurs.(b) There is a linear relation between the field E and the polarization.

It is clear that since free charges can be specified, a new system unlikethe original one can be used for determination of the electric field indielectrics.


Now, let us derive the equation for the potential U, corresponding toEqs. (1.328). From the first equation we have

E = -gradU

and substituting into the second equation we obtain

divt e grad L' ) = -00

or

(1.329)

Certainly this equation is more complicated than Poisson's equation

but its right-hand side is known, and therefore it can be used to find thepotential.

Of course, it is very simple to represent Eq. 0.329) as a Poissonequation. In fact, applying the operator V we have

or

00 V6' VUV 2U = -- - =-0

6 6(1.330)

where 0 is the total charge.Again we have obtained Poisson's equation with an unknown right-hand

side expressed in terms of the potential and free charge.As concerns surface analogies of the equations for U we have

andaU2 au,

6--6-=-02 an 1 an 0

Example 2 The Electric Field in a Conducting Medium

Since a constant electric field in a conducting medium is also caused byelectric charges, the system of field equations is the same as that in theprevious case.

curl E = 0

CurlE = 0

divE = 0

DivE = (J'

(1.331)

but the densities of the free and bounded charges are related to each

1.11 Vortex Fields 123

other and depend on the field E. Correspondingly, the right-hand side ofthe second equation is unknown, and therefore it is necessary to performcertain modifications of the system to find the field. With this purpose inmind, we have to introduce for consideration another field that accompa-nies the electric field in a conducting medium, namely the current densityj, and make use of two laws.

(a) The principle of charge conservation

and(b) Ohm's law

divj = 0 and Divj = 0 (1.332)

j = yE

Here l' is the conductivity of the medium.

Then, replacing the second equation of the system by Eq. 0.332) andtaking into account Ohm's law, we obtain a new system for the field Eonly.

curlE = 0

CurlE = 0

div yE = 0

DivyE = 0(1.333)

It is clear that the equation for the potential U is

V(yVU) = 0

1.11 Vortex Fields

(1.334)

In this section we will consider general features of fields caused by vorticesonly. For instance, the time-invariant magnetic field, the electromagneticfield due to induced and displacement currents only, and the field of shearwaves are examples of the vortex field. In accordance with Eqs. 0.207) thesystem of equations for this field in some volume V is

or, in integral form,

curlM =W

CurIM=Ws

divM = 0

DivM=O

JM'dS=O

(1.335)

(1.336)


That is, the flux of the field through any closed surface inside the volumeis always zero, while the voltage between two points in general depends onthe path of integration. This is one of the most fundamental differenceswith the source field, where the voltage is path independent. Of course, wecan find lines having the same initial and final points, such that along theselines the voltage remains the same. This occurs if the lines are located inparts of the volume where vortices are absent.

As follows from the second equation for the field, the vector lines arealways closed, because sources are absent. In general, the field M and itsvortices Ware not perpendicular to each other.

M· curlM =1= 0

and, correspondingly, normal surfaces do not exist. However, there areexceptions; for instance, as is shown in Section 1.6, the quasi-potentialfield

M = tp grad U

can be described with the help of normal surfaces.Now let us evaluate the extent to which the system (1.335) defines the

field. With this purpose in mind, we will make use of the same approach aswas used in the previous section.

Suppose that there are two arbitrary solutions M 1 and M 2 satisfyingthis system.

and

curl M, =W

CurIM I =Ws

curl M, = W

CurIMz =Ws

div M, = 0

DivM 1 = 0

divM, = 0

DivM2 = 0

(1.337)

Then, we will consider the difference between them.

M 3 =Mz -M1 (1.338)

As follows from Eqs, (1.337) the field M 3 satisfies the homogeneoussystem

curl M; = 0

CurIM 3 = 0

div M, = 0

DivM 3 =0(1.339)

In other words, M 3 does not have generators inside, and therefore it is aharmonic field within the volume V whose generators are located outside.Thus, we can say that system (1.335) defines the field up to a harmonic


field. As was shown earlier, this field can be described with the help of ascalar potential a, M3 = - grad a, and correspondingly,

M 2 = M 1 - grad aThis analysis demonstrates that Eqs, 0.335) uniquely defines that part

of the field caused by generators located inside the volume, as for the caseof a source field. At the same time, the system does not determine theharmonic field M3, caused by external generators, and therefore it isnecessary to introduce boundary conditions. Taking into account theresults derived in Section 1.5, we can formulate two boundary-valueproblems.

1. Dirichlet's problem(a)

curlM =W

CurlM =Ws

divM=O

DivM =0

and(b) M, = ({)l(P) is known on the boundary surface. Here t is a direction

tangential to the boundary surface and ({)l(P) is a given function.2. Neumann's problem(a)

curlM =W

CurlM=Ws

divM = 0

DivM=O

and(b) M n = ({)2(P) is known on the boundary surface, and n is the unit

vector normal to the surface.

As was shown earlier in the case of harmonic and source fields, thesystems of field equations can be replaced by either Laplace's or Poisson'sequations for the scalar potential a. However, in general, this replacementcannot be performed for the vortex field since in those parts of the volumewhere vortices are present, curl M is not zero. At the same time it ispossible to replace the system

curlM = W divM = 0

by one equation, and with this purpose we will make use of the equality

divcurl A == 0

Therefore, from the second equation of the field, it follows that the vortexfield M can be represented as

M = curl A (1.340)


The vector A is called the vector potential of the vortex field and is notuniquely defined from Eq, (1.340), inasmuch as we can add any gradient tothe vector A but it will still define the same field M.

M = curl A = curl(A + grad cp)

This ambiguity, which manifests itself even to a greater extent than inthe case of the scalar potential, can often be used to simplify the equationsfor potentials and, therefore, the solution of the forward problem. Thetransition from the source field to the scalar potential is obvious, since it ismuch easier to operate with the function U than with the vector field.However, it is not so clear why it is useful to introduce the function A,which is also a vector as well as the field itself. Notice that several reasonscan often justify introduction of the vector potential.

1. Sometimes it is possible to describe the field M with the help of oneor a maximum of two components of the vector potential, resulting ingreat simplification.

2. In general, systems of field equations of electromagnetic fields andelastic waves contain four equations with two unknown fields, and it turnsout that they can often be described with one or two vector potentialshaving a small number of components.

Bearing in mind these facts, we will derive an equation for the vectorpotential A and formulate boundary value problems in terms of thisauxiliary function. Substituting Eq. 0.340) into the first field equation wehave

orcurl curl A = W (1.341)

VXVxA=WThis procedure of replacement of the system of field equations is shownbelow:

Icurl M = W I I divM = 0 I

1 r 11 ~~~IM=CUrlAI

I curl curl A = wiInasmuch as

curl curl A = grad divA - V2A

instead of Eq. (1.341) we have

V2A = -W + grad divA


(1.342)

Often from consideration of the specific problem we can determine thedivergence of the vector potential A and, correspondingly, assume that

divA = f3 (1.343)

Here f3 is a known function.For instance, in the case of the time-invariant magnetic field from the

Biot-Savart law and the principle of charge conservation it follows thatf3;: O.

Thus, taking into account Eq. (1.343) we have

(1.344)

where

WI = W - grad f3

Then, considering components of the vector potential in a Cartesiansystem of coordinates, we can use the results derived in previous sectionsand formulate boundary value problems.

Further we will consider boundary value problems for A, proceedingdirectly from Eq, (1.341). First, suppose that inside the volume surfacevortices are absent and, correspondingly, Eq. (1.341) is valid everywhere inV. Let us assume that there are two different solutions of this equation, Aiand A2 ; that is,

Therefore, their difference

and VXVXA2=W

A3 =Az -AI

satisfies a homogeneous equation

(1.345)

Representing this equation as

V2A3 = grad divA3

and even assuming that divA3 = 0, we see by analogy with Poisson'sequation that Eq, (1.345) has an infinite number of solutions.

Now we are prepared to find boundary conditions for the vectorpotential A, which uniquely define the field. In principle, we will follow thesame pattern of derivation as that in the case of the source field. First, let


us introduce the vector

x = A 3 X V X A 3 = A 3 X M 3

Applying Gauss theorem to the vector X, we obtain

(1.346)

and performing a differentiation inside the volume integral and taking intoaccount the equality

div(a X b) = b . curl a - a . curl b

we have

Then, in accordance with Eq. (1.345) we obtain

(1.348)

It is obvious that if from the boundary conditions it follows that thesurface integral on the left-hand side of Eq. (1.348) equals zero, then thefield M 3 vanishes inside the volume V and in accordance with Eq. (1.338)the field M is uniquely defined. Now we will determine such boundaryconditions for two cases.

Case 1

Suppose that the tangential component of the field M is given at theboundary surface; that is,

nXM=nXP (1.349)

Here P is the given function on the surface So'Then, the difference of two arbitrary solutions with the same tangential

component satisfies the following equality on So:

n XM 3 = 0

Therefore, taking into account that

a . (b X c) = c . (a X b) = b X (c X a)

(1.350)


we can conclude that the surface integral in Eq. (1.348) equals zero, andcorrespondingly at every point of the volume M 3 =O. In other words, wehave demonstrated that Eq. 0.341) and the boundary condition 0.349)uniquely define the field M.

Case 2

Let us assume that the normal component of the field is known at theboundary surface; that is,

n'M=N (1.351)

where N is a given scalar function. Respectively, for the difference ofsolutions M 3 on this surface we have

n : M 3 =0 (1.352)

That is, the vector M 3 has only a tangential component. This means thatits vector potential A3 is directed along the normal n, and since

n '(A3 X M 3 ) = M 3 '(0 XA 3 )

the surface integral in Eq. (1.348) equals zero, and therefore

M 3 =0

Thus, we have proved that knowledge of either the tangential or normalcomponent of the field at the boundary surface is sufficient to take intoaccount the field caused by external generators. It is proper to notice thatwe have proved the theorem of uniqueness by using two different ap-proaches, and, of course, arrived at the same result.

Now we will suppose that there is a surface of singularity of the field S,with vortices w.. Then the system of field equations has the form

curlM =W

n X (M(2) - M(l») = Ws

divM = 0

n . (M(2) - M(1») = 0and in terms of the vector potential A the surface analogies of theequations are

O' V' X (A(2) - Nl)) = 0 (1.353)


where A(l) and A(2) are values of the vector potential at the back and frontsides of the interface, correspondingly.

By analogy with the case of the source field we will generalize Eq.(1.348) and obtain

+f n . (A(j) X M~l) - A~) X M~2)} dSSi

= fM~dVv

It is clear that the second integral at the left-hand side vanishes if

A(j) X M~l) = A(~) X M 3

or

Nj) X V' X Nr = N~) X V' X A(~)

and this occurs if we require continuity of the two vector functions

A(j) = Aj2) and n X V' X A(r = n X V' X A(~) (1.354)

By definition of the function A3 Eqs, 0.354) hold, provided that boththe vector potential A and the tangential component of the field M

nXV'XA

obey the following conditions at the interface Sj, where the vortices arelocated.

1.

2.

A(2) - A(l) = N,

n X (V' X A(2») - n X (V' X A(1») = Ps (1.355)

where N, and P are given functions.

Comparison with the system of field equations shows that

Ps=Ws

and this characterizes the distribution of surface vortices.Therefore, in our case a solution of the boundary-value problem con-

sists of three steps, namely,

1. Solution of equation V X V' X A = W.2. Determination of solutions of this equation that satisfy one of two

boundary conditions.3. Choice among these solutions of functions A that obey Eqs, 0.355).


Let us note that this result can be easily generalized to those cases inwhich there are also lines of singularities.

This analysis of boundary-value problems shows that there are relationsbetween the vector potential A, on the other hand, and vortices within thevolume V and boundary conditions on the other hand. In principle, thereare two approaches to deriving such equations. The first one is based onthe representation of A in Cartesian components.

A =Axi +Ayj +Azk

Inasmuch as the unit vectors i, j, k do not depend on coordinates of thepoint, each component satisfies Poisson's equation.

and we can make use of the results derived in the previous section for thescalar potential U.

At the same time we can derive an expression for the vector potentialproceeding directly from Eq, (1.341), and this approach is consideredbelow. Suppose that the vector functions Land N and their first andsecond derivatives are continuous in volume V and at the surface S. Then,applying Gauss theorem to the vector X,

we obtain

X = L X (\7 X N) (1.356)

(1.357)f div(L X curl N) dV = ~(L X curl N) . n dSv s

Taking into account the equality

dive a X b) = b X curl a - a X curl b

we obtain the vector analog of the first Green's formula (Stratton, 1941).

f (curl L: curiN - L· curl curl N) dVv

= ¢(L X curl N) . n dS

In the same manner we can write (Stratton, 1941)

f (curl L· curiN - N . curl curl L) dVv

=¢.(N X curlL) . n dSs

(1.358)

(1.359)


Then, subtracting Eq. (1.359) from Eq. (1.358) we have the vector analog ofthe second Green's formula.

f (N . curl curl L - L . curl curl N) dVv

= ~(L X curlN - N X curlL) . n d.S5

To find the vector potential, we assume here that

divA = 0

That is,

and introduce the notations

(1.360)

(1.361)

L = A(q) and N = G(q,p) (1.362)

where q is an arbitrary point within volume and p is the point where theGreen's function G has a singularity, l/r. The quantity r is the distancebetween the points q and p; that is,

As in the case of the source field, the Green's function can be chosen inmany ways, but there we consider one function only.

a oG(q,p) =-

r(1.363)

where a o is a vector that does not depend on the position of the point.Inasmuch as the Green's function has a singularity at the point p, we

will surround this point by the spherical surface 51 with a small radius r1•

Then, applying Eq, (1.360) to the volume confined by the surfaces 5 and 51we obtain

f {G' curl curl A - A· curl curl G} dVv

=~{A X curlG - G X curl A] . n d55

+~ {A X curlG - G X curl A} . n d551

(1.364)


First, consider the volume integral. In accordance with Eq. 0.341) wehave

j G ' curlcurlAdV= j ~. WdVv vr

Then, taking into account the equalities

curl !paD = cp curl a., + grad !p X aD

curl curl !paD = grad div !paD - V 2cp ao

div !paD = !p diva., + a, • grad tp

and the fact that aD= constant, we obtain

1curl G = grad - X aD

r

curl curl G = grad (8 0 ' grad ~ )

and

A· curl curl G = A' grad(30' grad~ )

(1.365)

since divA = O. Correspondingly, the volume integral can be representedas

fj :0 .W - diV{ (aD' grad~ )A} dV

W 1= a, ·f- dV - aD' ~(A' n) grad - dS

r 'Ys r

1- aD' ~ (A' n) grad - dS

Si r(1.366)

Now we will study the surface integrals in Eqs. (1.364), (1.366) aroundthe point p; and with this purpose in mind, let us make the following


transformations:

{A X curl G} . n = {A X (grad ~ X a o )} • n

= a , . {grad ~ X (A X n) }

and

MXn(GxcurlA)'n=a .--

o r

Here

M = curl A

Then taking into account Eq. (1.366) we can rewrite Eq. (1.364) as

W 1

f - dV = ,{, (A . n) grad - dSv r 'J;;+s, r

+ f {grad ~ X (A X n)} dSS+SI r

nXM+J --dS

S+S, r

On the spherical surface around the point p we have

1 r ograd - =-

r r 2

(1.367)

where r o is the unit vector directed to the point p along the radius r.Correspondingly, the integrals around p can be written in the form

1+-~ (nXM) dSr, SI

Making use of the identity

a X (b X c) = b( a . c) - c( a . b)

(1.368)


we can represent the second integrand in Eq. 0.368) as

roX(AXn) =(ro'n)A-n(ro'A) =A-ro(A'n)

since r 0 = n. Therefore, for the surface integrals around the point p wehave

1 1 1+ 2"¢(n X M) dS = 2: ~ A dS + - ¢(n X M) dS

T1 r 1 51 r 1

Then, making use of the mean value theorem and taking into account thefact that the second integral equals zero, we have

(1.369)

where NY is the average value of the vector potential.From Eq, (1.369) it follows that in the limit the sum of the surface

integrals around point P is equal to 4'7TA(p), and in accordance with Eq.(1.367) we have

1 W(q) 1 n X MA(p)=-f-dV--rf..--dS

4'7T v r 4'7T 'Ys r

1 1- -rf..(n· A) grad - dS

4'7T 'Ys r

- _1 rf.. {( n X A) X grad ~} dS4'7T 'Ys r

(1.370)

Thus, we have expressed the vector potential in terms of the vorticeswithin the volume V and the values of the field and its potential on theboundary surface. It is obvious that the surface integrals in Eq. (1.370)describe a vector potential caused by vortices located outside the volumeV. Here it is appropriate to make the following comments:

1. As was mentioned above, Green's functions can be chosen in differ-ent ways. In particular, they can be a solution of the same equation as thatfor the vector potential, and due to this fact, in some cases the volumeintegral in Eq. (1.364) vanishes.


2. Equation (1.370) has been derived under the assumption that surfacevortices are absent. Applying the same approach as in the case of thesource field it is not difficult to take into account the presence of surfacesor lines with field singularities.

3. In general, vortices that arise in a medium depend essentially on thefield and, correspondingly, they cannot be specified if the field is unknown.In other words, as in the case of the source field, we again experience theso-called closed circle problem, and this fact always requires very seriousmodifications to the system of field equations.

References

Alpin, L.M. (1966). "Theory of Fields." Nedra, Moscow.Stratton, 1.A. (1941). "Electromagnetic Theory." McGraw-Hill.

Chapter II The Gravitational Field

ILl Newton's Law of Attraction and the Gravitational Field11.2 Determination of the Gravitational Field

Elevation CorrectionThe Bouger Slab CorrectionTwo-Dimensional ModelThree-Dimensional Body

11.3 System of Equations of the Gravitational Field and Upward ContinuationIntegral Form of Equations for Gravitational FieldDifferential Form of Equations for Gravitational Field

References

II.! Newton's Law of Attraction and the Gravitational Field

The gravitational method is a study of the distribution of Newton'sattraction force F caused by all masses of the earth. One part of this forceprovides the uniform motion of a body around the rotation axis of theearth, and the other part originates the weight force. Thus,

F= Fa + P

where Fa is the centripetal force directed toward the rotation axis and P isthe weight. The directions of these forces are shown in Fig. II.la. It isessential that the centripetal force can easily be taken into account alongwith the force of the bulk attraction of the earth.

Because Fa can be calculated and removed very accurately, we will payattention only to that part of the attraction force F that obeys Newton'slaw. To formulate this law, we will suppose that there are two masses,ilm(p) and i1m(q), located in elementary volumes i1V(p) and i1V(q),respectively. The points p and q characterize the position of thesevolumes. The distance between the points is L p q (Fig. II.1b). It is properto notice that dimensions of volumes i1V(p) and i1V(q) are much smallerthan the distance L q p ; this fact is the most important feature of elernen-

137

138 II The Gravitational Field

Fig. 11.1 (a) Gravitational force and its components; (b) interaction between elementarymasses; (c) arbitrary distribution of masses; and (d) the field caused by an elementary mass.

tary volumes. Then, in accordance with Newton's law of attraction, theelementary mass I1m(q) acts on the elementary mass I1m(p) with a forcedF(p) equal to

I1m( q) Am( p)dF(p) = -1 L 3 L qp

qp

(11.1)

where 1 is a coefficient of proportionality, or gravitational constant, whichin the International System of Units (SI) is

1 = 6.6710- 11 m3 kg"! sec-2

L q p is the vector

and L~p is the unit vector directed from the point q to the point p alongthe line connecting these points. AV(q) and I1V(p) are elementary vol-umes. It is clear that only in such cases the force of attraction of themasses, located in these volumes, does not depend on the position of the

11.1 Newton's Law of Attraction and the Gravitational Field 139

points p and q. From this definition it follows that dimensions of elemen-tary volumes in different problems can change from very small values toextremely large ones.

Thus, in accordance with Eq. (II.1) the mass Ilm(p) is subjected to aforce dF(p), which is proportional to the product of both masses andinversely proportional to the square of the distance between them, and hasa direction opposite to that of L q p .

This extremely simple formula describing the basic physical law ofgravimetry may need some almost obvious comments, and they are

1. Elementary masses Ilm(q) and Ilm(p) can be different.2. Newton's law states that due to the presence of any mass Ilm other

masses experience an action of the force that tends to attract these massesto Ilm. This effect decreases with an increase of the distance from tsm.

3. Newton's law of attraction is a mathematical description of one ofthe most fundamental phenomena of nature. Notice, however, that it doesnot explain the mechanism of transmission of this force through a medium.

4. In accordance with the principle of superposition, the force ofattraction between two masses does not depend on the presence of othermasses provided that their position in space remains the same.

5. Gravitational masses, unlike electrical charges, have only positivevalues, and therefore only attraction is observed.

6. In the SI system of units the distance is measured in meters, mass inkilograms, and the force in Newtons.

7. As follows from Eq. (II.1) the force acting on mass Ilm(q) andcaused by mass Il mt;p) is

Ilm(q) Ilm(p)dF(q) = -r 3 L p qL p q

Inasmuch as

we have

L3 -L3qp pq and Lo - L O

qp pq

dF(q) = -dF(p)

Now we are ready to introduce the concept of a gravitational field.Taking into account the fact that the force dF(p) is proportional to the


elementary mass ~m(p), which is subjected to the action of this force, it isnatural to consider their ratio dg( p).

or

dF(p) ~m(q)dg( p) = ~ ( ) = - Y L 3 L q p

m p qp

~m(q) 0dg(p)=-y L2 L q p

qp

(11.2)

The function dg(p) is called the gravitational field at point p, caused bythe elementary mass ~m(q). It has the dimensions of force per unit mass;that is,

[g] = mysec '

In practice, the field is usually measured in milligals, which are relatedto meters per second by

cm1 Gal = 1-2 = 1000 mGal

secand

1 rnysec ' = 100 gal = 105 mGal

Note that unit "Gal" was introduced in honor of Galileo.It is proper to notice that for a given mass ~m(q), the function dg(p) is

a vector field inasmuch as its magnitude and direction depend on thecoordinates of the observation point only.

As follows from Eq. (II.1) it is a simple matter to find a force acting onany elementary mass when the field is known, and we have

dF(p) = ~m(p) dg(p) (11.3 )

It is useful to regard Eq. (11.2) as a relation between an elementary massand the gravitation field caused by this mass, which exists at any pointregardless of presence or absence of other elementary masses at this point.

Now let us assume that instead of an elementary mass there is somedistribution of masses in volume V (Fig. 1I.1c). To find the field g, causedby all the masses within this volume, we will use the principle of superposi-tion. With this purpose in mind, the volume V is mentally divided intomany elementary volumes, the size of which satisfies two conditions,namely,

(a) They are small with respect to the distance between this volumeand the observation point.

II.t Newton's Law of Attraction and the Gravitational Field 141

(b) The masses are distributed uniformly within every elementary vol-ume.

The latter allows us to present its mass, tJ.m, as

tJ.m(q) = Seq) dV (lIA)

where S(q) is the density of the mass, which can vary arbitrarily within thevolume V.

Inasmuch as the gravitational field caused by the elementary mass is

S(q)dVdg( p) = -y L 3 L q p

qp

the total field is the sum of fields due to all masses.

(11.5)

This equation describes the gravitational field at any observation pointp, whether it is located inside or outside of the masses. It is essential tonote that masses can be specified before we calculate the gravitationalfield caused by them. This fact dramatically simplifies the determination ofthe field, and in accordance with Eq, (11.5) it is reduced to numericalintegration only. Later, when considering other fields applied in geophysi-cal methods, we will show that determination of fields is usually a muchmore complicated problem,

Now making use of Eq. (1I.s) we will consider several examples illustrat-ing some features of the behavior of the field.

Example 1 Field of an Elementary Mass

We will study the field caused by an elementary mass Srn, located insideof the elementary volume tJ.V in the vicinity of the point q (Fig. IUd).Inasmuch as by definition observation points are located far away withrespect to the size of the elementary volume, we can neglect the change ofthe distance from the observation point to any point inside this volume.Correspondingly, instead of a real elementary mass we can consider thesame mass located at point q. This is of course physically impossible, but itis convenient for field calculation. Thus, a point mass is a useful conceptwhen the field is studied at distances greatly exceeding dimensions of the


elementary volume. With this definition of a point mass, the expression forits field is

(11.6)

For simplicity we will make use of a Cartesian system of coordinateswith its origin at point q. Then, for the vector L qp and its magnitude wehave

L q p = Lop ";"xi + yj +zk

L = _/X2+y2+ Z2qp V (11.7)

and the gravitational field has a direction opposite to that of the vectorLop. Because x, y, and z are projections of the radius vector Lop on thecoordinate axes, the components of the field are

m ymgx=g·i=-YL3 · L op·i=-L3 x

op op

. m . ymg y = g . J = - Y L 3 Lop' J = - L 3 Y

Op Op(II .8)

For illustration, consider the behavior of the field III the x -z plane.Then, in accordance with Eqs. (II.7), (11.8) we have

mxg = - y -----;;-=

x (x 2 + h2 ) 3/ 2 '

since y = 0

Here h is a constant for the profile characterizing the position of the mass.The behavior of the field components is shown in Fig. II.2a. The

horizontal component gx is positive for negative values of x, and itincreases with an increase of x, reaches a maximum, and then tends tozero as x ~ O. For positive values of x,

gAx) = -gx( -x)

That is, it is an odd function.The vertical component gz has simpler behavior; it is a symmetrical

function with respect to x

II.l Newton's Law of Attraction and the Gravitational Field 143

Fig.II.2 (a) Components gx and gz due to an elementary mass; (b) surface masses; (c) thenormal component of gn near a surface mass; and (d) field gz near a disk.

and reaches a minimum at x = 0, where the distance from the observationpoint to the mass is minimal.

Negative values of the vertical component are explained by the fact thateverywhere along the profile the vector projection of the field on the z-axishas a direction opposite to this axis. For the same reason, the horizontalcomponent gx is positive for negative values of x, and it becomes negativefor x > O. Now let us look at these formulas from a geophysical point ofview; that is, we will attempt to find the position of the mass and its value.First of all, the change of sign of gx and maximal value of magnitude ofthe component g z as x = 0, indicate that the mass is located at the z-axis.Then, in accordance with Eqs. (II.9) the ratio of the components is

s, Xor

gzh=-x

gx(11.10)

That is, this ratio allows us to determine the position of the mass. Finally,using the values of either gz or gx' the mass m can also be calculated. Ofcourse, both parameters m and h can be found from only one of thesecomponents. It may be seen that the point mass is an over-simplication of


the real distribution of masses. It is certainly true, however, with anincrease of a distance from masses, distributed within a volume of arbi-trary shape and dimensions, their field approaches that of the point mass,regardless of where inside the volume this point mass is placed. This resultdirectly follows from Eq. (II.S), since any mass can be treated as anelementary one if its field is considered far away from the mass. In fact,with an increase of the distance from the observation point to the volumemass we have

(11.11)

where qo is an arbitrary point within the volume V, and m is the totalmass in this volume.

Now, let us look in a little more detail at Eqs. (11.9). Taking the firstderivative of the horizontal component gx with respect to x, we find thatthe abscissas for their extrema are

hx=+--fi (11.12)

As is seen from Eqs. (11.9), with an increase of the coordinate x themagnitudes of both components approach each other, and they becomeequal if

x=h (11.13)

As x increases, the horizontal component becomes dominant; that directlyfollows from an analysis of the field geometry.

It is also proper to notice that Eq. (11.10) demonstrates the relationbetween the vertical and horizontal components of the field caused by anelementary mass. Proceeding from the principle of superposition we canexpect that for an arbitrary mass and an observation point there is also arelation between these components of the field which, of course, is morecomplex than Eq. (IUD). In this connection let us point out that suchrelations show that both components of the field measured along a profilecontain the same information about the distribution of masses, and this'means that in principle it is sufficient to measure only one of them.

11.1 Newton's Law of Attraction and the GravitationaJ Field 145

As is seen from Eq. (11.9) and was not noticed above, the horizontalcomponent, unlike the vertical one, is an odd function of x and therefore

(11.14)

Later we will show that in the general case, when the component gx is notan odd function, this equation still remains valid.

Example 2 The Nonnal Component of the GravitationalField Due to Plane Surface Masses

Suppose that masses are distributed within a plane layer whose thicknessis much smaller than the distances from these masses to the observationpoint (Fig. IL2b). In other words, the distance between the observationpoint and any point of the elementary volume is practically the same.Taking into account this fact, we can replace this layer by a plane surfacewith the same mass, located somewhere at the middle of the layer(Fig. lI.2c).

Inasmuch as every elementary volume contains the mass

dm =8(q)hdS

its distribution on the surface can be described by

dm = a ( q) dS

where

u(q) =o(q)h (11.15)

The function u(q) is called a surface density or density of surface masses.The latter, as well as point masses, is a pure mathematical conceptintroduced to simplify calculations. Of course, every element of thesurface S bears the same mass as a corresponding elementary volume ofthe layer. Our next step is to find the field caused by these surface masses.First of all, we will distinguish the back and front sides of the surface Sand choose the direction of the normal n(q) at every point of the surfacefrom the back to the front side.

In accordance with Eq. (11.2) the field caused by an elementary mass onthe surface dS is

dm o dSdg( p) = -y L 3 L qp = -yvLqp

qp qp(11.16)


For simplicity suppose that the surface is a plane and the masses aredistributed uniformly; that is, a = constant. At every point the field can berepresented as the sum of two components.

(11.17)

where nand t are unit vectors perpendicular and tangential to the surface,respectively. As is seen from Fig. 1I.2c the normal component of the fieldat the point p is

adS= -Y--zy-Lqp·n

qp(11.18)

(11.19)

where (Lqp,n) is the angle between vectors L qp and n.Applying the principle of superposition we obtain for the normal

component of the field due to all surface masses

( dS . L qpgn(p)=-ya), L 3

S qp

since dS = dS nTaking into account that L qp = -Lpq and L~p =L~q, we have for the

normal component of the field

(11.20)

(11.21)

As follows from Eq. (1.44) the integral at the right-hand side of thisequation describes the solid angle w(p).

( dS . L pqw(p) "). L 3

S p q

Thus, the normal component of the gravitational field caused by massesuniformly distributed on the plane surface can be expressed as

(11.22)

Making use of the results derived in Chapter I, we will describe thebehavior of the normal component gn' With an increase of the distancefrom the surface, the solid angle becomes smaller, and therefore thenormal component of the field decreases. On the other hand, in approach-


ing the surface the solid angle tends to its limiting values: - 27T and + 27Ton the front and back sides of the surface, respectively. Thus, for thenormal component gn on the surface S, we have

{

-27T'}'lTg =

n 27T'}'lT

on the front sideon the back side

(11.23)

and correspondingly, the discontinuity of the normal component of thefield on the surface mass is

where g~ and g;; are values of the field at the front and back sides of thesurface.

As we shall see in the next example, the field is everywhere a continu-ous function, but this discontinuity arises due to replacement of volumemasses by surface ones. Let us consider two special cases.

(a) A plane surface, infinite in extent. Then the solid angle under whichthe plane is seen from the front and back sides does not depend on theposition of the point p and is equal to =+= 27T, respectively. In other words,plane surface masses with infinite extension and constant density create auniform field in each half space.

(b) A plane surface in the form of a disk with radius a and the normalcomponent measured along the z-axis (Fig. 1I.2d).

In this case, making use of Eqs. (1.48) we have

where

gzC z) = =+= 27T '}' (1 - cos a )u

[z]

(11.24)

With an increase of the distance z, the angle a tends to zero, andtherefore the field gz decreases. Expanding the radical (Z2 + a2) 1/ 2 in aseries,

if z» a


we obtain

or

m ( 3 a2

)g7(Z)Z+Y- 1---- Z2 4 Z2

where m is the total mass of the disk.From this equation it is very simple to evaluate the minimal distance z

at which the field of masses located on the disk coincides with that of apoint mass.

In approaching the disk the field tends to its limit, equal +27TyO',which corresponds to the field due to the infinite plane surface withconstant density 0'. Applying again the expansion of the same radical, wehave

if a»lzl

Example 3 Field Caused by a Volume Distributionof Masses in a Layer with Thickness hand Density 8 (Fig. lI.3a)

First, introduce a Cartesian system of coordinates with its origin at themiddle of the layer and the axis z directed perpendicular to its surface.Let us note that the layer has infinite extension along the x and y axes.

At the beginning, suppose that the observation point is located outsidethe layer, that is, [z] > h/2. Then we mentally divide the layer into manythin layers which are in turn replaced by a system of plane surfaces withthe density 0' = 8 Sh, where Sh is the thickness of the elementary layer.Taking into account the infinite extension of the surfaces, the solid angleunder which they are seen does not depend on the position of theobservation point and equals either - 27T or +27T.

Correspondingly, each plane surface creates the same field.

hif z>-

2and

hif z < --

2


Fig. 11.3 (a) Model of a homogeneous layer; (b) observation point inside a layer; (c) fieldbehavior, gz; and (d) system of layers.

Therefore, after summation the total field due to all masses in the layer is

hgz= -2rryoh if z> -- 2

and (II.25)

hgz = 2rryoh if z< --- 2

It is interesting to notice that these formulas are used to calculate theBouguer correction in gravimetry.

Now we will study the gravitational field inside a layer when thecoordinate of the observation point z satisfies the condition

First, suppose that z > O. Then, the total field can be presented as a sumof two fields: one of them is caused by masses with thickness equal(h /2) - z, and the second one is caused by masses in the rest of the layer


having the thickness z + h/2 (Fig. 1I.3b). In accordance with Eqs. (I1.25)these fields are

and

Correspondingly, for the total field we have

(11.26)

and by analogy,

h hif - - <z <-2 - - 2

ifh

--<z<O2 - - (11.27)

The behavior of the field due to masses within the layer is shown inFig. lI.3c. Thus, for negative values of z the field is positive, since themasses in the upper part of the layer create a field along the z-axis, andthis attraction prevails over the effect due to the masses located below theobservation point. At the middle of the layer, where z = 0, the field isequal to zero. Of course, every elementary mass of the layer generates afield at the plane z = 0 in accordance with Newton's law, Eq. 0I.2); butdue to symmetry the total field turns out to be zero. For positive values ofz the field has the opposite direction, and its magnitude increases linearlywith an increase of z. As follows from Eqs. (11.25)-(11.27) the fieldchanges as a continuous function at the layer boundaries.

One more feature of the field behavior is worth noting. Inasmuch as thelayer has infinite extension in horizontal planes, the distribution of massespossesses axial symmetry with respect to any line parallel to axis z thatpasses through an observation point. For this reason it is always possibleto find two elementary masses such that the tangential component of thefield caused by them is equal to zero. Respectively, the field due to allmasses of the layer has only a normal component, described byEqs. 0I.25)-0I.27).

Now let us consider the field inside a layer when it is surrounded by twoother layers with densities 0l(Z) and 0iz), which are arbitrary functionsof z (Fig. IUd). In accordance with Eqs. (11.25-11.27) the field in the


middle layer can be represented as

ifh h

- - <z<-2 - - 2

where /3 is the constant density of the middle layer, while g lz and gzz arethe fields caused by the masses of the upper and lower layers, and boththese fields do not change within the middle layer.

Suppose we would like to find the density /3. Then, after determinationof the field at two points of the layer we have

Therefore,

and

/3 = gzC z,) -qzCzz)47TY(ZZ-ZI)

Note that this approach is used in measuring the density of rocks in aborehole.

Example 4 The Field Caused by Thin Spherical Shellwith Density <T

Taking into account the spherical shape of the body (Fig. lI.4a) we willchoose a spherical system of coordinates R, e, 'P with its origin at thecenter of the shell. We can in general expect three components of thefield, gR' go, and g'P' directed along coordinate lines R, 8, and 'P,respectively. However due to the symmetry, it is a simple matter to showthat components go and g'P are everywhere equal to zero. In fact,coordinate planes e= constant and 'P = constant are planes of symmetry,and therefore it is always possible for any elementary mass to find anothersymmetrically located with respect to it so that the field caused by bothmasses does not have components go and g'P' Correspondingly, the field ofa mass shell has only a radial component, gR' First, we will study the field


Fig. 11.4 (a) Measuring the field inside a shell; (b) measuring the field outside a shell;(c) field behavior due to shell masses; and (d) spherical mass with radius a and density 8.

inside the shell at any point p caused by two elementary masses (T dS j and(T dS z, as is shown in Fig. BAa.

In accordance with Eq. (If.Z) these fields are

Inasmuch as

«as,dg j R = -'Y~,

pql

(TdSzdg ZR = -'Y~

pq2

as, as, dw

L;ql = L;q2 = cos a

(dw is the solid angle) and the fields have opposite directions, the totalfield caused by these two elementary masses is equal to zero. Consideringthe spherical shell as a system of such pairs, we conclude that the fieldinside a uniform spherical shell is zero.

Now we will consider the field outside the shell (Fig. BAb). First let usdetermine the field caused by the mass within the elementary ring withradius x and width rdip, located on the shell surface. Each elementary


mass of the ring creates a field at the point p with magnitude

a r dtp dtdg = -Y--L-::-2 -

where r dc: dt is the area of the elementary mass, and L is the distancefrom the mass to the observation point. As is seen from Fig. BAb its radialcomponent is

R - r cos 'PdgR = dg cos a = dg-~-

L

or

a r dip dtdgR = - Y 3 ( R - r cos 'P)

L

where

xsin e = -

r

Since all elements of the ring are located at the same distance from theobservation point, we have the following expression for the radial compo-nent due to the ring mass:

Replacing L and x with L = (r 2+ R 2- 2rR cos 'P)1/2, x = r sin 'P, andintegrating, we obtain for the field caused by all masses of the shell

or

where

7T sin 'P ( R - r cos 'P )gR=-yu27Tr21 '12 d'P

o (r 2+ R 2 - 2rR cos 'Pr

7T sin 'P dipII = 1 3/2

o (r 2+R2-2rRcos'P)

7T sin 'P cos 'P d 'P12 = 1 3/2

o (r 2+R2-2-rRcos'P)

(IL28)


We will use the notations

b = 2rR

Then we have

tr sin 'P dipII = 1 3/2

a (a - b cos 'P )

_ (Tr sin 'P cos 'Pdsp12 - l, 3/2

a (a - b cos 'P )

Introducing the variable

z = a - b cos 'P

we obtain

dz = b sin 'P de:

and

2{II}= b (a - b) 1/2 - (a + b) 1/2

For the second integral,

2 {a a }= - - - + Va + b - va - bb? va + b va - b

Inasmuch as

we have

1/2(a+b) =R+r and 1/2(a-b) =R-r


Whence

(II .29)

if R"?r (11.30)

where m is the total mass on the spherical surface.

Thus, masses located on the shell with density a generate a field, whichoutside of this surface coincides with that of a point source with the samemass, placed at the center of the shell. The behavior of the field gR' insideand outside is shown in Fig. HAc. As follows from this analysis the normalcomponent gR' as in the case of the plane surface, has different values ateach side of the shell. In accordance with Eqs. (11.23) and (11.29) thisdiscontinuity is the same for both of these surfaces. This coincidence is notaccidental, and it can be shown that for arbitrary surface masses thedifference of normal components of the gravitational field is equal to- 47TYU(q); here ai.q) is the surface density at vicinity of point q wherethe field is studied.

The next example is a natural generalization of this case.

Example 5 Gravitational Field of the Sphere (Fig. HAd)

Applying the same approach as in the previous example we can concludethat the field has a radial component gR only; that is, go = gCP = O. Tocalculate this component, let us mentally divide the sphere, which has theradius a and volume density B, into a system of thin spherical layers withsurface density o = BdR, where dR is the layer thickness. First, supposethat the observation point is located outside the sphere R > a. Thenapplying the principle of superposition and making use of Eq. (11.29) wehave

47TBy a 47Ta 3 B mgR= -~ fa r

2dr= --3- y R 2 = -y R 2

where m = 17Ta3B is the total mass of the sphere.If the observation point is inside the sphere and its coordinate is R,

then as follows from the previous example, all spherical shells that haveradius r exceeding R do not contribute to the gravitational field at the


Fig. II.S. (a) Field due to spherical mass; (b) point p is the center of an elementary sphericalmass; (c) the secondary field; and (d) one-dimensional model of a medium.

observation point. Correspondingly, making use of Eq. (11.29) we obtain

47TO R 47TgR = -"7 fa r

2dr = - 3"oR (II.31)

Thus, inside the sphere the field magnitude increases linearly from zero toa maximum value on the sphere surface, equal to (47T j3)yoR, and then itdecreases inversely proportional to the distance R (Fig. II.5a).

It is a simple matter to generalize this result to the case in which thedensity 0 is a function of the radius. Then it is clear that inside the spherethe field decreases approaching its center, while outside the sphere itbehaves as the field of the point source. Let us emphasize that the mainpart of the gravitational field of the earth has this behavior.

It is proper to notice that as in the case of the layer, the field inside thesphere has everywhere finite values, and it is a continuous function. Wewill consider this fact proceeding from Eq. m.su With this purpose inmind, we will represent the total field at point p as a sum of two fields:The first part is caused by mass within the elementary spherical volumewith its center at point p, and the second part is due to other masses(Fig. II.5b). The latter are located at distances equal to or exceeding the

11.2 Determination of the Gravitational Field 157

spherical radius p, and therefore their field has a finite value at the pointp and is a continuous function near this point.

At the same time, in accordance with Eq. (11.31), the first part of thefield is equal to zero at point p. Thus, the total field has a finite value atany point inside the mass. Bearing in mind that on the surface of theelementary spherical mass its field is directly proportional to radius p,Eq. (11.31), we can conclude that with a decrease of this radius thecontribution of the first part of the field becomes negligible; this results incontinuity of the total field. Note that this consideration can also beapplied to an arbitrary distribution of masses.

As follows from Eq. (11.30) the behavior of the horizontal and verticalcomponents gx and qz along some profile outside the sphere is the sameas that for the point source studied in the first example. This means, inparticular, that both the direction and magnitude of the 'field do notchange if the radius of the sphere and its density vary in such a way thatthe total mass remains the same. Of course, this is true only if theobservation point is located outside the sphere. This consideration showsthat measuring the gravitational field along some profile or system ofprofiles we can determine only the total mass of the sphere-that is, theproduct }7TOa 3

, as well as the position of the sphere center-but it isimpossible to find out separately the density of masses 0 and the radius ofsphere a.

This example vividly demonstrates the simplest case of nonuniquenessin which different distributions of masses generate exactly the same fieldas for measurements performed outside the masses. Certainly, this is anegative factor that imposes some limitations in determining the massdistribution. Later we will discuss more complicated types of such ambigu-ity in various geophysical methods.

In this section we have obtained some insight into the field behavior bymaking use of Newton's law of attraction and the principle of superposi-tion. Now we consider their application to the theory of the gravitationalmethod.

IL2 Determination of the Gravitational Field

In this section we will continue the study of the behavior of the field, butmain attention will be paid to calculation of the gravitational field causedby masses located beneath the earth's surface.


Taking into account the fact that the field is defined by a distribution ofmasses, it is appropriate to give some information about the density ofrocks.

OilWaterSand, wetSand, dryCoalEnglish chalkSandstoneRock saltKeuper marlLimestone

(compact)QuartziteGneiss

90010001950-20501400-16501200-150019401800-27002100-24002230-2600

2600-27002600-27002700

GraniteAnhydriteDiabaseBasaltGabbroZinc blendeChalcopyri teChromitePyrrhotitePyriteHaematiteMagnetiteGalena

2500-270029602500-32002700-32002700-3500400042004500-48 046005000510052007500

Rock density in kg/m3 [after Parasnis (I 979)]

As is seen from this table, the densities of sedimentary rocks vary withina relatively small range, and they have lower density than those of igneousand metamorphic rocks. For example, the density difference between rocksalt and sandstone has a maximum of 600 kg/m3 or approximately 30% oftheir values.

The gravitational field of the earth, as is well known, depends onseveral factors, such as

1. the latitude of the observation point2. tides3. elevation4. topography of the area where the field is measured5. lateral changes of rock density

Taking into account these factors, we will present the total gravitationalfield as

G= gN+ g (II.32)

where gN is the normal field, which depends on the latitude, and g is theanomalous field, which depends on the elevation, topography, and lateralchanges of rock densities.

Inasmuch as our main purpose is the study of the anomalous field, letus only briefly describe the normal field. The earth's surface can be


presented as approximately spheroidal, slightly enlarged at the equatorand flattened at the poles: The ratio between the equatorial and polarradii is around 1.003.At every point of this surface, the normal field gN isperpendicular to the spheroid and can be described by the formula

(11.33)

where go = 978,0318 Gal, which is the normal field at the equator, a =0.0053024, {3 = - 0.0000058, and cp is the latitude.

About the normal field let us make several comments.

1. The spheroid surface can be considered as the first approximation tothe geoid surface, and the equation describing this spheroid surface can bederived assuming that the spheroid is a fluid that rotates around its polaraxis. The extreme differences between the spheroid and geoid surfaces are-105 m and +73 m.

2. Equation (11.33) describes the field caused by masses of the spheroidand the centripetal force. Notice that if the earth were a nonrotatingsphere, the gravitational field would not depend on the latitude. '

3. The change of the field due to the attraction of the moon and sun(tides) is usually very smooth and has a maximum value of approximately0.3 mGal. Assuming that the normal field is known, we will concentrateour attention on the anomalous or secondary field only.

Inasmuch as the vertical component of the field gz is the main subjectof measurements in the gravitational method, we will further study onlythis component. First we will compare magnitudes of normal and anoma-lous fields. With this purpose in mind, we will consider several numericalexamples.

1. Suppose that the earth is a sphere with radius R = 6370 km anddensity 8 = 5500 kg/m3• Then in accordance with Eq, (IUl) the field onits surface is

g = 6.671O- llj 1T 55006370 = 9.8 mysec?

= 980 Gal = 0.9810 6 mGal

This is the normal field.2. Here we will evaluate the contribution of sediments assuming that

their density and thickness are 0 = 2000 kg/m3 and h = 5 km, respec-tively. Replacing this spherical layer by a surface with density

(J = Bh = 2000 kgyrn:' 5 km = 107 kg/m2


and making use of Eq. (11.29) we obtain for the field magnitude

g = 4rry(T = 4rr 6.6710- 11107

= 0.8410- 2 mysec" = 0.84 Gal

In other words, sediments contribute very little to the normal field.3. Now we will calculate the field on the z-axis caused by a disk

(Fig. Il.Zd), Suppose that the radius and the thickness of the disk are 1 kmand 0.5 km, respectively, and that the density is 1000 kg zrn:'. Such a modelcan be considered the first approximation to some basement structure.Then letting z = 2 km and making use of Eq. (11.24) we have

g = 2rryoh(l- cos a) = 2rr' 6.67 '10- 1110 3. 500(1- ~ )

= 2.210- 5 mysec ' = 2.210- 3 Gal = 2.2 mGal

That is, this effect is extremely small in comparison with the normal field.In fact, in the practice of gravimetry even smaller signals are measured.

One more example follows.4. Suppose that the radius of the sphere is 1 m and its density is 1000

kgyrn:'. Then, the field on its surface is

g = ~rroyR = ~rr' 103. 6.6710- 111105 mGal ::::; 0.03 mGal

This example can serve as an illustration of the effect of some boulderlocated near an observation point where the field is measured.

Before we start to derive formulas for the vertical component of thefield, let us in the most general form outline the main features ofinterpretation of gravitational data. To simplify this subject we will beginwith the simplest model and gradually approach more complicated ones.

First suppose that measurements of the gravitational field are per-formed along a profile shown in Fig. II.5c. Here 0o(z) and 0 are densitiesof the surrounding medium and a body, respectively. We assume that thedensity 00 is in general a function of coordinate z, and in particular, themedium surrounding the body can be described by a horizontally layeredmodel. It is clear that in absence of the body, the gravitational field doesnot change along the profile and, correspondingly, the anomaly vanishes.

II.2 Determination of the Gravitational Field 161

Fig. II.6 (a) Equivalent model of nonuniform medium; (b) elevation correction; (c) Bouguercorrection; and (d) field calculations due to dimensional model.

In the presence of a body with a different density, however, the secondaryfield arises and its magnitude becomes greater as the density difference~o = 0 - 00 increases. This simple analysis shows that we can mentallypresent the original model of the medium as a combination of two simplermodels, namely,

1. A half space with density oiz), which does not create a secondaryfield (Fig. II.5d).

2. A body with density ~o = 0 - 0o(z) surrounded by free space, whichgenerates the anomalous or secondary field (Fig. II.6a). Undoubtedly thismodel is simpler than the original one, and in calculating the secondaryfield by Newton's law, Eq. (11.5), it allows us to perform the necessaryintegration only within the body volume.

Two more obvious conclusions follow:

1. On the surface of a horizontally layered medium the field does notchange; that is, information about the medium due to measurements atone point and along a profile is the same. However, the field depends onmany parameters of this model, such as the density and thickness of layers,


and therefore their determination from the gravimetry becomes impossi-ble.

2. To apply the gravitational method there must be lateral changes ofrock density.

Now we are ready to discuss some aspects of interpretation in gravime-try. Suppose that from measurements along a profile or a system ofprofiles, the anomalous field is known (Fig. II.6a). Then, the main purposeof interpretation is to determine the location, shape, dimensions, anddensity of the subsurface bodies. This task is often called the inverseproblem of the gravitational field theory, since it is necessary to find adistribution of masses when the field caused by them is known along someprofile or in some area. It is essential that the field is not known in avolume occupied by masses, since measurements are always performed atsome distance from them and thus interpretation usually becomes a rathercomplicated problem. In accordance with Newton's law the field can berepresented at every observation point as a sum of fields caused byelementary masses of the body; and their contributions depend on the sizeand location of the corresponding volumes.

In particular, those masses within the body located relatively far awayfrom the observation point only slightly affect the field magnitude. Strictlyspeaking, at every observation point the field is subjected to the influenceof all parameters of the body, although to different extents, and theirrelative effect varies from point to point since they have different positionswith respect to the body. Thus, in principle, all information about massesis contained in the measured field. Taking into account this simple butfundamental fact, let us formulate the main steps of interpretation.

First, we will make some assumptions about the distribution of masses,and correspondingly ascribe values to parameters of the body that charac-terize its geometry and density contrast. Such a step is usually called thefirst guess or the first approximation, and it is mainly based on specificgeological information. For example, if the gravimetry is carried out fordetecting salt domes, there is usually some information about the densityof both the surrounding rocks and the salt domes, as well as their shapeand location. Of course, the difference between the first approximationand the factual values of the body parameters can vary significantlydepending on our knowledge of the geology.

The second step of interpretation consists of calculating the verticalcomponent of the field along the profile, using the parameters of the firstapproximation and comparing the measured and calculated fields. Coinci-dence of these fields suggests that the chosen parameters of the model are


close to the real ones. If there is a difference between the measured andcalculated fields, all parameters of the first approximation or some of themare changed in such a way that a better fit to these fields is achieved. Thus,we obtain a second approximation of the mass distribution. Of course, inthose cases when even this new set of parameters does not providesatisfactory matching of fields, this process of calculation has to becontinued. As we see from this process, every step of interpretationrequires application of Newton's law. Let us note that this procedure,based on the use of Newton's law, is often called the solution of theforward problem of the gravitational field. In summary we can say that theprocess of interpretation for the simplest case, shown in Fig. Il.Sc, in-cludes two steps, namely,

1. Formulation of the first approximation for parameters of the body,the "first guess"; and

2. A solution of the forward problem that includes changing the param-eters of a model to provide a better fit between measured and calculatedfields.

Inasmuch as in the gravity interpretation process every step is reason-ably well defined, we may arrive at the impression that this procedure ofinterpretation is straightforward and does not contain any complications.Unfortunately it is not true even in the case when the field is caused byonly masses of the body-that is, when we deal with only the useful signal-that "noise" is absent.

First, suppose that both calculated and measured fields are known withinfinitely high accuracy. Then, by sequentially repeating the solution of theforward problem we can obtain a set of parameters such that differencebetween these fields will be extremely small. In this connection, thefollowing question arises. Does it means that by providing an unrealisti-cally ideal fit between the measured and calculated fields, it is alwayspossible to determine with infinitely small error the shape, dimensions,and density as well as the location of masses that create the given field? Asthe theory shows, the answer is negative and in general a solution of theinverse problem is not unique; that is, different distributions of masses cancreate exactly the same field along a profile or a system of them. In otherwords, in general, different bodies can generate a field that provides exactfitting with the measured field. The simplest example of such nonunique-ness is the case when the field is caused by different spheres with the samemass and' common center but with different densities and radii.

This phenomenon is hardly obvious. In fact, Newton's law of attractiontells us that a change of mass distribution should result in a change of the


field. However, from nonuniqueness it follows that different mass distribu-tions can create outside them exactly the same field, regardless of theaccuracy of measurement; that is, it is impossible to detect the differencebetween fields generated by such masses. Nonuniqueness is really anamazing fact that is more natural to treat as a paradox than as an obviousconsequence of gravitational field behavior.

Now let us look at this subject from a practical point of view andimagine that nonuniqueness is always observed when the inverse problemof the gravitational field is solved. Then, it is clear that interpretation ofgravitational data would always be impossible. In fact, having determinedparameters of a body that generates the given field, we have to alsoassume that due to nonuniqueness there are other distributions of massescreating exactly the same field.

Certainly we can say that such ambiguity would be a disaster forapplication of the gravitational method. Fortunately, nonuniqueness doesnot always manifest itself and there are such types of mass distribution forwhich the solution of the inverse problem for them becomes unique.Moreover, some of these cases are of great practical interest because theyallow with sufficient accuracy to approximate a real distribution of masses.For instance, the inverse problem is unique if a body can be presented as aprism or a system of prisms even if their density is unknown.

Another example corresponds to a more general shape of the bodywhen every ray drawn from any point within its volume intersects the bodysurface only once. It turns out that in these cases of so-called star-shapedbodies the inverse problem is also unique if the density is known. Further,we will pay attention to only such distributions of masses for which thesolution of the inverse problem is unique.

Now we are ready to make the next step and discuss some aspects ofinterpretation for real conditions when the gravitational field is measuredwith some error; that is, the numbers that describe the field are accurateto some number of digits only. From this fact it follows that the accuracyof the field calculation can be practically the same as that of the measuredfield, and because of it there is always a difference between these fields.For this reason any attempt to achieve fitting of the calculated andmeasured fields with an accuracy exceeding that of their determination hasno meaning.

Taking into account the fact that the difference between the measuredand calculated fields has always a finite value, which can sometimesconstitute several percent of the field, let us consider the influence of thisfactor on the interpretation. As was pointed out, the vertical component ofthe field can be represented at every observation point as a sum of fields


caused by different masses within the body, and their contributions essen-tially depend on the location and distance of these masses from theobservation point. In particular, masses located closer give a larger contri-bution, while remote parts of the body produce smaller effects. It isobvious that there are always such masses within the body that theircontribution to the total field is so small that with the given accuracy ofmeasuring it cannot be detected. For instance, we can imagine suchchanges of a shape, dimensions, location of the body, as well as the densityof some of its parts, that the measured field would remain the same. Inother words, due to errors in determination of the secondary field therecan be an unlimited number of different distributions of masses thatgenerate practically the same field at observation points.

Inasmuch as the secondary field is caused by all masses within the body-that is, an integrated effect is measured-some changes of masses inrelatively remote parts of the body can be significant; but their contribu-tion to the field would still remain small. At the same time similar changesin those parts of the body closer to observation points will result in muchlarger changes of the field. For this reason, in performing an interpreta-tion it is appropriate to distinguish at least two groups of parametersdescribing the distribution of masses, namely;

1. Parameters that have a sufficiently strong effect on the field; that is,relatively small changes of their values produce a change of the field thatcan be detected.

2. Parameters that have a noticeable influence on the field only if theirvalues are significantly changed. It simply means that they cannot bedefined from a field measured with some error.

Therefore, we can say that an interpretation or a solution of the inverseproblem consists of determining the first group of parameters of the bodyeven though usually they incompletely characterize the distribution ofmasses. It is clear that this "stable" group of parameters describes a modelof the body that differs to some extent from the actual one, but both ofthem also have common parameters. For instance, these can be the depthto the top of the body or the product of its thickness and its density, etc.

Before we continue let us make several comments, which allow us tosummarize out discussion and outline some features of a solution of theinverse problem. They are

1. In general the inverse problem of the gravitational field is notunique.

2. At the same time, there are types of bodies for which a solution ofthe inverse problem is unique. In particular, it is true for a body that can


be presented as a prism or a system of prisms, and this fact is one of thetheoretical foundations of interpretation of gravitational data.

3. In considering the problem of uniqueness it is assumed that the fieldis known exactly.

4. Interpretation is performed within a class of models for which theproblem is unique.

5. The most important factor, which in essence defines all features ofthe interpretation, is the fact that the measured field caused by somedistribution of masses is known with some error. Because of this, unique-ness itself does not guarantee that the error in determination of someparameter has a finite value. In fact, this error can be infinitely large. Suchinverse problems are called ill-posed or unstable ones.

6. In general, inverse problems in gravimetry, as well as in othergeophysical methods, are ill-posed. To illustrate this fact, let us write thefollowing relation between the change of the field Jlg and that of a bodyparameter JlPi'

Here k, is the coefficient of proportionality for the ith parameter of thebody, and Jlg is the change of the field within a certain range caused bythe presence of an error, which of course varies from point to point.

In the case of an ill-posed problem an upper limit of coefficient valueki , in principle, cannot be established. In other words, even unlimitedchange of some parameters of the body cannot produce a noticeablevariation of the field, and as a result of this, it is impossible to define arange within which these parameters vary.

Such ambiguity in determination of parameters of a body is an obstaclethat can be to some extent compared with non uniqueness. To overcomethis problem the interpretation is usually performed within a much nar-rower class of models of the body, where the parameter k , has a finiteupper limit; that is, the inverse problem becomes stable or well posed. Forinstance, if an approximation of the real distribution of masses is per-formed with the help of different prisms having the same number of sides,the inverse problem turns out to be well posed.

7. The transition from an ill-posed problem to a well-posed one iscalled the regularization of the inverse problem, and it is of great practicalimportance.

8. With an increase in the number of model parameters the approxima-tion of a real distribution of masses can in principle be better. However,the error with which some of these parameters are determined alsoincreases.


9. It is obvious that the interpretation of gravitational data is useful ifthe parameters of a model, approximating a real distribution of masses,are defined within a range of values, that is sufficient from a practicalpoint of view. Usually a choice of such a group of parameters is automati-cally selected making use of the corresponding algorithm of the solution ofthe inverse problem.

10. The interpretation of gravitational data is greatly facilitated by thepresence of additional information about sources of the field, usuallyderived from geology and other geophysical methods.

Now taking into account the obvious fact that a decrease of the error infield determination is of great importance, let us consider factors which,along with the error of measurement and interpolation between observa-tion points, define the accuracy of separation of the useful signal from thesecondary field. One factor is related to the change of a position of anobservation point with respect to the masses that create the normal field.If, for example, measurements are performed at different distances fromthe earth's surface, the normal field varies, in accordance with Eq. (11.29).To separate the secondary field, which is usually much smaller than thenormal field, its change has to be taken into account, and such a proce-dure is called the elevation correction.

The second factor arises due to the presence of surface structures, sincemasses within them generate a field that can also vary from point to point.For these reasons, it is necessary to determine this effect and then removeit from the measured field. This procedure is called the terrain correction;for very gentle topography it reduces to the Bouguer's correction.

A third factor also affects the anomalous field, and it is the field causedby lateral changes of density other than those that are sources of theuseful signal. This part of the field is usually called the geological noise,and it is mainly caused by sources located relatively close to the earth'ssurface. It is not simple to remove their influence. One of the methodsallowing us to reduce the geological noise will be described in the nextsection. Certainly, the second factor can also be treated as geologicalnoise.

It is clear that the accuracy with which all these factors can be takeninto account, along with that of measurement, defines the accuracy ofdetermination of the useful signal that in turn affects the quality ofinterpretation.

In connection with the first two factors, notice that procedures forcorrections do not involve a change of positions of observation points.

Now we will demonstrate the application of Newton's law in calculatingthese corrections and the useful signal.


Elevation Correction

As is seen from Fig. I1.6b, points "1" and "2" are located at differentelevations with respect to the earth's surface, and correspondingly thenormal field is different at these points. It is clear that at point "2" thisfield is smaller, and to take into account this change it is sufficient toassume that the earth is a sphere. Then its gravitational field at bothpoints is the same as that due to a point source. Making use of Eq. (11.29)we obtain the difference of vertical components of the fields due to anelevation change.

1(

h)-2 2h

(

h )2 = 1 + R ::::: 1 - R1+ -

R

where Rand h are the earth's radius and the elevation, respectively.Inasmuch as in reality h/R « 1, we have

R2

and therefore

(II.34)

Having assumed that the earth's radius R and the normal gravitationalfield gz are

R = 6378 km and s, = 980 Gal

we obtain

~g, mGalh:::::0.3086~ (11.35)

If, for example, point "2" has higher elevation than point "I" we add tothe measured field g/R + h), to compensate a decrease of the field due tothe elevation. Often, the anomalous field has a magnitude around 1 mGalor smaller. In such a case the elevation has to be known within a fewcentimeters. Let us notice that the standard correction formula is derivedusing the spheroid model.


The Bouguer Slab Correction

Again we will compare the fields at points "1" and "2," assuming that thelatter is located above some layer of rocks that is absent around point "1"(Fig. II.6c). It is clear that its presence leads to an increase of the field atpoint "2." Unlike the previous case, where the difference of normal fieldswas considered, here we pay attention to the change of the secondaryfield. If we assume that the influence of finite horizontal dimensions of thelayer on the field at point "2" is negligible, we can use the results fromSection 11.1 (Example 3), and in accordance with Eq, m.23) the fieldcaused by the layer with thickness Sh is

b.g z = 21TyO b.h

For instance, letting S = 2600 kqyrrr', we have

b.g zb.h =:: 0.11 mGaIjm

(II.36)

Equation 01.36) describes the Bouguer slab correction and, as well asEq. (I1.35), is commonly used in the practice of gravimetry. Of course, ifthe layer model is not able to describe adequately the effect caused by thetopography, more complicated formulas are used.

Next, we will derive formulas based on Newton's law.

b.o( q) dVg(p) = -y1 L3 L q p

v qp

(II.37)

To calculate the vertical component of the field caused by some distribu-tion of masses beneath the earth's surface, we will start with a two-dimen-sional case.

Before we discuss the reduction of geological noise, let us derive someequations allowing us to simplify the calculation of the useful signal.

Two-Dimensional Model

Suppose that a body is strongly elongated in some direction and, corre-spondingly, that it can be treated as two-dimensional. In other words, anincrease of the dimension of the body in this direction does not practicallychange the field at observation points. Thus, we will consider a two-dimen-sional body with an arbitrary cross section and introduce a Cartesiansystem of coordinates x, y, z, as is shown in Fig. II.6d, so that the body iselongated along the y-axis, It is clear that at any plane y = constant the


behavior of the field is the same. To carry out calculations we will performtwo procedures, namely,

1. The cross section of the body is mentally divided into many elemen-tary areas, and correspondingly we can treat the model as a system ofmany elementary prisms. Dimensions of every elementary cross section aremuch smaller than the distance between an observation point and anypoint in this area.

2. An elementary prism is replaced by an infinitely thin line directedalong axis y, and it bears the same mass per unit length as that of theprism.

These two steps have allowed us to replace the two-dimensional bodyby a system of infinitely thin lines, which are parallel to each other, andthe distribution of mass on them is defined from the equality

dm = il8(q) dS(q) dy =A(q) dy

since

A(q) = il8(q) dS(q) (11.38)

where A(q) is the linear density of mass on the line passing through pointq, and dS(q) is the area of the elementary cross section of the prism. Letus note that the density is a function of the coordinates x and z, but itdoes not depend on y.

Derivation of the formula for the gravitational field caused by aninfinitely thin line with the density A is very simple, and it is illustrated byFig. II.7a. We will consider the field at the plane y = O. Due to thesymmetry of the mass distribution, we can always find a pair of elementarymasses A dy and - A dy, which do not create a field component g y

directed along the y-axis, and respectively the field generated by allelements of the line has only component g" located in the plane y = O.Here r is the coordinate of the cylindrical system with origin at point 0*,and the line is directed along its axis. As is seen from Fig. II.7a, thecomponent dgr generated by the elementary mass located at the distancedy is

A dy r dydg; = -y R 2 • R = -yAr R3

or

Il.2 Detennination of the Gravitational Field 171

Fig.II.7 (a) Field caused by a linear mass; (b) model of a surface mass; (c) cross section of a3-D body; and (d) the solid angle subtended by an elementary layer.

We will assume for a moment that the line length is 2 t'. Then,integrating fields caused by all elements, we obtain

f+ t dy

s, = -')'Ar t (2 2)3/2- Y + r

Introducing the new variable ip,

y = r tan ip

we have

dy = r sec 2 ip dip

and

')'A f'P O 2')'Agr = - - cos ip dip = - -- sin ipo

r -'Po r

(11.39)


where(

tan 'Po = -r

Since

tan 'Psin e = 1/2

(1 + tan? 'P)

we obtain

2y,1 (g =--

r r ((2+ r 2)1/2

Because it is assumed that (» r, let us represent the latter in the formof a series.

(II 040)

The first term of this series describes the field caused by the infinitely longline, and in this case,

2y,1g =---

r r (11041)

Comparison of the last two equations allows us to determine the error thatoccurs when we replace a line of finite length with an infinitely long one,and then apply this result to a real elongated body. For instance, if thelength of the line is 10 times greater than the distance from an observationpoint, this error is less than one-half percent.

As follows from Fig. 1I.6d for the vertical component of the field, wehave

In particular, on the earth's surface where zp = 0,

(II 042)

and

(II 043)


Equation (H.42) describes the vertical component of the field caused bymasses within an elementary prism. Correspondingly, for the field gz dueto a two-dimensional body, we have

_ j Ao(q) dS(q)gz-2y 2 Zq

S rif zp = 0 (II.44)

(II.4S)

Thus, instead of a volume integral, the field is represented as a surfaceintegral, which simplifies calculations.

If the function Ao(q) is constant, we have

jZ q d S

gzCp) = 2yAo -2-s r

or

(11.46)

For those cases when the cross section of the body has a relativelysimple shape, the integrals on the right-hand side of Eq. (lIAS) are usuallyexpressed as elementary functions-for instance, the cylinder, the thinrod, the sheet, the fault, the thick prism, etc. However, in more compli-cated cases, determination of the field is performed by numerical integra-tion of Eq, (11.45). To further simplify calculations, suppose that thetwo-dimensional body is oriented along the y-axis, and its thickness ismuch smaller than the distance between the body and observation points;that is,

h(x) «rqp

where hex) is the body thickness. Both hex) and the density 0 can be afunction of the coordinates x and z. Then it is obvious that if the body isreplaced by a two-dimensional strip (Fig. II.7b) bearing the same mass andplaced somewhere inside the body the field would not change at theobservation point. Now Eq. (II.44) is greatly simplified and we have

jX 2Q <T ( Xq ) dxq

gz(p) = 2YZq 2+ 2x!q m Zq

where m = x p - x q , Zq is the strip coordinate along the z-axis,

!T(xq ) = A8(xq )h( xq )

is the surface density of masses within the strip, and x Iq and x 2q arecoordinates of terminal points of the strip. Note that such a model is


useful in calculating, for example, the gravitational field caused by two-dimensional structures (anticlines, depressions) on the surface of thebasement, wherein their amplitudes are small with respect to the sedimentthickness.

Now we will make one more simplification and assume that the surfacedensity a is constant. Then we have

Applying again the substitution

m = Zq tan 'P

the field g z due to masses of the strip is expressed as

(II .47)

where

if xlq<O,

In the limiting case when the strip becomes a plane, we again obtain theformulas for the Bouguer slab correction.

In fact, in this case 'PI = 7T"/2, 'Pz = -7T"/2, and therefore

Suppose that the observation point is located in the plane x p = 0 andx 1q = -XZq =X. Then, we can rewrite Eq. (11.47) in the form

Ixigz(x) =4yhdotan- l -

Zq

Assuming that Ixl» Zq we can expand tan-1lxl!zq in a series.

[x] 7T" Zq 1 z~tan -I - ::::: - - - + - -

Zq 2 Ixl 3 x 3

and then we obtain

(

Zq 2 z~ }g (p) ::::: 2Y7T"h do 1 - - + - -- ...z Ixl 37T" Ixl3

It is clear that this equation allows us to evaluate the difference between


the fields caused by a plane and a strip with a finite width. Of course, thisevaluation can be done directly proceeding from Eq, (lIA7).

Now we will show that, making use of this equation, we can calculatethe gravitational field caused by masses in a two-dimensional body with anarbitrary cross section. With this purpose in mind, let us mentally dividethe body cross section into a sufficient number of relatively thin layers withthickness h, (Fig. H.6d). Then, applying the principle of superposition andEq. (1.47) for every elementary layer, we have

N

gz( p) = 2y L h, .6.0;( 'PI; - 'P2;);=1

(liAS)

Here gz(p) is the anomalous field caused by all masses of the body,hi = Z i + 1 - z;, Z i + 1 and z, are vertical coordinates of the bottom and thetop of the i-layer, N is the number of elementary layers, and

where

_IXp-Xli'Pli = tan

ZOi

ZO; =2

(HA9)

X2;' Xli are horizontal coordinates of terminal points of i-layer, and .6.0;characterizes the distribution of masses in an elementary layer.

In particular, if .6.0; is constant within the body, we have

N

gz(p) =2y.6.0 L (Zi+I- Zl)('P2;-'Pli)

i=1

Note that with an increase in the number of elementary layers, theaccuracy of field determination increases too.

Three-Dimensional Body

Suppose that the gravitational field is caused by masses in a three-dimen-sional body (Fig. 1I.7c). In principle, the field can be calculated fromNewton's law, Eq. (I1.5); but even with fast computers numerical integra-tion over the volume for many observation points requires a lot of time,and it is natural to apply methods that allow us to simplify this integration.Two of them are described in this section. By analogy with the previouscase let us represent the three-dimensional body as a system of elementarylayers located in horizontal planes whose thickness is much smaller than


the distances from them to observation points (Fig. 1I.7d). In such a case,every layer can be replaced by a horizontal surface of finite dimensionswith the density

u(q) = Llo(q)h(q)

Correspondingly, the vertical component of the field caused by all massesof the body can be presented as a sum of fields giz due to elementarysurface masses.

(11.50)

As was shown in the first section, the field generated by surface massesis expressed through the solid angle wi(p) under which the surface is seenfrom the observation point p. Assuming that the density difference,Llui(q), is constant at the ith surface and making use of Eq. (11.22), wehave

(11.51 )

here

Thus, for the field gz we obtain

N

gAp) ='}' L LloihiWi(P)i=l

or

N

gAp) ='}' L LlO;(Zi+l- Z;)Wi ( p )i=1

(11.52)

where hi = Z i + 1 - z, is the thickness of the elementary layer.In particular, if the function Lloi is constant within the body we finally

have

N

gzCp) ='}'Llo L (Zi+l- Z;) Wi( P)i~1

(11.53)

Correspondingly, determination of the vertical component of the field dueto masses in a three-dimensional body consists of calculating a set of solidangles; this procedure is described in Chapter I.

IL2 Determination of the Gravitational Field 177

The other approach allowing us to simplify the field calculation is basedon the use of Eq, 0.89).

q1gradTdV=~TdSv s

where S is the surface surrounding the volume of the body, dS = dSn, andn is the unit vector normal to the surface element and directed outward. Tis a continuous function in the volume v.

Assuming that the function ~8 is constant and taking into accountEq. (1.80),

we can rewrite Eq. (II.37) as

1Lqpg(p) = -'Y~'Y -3- dV-c:

q 1= -'Y~81 grad-dV

v L q p

(II.54)

where the index "q" means that derivatives are taken with respect to thepoint q.

Now making use of Eq. 0.89) we obtain

dSg(p) = -'Yl18~- v ;

Respectively, for the vertical component of the field we have

where k is the unit vector directed along the z-axis.Inasmuch as

dS . k = dS cos f3

where f3 is the angle between nand k depending on point q of thesurface, we have

dS(q)gz(p) = -'Y~8~-- cosf3(q)

s L q p

(II.55)

17S II The Gravitational Field

Thus, instead of the volume integral Eq, (II.5), we have derived anexpression for the field that requires an integration only over the surface.The formulas described in these two sections allow us in many cases tosimplify the solution of the forward problem in calculating the usefulsignal. They are also used to take into account the topography effect, thecorrection for the change of elevation of observation points, and tointroduce the Bouguer slab correction.

At the same time, as we stated earlier, there is one more factor thatstrongly affects the quality of interpretation. Geological noise is mainlycaused by the lateral change of rock density near the earth's surface. Ofcourse, separation of the geological noise and the useful signal cannot bedone without some error, and very often the latter ultimately defines thedegree of ambiguity of interpretation. If we had some reasonable informa-tion about the distribution of masses, which characterizes the geologicalnoise, then the use of Newton's law would be the most natural way toevaluate its contribution. However, such information is usually absent, andcorrespondingly it is impractical to perform this separation by solvingforward problems.

Note that sources of the useful signal are located, as a rule, deeper thansources of the geological noise, and this fact results in a difference ingeometries of these two parts of the anomalous field. For this reason, thereduction of the geological noise is based on study of geometry of the fieldcaused by sources located at different distances from observation points; inthe next section we will describe the theoretical basis of one such ap-proach. To accomplish this task it is necessary to derive a system ofequations of the gravitational field, introduce its potential, and make useof Green's formula (Chapter 1).

11.3 System of Equations of the Gravitational Fieldand Upward Continuation

As is demonstrated in Chapter I, field equations show the relationshipbetween a field and its generators. In the case of the gravitational fieldthere is only one type of generator, namely sources (massesj-i-the vortextype of generator is absent. Proceeding from this concept, we will derivethe system of equations for the field. First, we shall consider an elemen-tary mass located at point q and calculate the flux of this field through anelementary surface at point p, as is shown in Fig. II.Sa. Applying Newton'slaw we have

dm L q p ' dSg • dS = - y 3 = - y dm dw

L q p

(II.56)

11.3 Systems of Equations of the Gravitational Field Upward Continuation 179

Fig. 11.8 (a) Flux through an elementary surface; (b) flux of surface masses; (c) evaluation ofmass generating secondary field; and (d) voltage due to an elementary mass.

where dto is the solid angle under which the surface dS is seen frompoint q.

It is obvious that the flux through an arbitrary surface S presents a sumof elementary fluxes, and therefore

1g . dS = - y dm ws

where w is the solid angle under which surface S is seen from the point q.In particular, the flux through an arbitrary closed surface surrounding theelementary mass dm is

~ g : dS = -4'lT')' dms

(11.57)

since the solid angle under which the closed surface is seen from the point


q is always equal to 47T, regardless of the surface shape and the position ofthe point q in the volume surrounded by the surface S.

Now making use of the principle of superposition and assuming thatinside the volume V there is an arbitrary distribution of masses, we obtain

~ g : dS = -47T'}'ms

(11.58)

where m is the total mass in the volume V.Equation (11.58) is called the second equation of the gravitational field

in the integral form, and in this regard let us make two comments.

1. The theory of fields described in Chapter I shows that the flux of anyfield through a closed surface characterizes the quantity of sources in thevolume surrounded by the surface S. Therefore, it is natural that the massm is present in the right-hand side of this equation. At the same time thecoefficient - 47T'}' follows directly from Newton's law, and its value isdefined by the system of units.

2. Masses located outside the volume have an influence on the fieldeverywhere, including points of the surface S surrounding this volume. Atthe same time the field caused by these masses does not contribute to theflux: through this surface. This fact is proved in Chapter I for any fieldregardless of its nature; but it also follows from Eq. (11.56), since the solidangle under which a closed surface is seen from a point located outside itis always equal to zero. This is a remarkable fact that is difficult to predictif we do not know that the flux of the field through any closed surface isdefined only by masses within the volume surrounded by this surface.

Assuming a volume distribution of masses characterized by density 8,we will present Eq. (11.58) as

(11.59)

Now we are ready to derive the differential form of this equation. Inaccordance with Gauss' theorem, Eq. 0.138), we have

~ g . dS = f div g dV = - 47T'}'f 8 dVs v v

(11.60)

Inasmuch as this equality holds for any volume, the integrands are alsoequal.

div g = -47T'}' 8 (11.61)


This is the differential form of the second equation of the gravitationalfield, which is valid for regular points, where the first derivatives of thefield g exist. In particular, outside of masses this equation is essentiallysimplified, and we have

div g = 0 (II.62)

In reality there are always only volume distributions of masses withfinite values of 8(a). However, as was demonstrated in the first section, forcertain conditions it is useful to introduce surface masses with densityai.q). In such cases, making use of Eq, (I.135), it is easy to derive a surfaceanalogy of the second equation. In fact, we shall assume that there is asurface distribution of masses shown in Fig. Il.Sb, and imagine a cylindri-cal surface around point q. Then, applying Eq, (II.58) we have

gz·dSz+gj·dS+ f g·dS= -47TyudSSf

where

as,=dSn, dS I = -dSn

Sf is the lateral surface of the cylinder, n is the unit vector directed fromthe back to the front side of the surface, and tr dS is the elementary massinside the cylinder.

In the limit, when the cylinder height tends to zero, we obtain

(II.63)

where gZn and gin are normal components of the field at either side of thesurface. Therefore, the difference of the normal components of the fieldnear the surface mass is defined by the surface density at the same point.

Equation (II.63) represents the surface analogy of Eq. (Il.Sl) in thevicinity of points where singularity in the field behavior is observed. Thus,we have derived three forms of the second equation of the gravitationalfield.

¢g. dS = -47Tym

div g = - 47Ty8(II.64)

Before we derive the next equation of the field, let us illustrate onerpplication of Eq, (II.58) in interpretation of gravitational data. Suppose


that measurements of the field are performed over some areas andcorresponding corrections are introduced. Also, the useful signal gz prac-tically vanishes at the boundaries of this area (Fig. II.8c). Then, the halfspace is a volume where all sources of this field are located. This volume issurrounded by the area of measurement and a half-spherical surface Sowith relatively large radius where the field can be considered to be that ofthe point source. Correspondingly the flux through this surface is

f g : dS = -21T"ymSo

For this reason, the flux through a closed surface surrounding thisvolume is expressed in terms of a surface integral over the observationarea only. Therefore, we obtain

(11.65)

since

s : dS = -gz dS

Thus, we have found the total mass causing the useful signal provided thatgeological noise is absent, and it is equal to

(II .66)

Now we shall derive the first equation of the gravitational field makinguse of two approaches. The first one is based on results described inChapter I where it is shown that the circulation of any field characterizesthe amount of vortex generators. Since the gravitational field is caused bysources (masses) only, we can instantly write all three forms of the firstequation of the field.

curl g = 0 (11.67)

where n is the normal to the surface and q2 and ql are fields from thefront and back sides of the surface, respectively.

The second approach is based on Newton's law, and we will describe itin detail. Suppose that there is an elementary mass at point q, and


consider the voltage

between two points along the path bb' and b'a, where bb' is an arc andb'a is a displacement along the radius (Fig. 1I.8d). In the case of theelementary mass it can be presented as

r rb r ra dtl, g' d/ = I, I g . d/ + l, g' d/ = - y dm l. -2-b b b, b, L q p

r dL= -ydmJ, -2-

b L q p

(11.68)

since along the arc bb I the field g and displacement d/ are perpendicularto each other and, correspondingly, dot product g . d/ is zero, while alongthe path b I a displacements dl' and dL coincide. Performing the integra-tion in Eq. (II.68) and taking into account that L q b l = L q b we obtain

a [1 111g'd/=ydm - --b L q b L q a

(11.69)

Now we shall represent an arbitrary path between points b and a as asystem of elementary arcs and small displacements in a radial direction(Fig. II.9a). Then, taking into account the fact that integration along arcsdoes not give a contribution to the voltage, we obtain

+ ... __1)= y dm{_I __1}i; i-; i.; (11.70)

where L q b and L q b are distances from an elementary mass to terminalI 1+1

points of corresponding displacements along a radial direction. As followsfrom Eqs. (11.69), (I1.70) the voltage does not change when the path ofintegration varies, but it depends on the position of the terminal points


Fig. 11.9 (a) Voltage along an arbitrary path; (b) circulation of the gravitational field; and(c) continuity of tangential components.

(a, b). In other words, the voltage of the gravitational field is pathindependent. This well-known result directly follows from Newton's lawand reflects the fact that only masses generate the gravitational field.

I think it is natural to be impressed by this amazing feature of the field.Indeed, suppose there are two points at a distance 1 m apart. Calculatingthe voltage between two points along a straight line with length 1 m, weobtain its value. Then, let us choose a completely different path betweenthe same points, which has a length of thousands of kilometers and goesthrough mountains, valleys, oceans. Of course, the field g varies in magni-tude and direction from point to point of this path. But what is reallyremarkable is the fact that in both cases the voltage remains the same.

Only one step is left to derive the first equation of (II.67). We willconsider two arbitrary paths 2 1 and 2 2 between points a and b(Fig. 1I.9b), and due to independence of the voltage of the path, we have

1 g·d.!= 1 g·d.!.2"1 .2"2

or

f god.!= f god.!acb adb


Inasmuch as a change of the direction dl" to the opposite one results in achange of the sign of the voltage, we can write

f g'dl'=-L g'dl'acb bda

or

f g : dl' +1 s : dl' = 0acb bda

Finally we have

(11.71)

where 2' is an arbitrary closed path. Thus, we have proved that thevoltage along a closed path (circulation) is always zero for the gravitationalfield, and Eq, (11.71) is called the first equation in the integral form. Ofcourse, this result is valid for any closed path that can, in particular, passthrough media with different densities. Let us emphasize again thatEq, (11.71) does not require a proof as soon as it is known that thegravitational field is caused by sources (masses) only.

Now applying Stokes' theorem at the vicinity of regular points of thefield, we have

¢C g . dl' = I. curl g . dS = 0'Sf' S

or

curl g = 0 (11.72)

where S is an arbitrary surface bounded by the contour 2', and thedirections dl" and dS are related to each other by the right-hand rule.

Equation (II.72) represents the first equation of the gravitational fieldin differential form, which is valid at points inside and outside masseswhere the first derivatives of the field exist. If we suppose that there arealso surface masses, then it is necessary to derive a surface analogy ofEq. (11.72). This is related to the fact that this equation cannot be appliednear masses where the normal component of the field, gn' is a discontinu-ous function, Eq. (11.63).

To derive this analogy let us calculate the voltage along the path shownin Fig. II.9c. Making use of Eq, (I1.71) we obtain

(11.73)


since the displacements de' are small, and the integrals can be replaced bydot products of the field and the displacement while the voltage alongpath dh, perpendicular to the surface, vanishes when dh tends to zero.


dt; = -dl;we obtain

or

(11.74)

where g 11 and g 21 are the tangential components of the field.Equation (11.74) is the surface analogy of the first equation of the

gravitational field, and it shows that the tangential component of the fieldg is a continuous function. Thus, we have derived three forms of the firstequation.

curl g = 0

and all of them contain the same basic information, namely, that thegravitational field is caused by masses.

Now we are prepared to present the system of equations of thegravitational field, derived from Newton's law, as well as of course, theprinciple of superposition; both integral and differential forms of thissystem are shown below.

Integral Form of Equations for Gravitational Field

Newton's law

()( 8(q)Lq p dV

g p = -yJT 3V L q p

/fg' d/ ~ 0 I 1_11

--'

Here y is the gravitational constant and m is the mass located inside thevolume surrounded by the surface S.


Arrows show that both equations, which are valid everywhere, arederived from Newton's law.

Differential Form of Equations for Gravitational Field

Newton's law

B(q)g(p) = -y1-3-Lqp dV

v L qp

(11.75)

divg = - 47TyB

gZn - gln = Divs = '-47TyB I

curl g = 0I

on S on S

Here (J" is the surface density of mass, and the last pair of equationsdescribe the field behavior in the vicinity of points where surface massesare present.

Inasmuch as the field equations are derived from Newton's law and theprinciple of superposition they do not contain more information about thefield behavior than these laws themselves. However, they allow us tounderstand better some of the features of the field and, in particular, todevelop a method for reduction of geological noise. With this purpose inmind, let us first of all introduce a new scalar function called the potentialof the gravitational field. It can be done making use of the first equation indifferential form. As is shown in Chapter I, the solution of the equationcurl g = 0 is

(11.77)

g = grad U (11.76)

and this result is verified by direct substitution of Eq. (11.76) intoEq. (lI.72).

Thus, we have expressed the vector field g through a scalar function,U(p), with the help of a relatively simple operator, Eq. (1.64).

1 au 1 au 1 aug=--i l + --iz +--i3h , aXl hz axz h3 aX3

where hl' hz, and h3 are metric coefficients; xi' x2, x 3 are coordinates of


the observation point; and iI' i 2, and i 3 are unit vectors of the coordinatesystem.

It is clear that Eq, (II.76) defines the potential U up to a constant; thatis, an infinite number of potentials describe the same field g. For thisreason it is natural to treat the potential as an auxiliary function intro-duced with only one purpose, namely to simplify the analysis of the morecomplicated vector field g.

Inasmuch as the field g is expressed through the potential U, it isproper to derive an equation describing its behavior. We already used thefirst equation, curl g = 0, to introduce the potential; now, substitutingEq. (11.76) into the second field equation, Eq. (11.61), we obtain

div grad U = -4'lTrc5

or (II.78)

Thus, we have obtained Poisson's equation, which in accordance withEq, (1.236), can be represented in an orthogonal system as

At the same time, outside masses Eq. (II.78) is simplified, and we obtainLaplace's equation.

(II.79)

Both relations (11.78) and (II.79) describe the behavior of the potential atregular points where the first and second equations of the field are valid.

To characterize the behavior of the potential near surface masses, let usmake use of Eq. (11.76) according to which any component of the fieldalong some direction t is equal to the derivative of the potential in thisdirection; that is,

(II.80)

(II.81)

where t is a line along any direction. Thus, instead of surface analogies offield equations (11.74) and (11.63) we obtain

aU2 eu,---=0at at

(II .82)


and

auz auz- - - = -47TyCTan an

where U, and Uz are values of the potential at the back and front sides ofthe surface, respectively.

It is obvious that continuity of tangential derivatives of the potentialfollows from continuity of the potential itself, and correspondinglyEq. (II.8l) can be replaced by

(II.83)

Thus, the behavior of the potential is described by the system of equationsgiven below.

Newton's law

IVzU= -47TyO Iif 8 * 0

on S

(II.84)

IVzU= 0 I0=0

auz eo.- - - = -47TyO"an an

on S

Now we shall find the relation between the potential U and masses.First consider an elementary mass dm = 0 dV. In accordance with Newton'slaw we have

since

dm p 1g( p) = - y-3-Lqp = Y dm grad-

L q p L qp(II.85)

where the index p means that the gradient is considered in the vicinity ofthe point p.

Comparing Eqs. (II.76) and (II.85) we can conclude that the function Ucorresponding to the field caused by elementary mass 0 dV located at


point g is

mU(p)=8-+C

i;(II.86)

since if gradients of two functions are equal, then the functions themselvesdiffer in general by a constant.

Taking into account the fact that the field g caused by mass dm tends tozero at infinity, it is natural to assume that its potential also vanishes asL q p ~ 00. Then from Eq. (11.86) it follows that C = 0, and we obtain

mU(p) = y-

-;(II.8?)

Now applying the principle of superposition we derive an expression forthe potential U caused by a volume distribution of masses.

f8(q) dV

U( p) = yv L q p

(II .88)

Comparison of Eqs. (11.5) and (11.88) clearly shows that the potential isrelated to masses in a much simpler way than g, and this fact is one of thereasons for its introduction.

If, along with volume masses, all other types of masses are considered,we have

[f 8( q ) dV j(T(q)dS "m j 1.A(q)dt]U(p)=y + +,-,-+ (II.89)

v L q p S L q p j~ I L q p Sf' L q p

where m j is a point mass and A(q) is the linear density.Let me show one more application of the potential, and with this

purpose in mind, we will consider the change of this function in thevicinity of some point dll. As is well known, this can be represented as

(II.90)

where dtp dtz' and dt3 are elementary displacements along the coordi-nate lines x p x z, and x3 •

I1.3 Systems of Equations of the Gravitational Field Upward Continuation 191

It is easy to see that the right-hand side of this equation can be writtenas a dot product of two vectors, namely,

and

Here

(II.91)

Therefore,

dU = dl' . grad U = g . dl' (II.92)

After integration of this equation along an arbitrary path with terminalpoints a and b we obtain

U(a) - U(b) = fa g : dl'b

(II.93)

Thus, the voltage along some path is expressed by the difference ofpotentials at terminal points of this path. Certainly it is much simpler totake a difference of the scalar at two points, Ui a) - U(b), than to performan integration, f:g .dl; and this fact demonstrates another advantage ofusing the potential.

Let us return to Poisson's and Laplace's equations, which describe abehavior of the potential inside and outside masses.

f1U = -47TyO and f1U=O

At the same time we have already derived an explicit expression for thepotential U that allows us to find this function if masses are known[Eq. (II.88)]. This means that

fo(q)dV

U( p) = Y Lv qp

is the solution of Poisson's equation inside of masses, and it satisfiesLaplace's equation outside of them.

The potential is also useful to establish relations between differentcomponents of the gravitational field. In accordance with Eq. (II.76) we


have

and

and taking into account the fact that

/ 2 ( 2 2Lqp=V(xp-Xq) + Yp-Yq) +(Zp-Zq)

we obtain from Eq. (11.88)

j v; - v,s, = y 0-3- dV,V L qp

and

(11.94)

Of course, these equations follow directly from Newton's law, too.Then, taking corresponding derivatives, we have

(11.95)

The equations indicate that the information contained in one componentof the field about its sources cannot be in principle increased by measuringother components. In fact, this result was established earlier in consider-ing the field of an elementary mass.

Now we will demonstrate another merit of the potential that is impor-tant for interpretation of the gravitational data. Suppose that the anoma-lous field measured on the earth's surface consists of two parts: One iscaused by relatively deep structures and represents the useful signal, whilethe other is generated by lateral changes of rock density near this surfaceand characterizes the geological noise (Fig. II.10a).

Let us assume for a moment that the field is measured at differentdistances above the earth's surface. Then it is obvious that with anincrease of the distance both parts of the field begin to decrease. However,there is one important difference in their behavior; namely, the usefulsignal decreases more slowly because distances between deep structuresand observation points alter by a relatively small amount, while the signalcaused by the geological noise varies more rapidly because sources of thisfield are closer to the observation points. Consequently, the contributionof the geological noise to the anomalous field decreases with an increaseof an elevation of the observation point. This tendency is observed until


Fig. n.IO (a) Useful signal and geological noise, and (b) analytical continuation upward.

dependence of both fields on the distance to their sources becomespractically the same.

Taking into account this behavior of the useful signal and the geologicalnoise, we will describe a method to calculate the field in the upper halfspace when its values on the earth's surface are known. It is clear thatsuch a procedure will allow us to reduce the influence of the geologicalnoise.

First of all we know that in the upper half space, where masses areabsent, the potential U satisfies Laplace's equation.

.lU=o

Suppose that on the earth's surface the vertical component of the gravita-tional field, that is, the derivative au/az is known.

Inasmuch as only the anomalous field is considered, at sufficiently largeelevations the field becomes small and, correspondingly, we can assumethat the potential U is equal to zero on some half-spherical surface with arelatively large radius R (Fig. 11.10b).

Thus, we have a volume V, surrounded by the earth's surface, wherethe derivative au/az is known, and the hemispherical surface with radiusR, where the potential is zero, and our task is to find the potential U aswell as the field g at every point of this volume. This means that we haveto solve Dirichlet's value-boundary problem (Chapter 0, which uniquelydefines both the potential U and the field g. This can be written as theproblem of field determination within the volume V if the following isknown:

1. Above the earth


2. On the earth's surface

where gz is a known function3. V ~ 0, as r ~ 00

To derive formulas allowing us to calculate the field in the volume V,we will make use of the second Green's formula, Eq. (1.289).

(11.96)

where both functions cp(q) and l/J(q) are continuous together with theirfirst derivatives, and n is the unit vector directed outside the volume. Inessence, Eq. (11.96) is Gauss' formula and consequently it establishes arelation between values of scalar function inside the volume V on onehand and those of functions and their derivatives on the surface S on theother hand. For this reason it is natural to apply Eq. (11.96) to find thepotential U in the upper half space above the earth's surface. Followingvery closely the derivatives in Chapter I, we assume that function l/J(q) isthe potential of the gravitational field Ut.q) and introduce the notations

l/J(q)=U(q) and cp(q) = G(q)

(11.97)

Then, Eq. (II.96) can be rewritten as

f U V 2GdV = rr. (vaG - G av) dSv '.fs an an

since V2V = 0 in the upper half space.Our task is to derive an explicit expression for the potential U proceed-

ing from Eq. (II.97); that is, we have to take the function V out of integralsin this equation. To realize this, let us choose a function G that satisfiesthe following conditions:

1. Everywhere inside the volume V it is a solution of Laplace's equa-tion

V2G =0

except at the observation point p.2. In approaching point p function G has a singularity l/L p q ; that is,

1G(q,p) ~L'

qp


3. With an increase of the radius R of the hemispherical surface(Fig. II. lflb), the function G decreases at least inversely proportional todistance L q p • In other words, the function G, which is often calledGreen's function, is harmonic except at the point p, where the potential iscalculated.

Inasmuch as Gt.q, p) has a singularity at the point p, we can applyGreen's formula 0.97), provided that the point p is surrounded by asurface S* with very small radius r 1 (Fig. II. l Ob).

Then, applying this equation to volume V surrounded by surfaces SandS* and taking into account the fact that V2G = 0, we obtain

(so au ) (aG au ),.{.. U--G- dS+,.{.. U--G- dS=O

'f:., an an 'f:.,. an an

since the volume integral vanishes.We will represent the first term of this equation as

(11.98)

(ec au) (aG au) (aG au ),.{.. U--G- ds=f U--G- <:! U--G- dS

'f:., an an Su an an SR an an(11.99)

where So is the earth's surface, the z-axis is directed downward, and SR isthe hemispherical surface with radius R.

Since with increasing radius R both functions U and G decrease asIjR, their first derivatives aujaR and aG jaR tend to zero as 1jR2

•

Correspondingly, the integrand of the second integral at the right-handside of Eq. (11.99) decreases as IjR3

, and making use of the mean valuetheorem we obtain

(ec au) c c 2

f u- - G- dS --'> -f dS = -21TR --'> aSR aR aR R3

SR R3

where C is some constant.Therefore, Eq. (11.98) can be rewritten in the form

as R --'> 00

{eo au } {aG au }

f U--G- dS+,.{.. U--G- dS=Os, az az 'f:.,u an an (II .100)

Now we will consider the integral over the spherical surface S* aroundthe point p with radius r l' which in the limit tends to zero. Let us make

as r 1 ~ 0 (II.lOl)


two comments about the behavior of the integrand, namely,

1. The potential U(q) and its normal derivative on the spherical surface

au au-=--=-gan ar r

have finite values since they describe a real field.2. With a decrease of radius r function G(q, p) behaves as l/r, and its

derivative aG/an = -aG/ar increases proportionally to l/r 2•

Then applying again the mean value theorem, we obtain

(eo au) ea au

rf.. U--G- dS=-rf.. U-dS+J G-dS'f'.s. an an 'f'.s. ar s; ar

1 1 aU(p)= U(p) 2417rf + - --41Trf = 417U(p)

r 1 r 1 Br

Thus, due to the fact that the chosen Green's function has a singularityl/L q p at the point p, we have been able to take function U(p) out fromthe integrand and obtain its expression in an explicit form.

In fact, from Eqs. (11.100), (11.101) we have

{aG(q,p) au(q)}

417U(P)=J U(q) -G(q,p)-- dSSo az az

or

1 { aG(q,p) }U(p) = -4J G(q,p)gzCq) - U(q) a dS (II.102)

17 ~ z

sinceau

s, = a;Therefore, we have derived a formula for calculation of the potential ofthe field everywhere in the upper half space if both the potential U andthe vertical component of the gravitational field are known at the earth'ssurface SQ.

Having taken the derivative from both sides of Eq. (11.102) we obtainfor the vertical component of the gravitational field at the point p

(II.103)


where zp and Zq indicate that derivatives of the Green's function aretaken with respect to the coordinate Z near points p and q, respectively.

We can imagine an infinite number of Green's functions satisfying theconditions formulated earlier. The simplest of these is

1G=-

c; (11.104)

Indeed, it satisfies Laplace's equation everywhere except at the point p,since it describes up to a constant the potential of a unit mass located atthe point p. Also, it has a singularity at this point and it provides a zerovalue of the surface integral over the hemisphere when its radius R tendsto infinity. Correspondingly, we can write

However, this equation is impractical since the potential U is notmeasured on the earth's surface, and therefore we have to choose aGreen's function such that its derivative aGjaz on the earth's surfacewould be zero. In this case Eq. (11.102) is greatly simplified.

1U(p) = 4 f G(q,p)gz(q) dS

1T So(11.106)

where gz(q) is the measured vertical component of the field on the earth'ssurface, and Gi q, p) is an unknown function that satisfies the followingconditions:

1. In the upper space, z < 0, Green's function G is a solution of theLaplace equation

everywhere except at the observation point p.2. It has a singularity of type IjL q p , that is, in approaching point p,

1G(q,p) ~-

-;


3. At the hemispherical surface it decreases at least as I/R with anincrease of radius R; and finally,

4. At the earth's surface the derivative aG(q, p)/az vanishes.

aG-=0an

As was demonstrated in Chapter I, determination of the function Gsatisfying all these conditions presents in essence a solution of the bound-ary value problem; and in accordance with the theorem of uniqueness,these conditions uniquely define the function G(q, p). In general, asolution of this problem is a complicated task, but there is one practicalimportant case of the plane surface So when it is very simple to find theGreen's function.

Let us introduce the point s, which is a mirror reflection of the point pwith respect to the plane of the earth's surface (Fig. 1I.10b) and considerthe function GI(p, s, q) equal to

(11.107)

where q is a point at the earth's surface, and

{2 2 2} 1/2

L qs = ( x q - x s) + (Yq - Ys) + ( Z q - Zs)

Taking the derivative aGIIaz, we obtain

aG I = _ Zq-Zp _

aZ q L~p

Inasmuch as at every point at the earth's surface Zq = 0 and zp = -zs' but

L qp = L qs

the derivative aGI/azq is equal to zero.It is obvious that the other three conditions are also met. Therefore, in

accordance with Eq. (11.106) we have

References 199

The latter allows us to calculate the vertical component of the field in theupper half space when it is known at the earth's surface. Correspondingly,this transformation is called upward continuation and is used to reducethe influence of geological noise.

References

Garland, G.D. (1979). "Introduction to Geophysics." W.B. Saunders, Philadelphia.Grant, F.S., and West, G.P. (1965). "Interpretation Theory in Applied Geophysics."

McGraw-Hill, New York.Green, R. (1986). The use of the subtended solid angle for calculating the magnetic anomaly

over structurally complex bodies. Geoexploration 2461-69.Parasnis, D.S. (1979). Principles of Applied Geophysics 3e. Chapman and Hall.

Chapter III Electric Fields

111.1 Coulomb's LawNormal Component of the Electric Field Caused by a Planar Charge DistributionEffect of a Conductor Placed in a Free Space and Situated within anElectric Field

111.2 System of Equations for the Time-Invariant Electric Field PotentialIntegral Form of Equations for Time-Invariant Electric FieldDifferential Form of Equations for Time-Invariant Electric FieldA Conductor Situated in Free Space

III.3 The Electric Field in the Presence of DielectricsIlIA Electric Current, Conductivity, and Ohm's LawIII.5 Electric Charges in a Conducting MediumIII.6 Resistance111.7 The Extraneous Field and Its Electromotive Force111.8 The Work of Coulomb and Extraneous Forces, Joule's LawIII.9 Determination of the Electric Field in a Conducting Medium

III.10 Behavior of the Electric Field in a Conducting MediumReferences

In this chapter we will develop the theory of electric fields, used indifferent electrical methods. This theory is based on Coulomb's law,Ohm's law, and the principle of charge conservation. Let us begin withCoulomb's law.

III.I Coulomb's Law

Experimental investigations carried out by Coulomb in the 19th centuryshowed that the force acting on an elementary charge situated at the pointp due to the presence of an elementary charge situated at the point q is

m.l Coulomb's Law 201

described by an extremely simple expression.

I de(q)de(p)F(p) = -- 3 L q p

47TEO L q p

where L q p is the vector

(III.I)

Here L q p is the distance between points q and p; L~p is a unit vectordirected along the line connecting points q and p; and EO is a constantknown as the dielectric permeability or electrical permittivity of freespace. In the International System of Units (SI) this constant is

I1'0= --10-9 Fyrn

367T

By definition elementary charges occupy volumes much smaller than thedistance L q p between them, and

de(q) = 8(q) dV, de ( p) = 8( p) dV

where 8 is the volume density of charges.Equation (III.1) can also be written as

I de(q) deep) 0F(p) = -- 2 L q p

47TEO L qp

and it is obvious that

F(p) = -F(q)

(111.2)

The electric force of interaction between two elementary charges isdirectly proportional to the product of the charge strengths and inverselyproportional to the square of the distance between them. Unlike gravita-tional mass, electric charges can be both positive and negative, and for thisreason the electric force F( p) has the same direction as the unit vectorL~p when charges have the same sign, and it has the opposite directionwhen the product of charges is negative (Fig. III.1a). This simple expres-sion is valid of course only as long as the distance between charges is far

202 III Electric Fields

a b

p

q

c dq

Fig. 111.1 (a) Interaction between elementary charges; (b) electric field of volume charges;(c) normal component of charges on a plane surface; and (d) normal component of chargeson an arbitrary surface.

greater than the dimensions of the volume within which the charges aresituated.

To define the electric force of interaction between charges when one ofthem is contained in a volume having a dimension comparable to thedistance between charges, we must make use of the principle of superposi-tion, as was done in the case of the gravitational field. According to thisprinciple the force of interaction between two charges is independent ofthe presence of other charges. Using this principle, the force between anelementary charge at point p, dee p), and an arbitrary volume distributioncan be written as (Fig. HUb)

_ de(p) (o(q) dVF(p) - 4 if L3 L q p

17£0 V qp(III.3)

Extending this approach to a more general case in which all types ofcharges are present (volume, surface, linear, and point charges) and againapplying the principle of superposition, we obtain the following expressionfor the electric force of interaction between an elementary charge deep)

III.I Coulomb's Law 203

and a completely arbitrary distribution of charges.

_ deC p) [j8(q) dV j I(q) dSF(p)- 47TE L3 L qp + S L3 L qp

o v qp qp

j, A( q ) dt . "ei(q) 1+ 3 Lqp + z: -3-LqpL L qp L qp

(IlIA)

where 8 dV, I dS, A dt are elementary volume, surface, and linear chargeswith densities 8, I, and A, respectively, and e, is a point charge, that is, anelementary charge regarded as if it is located at some point. Unlikevolume charges, the others are mathematical models of real distributionsof charges that in many cases drastically simplify calculation and analysis.

At this point we will define the strength of the electric field E( p) asbeing the ratio between the force of electrical interaction F and the size ofthe elementary charge (considered to be a test charge) at the point p,

(IlLS)

For convenience the strength of the electric field is usually referred tomerely by the term "electric field." It does not have the same dimensionsas the force, but in the SI system of units it is measured in volts per meter.The electric field E can be thought of as the electric force acting on a testcharge de, inserted into a region of interest, and normalized by thischarge. Under this action a positive charge moves in the direction of thisfield and a negative charge moves in the opposite direction. Of course, ifthe electric field is known, it is a simple matter using Eqs. (IlLS) tocalculate the force of interaction F. As follows from Eq. (IlI.1), chargeshaving opposite signs attract each other while charges with the same sign,unlike gravitational masses, repel. In accordance with Eq, (IliA) theexpression for the electric field can be written as

1 [8dV IdSE(p)=--j-3-Lqp+!.-3- Lqp

4'17"EO v L qp S L qp

(III .6)

If the distribution of charges is given, the function E depends only oncoordinates of an observation point p. Because of this the function E, in


the same manner as the acceleration g, is termed a "field." Here it isappropriate to make the following comments:

1. Electric charges are the sole sources of an electric field that does notvary with time.

2. Coulomb's law describes the dependence of this field on charges. Itis essential to note that the electric field caused by a given distribution ofcharges is independent of the physical properties of the medium. In otherwords, the electric field due to the same distribution of charges remainsthe same whether it is considered in free space or in a nonuniformmedium. This follows from the fact that neither the dielectric constant northe conductivity of a medium are present in Eq, 011.6).

3. Coulomb's law, like Newton's law, was not derived from otherequations; and in this sense it is the fundamental physical law thatdescribes the behavior of the constant electrical field. Consequently, thebasic field equations will be obtained from Coulomb's law.

4. Under certain conditions of great importance for geophysical appli-cations, Coulomb's law remains valid even when electric fields change withtime.

5. As in the case of the gravitational field when distribution of electriccharges is known, calculation of the field E, using Eq. (III.6), presents noserious difficulties. However, unlike the gravitational field, in most practi-cal cases it is impossible to know all the charges prior to calculation, andcorrespondingly Coulomb's law becomes useless from a practical point ofview.

Now we shall consider two examples of fields caused by specific distri-butions of charges.

Normal Component of the Electric Field Causedby a Planar Charge Distribution

Suppose that there is a distribution of charges with density !,(q) on theplane surface as shown in Fig. 1I1.1c. Introduce the vector

dS =dSn

where n is the unit vector directed away from the back side of the plane(1) toward the front side of the plane (2) on which the charge is dis-tributed. We consider only the normal component of the field, that is, thecomponent perpendicular to the surface. In accordance with Coulomb'slaw, as expressed by Eq. (111.6), every elementary charge !,(q) dS creates a

111.1 Coulomb's Law 205

field described by the equation

1 I(q) dSdEep) = -4- L 3 L qp

7TE O qp

Therefore, the normal component of the field is

(111.7)

(111.8)

Here (L qp, n) is the angle between the directions of L~p and n.It is clear that the product dS L qp cos(Lqp, n) can be written as a dot

product as follows:

(III.9)

since L qp = - Lp q • Thus, the normal component of the electric field canbe written as

1 dS . L p qdEn(p) = - - 3 I(q)

47Teo L pq

Inasmuch as the expression

represents the solid angle dw(p), subtended by the element dS from thepoint p, we have

(1II.10)

In a similar fashion, for the normal component caused by all surfacecharges we obtain

(111.11)

In particular, if the charge is distributed uniformly on the surface CI is

206 II1 Electric Fields

constant), we have

(III.12)

where w(p) is the solid angle subtended by the surface S when viewedfrom the point p, As was shown in Chapter I, the solid angle is eitherpositive or negative depending on whether the back or the front side ofthe surface is viewed. With increasing distance from the surface S thesolid angle decreases, and correspondingly the normal component of thefield becomes smaller. In the opposite case, when the point p is consid-ered to approach the plane surface S, the solid angle increases and in thelimit becomes equal to - 271" and + 271" when the observation point p islocated on either the front side (2) or the back side (1) of the planesurface, respectively. Thus we have the following expressions for thenormal component of the electric field on either side of the surface:

IE(I)= --

n 2£0(III.13)

These two expressions, as well as similar ones for the case of thegravitational field, indicate that the normal component of the electric fieldis a discontinuous function across the surface S. Let us examine thisbehavior of the normal component in some detail. The normal componentof the electric field can be written as the sum of two terms.

(III.14)

where E~ is the part of the normal component caused by the elementarycharge I(q) dS located in the immediate vicinity of the point q, and E~-q

is. the part of the normal component contributed by all of the other surfacecharges. It is clear that

where ws-q(p) is the solid angle, subtended by the plane surface Swithout the element of the surface dS(q), as viewed from the point p.

Letting the point p approach the elementary area dS(q), the solidangle subtended by the rest of the surface, WS-q(p), tends to zero, and thenormal component is defined only by the charge located on the elementarysurface dS(q).

as p ~q


At the same time the solid angle subtended by the surface element dS(q),no matter how small that area is when viewed from an infinitely smalldistance from point q, tends to ±27T.

as p ~q

Therefore, the normal component of the field on either side of the surfaceis determined only by the elementary charge located in the immediatevicinity of the point q.

(111.15)

The difference in sign of the fields on either side of the surface reflectsthe fundamental fact that the electric field vector shows the directionalong which an elementary positive charge will move under the force ofthe field. Thus, the discontinuity of the normal component of the field, asan observation point passes through the surface, is caused only by theelementary charge located near this point. For example, if there is a holein the surface, the normal component on either side of the surface isE~-q, and therefore the field is continuous along a line passing throughthe hole.

We can generalize these results to the case in which the surfacecarrying the charge is not planar. Making use of the same approach basedon the principle of superposition and the definition of solid angles, wearrive at the following expressions for the normal components on eitherside of a surface:

(III.16)

if p ~q

In contrast to the previous case, the normal component E~-q(q) causedby charges located at the surface but outside the element dS(q) is notnecessarily zero at the point q (Fig. IIUd). However, we can readilyrecognize a very important feature of this part of the field. Inasmuch asthese charges are located at some distance from the point q, theircontribution to the field is a continuous function when the observationpoint p passes through the element dS(q), and therefore

(111.17)


Correspondingly, Eq. (I1I.16) can be written as

(IIU8)

This means that the discontinuity in the normal component, as before, is

(III.19)

and it is caused only by charges located within the elementary surfacedS(q).

It should be stressed that Eq. (I1I.19) is a fundamental equationdescribing electromagnetic field behavior, and it is valid for any rate ofchange of the field with time. In essence, we might say even though we riskgetting ahead of ourselves that Eq, (I1I.19) is a surface analogy of Maxwell'sthird equation.

Unlike the gravitational field, the surface distribution of charges playsan essential role. In fact, in real conditions of electrical methods practi-cally only this type of charge distribution occurs. For this reason, regard-less of how surface charges arise, Eqs. (HU8), (III.19) will be used oftenin this chapter.

Effect of a Conductor Placed in a Free Spaceand Situated within an Electric Field

We will now consider a second example illustrating electrostatic induction.Suppose that a conductive body of arbitrary shape is situated within theregion of influence of an electric field Eo, as shown in Fig. IIL2a. Underthe action of the field the positive and negative charges residing inside theconductor move in opposite directions. As a consequence of this move-ment electric charges develop on both sides of the conductor. In doing so,these charges create a secondary electric field directed in opposition to theprimary field inside the conductor. The induced surface charges distributethemselves in such a way that the total electric field inside the conductorwill disappear; that is,

(I1I.20)

where E, is the electric field strength within the conductor. This process is

a b

III.l Coulomb's Law 209

c=:>e2 ~e3air

earthsurfacef_,tiJt41\iI:'li;W4@UiiM!'lIlilJkliiil§_

c d

a

Fig. 111.2 (a) Electrostatic induction; (b) influence of charges in air; (c) flux of surfacecharges: and (d) voltage of the electric field.

termed "electrostatic induction." At this point it is important to make twocomments.

1. In our description of this phenomenon we have given a very approxi-mate picture of the process in which only the electrostatic field is consid-ered to be present. In fact, the process of accumulation of surface chargesinvolves other phenomena of the electromagnetic field including, in partic-ular, a change of the magnetic field with time. Very often this process lastsa relatively short time, and afterward the constant electric field is gov-erned by Coulomb's law.

2. The phenomenon of electrostatic induction is observed in any con-ductive medium regardless of the electric resistivity, provided that theconductor is surrounded by an insulator and that the sources of theelectric field Eo are located outside of the conductor. For example,the conductive body could be composed of metal or an electrolyticsolution, minerals, or rocks. One can show that the magnitude of theresistivity plays a role in determining the time required for the electricfield inside the conductor to disappear, but it does not change the final


result of the electrostatic induction, namely that the internal electric fieldwill go to zero (E j == 0).

It should be obvious that the secondary electric field contributed by thesurface charges can be defined from the equation

(I1I.21)

where !,(q) is a surface density of charges. Consequently, condition(I1I.20) can be rewritten as

(I1I.22)

(111.23)

where Eo is the primary field due to charges located outside the conduc-tor, S is the conductor surface, and q is an arbitrary point on this surface.

For instance, if a single point charge e is situated outside the conductorat the point a, its electric field at any point b inside the conductor is

I eEoCb) = ---3-Lab

47TE O Lab

That is, it is the same as if the conductor were absent. As was stressedabove, the electric field caused by a given system of charges does notdepend on the electrical properties of the medium; and if the fieldchanges, this means that charge distribution has been changed, or newcharges have arisen.

In our case, positive and negative charges appear on the surface of theconductor. At the same time, the total charge es of the conductor, which isnot charged, remains zero.

(I1I.24)

In other words, when the electric field Eo is absent, within every elementdV and dS there is an equal amount of positive and negative charge sothat the presence of such a conductor itself does not create an electricfield inside or outside the conductor. In contrast, due to the presence ofthe electric field Eo, positive and negative charges occupy different partsof the conductor surface and consequently the secondary electric field E,arises, even though the total charge es is still equal to zero.

If we knew the density of surface charges, calculation of the electricalfield outside the conductor could be easily performed from Eq. (III.2l).However, the distribution of charges caused by electrostatic induction is


not known beforehand, and this fact reflects the fundamental differencebetween solutions to the forward problems of the gravitational and electricfields.

Inasmuch as Coulomb's law cannot be used we are forced to developspecial methods for field calculations that do not require a knowledge ofcharge distribution. Now we will describe one approach considering theeffect of electrostatic induction. With this purpose in mind, we will derivethe so-called integral equation with respect to surface density I(q). Then,knowing this function, we can make use of Coulomb's law, Eq. (III.21),and calculate the secondary field E s ' First, from Eq. (111.19) we have

E~(p) -E~(p) = I(p)EO

(III.2S)

where E~ and E~ are normal components of the electric field from theexternal and internal sides of the conductor surface near the point p,respectively. Because of the electrostatic induction the electric field insidethe conductor vanishes, and therefore E~ == O. As a consequence,Eq. (III.2S) is simplified.

E~(p) = I(p)EO

(III.26)

Applying the principle of superposition, let us write E~ as the sum ofthree terms.

(111.27)

where E~ is the normal component of the primary field at the point p, Egis the normal component of the field caused by the elementary surfacecharge I( p) dS situated in the immediate vicinity of the point p, andE~-P is the normal component contributed by the rest of the surfacecharge.

In accordance with Eq. (nUS),

I(p)EP=--

n 21'0

where n is the unit vector directed outward from the conductor surface.Inasmuch as

1 I(q)ES-P = --f --L dS

4 L3 qp7TEO S qp

if q"*p


we have

1 I(q)(L· n )ES-P =--f qp p dS

n 4'lTeo s L~p(111.28)

where n p is the unit normal vector at the point p and L qp • up is the dotproduct of vectors L qp and np'

L qp• n p = L qpcos(Lqp, np)

Collecting all terms in Eq. (I1I.27) and taking into account Eq. (III.26) weobtain

I(p) I(p) 1-- =E~+ -- + --fI(q)K(p,q) dS

EO 2eo 4'lTEO s

or

(111.29)

where

and 2EOE2are known functions.Equation (III.29) presents a Fredholm integral equation of the second

kind with respect to the unknown density at any point p on the conductorsurface. In practice we can conceptually replace the surface of the conduc-tor with a system of small cells, within each of which the charge density ispractically constant. In doing so, the integral equation (III.29) can berewritten as an approximation.

if q"* p (11I.30)

Having written this equation for every cell, we obtain a system of Nlinear equations with N unknown terms. When the charge density isknown, then, using Eq, (III.2l) the secondary field is calculated in terms ofthe surface integral. I hope that this example vividly demonstrates howdetermination of the field becomes much more complicated with respectto the use of Coulomb's law, if some of the sources are unknown. This isthe main reason, unlike the case of the gravitational field, for deriving thesystem of field equations and considering the boundary-value problems.

Ill.2 System of Equations for the Time-Invariant Electric Field Potential 213

Let us look one more time at electrostatic induction. Suppose that weconsider a model of a conducting earth that consists of a sequence oflayers each characterized by its own resistivity (Fig. III.2b). Above thesurface of the earth electric charges due to atmospheric processes arepresent. In accordance with Coulomb's law they create the same field inthe conducting medium as would have been observed if that volume werefree space. But because of electrostatic induction, induced charges appearon the earth's surface that exactly compensate the primary field inside theconducting medium. Because of surface charges that accumulate on theinterface between the upper half space and the conducting medium,the charges situated above the earth's surface do not have any effecton the electric field within the earth, and therefore, the constant fieldwithin the conducting medium can only be caused by sources existingwithin the medium. Fortunately, due to electrostatic induction, electricalmethods can be used in geophysics, based on measuring the constantelectrical field.

Unlike the conducting medium, charges induced on the earth's surfacecreate an electric field in the upper space that has only a verticalcomponent; and near the earth's surface it is equal to 100v1m = 105mv1m,which is much greater than the horizontal component of the field, mea-sured on the earth's surface by electrical methods.

111.2 System of Equations for the Time-InvariantElectric Field Potential

Due to the mathematical similarity of Coulomb's and Newton's laws wewill follow exactly the same path in deriving the equations for the time-invariant electric field as in the case of the gravitational field. Let us startfrom the second equation and with this purpose in mind, we will consideran elementary charge de at the point q and its field.

1 dee q)E( p) = 4'7Tc u-L qp

o qp

Then the flux through an elementary surface dS(p), caused by this field, is

de(q) L q p ' dSE·dS=------:.:.~-

4'7Tco L~p

or1

E' dS = --de(q) dw(q)4'7Tco

(IIL31 )


where dw{q) is the solid angle under which the surface dS is viewed frompoint q. Consequently, for the flux through a closed surface surroundingthe charge dei.q), we obtain

de(q)~E.dS=-s eo

Now applying the principle of superposition we generalize this equationfor an arbitrary distribution of charges inside the closed surface S.

~E'dS=~[f8dV+ fIdS+ fAdt'+Ie i ]s eo v S L

(III.32)

(111.33)

where 8, I, A are the volume, surface, and linear density of charges; ej is apoint charge; and all of these are situated inside of the surface S. The fluxof the field caused by charges located outside of the surface is zero and, asin the case of the gravitational field, this follows from the behavior of thesolid angle.

Inasmuch as in the vicinity of regular points only volume charges arepresent, Eq. (1II.32) is simplified and we have

1~E.dS=- f 8dV

S eo v

This is traditionally called the second equation of the field in integralform, describing the field at regular points.

Taking into account that inside the volume V we can expect positive aswell as negative charges, the flux of the electric field through a closedsurface, unlike the gravitational field, can be zero in spite of the presenceof charges in the volume. To characterize in more detail the distribution ofcharges, we will make use of Gauss' theorem. Then we obtain

1~E.dS= f divEdV=- f 8dV

S v eo v

Consequently,

8divE= -

eo(III.34)

This is the second field equation in differential form, valid only for regularpoints.

Now we will suppose that there is a surface distribution of charges(Fig. III.2c). As was shown in the previous section, the normal component

111.2 System of Equations for the Time-Invariant Electric Field Potential 215

of the electric field is a discontinuous function of the spatial variables inpassing through a surface charge. Correspondingly, the derivative JEn/Jndoes not exist, and therefore Eq. (III.34) cannot be used. Then, applyingEq. (111.32) to an elementary cylindrical surface enclosing a small piece ofthe interface (Fig. III.2c) and keeping only the term containing the surfacecharge I dS, we obtain the third form of the second field equation.

(111.35)

By starting with Coulomb's law we have obtained three forms of thesecond equation.

sdivE= -,

EO(III.36)

Each of these characterizes the relation between charges and the electricfield. In particular, if in the vicinity of some point charges are absent, wehave

divE = a and E(2) =E(I)n n

Next we will derive the first equation of the electric field.Inasmuch as both the electric and gravitational fields are caused by

sources only, we can make use of the results of Chapter I and write thefirst equations of the electric field.

curlE = a

However, taking into account the importance of the concept of voltage inthe theory of electric fields, let us derive the above equations. First of all,the integral

t E· dl'= t Edtcos aay Q2'

(III.37)

is the voltage between points a and b measured along some arbitrary path2' (Fig. Ill.Zd) and caused by the electric field. Here a is the anglebetween the electric field vector and the tangent to the path 2' at everypoint. It is clear that the product E . dl'is the elementary work performedby the electric field in transporting a unit positive charge along thedisplacement d.l. This product has the dimensions of work per unit charge,and in the practical system of units has dimensions of volts. Therefore, theintegral in Eq. (111.37) represents the work or voltage done in carrying a


a b

b

~a

de(q)

c a d

Lacb

Fig. IlL3 (a) Voltage along a radius-vector; (b) voltage along a radius-vector and arc; (c)voltage along an arbitrary path; and (d) circulation.

charge between two points a and b. In the general case of an alternatingelectromagnetic field, for a given function E this integral depends on theparticular path of integration 2' that is chosen, and on the terminal pointsa and b of the path.

Starting from Coulomb's law we will show that the voltage of theelectric field caused by charges only is independent of the path ofintegration, as is that of the gravitational field. Assume that the source forthe field is a single elementary charge de, then its electric field is

1 dee q)E(p) = ----3-Lqp

41TEO L q p

If both terminal points a and b are situated at the same radius vector L q p

and the path of integration is along this radius (Fig. III.3a), the voltagebetween these points is very easily calculated.

III.2 System of Equations for the Time-Invariant Electric Field Potential 217

because

d/'· L q p = dtL q p cos 0 = L q p dL

Carrying out the integration as indicated, we obtain

(III.38)

Now suppose that the points a and b are situated on two differentradius vectors L q a and L q b , as shown in Fig. III.3b. Let us choose a path

.2"1' which consists of two parts. The first part is a simple arc ab', and thesecond element of the path is along the radius vector L q b . In this case thevoltage can be written as

1 [I d/'· L q p Jb d/'· L q p 1V = -- deeq) 3 + 341TE O arcab' L q p b' L q p

The integral along the arc ab' is clearly zero since the dot product is

d/'· L q p = 0

Thus, the voltage between points a and b is again equal to

(III.39)

That is, it remains the same in spite of the fact that the path of integrationhas been changed. If instead of the path .2", we consider an arbitrary path

.2"2' it should be clear that this path could be represented as a sum of arcsand elements of radius vectors, as is seen in Fig. III.3c. All the integralsalong simple arcs are zero, while the sum of integrals along the radiusvectors is

deeq) [1 1]---- -----41TEO L q a L q b

and this is equal to the voltage along the path .2"t.We have established the second fundamental characteristic of the

electric field, namely, that the voltage between two points does not dependon the particular path along which integration is carried out, but is


determined by the terminal points only. This fact can be written as

fb E . dl'= fb E . dl'= .. , = t E' dl'0.2'1 aY2 Q..:lfn

( III AD)

Making use of the principle of superposition, this result can be general-ized to a field caused by any distribution of charges. It must be stressedagain that this result is valid only for the electric field caused by constantelectric charges; it cannot, in general, be applied to time-varying fields. Letus also notice that unlike the gravitational field, the voltage plays a muchmore important role because the basic measuring device, the voltmeter,measures the value of the integral

that is, voltage, and if within this path the field E does not vary, then weare able from the voltage to find the electric field.

The independence of the voltage of the path of integration can bewritten in another form. Consider a closed contour .2', as is shown inFig. III.3d, as consisting of two other contours, .2'acb and .2'bda' Inaccordance with Eq. (IIIAO) we have

f E' dl'= f E' dl'acb adb

(IIIA1)

In these integrals the element dl'is directed from a to b. Changing thedirection in the integral on the right-hand side of Eq. (III.4l), we can write

f E' dl'= - f E· dl'acb bda

That is,

f E· dl'+ f E· dl'= 0acb bda

or

(IIIA2)

Thus, the voltage along an arbitrary closed path is zero.Sometimes the quantity ¢E . dl'is called the "circulation" of the elec-

tric field or the electromotive force. The path .2' can have an arbitraryshape, and it can intersect media with various physical properties(Fig. III Aa). In particular, it can be completely contained within a con-


al...- _ .

• (,~#g:'

./.....•...._ ./

c

Ii 1111

b

d

Fig. IlIA (a) Circulation through a conducting medium; (b) continuity of tangential compo-nents; (c) electric dipole; and (d) double layer.

ducting medium. Because the electromotive force caused by the electriccharges is zero, Coulomb's force E C cannot alone cause an electric current;and this is the reason that non-Coulomb forces must be introduced toprovide a current flow. This question will be examined in detail in the nextsection.

Equation (111.42) is the first equation for the electric field in integralform. Applying Stokes' theorem for regular points of the medium, we have

i E' d/= f curl E> dS = 0:2' S

or

curlE = 0 (IIIA3)

The latter is the differential form of the first equation, showing that theelectric field is not generated by vortices, and it applies at those pointswhere the first derivatives of the electric field exist.

To obtain a differential form of Eq. (111.42) near surface charges wherethe normal component En is discontinuous we will apply this equationalong the elementary path shown in Fig. II1.4b. Considering that elements


dl" and dz" are separated by a distance dh, which tends to zero, weobtain

E . dl''' + E . dl" + E . dh = 0

or

E(2) dt - E(1) de' = °t t

and finally

or (III.44)

That is, tangential components of the field are continuous as the pathpasses across a surface charge. Let us note that there is one case, namely anonuniform double layer, in which the tangential component E, is adiscontinuous function.

We have now derived three forms of the first equation based onCoulomb's law.

¢E. dl'= 0, curiE = 0, E(2) = E(1)t t (III.4S)

Each of them expresses the same fact; that is, the electromotive forcecaused by electric charges is zero, or in other words, the voltage betweentwo arbitrary points does not depend on the path of integration.

We must make an important comment about Eqs. 011.45). The first tworelationships are not valid when the field is time-varying, since the secondtype of generators (vortex), that is, the change of magnetic field intensitywith time, is not taken into account. On the other hand, the surfaceanalogy for these equations is valid for any electromagnetic field. Thisreflects the fact that in the development of this particular form of theequation, it was assumed that the area surrounded by the integration pathwas zero, and therefore the flux of the magnetic field through this areavanished.

Let us note one further feature of the field. Although the equations~E· dl'= 0 and curlE = 0 are not valid for time-varying electromagneticfields, this does not mean that Coulomb's law is always inapplicable insuch cases.

Now we are ready to write the system of equations for the time-invariant electric field in two forms.


Integral Form of Equations for Time-InvariantElectric Field

Coulomb's law (III,46)

II ~E. dS = ~s So

where e is the total charge inside the volume V, surrounded by the surfaceS; and .2' and S are an arbitrary contour and a surface, respectively,which can intersect media with different electrical properties.

Differential Form of Equations for Time-InoariantElectric Field

Coulomb's law

I curlE = 0 II8

divE =-So

(I1IA7)

Here it is appropriate to make the following comments:

1. Any electric field caused by charges satisfies Eqs, (IIIA6), (I1IA7);that is, it is itself a solution of the system of field equations in the integraland differential forms.

2. These equations contain the same information about the field asCoulomb's law, but as will be shown later they allow us to find the field inthose very practical cases when Coulomb's law turns out to be useless.

3. Comparison with the system of equations for the gravitational fieldshows the two systems are identical, and this happens because these fieldsare caused by sources only. Moreover, in accordance with Newton's andCoulomb's law, in both cases the field possesses the same dependence on


distance provided that masses and charges are distributed in a similarmanner.

4. Systems (III.46) and (III.47) are valid everywhere, and correspond-ingly they correctly describe the electric field in the presence of anyconducting and polarizable medium. In particular, the time-invariantelectric field, measured in all electrical methods used in geophysics, isitself a solution to these systems.

5. In spite of the identity of the systems for the gravitational andelectric fields, stilI there is one fundamental difference. In the case of thegravitational field the right-hand side of the second equation-that is, thedensity of masses-can usually be specified, while the density of electriccharges is usually unknown. There is, however, one exception: when theelectric field of a given distribution of charges is considered in free spaceand conductors and dielectrics are absent. In such a case, to calculate theelectric field we can use Coulomb's law directly in the same manner asNewton's law is applied in solving forward problems of the gravitationalfield. But this case has hardly any practical interest in applied geophysics.Considering the electric field in the presence of conductors and dielectrics,we cannot in principle specify the density of charges before the electricfield is calculated. The example of the previous section describing electro-static induction has been designed to illustrate this problem.

Thus, the system of field equations, for instance Eq. (III.47), containsseveral unknowns such as the electric field E and the density of charges 0and k. Because of this we will be forced to replace this system by adifferent one that contains only one unknown, namely the electric field E.

Now let us, proceeding from the first equation curl E = 0, introduce ascalar function U.

since

E = -gradU (III.48)

curl grad U == 0

The scalar function U is called the potential of the electric field.In accordance with Eq. (III.48) the electric field E coincides with the

direction of maximum decrease in potential, and any component of thefield can be expressed in terms of the potential as follows:

auE---r : at (III.49)

Now we will demonstrate three main reasons why it is useful tointroduce the potential U. First, we will write an expression for the voltage


using the potential. It is clear that

audU = - dt = dl'" grad U = - E . dl"

at (111.50)

where dl" = dti o and i o is a unit vector.Integrating the last of these terms along any path between two arbitrary

points and taking into account the fact that the voltage is path indepen-dent, we obtain

t E ' dl"= - t dU= U(p) - U(b)p p

(111.51 )

That is, the voltage of the electric field along any path with terminal pointsp and b can be written as the difference of the potential between thesepoints. Therefore, knowing the potential, it is very simple to calculate thevoltage, and this is the first reason for the introduction of this function.

Next we will use Eq. (IlL51) to define the potential caused by anarbitrary distribution of charges. From this equation we have

U(p) = U(b) + JbE' dl"p

(III.52)

It is obvious that at great distances from charges the field E is very smalland therefore the potential also vanishes. Then, letting b equal infinity inEq. (IlL52) and assuming that the potential at this distance is zero, wehave

U(p) = f'E' dl"p

(III .53)

(111.54)

Suppose that the source of the electric field is a single elementary chargede, situated at the point q. By using Eq. (IlLS) and Eq. (III.53) we obtain

de 00 dL deU(p)=-J -=--

47TEO P L247TEOL q p

Making use of the principle of superposition for an arbitrary distributionof volume, surface, linear, and point charges, we arrive at the followingexpression for the potential:

1 [ /5 dV "i dS Adt e. ]U(p)=-j-+j-+l-+L:- (III.55)

47TEO v L q p S L q p '2' L q p L q p


Comparison with Eq. (III.6) clearly shows that the potential U isrelated to the charges in a much simpler way than the electric field. Thissimplicity is the second reason for using the potential. It is obvious thatthe electric field can be easily found applying Eq. (I1IA9), if the potentialis known. Let us consider two examples.

Example 1 The Potential and the Fieldof an Electric Dipole

Suppose we have two elementary charges equal in magnitude and ofdifferent sign, as is shown in Fig. IlIAc. Then, the potential U due to thesecharges is

(III .56)

where ql and q2 are points in the vicinity of which the negative andpositive charges with magnitude e are located. We will consider the fieldonly at distances greatly exceeding the distance between charges; that is,the displacement dt of one charge with respect to another is much smallerthan the distance L q p , where q is midpoint between the charges.

(I1I.57)

In this case the system of charges is called an electric dipole. Taking intoaccount the inequality (111.57), the difference in the right-hand side ofEq. (111.56) can be replaced by

1 1 a 1---=--dtL q 1P -; at i;

and in accordance with Eq. (111.50) we have

e ale q 1U(p) = ----dt= --dl"grad-

417"£0 at L q p 417"£0 L q p

where the index "q" means that only the point q changes. Since

(I1I.58)


Eq. (III.58) can be presented as

1 M' L qpU(p) = -- 3

41TBO L q p

(III.59)

where M = e dl' is the moment of the electrical dipole, which is a vectorshowing the direction of displacement of the positive charge with respectto the negative one, and whose magnitude equals the product of thepositive charge and the distance between them, de.

As follows from Eq. (III.59) we have

1 M cos eU(p) = -.- 2

41TBO L qp

(III.60)

where () is the angle between the direction of the moment M and that ofthe radius vector from the dipole center q to the observation point p. Inaccordance with Eq, (III.60) the potential of the electric dipole decaysmore rapidly along the radius vector than the potential due to a singlecharge and essentially depends on the angle e. In particular, if e< 1T/2 itis positive, since the positive charge is closer to the observation point; atthe equatorial plane e= 1T/2, it is equal to zero, because at all points ofthis plane a distance to both charges is the same; and finally, whene> 1T/2, the potential is negative. Now let us introduce a spherical systemof coordinates R, e, cp with z-axis directed along the dipole moment. Thenthe potential U can be written as

1 M coseU(p) = -4- R2

1Teo

and taking into account Eq, (III.49) the components of the electric fieldare

2M cos (JE -----

R - 4rrBo R3 '

M sin eE -----

(J - 4rreo R3 (III.61)

Thus, the electric field of the dipole decays more rapidly than that ofthe single charge. Along z-axis, as follows from Coulomb's law, the field isdirected parallel to the dipole moment, regardless of whether the angle (J

is equal to 0 or tr. In the equatorial plane the field has a directionopposite to that of the electric dipole. It is interesting to note that thedirection of the vector E along a given radius vector e= constant, remains


the same. In fact, from Eq, OIl.6l) we have

Eo 1tan cp = - = - tan (J

ER 2(III.62)

where cp is the angle between the field E and radius vector R.In deriving formulas for the potential and the field of the electric dipole

we have made only two assumptions, namely,

1. The sum of charges is equal to zero, and2. The field is considered at distances that greatly exceed the displace-

ment between the charges.

For this reason we can say that Eqs. (111.60), (III.6l) describe thepotential and the field of an arbitrary but neutral system of charges,provided that the observation points are located far away with respect tothe dimensions of the volume where charges are situated. For instance,due to electrostatic induction, charges of both signs appear on a conductorsurface, and with an increase in distance the electric field of these chargestends to that of the electric dipole. It is essential to note that this behaviorof the field takes place regardless of the shape and size of the conductor.

Example 2 The Potential and the Fieldof a Double Layer

Let us imagine that positive and negative charges with density

and (III.63)

are distributed on two surfaces S + and S _, respectively, and that theseparation t between them is much smaller than the distance from theobservation point to these surfaces (Fig. IlIAd). As follows fromEq, (IIl.63) every pair of surface elements located opposite to each otherhave charges with the same magnitude and different signs. Such a systemof charges is termed a double layer, and double layers are widely appliedin the theory of self-potential and induced polarization methods of electri-cal prospecting.

For convenience we will choose some surface S, which is locatedbetween the surfaces S + and S _, whose normal n points in the directionfrom the surface S _ to the surface S +. Let us consider some point q of

IIl.2 System of Equations for the Time-Invariant Electric Field Potential 227

this surface and two corresponding charges located at distance t fromeach other.

de2 = de = ~(q) dS and de, = -de = -~(q) dS

It is clear that these charges form an electric dipole with a moment dMequal to

and

dM=de/ or dM = ~(q) dS tn

where

dM = 7J( q) dS

TJ(q) =!.(q)t(q)

(III.64)

(III.65)

which characterizes the density of dipole moments, and correspondingly itis called the double-layer density. In accordance with Eq. (III.59) thepotential of the electric field caused by this dipole is

1 dS· L q pdU(p)=-4- TJ ( q ) L3

7TEO qp

or

1 dS· L p qdU(p)=---7J(q) 3

47TEo L q p

(111.66)

where dS = dS n is an element of the middle surface S. In essence, wehave replaced the real distribution of charges at both surfaces S _ and S+by the mathematical model of the double layer placed on the surface S. Itis obvious that if in both cases the density TJ(q) remains the same, and theobservation points are located far away, this replacement does not alterthe field. Now making use of the results derived in Chapter I we canrewrite Eq. (111.66) as

7JdU(p) = - -dw(p)

41TEO

(III.67)

where dcoi p) is an elementary solid angle.Therefore, for the potential caused by charges of a double layer we

obtain

1U(p) = - -fTJ(q) dos

41TEO S(III.68)

Taking into account the fact that the procedure for calculation of the solid


angle is known, Eq, 011.68) allows us to find the potential and, corre-spondingly, the electric field caused by an arbitrary double layer. Inparticular, if the double layer is uniform, that is, 1) = constant, then wehave

1)U(p) = - -4-w(P)

7TSo(III.69)

where w(p) is the solid angle subtended by the layer surface S whenviewed from the point p. In the case where the uniform double layer isclosed, we have for points in the volume confined by the surface S, andoutside,

. 1)U'(p) =-

Soand (III .70)

respectively, and since the potential does not vary, the electric field equalszero.

Next we will study the behavior of the potential and the electric fieldnear the double-layer surface, when its density 1)(q) is an arbitraryfunction. The potential U(p) at every point can be represented as the sumof two terms.

(I1I.71)

where uq(p) is the potential caused by the element dS(q) of the doublelayer in the vicinity of the point q, and US-q(p) is the potential due to therest of the layer. When the observation point approaches the point q thesolid angle subtended by the surface dS(q) tends to +27T, and corre-spondingly, for the potential caused by this element we have

U( ) = + 1)(p)p - 2

Soif p ~q

Therefore, the potential U at the front and back sides of the doublelayer is

if p ~q

Inasmuch as all charges located outside the element dS are at somedistance from the point p, the potential US-P is a continuous function in


the vicinity of the point p; that is,

Therefore, the difference of the potential on either side of the doublelayer is

(111.72)

Thus, the potential is a discontinuous function across the double layer,and this discontinuity is defined by the density 7J near the point ofobservation. Let us notice that such behavior of the potential is anexception, and it is inherent only to the model of the double layer. If weconsider the potential between surfaces S _ and S + this. discontinuitydisappears.

Having taken the derivative at both sides of Eq. (1II.72) in any directiont tangential to the double layer, we obtain

or

(III.73)

That is, the tangential component of the electric field is a discontinuousfunction at those points of the double layer where the density 7J is notconstant.

Unlike the case where the charges are distributed on one surface, thenormal component of the field due to a double layer turns out to be acontinuous function. Indeed, in accordance with Eq. (lIUS), two elemen-tary surfaces with charges - I(p) dS and I(p) dS create inside the doublelayer a field equal to I(p)/eo, but outside at both surfaces S _ and S + itis zero. At the same time other elements of the double layer, being atfinite distances from the point p, generate a field that is a continuousfunction near this point. Thus, the normal component of the electric fieldhas the same values on both sides of the double layer.

(111.74)

230 rn Electric Fields

As follows from Eqs. (IlI.S!) and (IlI.72) we have

(IlI.75)

where E i is the field inside the double layer, and the indices" +" and" - " mean that the points belong to the surfaces S + and S _, respectively.

Now let us consider the third and perhaps the most important reasonjustifying the introduction of the potential U. Earlier, from the first fieldequation curl E = 0 we obtained

E = -gradU

Substituting this into the second equation we see that the potentialsatisfies Poisson's equation.

(III.76)

The procedure for derivation of this equation is shown below

IcuriE = 0 Is

divE = -EO

11jIE = - grad U I---+~=====~_~ -----,

sdivgrad U = V' 2U = --

EO

In particular, in the vicinity of points where charges are absent, thepotential obeys the Laplace equation.

(IlI.77)

a


bU = constant

cU = constant

dU =constant

Fig. 111.5 (a) Potential behavior near surface charges; (b) potential behavior on conductorsurface; (c) electrostatic screening; and (d) charges inside a conducting shell.

The analogy with the potential of the gravitational field is obvious andfollows from the fact that both fields are caused by sources only.

Equations (III.76), (I1I.77) are valid at regular points only. For thisreason let us study the potential behavior near surface charges, where thenormal component of the field is a discontinuous function. In accordancewith Eq. (III.53) the potential on either side of the surface (Fig. IIL5a) is

and

or

(III.78)

Taking into account the fact that the field on both sides of the surface isfinite, but the distance between points PI and Pz is vanishingly small, thedifference in potential from both sides tends to zero. Therefore, thepotential of the electric field across any surface carrying a charge with


density ~ is a continuous function.

(1II.79)

Now we are ready to replace the system of field equations by anothersystem, which describes the behavior of the potential of the electric field.

Coulomb's law

BaU2 aUl---an an

on S

(1II.80)


1. System (I1I.80) consists of three parts; one of them is Poisson'sequation, which describes the behavior of the potential at usual pointswhere the first derivatives of the field exist, and the two others character-ize the potential in the vicinity of surface changes. Note that there is oneexception, namely the surface of a double layer, where this system cannotbe applied.

2. In accordance with Eq. (III.55), the potential caused by a volumedistribution of charges is

1 j8(q) dVU(p) =-

47TE O V L q p

(III .81)

On the other hand, the potential is described at regular points byEqs. (1II.76), (I1I.77). Therefore, the expression given in Eq. (1II.8l) is thesolution of Poisson's and Laplace's equations inside and outside of volumecharges, respectively.

3. As was shown in Chapter I, Poisson's equation can be written as

6 8- [U( p) - U av] = -h2

EO(1II.82)


since

where 2h is the length of the side of an elementary cubic volumesurrounding the point p, and U(p) and or:» are the value of thepotential and its average value at this point, respectively.

As follows from Eq. 011.82), outside of charges,

U(p) = U'" or

That is, the potential is a harmonic function. In the vicinity of pointswhere the charge is positive, the potential exceeds the average value.

U(p»u av

while at places where negative charges are distributed, the oppositerelation holds.

U(p) < o-:

4. Conditions A and B of the system 011.80) provide continuity of thetangential component and discontinuity of the normal component of theelectric field across the surface charges, respectively.

5. The system (III.80) is identical to that for the potential of thegravitational field, and transition from the electric field E to the scalarfield U often essentially simplifies the solution of the forward problems ofthe theory of electrical methods.

This fact constitutes the third merit that justifies the use of potential U.However, introduction of the potential itself does not allow us to removethe basic difficulties of field determination, such as absence of knowledgeof induced charges if the electric field is unknown. In this and the nextsections we will consider several cases that vividly illustrate this problem.Let us start from the following model of a medium.

A Conductor Situated in Free Space

Suppose an arbitrary conductor is placed in the electric field Eo caused bya given distribution of charges with density [) in free space (Fig. III.5b).Due to electrostatic induction, the electric field E, vanishes inside of theconductor. Correspondingly, the derivative of the potential U, in any


direction within the conductor is zero.

au_'=0at

That is, the potential U, does not vary, and in particular, the conductorsurface is an equipotential surface.

on S (III.83)

where C is a constant, usually unknown.Let us write the potential as the sum of two terms.

U= Uo+ Us

where U O is the potential caused by a given distribution of charges,

(III.84)

and Us is the potential due to surface charges with density 2, whichappear on the conductor surface.

Us = _1_j2(q) dS47TSo v L q p

(III.85)

where q is a point on the conductor surface.It is useful to make three comments.

1. Surface charges with density 2 are distributed in such a way that theelectric field within the conductor disappears.

2. Equations (nI.84), (nI.8S) describe the potential everywhere as ifthe conductor were absent.

3. Of course, these equations are similar, but there is one essentialdifference, namely, the sources of the field Uo are given, while the density2(q) is unknown. Considering the electrostatic induction in the previoussection we have been able to establish a relation only between the field atthe external side of the conductor surface E~ and the density 2(q).

(I1I.86)

but

E~=E~(q) +E~(q)

where E~(q) and E~(q) are the normal components of the field on the


external side of the conductor surface caused by the given and unknowndistributions of charges, respectively.

As follows from Eq. OII.85) the potential Us can be found if the densityof induced charges I is determined. On the other hand, in accordancewith Eq. OII.86) these charges can be specified, provided that the electricfield

E S = -grad Us

is known. Thus, as was stressed in the previous section, we are faced withthe problem of "the closed circle," which vividly shows that the fieldcalculation cannot be performed by using Coulomb's law. This is thegeneral problem inherent for practically all fields in applied electricalmethods. Consequently, to find the electric field E we have to make use ofa completely different approach based on the theorem of uniqueness, andformulate boundary-value problems. Since the field inside a conductor isknown (E i == 0), our attention is paid only to the potential in a volume offree space, confined by a conductor surface S, and a spherical surface witha very large radius that in the limit tends to infinity. As we know fromChapter I the electric field is uniquely defined if the potential U satisfiesthe following conditions:

1. Outside the conductor or conductors, the function U is a solution ofPoisson's equation in every point of free space.

where 0 is the volume density of charges, which is specified. Of course, inthe vicinity of points where the density is zero, the potential is a harmonicfunction.

2. Since both fields, caused by the volume and surface charges, de-crease with distance the condition at infinity is

U--+O as L --+00qp

3. At the conductor surface the potential must be constant and, de-pending on the type of available information we have, also must have


either

or

or

(III.87)on S

where 'Pl(q) and 'Piq) are given functions; Q is the total charge on theconductor surface; and in the case of a neutral conductor, Q = O.

There are various approaches to solving the forward problem, includingthe trial-and-error method. In particular, we have illustrated the idea ofthe integral equation method, when electrostatic induction was consid-ered. Certainly all of these methods are much more complicated than thedirect application of Coulomb's law, but it is natural "retribution," sincean essential part of the source, namely the induced charges on theconductor surface, is unknown until the field is calculated.

Now we will demonstrate the use of the theorem of uniqueness indescribing the electrostatic shielding effect. First suppose that some vol-ume V is surrounded by a conducting surface S-for instance, a metal foil-and that the sources of the field Eo are located outside this volume(Fig. III.5c).

Due to electrostatic induction, positive and negative charges arise onthe surface, and they are distributed in such a way that the potential doesnot change within the conductor. Our task is to find the field inside thevolume V. Taking into account the fact that charges are absent in thisvolume, but the surface S is an equipotential surface, we have

\12U=0

U=c

where C is an unknown constant.In accordance with the theorem of uniqueness (Chapter I) we have

formulated Dirichlet's problem, and so Eqs. 011.87) uniquely define thefield, even if the constant is unknown. We will look for a solution with thehelp of the trial-and-error method.

Suppose that within the volume V the potential is also constant, whichequals C; that is,

u=C m V (111.88)

TII.2 System of Equations for the Time-Invariant Electric Field Potential 237

It is clear that this function U automatically satisfies the boundarycondition as well as Laplace's equation, since even the first derivative fromconstant is zero. Therefore we have found the potential inside the volumeand there is no other solution to this problem. Inasmuch as the potential isconstant, the electric field in the volume, surrounded by an arbitraryconductive surface, vanishes.

£=0 (111.89)

In other words, we have described the effect of electrostatic screening andproved it by making use of the theorem of uniqueness.

Next assume that the sources of the field Eo are surrounded by aconducting surface S, but we are interested in the behavior of the fieldoutside of this surface (Fig. III.5d). Again, due to electrostatic induction,charges of both signs appear at the internal and external sides of thesurface S, which becomes an equipotential surface. Taking into accountthe fact that the electric field within a conductor is zero, the total flux ofthe field through any closed surface S, of the conductor is also zero.

~E' dS = 0Si

This means that the induced charges at the internal surface of theconductor are equal in magnitude and opposite in sign to the chargeslocated in the volume V. Correspondingly, at the external surface the samecharge as that in the volume V appears. It is obvious that this system ofsources creates a field outside of the conducting surface that is not zero.In other words, electrostatic shielding does not work in both directions.Let us show this again using the theorem of uniqueness.

In this case the potential outside of the conductor satisfies the followingconditions:

1. At regular points

2. On the conducting surface

U=c

3. At infinity the potential tends to zero

U~O

It is obvious that any constant value of the potential does not simulta-neously satisfy both boundary conditions if C i= 0, and consequently thepotential changes from point to point. Therefore the electric field differs


from zero although charges in the volume V are surrounded by theconducting surface. At the same time, in accordance with the theorem ofuniqueness, if the potential U on the surface S becomes equal to zero dueto grounding, the potential vanishes everywhere outside of the surface S.

U=o (111.90)

From this consideration it also follows that, surrounding some volume Vby two conducting surfaces that are connected with each other, weperform the electrostatic shielding of this area from the field caused byexternal sources even without grounding. This happens due to the contactbetween these surfaces, since the potential becomes the same on both ofthem. Correspondingly, inside the volume the potential also equals aconstant that coincides with that on surfaces.

111.3 The Electric Field in the Presence of Dielectrics

In the previous section we have shown that any conductor, placed in anelectric field, produces a change of the field. This happens because everyelement of the conductor contains an unlimited amount of charge, whichunder the action of an electric force moves freely through the conductorand, due to electrostatic induction, surface charges appear. Now we willconsider another type of medium, which is also able to change the field.This medium is called a dielectric, and its influence can be approximatelydescribed in the following way.

One would think that every element of the dielectric volume does nothave free charges that can move significant distances, as in a conductor,and each element contains an equal amount of positive and negativecharges, bound to each other. This description is based on the fact thateither molecules or areas around ions, located at lattice nodes, are aneutral system of charges. Due to the external electric field, bound chargesslightly change their position; but unlike free charges, they do not leavetheir elementary volume. Due to the external electric field, positive andnegative charges move in opposite directions, and therefore an elementaryvolume acquires a dipole moment; this phenomenon is called polarization.Now we are ready to describe the field caused by polarization within eachelementary volume. In fact, performing a summation of these fields overthe entire volume of the dielectric we obtain the secondary field which,together with the primary or external field, forms the total observed field.

ID.3 The Electric Field in the Presence of Dielectrics 239

Before we continue, let us make one comment, namely, the dielectric isnot antipodal to the conductor. In reality every medium possesses bothconducting and polarizable features and it is only for simplicity and toemphasize the polarization effect that the presence of free charges is nottaken into account here. In fact, the insulator is the antipode to theconductor.

To find the secondary field, first we will consider an elementary volumedV, which contains an equal amount of positive and negative charge,slightly displaced from each other and therefore having a dipole momentdM.

dM=PdV (111.91 )

where P is the polarization vector or, shortly, polarization. This is thedipole moment of the elementary volume referred to in this volume; thatis, it is the density of dipole moments.

dMP=-

dV

The greater the amount of charge and the greater the distance betweenthe positive and negative charges, the greater the polarization vectordirected toward the positive charges. In most cases we can assume that thepolarization P is directly proportional to the field E; that is,

P=aE (III.92)

where a is the polarizability, which characterizes the ability of a mediumto display polarization-in other words, the number of dipoles orientedalong the field. On the right-hand side of Eq. (III.92) E is the total fieldcaused by all sources, including dipoles in neighboring parts of thedielectric. Correspondingly, it consists of the primary and secondary fields

(III.93)

Taking into account the fact that charges within every elementaryvolume dV(q) create a field equivalent to that of an electric dipole andmaking use of Eq. (III.59), the potential dU/p) outside of the polarizedelement is

(III.94)

where L q p is the distance from the point q that defines the position of theelement dV to the observation point p. Therefore, the polarized dielectric


in the volume V generates an electric field E, whose potential is

1 P(q) . LUs(p) = --f 3 qp dV

4'77"£0 V L q p

(111.95)

(111.96)

This expression is relatively simple, but it cannot be used for calculationof the potential of the secondary field. On the one hand, in accordancewith Eq. (111.95), to find the potential U; we have to know the polarizationvector P. On the other hand, the latter is defined by the total field,Eq. (111.92), including the secondary field, E, = - grad u". Thus, we areagain faced with "the closed circle" problem illustrated in the previoussection. Correspondingly, the system of field equations (I1I.47), describingthe constant electric field everywhere including dielectrics, contains threeunknowns: the electric field E and the sources 0 and 2. Because of thisinterdependence we have to modify this system in such a way that a newsystem has only one unknown, namely the field E. With this purpose inmind, we will perform some simple transformations on the right-hand sideof Eq. (111.95) and introduce a new vector, the electrical induction, as wellas bound charges.

Let us represent the potential due to polarization as

1 q 1Us(p) = -1.P(q). V-dV

4'77"£0 v L q p

since


q 1 q P 1 q

P(q)·V-=V---VPi.: -: -:

instead of Eq. (III.96) we have

1 P 1 divPUs(p) =-f div-dV- -1. -dV

4'77"£0 v L q p 4'77"£0 V L q p

(III.97)

To simplify this equation we are going to make use of Gauss' theoremand replace the first integral by a surface integral. Also, let us supposethat in the volume V, surrounded by the surface S, there is some surfaceSj where either the polarizabiIity a or the field E or both of them arediscontinuous functions, and therefore the polarization vector P is also

111.3 The Electric Field in the Presence of Dielectrics 241

c

b

d-E

e--- --0 --0 ;'e--- --01,

e-----0 e--- --0\.

Fig. III.6 (a) Polarization in a dielectric; (b) negative bounded charges; (c) positive boundedcharges; and (d) movement of charges in a conductor.

discontinuous. Having surrounded S, by the "safety" surface 50(Fig. III.6a), we obtain the volume VI which is bound by surfaces 5 and 50and where Gauss' theorem can be applied. Then, in the limit the surface50 coincides with the front and back sides of 5 i , but the volume VI tendsto the original volume V. Therefore, the volume of integration remains thesame, but the surface of integration includes the surface 5 and both sidesof 5 i , where the vector P is a discontinuous function.

Now applying Gauss' theorem to the first volume integral in Eq. OII.97)we obtain

1 ( P . dS 1 1-divP+--},--+-- dV

417"£0 S L q p 417"£0 V L q p

(III.98)

where p(1) and p(2) are the values of the polarization vector at the backand front sides of the surface 5 i , respectively.

Taking into account that in performing the integration over the surfaces51 and 52 the normals have to be directed outside the volume (Fig. III.6a),


we have

I pO) . dS I p(2) . dS I pO) • 01 + p(2) • 02

-f +-f =-f dS47TE O S, L q p 47TE O S2 L q p 47TE O s, L q p

(III.99)I pO) _ p(2)

=-f n n dS47TE O s, L q p

where 01 = n and 02 = -0, and n is the normal directed from the back tothe front side of the surface Sj'

Now we will show that the integral over the surface S, which envelopesall possible dielectrics, vanishes. In fact, at infinity the electric field andtherefore the polarization vector P decreases at a rate inversely propor-tional to the square of the distance. Consequently, taking a sphericalsurface S of very large radius R, and making use of the mean valuetheorem, we have

as R ~ 00

where C is some constant.Thus, we have derived a new expression for the potential III the

presence of dielectrics.

I - div P 1 - Div PUs(p)=-f dV+-f dS (III.100)

47TEO v L q p 47TEO S L q p

where

(III.lOI)

is the difference of the normal component of the polarization vector ateither side of the surface of singularity Sj'

Considering the volume V as the whole space, we include all dielectricsas well as those parts of the volume where dielectrics are absent; and it isobvious that integration over such parts does not affect the field, sincep=o.

Both Eqs. OII.95) and (III.lOO) describe the same potential, caused bypolarized dielectrics; however, the latter turns out to be much more usefulfor understanding the charge distribution within dielectrics as well as forformulation of boundary-value problems. First let us write down expres-

(III.102)

III.3 The Electric Field in the Presence of Dielectrics 243

sions for the potential caused by free charges and those that arise due topolarization.

1 - div P 1 - Div PU=-f dV+-{ dS

s 41TEO V L qp 41TEO Js L q p

The striking resemblance between these equations is obvious. Fromtheir comparison we can conclude that, due to polarization, volume andsurface charges appear, and their density is defined by

Ob= -divP, lb= -DivP=ppl_ppl (III.103)

We use the index "b" to distinguish these charges from the free ones Doand lo.

Let us now investigate these polarization-induced charges in moredetail. The presence of this type of charges has been established analyti-cally due to the transformation of Eq. (111.95), but we can arrive at thesame conclusion from a physical point of view. Indeed, if the polarizabilityvaries from point to point, the density of dipole moments also changes,and correspondingly we can expect the appearance of charges. Thisphenomenon is illustrated in Fig. III.6b and c, when either positive ornegative charges arise. It is especially obvious when the boundary betweenthe dielectric and free space is considered. These charges, unlike freeones, can only slightly change their position, and this is why they are calledbound charges. The different extent of movement of the free and boundcharges is the sole difference between them, but what is most essential isthe fact that they both create an electric field in the same way,Eqs. (H1.102).

Thus, in accordance with Eqs, 011.102), (HU03) we can say that due tothe polarization the bound charges arise in a dielectric volume and at itsinterfaces, and these charges create an electric field E, that obeysCoulomb's law. Correspondingly, the potential and the electric fieldcaused by the free and bound charges are

1 0 dV 1 l dSU(p)=-f-+-f-

41TEo v L qp 41TE O Sj L qp

and (III.104)


where the total densities of charges are

(III.lOS)

and 80 and ~o are volume and surface densities of free charges that areassumed to be known. In general, in the vicinity of some points, there canbe only free charges-for instance, outside of dielectrics-while at otherpoints inside dielectrics either bound charges or both can exist.

As soon as bound charges are taken into account, we arrived inaccordance with Eqs. (111.104) at the same expression for the electric fieldas that in a free space. In particular, when there is only a volumedistribution of charges, the expression for the electric field is

1 f. 8 dV L q pE(p) =-- 3

4'7T£o V L qp

(III.106)

From a mathematical point of view this equation is exactly the same asthat derived in the first section for free charges only. By applying the sameapproach, we can write the equations of the field in integral form

e~E'dS=-s £0

(111.107)

where e = eo + eb is the total charge within volume V surrounded bysurface S, and it includes both the free and bound charges.

The first equation states that only sources create the time-invariantelectric field in the presence of dielectrics; the second one demonstratesthat both free and bound charges cause the electric field, and this occursin the same manner. Now applying Stokes' and Gauss' theorems forregular points, we obtain the differential form of Eqs. (111.107).

curlE = 08

divE = -£0

(111.108)

If there is an interface of discontinuity, then making use of the sameapproach as that in the previous section we have

E(1) = E(2)t t' (III.I09)

That is, the tangential component of the electric field is a continuousfunction. While the discontinuity of the normal component across theinterface is defined by a surface density of total charge, ~.


Thus, the system of field equations in differential form can be presentedin the form

Coulomb's law

I I curlE = 0 I

(III. 110)

BII divE =-

eo

This system correctly describes the field within and outside dielectrics.However, as in the case of conductors, the right-hand side of the secondequation contains an unknown quantity, namely the density of boundcharges s, and k b •

To eliminate these functions from the system, we will introduce a newvector related to the electric field, and for which this linkage is known forevery given dielectric. With this purpose in mind, let us represent theequation

BdivE = -

eo

as

Then taking into account Eq. (11.103) we have

Bo divPdivE=----

eo eo

or

. ( P) Bodiv E+ - =-

eo eo

We will introduce the vector

(111.111)

(111.112)


Correspondingly, instead of Eq. (IIUll) we obtain

divD = 00 (III.l13)

(II1.1l4)

since EO is constant.The vector D is called the vector of electric induction, and making use

of Eq, (III.92) we can express it in terms of the electric field.

D=EoE+aE=Eo(l+ :o)E=EO(1+{:l)E

where {:l = a/Eo is the dielectric susceptibility that characterizes the di-electric. Along with this parameter we will introduce the dielectric con-stant E.

E = 6 0(1 + (:l)

and correspondingly, instead of Eq. (lII.1l4) we obtain

D=6E

(III.1l5)

(II1.116)

Thus, knowing the dielectric constant, it is an elementary procedure tocalculate the vector of the electrical induction from the electric field andvice versa. In this relation it is appropriate to note the following.

Equation (111.116) is very simple and this fact can easily produce thewrong impression that both vectors E and D have the same physicalmeaning, since they differ only by the scalar 6, which is often constantwithin a dielectric. However, the vector of the electric induction is a sumof two completely different vectors, Eq. (111.112); namely, one of them isthe electric field E, and the other is the density of dipole moments, that is,the vector polarization P. Moreover, from the equality

curl D = curl E E = ECUri E + (grad 6 X E)

= grad EX E

it follows that in general the field D is caused by vortices as well as by freecharges. In particular, at the interface of media with different dielectricconstants the tangential component of an electric field is a continuousfunction.

E~l) =E~2)

and therefore the tangential component of the vector D has a discontinuity

that indicates the presence of surface vortices (Chapter I),

III.3 The Electric Field in the Presence of Dielectrics 247

After these comments let us continue the transformation ofEqs, (111.110). In accordance with Eq. (I1I.103) the surface analogy of thesecond equation can be represented as

I p(1) - p(2) IE(2) _ E(1) = ~ + n n

n neo eo eo

or

[

p(2) ] [ p(1) ] 'IE~2) + _n_ _ E~1) + _n_ = ~eo eo eo

or

(I1I.1l7)

That is, the normal component of the vector D is a discontinuous function,but unlike the normal component of the electric field, this discontinuity isdefined by the value of the density of free charges only.

Now we are ready to write down the system of field equations where onthe right-hand side only the known density of free charges is present.

Coulomb's law

I II divD = 00 II I curlE = 0 I

E(2) - E(1) = 0 It t

D=eE (I1I.llS)

Therefore we have modified the system of field equations (111.110) in sucha way that it no longer contains unknown bound charges. We were able todo this because of the assumption that there is a linear relation betweenthe polarization and the field, and that the polarizability a is known.

For a better understanding of the field behavior let us study thedistribution of bound charges. In accordance with Eqs. (111.113) and(111.116) we have

div sE = s divE + E· grad 10 = 00


and since

00 + DbdivE=---

Co

we obtainC

00 = -(00 + Db) + E· grad sCo

or

where Cf

= clco is the relative dielectric constant.It is convenient to distinguish two types of bound charges.

Db = 0h1) + 0h2

)

where

(III.119)

(III.120)

1-c,,(1) r"Ub - Uo

c r

and

The first type of bound charges arises in the vicinity of points where freecharges are present; and since c

f> 1, these bound charges always have a

sign opposite to that of the free charges.The second type of charges can arise in the vicinity of points where the

dielectric parameter e varies. If there is a projection of the electric fieldalong the vector grad e negative charges arise, but if the vectors include anangle exceeding 7T/2, positive charges appear. In particular, in the casewhere the electric field is perpendicular to the direction of maximalchange of parameter e, this type of bound charge does not arise.

Now we shall consider surface bound charges proceeding the equation

D~2) - D~l) = I o

which can be represented as

D(2) - D(1) = e E(2) - e E(1)n n 2 n 1 n

Inasmuch as


we have

or

(III.12l)

where

E(l) +E(2)E av = n n

n 2

whence

(III.122)

(III.123)

(III.124)

Again, as in the case of volume distribution, we .have distinguished twotypes of charges.

Let us illustrate these results by one very simple example. Suppose anelementary free charge with density Do is placed in a uniform medium withthe dielectric constant 13. In accordance with Coulomb's law the freecharge creates in the dielectric the same electric field as in free space.

1 ooL qpEo = -----dV

41T13o L~p

where the point q characterizes the position of the free charge.Since the medium is uniform, there is only one type of bound charge

1-13O~l) = __r Do

13 r

which arises around the free charge, and its field is

E = _1_ o~l)Lqp dV = _1_ ( 1 - 13 r ) 00 dV Lb 41T13 o L~p 41T13 o 13 r L~p qp

Therefore, the total field caused by both charges is

1 ( 1 ) Do dVE = Eo + E b = -- 1 + - - 1 -3--Lqp41T13o 13 r L qp

Jr

1 oodVE(p) = -4--L3 L q p

1T13 qp

(III.12S)

(III.126)


Comparing Eqs. (1II.l23) and (III. 126) we see that when a free charge isplaced in a uniform dielectric medium, the electric field turns out to be Sf

times smaller than if this charge were in free space, and this occursbecause the free charge is accompanied by bound charge having theopposite sign.

Now we will derive the system of equations for the potential. As before,from the first equation curl E = 0 it follows that

E = -gradU

and substituting this into the second equation of the system 011.118) wehave

div sgrad U = - 00

or

V(sVU) = - 00 (III.127)

Certainly, this is neither Poisson's or Laplace's equation, but if in thevicinity of some point. a dielectric is uniform, then we again obtainPoisson's equation.

or

(111.128)

If in addition free charge is absent, then at such points the potentialbecomes a harmonic function, and correspondingly we have

Performing differentiation on the left-hand side of (III.127) we obtain

or

00 VU' VsV 2U= - - - ---

s s(111.129)

and comparing this with Eq. (111.121) we see again that the Laplaciancharacterizes the density of total charge located in the vicinity of a point.

Inasmuch as the continuity of tangential components of the field isprovided by continuity of the potential and taking into account the fact

IlIA Electric Current, Conductivity, and Ohm's Law 251

thatau

D =EE = -E-n n an

the system of equations for the potential is

Coulomb's law

I U(1) = U(2) I

on Si

1IV(EVU) = -80 I (III.130)

aU2 so,E--E-=I

2 an 1 an 0

We can show that Eqs. (III.130) together with boundary conditions consti-tute boundary-value problems for the time-invariant electric field in thepresence of dielectrics.

In conclusion of this section let us make an additional comment.Various types of rocks have different dielectric constants; in particular,

the parameter Ef

for water is 81, and oil is approximately characterized by2-3. Such differences can, in principle, be used to distinguish formationsthat are saturated by oil or water, for example, in measuring the electricfield in wells. However, due to the fact that rocks are both dielectrics andconductors, it is impossible to measure their dielectric constant by makinguse of a time-invariant electric field. To illustrate this, suppose that acharge is placed in a nonconducting borehole. Then induced charges ariseon the surface of the borehole that cancel the primary electric field in therocks (the electrostatic induction), and correspondingly the polarizationinside the rocks vanishes. In the next section we will also show that in thecase of constant currents, the dielectric constant of rocks does not haveany influence on the behavior of the field.

IlIA Electric Current, Conductivity, and Ohm's Law

In considering electrostatic induction we have established that a fieldcaused by charges located outside of a conductor induces charges on itssurface in such a way that the total field vanishes at every point within the


conductor. We can also say that the field of external charges is not able tocause ordered motion of either electrons or ions, and consequently theymove at random as if these charges were absent.

Now we will suppose that with the help of other electric chargessituated inside and on the conductor surface there arises a constantelectric field, which in general can change within the conductor, but itdoes not vary with time. Of course, as before, this field E obeys Coulomb'slaw. If we knew the distribution of these charges, then the electric field atevery point within the conductor could be calculated from

1 sdV 1 IdSE(P)=-4-f-

L3 L qp + -

4-f-L3 L q p

7TE O V qp 7TE O S qp

Later we will formulate conditions that guarantee existence of constantelectric field within a conducting medium. For now let us accept this factand begin to study the movement of charges with a constant velocitythrough a conductor. This is the third phenomenon, after electrostaticinduction and polarization of dielectrics, which we are going to consider indetail.

It is obvious that if some charge is subjected to a constant electric field,it will start to move. Since the field E does not change with time, anordered motion of charges will be observed. This phenomenon is calledthe electric current. In metals the current is a motion of electrons, while insedimentary rocks, the pores of which are saturated by electrolytes, thecurrent is composed of ions. Consequently, we will distinguish betweenelectronic and ionic conductivity. Note, however, that in both cases rathercomplex, random movement of microcharges is accompanied by an or-dered motion in some direction defined by the electric field.

Perhaps it is proper here to notice that the motion of each microchargeis determined by the magnitude and direction of electric field in thevicinity of the point where this charge is located, and in this sense, it isindependent of the field in other places in the conducting medium.

The movement of microcharges in rocks is very complicated becausetheir structure is complex, and they consist of elements having both typesof conductivity with different values. To simplify the phenomenon of theelectric current we will imagine that all microcharges of every sign are thesame in the vicinity of some point, and they move with the same velocity.This conventional approach for performing the transition from micro tomacro scale allows us to consider the current as the motion of chargesdistributed continuously at certain places in a conducting medium. In thesame manner as we have described the distribution of sources of thegravitational and electric fields, let us introduce the current density j,

lIlA Electric Current, Conductivity, and Ohm's Law 253

which characterizes the movement of charges. This vector shows thedirection of charge movement in the vicinity of the point q. It is equal tothe amount of charge passing through a unit area with center at the pointq, and which is perpendicular to j per unit time. Here there are threeconditions, namely,

(a) The area has unit value.(b) The area is perpendicular to the direction of charge movement.(c) The time has unit value.

All of these conditions are essential. If one of them is not satisfied, thecurrent density vector loses its meaning.

In accordance with this definition, the vector j can be written as

dej(q) = dSdt'io(q) (III.l31)

where dS is the area perpendicular to the direction of the charge move-ment. This area is sufficiently small to assume that the same amount ofcharge passes through every element of this area for the same time. Alsode is the charge that passes through this area during the interval dt.Finally i o is a unit vector, showing the direction of charge movement.

It is obvious that the dimension of the current density is

coulomb[j] = m2 sec

Also in accordance with Eq. (111.131) we an conclude that the direction ofthe vector j coincides with that of movement of positive charges and isopposite to the direction of motion of negative charges, since then de < O.

In general, charges of both signs can move in different directions, forinstance in opposite ones, and correspondingly the total vector j is

(III.132)

where j + and j - are the current densities of the positive and negativecharges, respectively.

As usual, the vector field j can be described with the help of thegeometric models introduced in Chapter I, and in this case they are calledline currents, surface currents, and tube currents.

Now we will express the current density j of the positive and negativecharges in terms of their densities and velocities. Suppose that dS is thecross section of an elementary tube current, shown in Fig. III.6d, andcharges of both signs move along this tube. Then during the timedtcharges located at distances from dS,. which are smaller than the product


W dt, cross this area. Here W is the magnitude of the charge velocity.Correspondingly, the amount of charge is

de = 8 dV = 8 dS W dt

and in accordance with Eq. OII.13l) the current density is

j=8W

Here we see that the current density vector and the velocity have thesame direction if 8 > 0, while they are opposite to each other if 8 < O. Ingeneral, both types of charge are involved in movement and then we have

or

j+=W+8+,

(III.133)

j = W+5+ - W-15-1

where 5+, W+ and 5-, W- are densities and velocities of the positive andnegative charges, respectively, and 15-1 is the absolute value of thenegative charge.


1. The field j does not change under a simultaneous change of the signof both velocity and density. For instance, replacing a movement ofnegative charges by that of positive ones with the same magnitude butopposite direction, the field j remains the same. This shows that themovement of charges can always be reduced to the movement of positiveones only.

2. In metals, electrons form the current and j = W-8-. In electrolytes,however, both positive and negative charges move, in opposite direction.Therefore,

where i o is a unit vector, showing the direction of movement of positivecharges.

3. In accordance with Eq. (IILl33), the current density is not equal tozero if

(III.134)

This means that the current density j can vanish provided that bothcharges move in the same direction, but the equality j = 0 does not requirethat densities should be equal.


4. The total charge density inside of every elementary volume of aconductor is

It is interesting to notice that often in the presence of current flow thistotal charge equals zero. In a uniform part of a metal, for instance, thesum of moving electrons and unmovable positive charges is zero. This isalso true for uniform parts of electrolytes. It is clear that in such casesthese elements of the volumes do not create an electric field.

Now we will establish the relationship between the current density andthe electric field. As the field E is applied to charges, they begin to movewith some acceleration. However, the medium hampers their movement,essentially limiting the increase in velocity. The mechanism producing thisphenomenon can vary. In metals, for instance, this occurs due to thecollision of electrons with ions of the crystal lattice. But regardless of themechanism, the velocity of microcharges becomes proportional tothe electric field. Correspondingly, we have

(111.135)

Where u + and u - are positive coefficients called the mobilities of thepositives and negative charges, respectively. Note in particular that inmetals u + = O.

Substituting these expressions into Eq. (111.133) we obtain

or

and

j = yE (111.136)

(111.137)

Equation (III.136) is called Ohm's law in differential form, and thecoefficient of proportionality y is the conductivity of the medium.

Here let us make the following comments:

1. Ohm's law establishes the relationship between two completely dif-ferent fields, namely the electric field E and the current density describingthe movement of charges.

2. In accordance with Eq. (111.136) the electric field causes a currentdensity field, but not vice versa. For example, in a nonconducting medium


there can be an electric field while current is absent. Let us illustrate thisfact.

1 charges I ~ electric field E ~ current density j

which clearly shows that charges create the electric field E, and in aconducting medium this field generates movement of charges, that is, thefield j.

It is appropriate to notice that sources of the electric field do notusually have any relationship to the charges that constitute the currentdensity field.

3. It turns out that Ohm's law in differential form can also be appliedto alternating electromagnetic fields.

4. In an anisotropic medium the conductivity depends on the directionof charge movement. In this case, y is a tensor. However, in this chapterwe will consider only isotropic media and for this reason y is described bya scalar.

5. As follows from Eq. (111.137) the conductivity is directly proportionalto both the charge density and mobility, and the latter characterizes thevelocity of charges. For instance, the electron mobility in copper isapproximately

mysecu-"'" 4.4 10- 3 - -

Vim

This is a surprisingly small number, especially when we take into accountthe fact that the electric field in metals is very small. If we suppose that

lmv VE= -- = 10- 3 -

m m

then the velocity of electrons is

This shows that it takes almost 70 hours for the electron to travel 1 meter.Certainly when energy is transformed from generators to loads, it does

not travel through wires, otherwise it would take years and years instead ofthe practically instantaneous propagation observed in reality. Thus, thehigh conductivity of metals is mainly caused by the high density ofelectrons that compensates for their low mobility.

(III.B8)

llI.4 Electric Current, Conductivity, and Ohm's Law 257

Ions of electrolytes have an even smaller mobility, approximately 10- 9

m2/Vsec, and this fact along with their lower density results in muchsmaller conductivity for electrolytes than for metals.

6. Ordered movement of charges under the action of the electric fieldis always accompanied by a random motion with a relatively high velocity.

m m10 2

- < W< 106 - ,sec sec

which exceeds by many orders of magnitude the velocity of chargesforming the current.

Due to this random motion, even in the absence of the electric field wecan observe a very weak current, which is called the fluctuating current. Itis obvious that the existence of such currents defines the limits of sensitiv-ity of devices measuring the current.

7. It is convenient to mention along with conductivity 'Y the specificresistivity or simply resistivity p, which is related to 'Y by

1p= -

'Y

The dimensions of these quantities are

coulomb[ 'Y ] = V m sec and

These units will be presented as

Vm sec[p]=--

coulomb

[y] = mhoyrn and [p] = ohm' m

The values of rock resistivities for metals, sedimentary, and other typesof rocks are presented in the following table. The most important featureof this table is the extremely wide range of the resistivity.

Rocks and sediments Ores

Limestone (marble) > 1012 Pyrrhotite 10 5_10 3

Quartz > 1010 Chalcopyrite 10-4_10- 1

Rock salt 106_107 Graphite Shales 10- 3_10 1

Granite 5000-106 Pyrite 10-4-10 1

Sandstones 35-4000 Magnetite 10- 2_10 1

Moraine 8-4000 Haematite 10- 1_102

Limestones 120-400 Galena 10- 2-300

Clays 1-120 Zinc blende > 104

[After Parasnis (1979)]


a

c

b

d

(2)

(1)

s

-(2) LE =-

n 2EO

-(1) LE =--n 2E

O

Fig. 111.7 (a) Current density flux; (b) current density flux near surface charges; (c)continuity of the normal component jn; and (d) Kirchhoff's law.

In the previous sections we have described in detail basic features ofthe electric field caused by charges. By analogy let us study the behavior ofthe current density field.

First of all, knowing the vector j, it is a simple matter to calculate theamount of charge passing per unit time through an elementary surface dS,arbitrarily oriented with respect to j. As is seen from Fig. 1II.7a this isdefine by

dI <J: dS

or

dI = j dS cos(j, dS)

(111.139)

where dS = dS n, and n is the unit vector directed from the back to thefront side of the surface dS. It is clear that this amount of charge ispositive if the charges move along the normal n, and it is negative if thecharges pass through the surface in the opposite direction. In particular, ifthe current density vector is tangential to the surface, the amount ofcharge dI equals zero. As follows from Eq. (111.139), the flux of the vector,j, or the amount of charges passing through an arbitrary surface S per unit


time, is

(III .140)

where i; is the normal component of the current density.In accordance with Eq. (111.140) the flux I is an algebraic sum of

elementary fluxes through various parts of the surface, and it is called thecurrent. As follows from the definition of current density, the dimensionsof current are

sec

coulomb[1] = --- = ampere

To visualize better the movement of charges it is natural, as was alreadymentioned, to use the concept of vector lines. Every vector or current lineshows the direction of the vector j; that is, along such lines charges movein a conducting medium. Also it is convenient to consider current tubes,whose lateral surfaces are called the current surfaces. By definition theflux of current density through current surfaces is always equal to zerosince the vector j is tangential to these surfaces. In general, the crosssection of a current tube can vary from point to point.

Now we are prepared to formulate the basic law that characterizes thebehavior of the current density field, namely the principle of chargeconservation. For the time-invariant field it has the form

~j. dS = 0s

(111.141)

That is, the flux of the current density j through any closed surface is equalto zero.

Inasmuch as the current density vector describes the amount of movingcharge, Eq. 011.141) allows us to make the most important conclusionabout its behavior: The amount of charge arriving at any volume V isalways equal to the amount of charge that leaves this volume in the sametime; that is, their sum equals zero. This fact is independent of the sizeand shape of a surface S, surrounding an arbitrary volume V. It also doesnot depend on whether a uniform or nonuniform medium is considered.In particular, the surface S can intersect media with different conductivi-ties. Moreover, Eq, (111.141) remains practically valid for a wide range ofelectromagnetic fields when the electric and magnetic fields, as well as thecurrent density, change with time.

As follows from Eq. (111.141) at any closed surface there are placeswhere the normal component jn is directed outward, as well as places

260 Il1 Electric Fields

where the normal component is directed inward, and this provides flux inboth directions.

Suppose for a moment that Eq. (111.141) were incorrect. In this case theamount of charges arriving into the volume would not be equal to theamount leaving it. Then we can imagine two cases.

1. The amount of positive charge that arrives at the volume exceeds theamount leaving; that is, the magnitude of negative flux of the currentdensity is greater than the positive flux magnitude. Then, during everysecond new positive charge arrives in the volume and accumulation ofcharges takes place. This means that the electric field as well as thecurrent density would change with time.

2. The amount of positive charge leaving the volume is greater than theamount that arrives, and correspondingly the total charge within thevolume would decrease. Again the electric field caused by this charge, aswell as the current density, becomes a function of time.

This analysis vividly demonstrates that Eq. (III.141) has to be valid,otherwise the electric field cannot be time-invariant. Equation OII.141)can be treated as the second field equation of the vector field j in integralform, but it is usually called the principle of charge conservation. There isa very good reason for this name. Indeed, if inside the volume we have acharge, then in accordance with Eq. (III.141) it remains the same at alltimes, as long as a time-invariant electric field exists. In particular, thischarge can be equal to zero in spite of the presence of flux of currentdensity.

Let us again emphasize two essential features of the principle of chargeconservation, Eq. (III.14l).

1. Because the flux of the vector j through any closed surface S equalszero, it is possible to preserve the constant electric and current densityfields in a conducting medium.

2. Equation (111.141) does not mean that within a conducting mediumconstant charges are absent. This occurs if the equality

is valid for any surface, but the latter does not have any relation to theprinciple of charge conservation.

Now we will derive other equations describing this principle.Applying Gauss' theorem for regular points of a medium we have

~ j . dS = Jdivj dVs v

111.4 Electric Current, Conductivity, and Ohm's Law 261

Since this is valid for any volume, we obtain

divj = 0 (III.142)

This equation also expresses the principle of charge conservation, but itcan be applied only in the vicinity of points where the first derivatives ofcurrent density components exist. For instance, in the Cartesian system wehave

To derive a surface analogy to Eq. (III.142) we will consider the flux of thecurrent density through a closed cylindrical surface as shown in Fig. III.7b,which consists of three parts.

Since dS l = - n dS, dS z = n dS, and the third term disappears as thecylinder height tends to zero, we obtain

or

(III.143)

That is, the normal component of the current density is a continuousfunction across an interface between media with different conductivities. Itis proper to emphasize the importance of this condition, which allows us topreserve the time-invariant electric field in a conducting medium. In fact,if the normal components of the current density were not equal at eitherside of the interface, then during every second we would have either anincrease or a decrease of surface charge, and correspondingly the electricfield would not be time-invariant.

Also from Eq. (111.143) the presence of surface charge becomes evident.In other words, without the field caused by these charges it is impossibleto provide continuity of the normal component of the current density andthe principle of charge conservation would not be valid. To show this wewill imagine that in the vicinity of some point q of the interface betweenmedia with conductivities "1 and "z, the surface charge is absent(Fig. III.7c). Then, in accordance with Ohm's law the normal cornponents


of the current density are

j~l)( q) = 1'1E~l)( q),

where E~l) and E~2) are the normal components of the electric fieldcaused by all charges located outside of this surface as well as thosesituated on this surface, except at the point q, since we have assumed thatin the vicinity of this point charge is absent. Inasmuch as all these chargesare located at finite distances from point q, the field caused by them is acontinuous function across the interface; that is,

E~l)( q) = E~2}( q)

Then, taking into account the fact that 1'1 =F 1'2 we have to conclude that itis impossible to provide the continuity of the normal component of thecurrent density at some point of the interface if surface charge is absent inthe vicinity of this point.

Thus, we have derived three forms of the second equation of thecurrent density field, and all of them express the principle of chargeconservation.

divj = 0 "(2) _ ·(l)I n - I n (111.144)

Let us also consider one special case of a current distribution: conduc-tors are connected together at some place as shown in Fig. III.7d. This isoften observed in electrotechnical practice. Then, applying Eq, (111.141) tothe surface S, surrounding the point of connection of conductors andbearing in mind the fact that the flux of current density outside conductorsvanishes, we have

(I1I.145)

where S, is the cross section of i-conductor, and I, is the current.As is well known, Eq. (I1I.l45) expresses the first Kirchhoff law.Now we will derive the first equation of the field j. Making use of

Ohm's law we have

curlj = curl yE = I' curl E + VI' X E

Since curl E = 0 we have at regular points of the medium,

curlj = grad I' X E (111.146)

That is, vortices of the field j are located at places where the electric fieldhas a component in the direction perpendicular to grad y.

rnA Electric Current, Conductivity, and Ohm's Law 263

Taking into account the continuity of the tangential component of theelectric field and Ohm's law for the surface analogy of Eq. (III.146), wehave

£(2) = £(1) = Et t t

or

Thus

j[Z) jP)

Yz Yl(III.147)

(lII.148)

and the right-hand side of this equation characterizes the distribution ofsurface vortices.

In summary we will present the system of equations of the currentdensity field j.

Principle ofcharge conservation

Ohm'slaw

Coulomb'slaw

I

1

curlj = V'y X E

1

divj = 0 I

(III.149)

Proceeding from Ohm's law we have

j = -y grad U

and correspondingly the current density field is a special case of the vortexIeld, called a quasi-potential field (Chapter I).

Thus in a conducting medium there are simultaneously two completelyIifferent fields, namely, the electric field caused by electric charges andhe current density field arising due to the electric field. Inasmuch as in allelectrical methods of applied geophysics the study of the electric field in a.onducting medium is of great practical interest, we will consider this fieldn detail later in the chapter.


Taking into account the fact that the field E is caused by charges only,let us make use of results derived in previous sections. Therefore, thesystem of equations for both the field and the potential is

Coulomb's law

II

on S on S

au(2) aU(I) ~U(2) - U(I) = 0

on S

U(a) - U(b) = tE. dl'a

an an

on S

(111.150)

111.5 Electric Charges in a Conducting Medium 265

where 0 and 4 are the total density of volume and surface charges thatinclude both the free and bound types of charge. In the next section wewill study their distribution.

IlLS Electric Charges in a Conducting Medium

Now that we have discussed the principle of charge conservation, let usinvestigate the distribution of charges in a conducting medium, whichalways accompanies the current density field. Making use of Ohm's law wehave, for regular points,

divj = div yE = y divE + E· grady = 0

Taking into account the second equation of the electric field

sdivE =-

80

(111.151 )

and substituting into Eq. (111.151) we obtain an expression for the volumedensity of charge.

E . grad y0= -80----

Y(III.152)

Thus volume charges arise only in places where the conductivity of themedium varies, and where the direction of its maximal rate of change isnot perpendicular to the electric field. In accordance with eq. (111.152) thesign of charges depends essentially on the angle between the field E andgrad y. The most typical model of a conducting earth, used in appliedgeophysics, is a system of uniform formations each having a differentconductivity. In this case, the volume density of charges equals zero.

Next we are going to study surface charges. We will make use of thesurface analogy of the second equation, OIl.149).

Again applying Ohm's law we will represent this equation as

=t[(YI +Y2)(E~2)-E~1))

+(Y2-Yl)(E~2)+E~1))] =0 (111.153)

(III.154)


Inasmuch as

from Eq. (III.153) it follows that

~(q) = 2so1'1 - 1'2 E~v (q) = 2s/2 - PI E~v (q)1'1+1'2 P2+PI

where E~V(q) is the average value of the normal component of the electricfield in the vicinity of the point q.

E(l)(q) +E(2)(q)E av ( ) = n n

n q 2

Taking into account Eq, (111.13) we have

E(1)( )=E'-q-~(q)n q n 2'

So

where E~-q is the normal component caused by all charges except theselocated in the vicinity of the point q. Whence,

(III.l55)

(III.l56)

Thus the density of the surfacecnal'ge is directly proportional to thecontrast coefficient.

K.·• _PZ-PI12-

pz +PI

and the normal component E:.-q andEq. (UI.154) can be rewritten as

(III.157)

This study shows that in. most :practical cases the field E, measured byelectrical methods, is mainlYQIDsetl by charges arising at interfacesbetween media havingdIl~~~t ~si8tMties. This fact is the physicalfoundation for applicatioos of11k"t: "l¢¢trieal methods in geophysics.

Here it is appropriate to .~ .•~ •.comments.

1. In general in tpe \1idmi~ m::lim.Y point of a conducting medium it isnatural to distinguish _ ~ ·:(if ~tra1;ges, namely,

(a) Positiveandn¢~;')," l~~:tQr instance, ions in electrolytes-which constitute the ~c~ltd9Ve in opposite directions. In metals,however, onlynegathre e:~ ,fe'1'ettrons) are involved in this movement,

III.S Electric Charges in a Conducting Medium 267

while the positive ones remain at rest. It is essential to note that in bothcases the sum of these charges in every elementary volume of a conductorequals zero; and correspondingly, as we already pointed out, these chargesdo not contribute to the electric field.

(b) Volume and surface charges, which usually appear in the vicinity ofpoints where the conducting medium is not uniform. If the electric fieldremains constant these charges do not also change with time, and they arethe sale sources of the electric field.

2. Charges, as well as the electric and current density fields, do notarise instantly. It always requires some time for their establishment, andduring this time interval the behavior of the field is governed by electro-magnetic laws.

Now we will consider two examples illustrating the distribution ofcharges.

Example 1 Current Electrode in a Uniform Medium

Suppose that an isolated wire with current I is connected with a uniformmedium through a spherical electrode, as is shown in Fig III.Sa. Theresistivities of the electrode and the surrounding medium are Po and p,respectively. Inasmuch as both media are uniform, volume charges areabsent, but there is one interface between the electrode and the surround-ing medium where surface charges arise. It is clear that due to thespherical symmetry, the current density, as well as the electric field, hasonly a radical component and in accordance with Eq. OII.19) we have

(III.158)

where E~2) and E;1) are radial components of the electric field at the frontand back sides of the electrode surface, respectively, while I is the totaldensity of the charge at this surface.

Taking into account the continuity of the normal component of thecurrent density, Eq. (III.143), and the Ohm's law, we obtain

or

(III.159)


Fig. 111.8 (a) Current electrode in a uniform medium; (b) illustration of a charge densitycalculation; (c) distribution of charges as a function of radius;and (d) current tube.

Therefore, the total charge on the electrode surface is

or

(III.160)

since all current I, coming from the wire, leaves through the electrodesurface except at the place where the wire is connected to the electrode.In accordance with Coulomb's law this charge creates an electric field withonly a radial component, and its magnitude is

where Lop is the distance from the electrode center to the observationpoint p. Inasmuch as the resistivity of a formation is usually many orders

(III.161)

111.5 Electric Charges in a Conducting Medium 269

greater than that of the electrode, p » Po' we can write

e = EopI

and

pIE(p) = --2-

47TLop

Thus we have demonstrated that the charge situated on the electrodesurface and surrounded by a conducting medium creates exactly the samefield as if it were located in free space. It is also obvious that under theaction of this field, regardless of the distance from the electrode, ions inrocks move along the field and the current density is

Ij=yE=--

47TL~p(III.162)

Of course, this result directly follows from the symmetry and the factthat all current I goes through the spherical surface with center 0,located at the middle of the electrode. In essence we have described bothfields, E and j, and illustrated again the known relation

~h the electric the current~ ~ field ~ field j

It is obvious that the xitential of the electric field is

pIU(p) = 4L

7T Op(III.163)

and, correspondingly, equipotential surfaces are spherical, including theelectrode surface. Thus the difference of potentials, that is, the voltagebetween two arbitrary points, is

pI [1 1]U( a) - U( b) = - - - -47T LOa Lab

(III.164)

Now let us resolve one paradox. As follows from Eq. (IILl60) thecharge on the electrode surface is defined by the current I and resistivityof a medium p (p »Po), but it is independent of its dielectric constant.This means that if the dielectric constant of the medium is changed, thecharge e, and therefore the electric field E, does not change. At the sametime, the expression of the density k, Eq. (III.159), includes both the free


and bound charges and, as was shown in the previous section, the boundcharges depend on the dielectric constant. It seems that these two factorscontradict each other. However, unlike insulators, in a conducting mediumboth the free and bound charges are subject to the influence of thedielectric constant, and this affects the charges in such a way that the totaldensity of charge k becomes independent of 6.

We will demonstrate this interesting phenomenon, assuming that theresistivity of the electrode is very small. Then instead of Eq. (111.159) wehave

(1II.l65)

where k = k o+ kb is the sum of the free and bound charges.Next we will express each type of charge through the current density j,

and the electrical parameters P and 6. In accordance with Eq, (l1I.12l) wehave, for the free charge,

where 6 I and 62 are the dielectric constants of the electrode material andthe surrounding medium, respectively.

Then, making use of Eq, (111.165) we obtain

k = 6avpJ' + (6 - 6 )Eav

o r 2 1 r

or

since

~ av , .' ( ) P . ."""'0=6 PJr + 6 Z-6 1 "2Jr=6zPJr

E av = pjrr 2

(III.166)

(1II.l67)

Thus the density of free charge depends on the dielectric constant of thesurrounding medium only. But there is also bound charge and, as followsfrom Eq. (III.122),

III.S Electric Charges in a Conducting Medium 271

Or, taking into account Eqs. (III.16l), (III.167), we obtain

Thus

(III.168)

Therefore, the densities of the free and bound charges are subjected tothe influence of the dielectric constant of the surrounding medium, butthe density of the total charge, which defines the electric field, dependsonly on the resistivity of the medium for a given current. It is easy to seethat this result is directly applicable to an arbitrary medium and anyarrangement of current electrodes. The fact that the electric field in aconducting medium is independent of its dielectric constant is of greatimportance for the use of electrical methods in geophysics.

Example 2 Induced Charges on the Plane Interface

We will assume that the current electrode A is surrounded by a conduct-ing medium with resistivity PI' and at distance d there is a plane boundarywith another medium having resistivity pz (Fig. Ill.Sb), The current of theelectrode is 1. As we know, charges arise on the electrode surface and onthe boundary, and both of them create an electric field at every point ofthe medium. Let us restrict ourselves to a study of the charge distributiononly. In accordance with Eq, (111.160) the charge on the electrode surfaceis

272 I1I Electric Fields

To calculate the charges on the plane interface we will make use ofEq. 011.157).

(111.169)

where K 12 = (P2 - PI)/(P2 + PI) and E~-q is the normal component ofthe field caused by all charges except the charge in the vicinity of point q.Now a great simplification takes place because the interface is planar. Allinduced charges on this surface that are located outside the point q do notcreate a normal component of the field at this point, as directly followsfrom Coulomb's law. In other words, E~-q is defined by the charge on theelectrode surface, only.

We will choose a cylindrical system of coordinates with its origin at thecenter of the small spherical electrode A and with its z-axis perpendicularto the interface (Fig. III.8b). Then, in accordance with Coulomb's law thenormal component of the field at any point of the boundary is

(III.170)

where rand d are cylindrical coordinates of the point q.Correspondingly, for the surface density k we have

K 12 eA dk(q) = ~ (r 2 + d 2)3/ 2 (111.171)

As follows from this equation the maximal density is observed in thevicinity of the point r = 0, and then it decreases with an increase of r. Atthe beginning, as r < d, the density decreases slowly; but later, as thedistance r exceeds d, this process occurs very rapidly, and k decreasesalmost as l/r 3• Also from Eq. (111.171) it follows that the closer theelectrode A approaches to the interface, the higher the density becomes inthe vicinity of the point r = 0 and the more rapidly it decreases with anincrease of r (Fig. III.8c). When the current electrode is placed on theinterface, the density tends to infinity at the point r = 0, z = 0, while itvanishes at other points.

Now we will calculate the total charge induced on the plane interface.Making use of the axial symmetry, it is convenient to first calculate theelementary charge des' induced on the ring with radius r and thickness dr.

IlLS Electric Charges in a Conducting Medium 273

Since its area is

dS = 211"rdrwe have

Integrating with respect to r from zero to infinity we obtain

Thus, the total charge arising at the interface is

(III.In)

(III.I74)

That is, its value does not depend on the distance of the current electrodefrom the interface, which influences only the charge distribution on thesurface. As the electrode approaches the surface, induced charges becomemainly concentrated near point r = O. In the limit as d tends to zero, bothcharges eA and es are located on the electrode, and its total charge isequal to

while the charge at the plane interface disappears.In accordance with Eq. (III.156) the coefficient K 12 varies within the

region

and consequently the surface charge es cannot exceed in magnitude theelectrode charge eA" If the medium where the electrode is situated is moreconductive, then both charges eA and es have the same sign but in theopposite case, where PI> P2' they have different signs. If the secondmedium is either an ideal conductor P2 = 0, or an insulator, pz = 00, thesurface charge equals -eA and eA' respectively. Therefore, if the elec-trode is placed at the interface with an insulator such as air, its charge isdoubled, and it equals

PI!e =-

A 211"

In all electrical methods, as current electrodes are located on theearth's surface, their charges are defined by Eq. (III.174).


111.6 Resistance

In this section we will continue to study the electric and current densityfields and the relationship between them. Bearing in mind that in geo-physics, unlike electronics, we mainly deal with volume conductors, let usdiscuss in detail the concept of resistance.

We will consider an arbitrary element of tube current C, which isconfined by two cross sections 51 and 52 and the lateral surface 5f(Fig. III.8d). In accordance with Eq. (III.164) the difference of potentials,or the voltage, between two equipotential surfaces 51 and 52 is

or

VIZ = U(1) - U(2) = fE' d/1

VIZ = U(l) - U(2) = fpj· d/I

(III.175)

It is obvious that since the electric field is a source field, the integral onthe right-hand side of Eq. (III.175) is path independent, and the integra-tion can be performed along any path t having terminal points onsurfaces 51 and 52' Notice that along these current lines t , which arelonger and pass through more resistive parts of a medium, the magnitudeof the current density is smaller than that along shorter lines passingthrough more conductive regions (Fig. III.9a). In other words, in generalthe current density is not uniformly distributed over the cross section ofthe current tube.

Let us represent Eq. (III.175) as

where

U(1) - U(2) =IR I 2

2 J: dl'R 12 = ( p--

J] I

(III.176)

(III.I77)

R 12 is called the resistance of the current tube element C as the currentgoes from the surface 5] to 52' It is clear that the dimensions of Rare

[R] = ohm

Now we will make several comments, elucidating the concept of resistance.

1. The resistance is always positive, and as the path of integration tshows the direction from 5] to 52' we assume that I> O. If the opposite

III.6 Resistance 275

Fig. III.9 (a) Distribution of current density in a conductor; (b) electric field in a uniformcylinder; (c) system of uniform cylinders; and (d) change of position of equipotential surface.

direction of t is chosen, then it is necessary to change the sign of thecurrent.

2. The resistance R 12 , defined by Eq. (III. 177), makes physical senseonly if both surfaces S1 and S2 are equipotential surfaces and St is thelateral surface of the current tube. Indeed suppose that the surface S isnot an equipotential surface. Then changing the position of a point "1" onthis surface, the voltage and therefore the resistance of the conductor Cwould change, and R 12 would become meaningless. This also happens ifthe surface St is not a lateral one, since in this case the current I dependson the cross section S of the conductor, while in Eq. (III.17) it is assumedthat I is constant.

3. As follows from Eq. (nU77) the function R 12 characterizes theability of the conductor C to resist the passage of the current. It isnumerically equal to the difference of potentials V(l) - V(2), as thecurrent magnitude has unit value.

4. To determine the parameters that define the resistivity R 12 , we willmake use of the theorem of uniqueness, assuming that the conductor isuniform. However, later we will demonstrate that our conclusions remainvalid even in the general case, where resistivity changes arbitrarily within aconductor.

on 5 f


It is clear that the potential of the electric field satisfies the followingconditions:

(a) U is a solution of Laplace's equation inside the uniform conductor

V 2U = 0

(b) On the lateral surface 5f' the normal component of the electricfield is zero, and therefore

au-=0an

(c) On the equipotential cross sections 5] and 52 the potential is suchthat

1 au- ( -d5= -I,p ls, an

1 au-f -d5=-IP S2 an

As follows from the theorem of uniqueness, all three conditions uniquelydefine the electric field E inside the conductor C. In other words, therecan be only one distribution of the field if the current is given. Supposethat the current equals I] and correspondingly the electric field is E],which of course satisfies all three conditions. Now let us increase thecurrent m times, that is, 12 = ml., It is easy to see that such a changeresults in an increase of the magnitude of the electric field by a factor ofm at every point of the conductor, but it does not change direction. Thenew field is

E2=mE]

In fact, the potential of the field E2 is

U2 =mU]

and U2 as well as U] is a solution to Laplace's equation.At the same time this field obeys boundary conditions on the lateral

and cross-section surfaces since

aU2 au]-=m-an on

Thus, in accordance with the theorem of uniqueness we can say that thefunction E 2 describes the electric field in the conductor C if the currentthrough its cross section equals 12 . This analysis shows that the ratios

E

Ior

pj

I


are independent of the current magnitude. In other words, the field of thevector j/I in the volume V is determined entirely by its boundaries5t , 51' 52' and the distribution of resistivity p. Thus, in accordance withEq. (III.177) the resistance R of the conductor C is completely defined byits dimensions, shape, and resistivity. It does not, however, depend on thecurrent or any changes in other parts of the circuit if surfaces 51 and 52remain equipotential surfaces. This means that by defining the resistanceR 12' we will imply only such ways of introducing the current into anelectrical circuit that preserve constant potential on these surfaces.

5. To determine the resistance it is necessary to know the currentdensity field, and in general it is a complicated task related to the solutionof a boundary-value problem.

6. Equation (III.176) is the integral form of Ohm's law. Thus we havederived two forms of this law.

j = yE and U(I) - U(2) = R 121

As will be shown in the next section these equations are valid only forso-called external parts of the electric circuit.

7. If the magnitude of the vector j has the same values within everysection of the conductor, perpendicular to the current, then we have

J: d/ idt dt'--=-=-

I I 5

and

In particular, if the conductor is uniform,

f2 dt'

R'2=P 1 S

(III.178)

(III.179)

It is appropriate to notice that Eqs. (III.178), (III.179) can be used only ifthe geometry of the current density field is known, since 5 are-sections ofa conductor perpendicular to the vector j.

8. Suppose that the conductor C is a uniform cylinder of arbitraryshape, confined by a lateral surface and two equipotential cross sections 51and 52 (Fig. III.9b). In this case the electric field behavior is remarkably

278 ill Electric Fields

simple; namely, it is constant within the conductor and tangential to itslateral surface. Indeed, the potential of such a field is a linear functionand therefore satisfies Laplace's equation as well as the boundary condi-tions.

au-=0an

and

Thus, in accordance with the theorem of uniqueness, this function E is auniform electric field. Correspondingly, we can conclude that the currentdensity is also uniform and the expression for the resistance, Eq, (111.179),is drastically simplified.

(111.180)

where t and S are the length and the cross section of the cylinder,respectively.

As follows from the previous section, the electric field E within theuniform cylinder is caused by charges located on its surface. As we justproved, they are distributed in such a way that the electric and currentdensity fields are uniform and parallel to the lateral surface.

Often a conductor consists of several parts, and each one of them is auniform cylinder with a length that significantly exceeds the dimensions ofits cross section. Such elements of a circuit are called quasi-linear, and it isobvious that their resistance can be calculated from Eq. (111.180)(Fig. III.9c)

9. It is clear that if the resistivity of the cylinder, p, does not changewithin each cross section but varies along the conductor, then the currentdensity field j still remains uniform and tangential to the lateral surfaceSt. At the same time the electric field changes along the cylinder butremains constant within every cross section, and it is directly proportionalto the resistivity.

IE=pj=p-

S

Correspondingly, instead of Eq. (111.178) we have the following expression


for the resistance of a cylindrical conductor confined by equipotentialsurfaces 51 and 52'

I f2R 12 = - pdt5 1

(III.18I)

10. As was emphasized earlier, the resistance of the conductor C hasmeaning if surfaces 51 and 52 are equipotential surfaces, regardless of anychanges in other parts of the circuit. At the same time it is very simple toperform modifications outside of the conductor C that result in a changeof the potential over surfaces 51 and 52' or one of them.

For instance, let us introduce an inhomogeneity T, shown in Fig. III.9d.As we know, electric charges arise on its surface, and they create a fieldthat is certainly not perpendicular to the surface 52' It is also clear that inapproaching the inhomogeneity to 52' its influence becomes stronger.However, there are cases when the surfaces 51 and 52 can be onlyequipotential surfaces. For instance, this occurs if they are either surfacesof ideal conductors or cross sections of linear parts of the circuit, as shownin Fig. IIUOa and b. Of course, this conclusion remains practically valideven in those cases when the volume V is confined by conductors thathave great conductivity.

Now we will consider three examples illustrating the concept of resis-tance.

Example 1 Grounding Resistance of a Spherical Electrode

In the previous section we have shown that the electric field caused bycharge on the electrode surface, possesses spherical symmetry, and itspotential at any point p is

pIU(p) = 4L

tr Op(III.I82)

Consider two spherical and equipotential surfaces with radii LOa and LOb'

respectively (Fig. IIUOc). Then the difference of potential between themis

pI [1 I]U(a)-U(b)=- ---4rr LOa LOb


Fig. 111.10 (a) Resistance of volume when its cross sections have contact with a veryconductive medium; (b) resistance of volume when its cross sections have contact withquasi-linear conductors; (c) illustration to calculation of grounding resistance; and(d) potential distribution in a uniform half space.

Therefore, the resistance of this spherical layer is

R b = U(a) - U(b) = ~ [_1 1_]a I 47T LOa LOb

(III.183)

and it depends essentially on the layer thickness and the distance from theelectrode, as well as, of course, the resistivity p.

If we mentally represent the whole medium around the electrode as asystem of spherical layers with its center at the point 0 (Fig. III.IOc), andtake into account the fact that the current I goes through all of them, thenwe can say that they are connected in series. Correspondingly, the resis-tance of the whole uniform medium to the current, which moves from theelectrode, can be presented in the following way:


or

R=_P-e 47Ta

(III.184)

(III.18S)

where a is the electrode radius.Usually R; is called the electrode or grounding resistance, and by no

means does it characterize the resistance of the electrode material or thatof the surface between the electrode and the medium, since the influenceof both these factors is negligible. In contrast, the grounding resistanceshows how all space resists the current leaving the electrode. This is a veryuseful concept and, in accordance with Eq. (III. 184), the electrode resis-tance strongly increases with a decrease of its radius. Such behavior can beexpected since with a decrease of electrode surface area the currentdensity increases in its vicinity, and correspondingly the potential de-creases more rapidly near the electrode.

Substituting in Eq. (III.183) LOa = a, we represent the resistance of thespherical layer bounded by the electrode surface and that with radius LOb

as

R=Re(I-~)LOb

The latter shows that if LOb» a, the resistance of this part of the mediumdiffers only slightly from the grounding resistance. For instance, if LOb =

lOa, this spherical layer around the electrode contributed 90% of R e , andusually in applied electrical prospecting the thickness of such a layer doesnot exceed 1-2 m. This consideration allows us to understand that in mostcases application of two electrode arrays in electric methods is useless, andwe will return to this subject in the next section.

The concept of grounding resistance has been described for the case ofa spherical electrode placed in a uniform medium. Of course, the ground-ing resistance arises if the medium is not uniform and the electrode has anarbitrary shape. In particular, if due to electrochemical processes there isa thin layer around the electrode surface with a relatively high resistivity, itcan give the main contribution to the grounding resistance.

Example 2 Influence of Grounding Resistancein Measuring Voltage

Until now we have assumed that the electrode is the part of a circuit thatsupplies a conducting medium with a current I. Now we will consider therole of grounding resistance in measuring the voltage. Suppose we intend


to determine the difference of potentials between points M and N locatedon the earth's surface.

VMN = UM - UN

The distribution of the current density field as well as the equipotentialsurfaces are shown in Fig. III.IOd. To find the voltage in a conductingmedium we will insert two electrodes in the vicinity of points M and N,connected to each other through a device (voltmeter) measuring thevoltage.

Here it is appropriate to make one comment; namely, the voltmetermeasures the value of the integral

(111.186)

At the same time the voltage can be described with the help of thepotential difference UM - UN'

Inasmuch as the integral in Eq. (111.186) does not depend on the pathof integration, it can be written as

(III.187)

where E is the component of the field directed along the straight linebetween these electrodes. Very often the distance MN is sufficiently small,and we can assume that the electric field does not change within thisinterval. Then Eq. (111.187) is simplified,

VMN=EMN

and the measured value of the voltage of the electric field is easilycalculated.

Now we will continue to describe the distribution of the potential in theconducting medium in the presence of receiver electrodes M and N. As isseen from Fig. III.lOd the measuring circuit is connected in parallel withthe conducting medium. Near electrodes M and N one part of the currentgoes into this circuit, while the other moves through the conductingmedium. In the previous section we demonstrated that if the currentdensity field j intersects the boundary between media with differentresistivities, surface charges arise. Therefore, negative and positive chargesappear on the surfaces of the electrodes M and N, respectively. Thismeans that due to the current in the measuring circuit the potential of theelectrode M decreases, while it becomes greater for the electrode N. Inother words, the presence of receiver electrodes connected with each

111.6 Resistance 283

Fig. UI.11 (a) The influence of grounding resistance on measurements; (b) potential of thecharge electrode; (c) extraneous force mechanism; and (d) electric circuit.

other, results in a distortion of equipotential surfaces III their vicinity(Fig. HUla); that is,

(III.188)

where U/:1 and U~ are the potentials of the electrodes M and N, respec-tively, if some current goes through the receiver circuit. Thus in thevicinity of electrodes, when the current moves from the volume to linearconductor or vice versa, we observe a relatively strong decrease of thepotential.

and

By definition of the resistance, Eq. (III. 176), these differences can beexpressed as

(111.189)

where I, is the current in the measuring circuit, while R M and R N are thegrounding resistances of the electrodes M and N, respectively, which are


usually unknown. Performing a summation of these equations we obtain

or

(III.190)

Thus, the voltage V;;N' which is the input to the measurement circuit, andthe voltage VMN characterizing the field in the conducting medium, differfrom each other, and this difference is proportional to the sum of theirgrounding resistances.

For this reason it is necessary to take into account the influence of thegrounding resistances, and in essence this goal constitutes the basis of allmethods of measuring the voltage in a conducting medium. For illustrationlet us briefly discuss two such methods.

The First MethodSuppose that at the instant when the input V;;N is measured, the

current through electrodes If equals zero. Then in accordance withEq, (111.190) the influence of the grounding resistance vanishes and

(III.191)

(111.192)

To achieve this compensation the receiver circuit has a source of anothercurrent, Ie' which at the instant of measurement has the same magnitudeas If but the opposite direction. This is why this method is called thecompensating method.

The Second MethodLet us represent the input V;;N as

V;;N = IfR

where R is the total resistance of the receiver circuit, which includes thewire resistance R w and the internal resistance of the voltmeter.

R =R w +R j

Then Eq. (III.190) can be rewritten in the form

VMN«t.cn,+Rw +RM+RN)

where I.R, is the voltage between the terminal points of the voltmeter.If the resistance R, is chosen so that the inequality

Rj»RM+RN+R w


is met within a range of possible change of the grounding resistance, wehave

(111.193)

That is, the voltage measured by the voltmeter is practically equal to thedifference of potentials between the points M and N, as the influence ofgrounding resistances is absent. Thus using voltmeters with high internalresistance allows us, in essence, to eliminate the effect of R M and R N •

Example 3 Three-Electrode Array

We will consider an array that consists of one current electrode A and tworeceiver electrodes M and N placed along the line (Fig. III.11b). As thecurrent fA goes through the electrode A, a charge

eA = COPIA

arises on its surface, which creates a primary electric field in a conductingmedium.

where Lop is the distance between the current electrode and the observa-tion point p. Also, if there are interfaces between media with differentresistivities, surface charges appear, which in their own turn create asecondary electric field Es ' Thus, the total field at any point p is a sum ofthe primary and secondary fields

E = E p + Es

This vector field is described by a certain distribution of its potential,and the points M and N are located on corresponding equipotentialsurfaces. Correspondingly, with the help of receiver electrodes of thearray, the voltage between them

VMN = UM - UN

is measured.To eliminate the influence of a change of current lA' the ratio

VMN UM - UN

t; fA

is calculated, and in accordance with Ohm's law it is equal to theresistance R M N of the medium, confined by equipotential surfaces passing

(III.194)


through points M and N, respectively.

VM NR M N = --fA

Generalizing this result, we can say that any array in electrical methodsmeasures the resistance of the medium between equipotential surfaceswith potentials UM and Uw

We will consider two special cases where the three-electrode arrayplays an important role in electrical logging.

First, suppose that the electrode N is located far away from the currentelectrode A so that we can let UN = O. Then we have a two-electrode arrayAM, which is usually called the normal or potential probe, and in accor-dance with Eq. (111.194) it measures the resistance of the conductingmedium, confined by the equipotential surface,

U=UM

passing through the electrode M and a spherical surface of infinitely largeradius. Of course with an increase of the distance from the electrode Athe contribution of different parts of the medium, confined by equipoten-tial surfaces and having the same thickness, becomes smaller.

We will consider the opposite case, when the distance between receiverelectrodes is sufficiently small, and we assume that the electric field withinthis interval is practically constant. Then we have

VMN MNR = - = -E (III.195)

MN J JA A

and such an array is often called a gradient probe.It is proper to notice that in a uniform medium a three-electrode array

measures the resistance of the spherical layer with thickness MN andcenter at the point A, but in the general case of a nonuniform medium ashape of this layer can be very complicated.

III.7 The Extraneous Field and Its Electromotive Force

Proceeding from Coulomb's law and the principle of charge conservation,we have established that at the regular points of a conducting medium, forthe electric and current density fields the equations

curlE = 0 divj = 0 (111.196)

111.7 The Extraneous Field and Its Electromotive Force 287

hold. Also it was assumed that everywhere in the medium a linear relation(Ohm's law),

j = yE (III.l97)

holds true. It is essential to note that in Eqs. (III.l96), (IIU97) E meansthe field that is caused by electric charges only, and to emphasize this factwe will often use the index "c": E = EC

•

However, a study of these equations shows that it is impossible tomaintain a constant movement of charges in a conducting medium if theelectric field is caused by charges only. To account for the constant chargemovement we need to study the extraneous field. Let us discuss thissubject from various points of view. First, from Ohm's law,

j = 'Y grad U

it follows that

(III.198)

where j t is the projection of the current density vector on some line t.Suppose t coincides with one of the current lines that are, in accordancewith the principle of charge conservation, always closed. Consider thechange of the potential along this line.

Inasmuch as in this case the ratio j II 'Y is positive, we have to concludethat the potential continuously decreases along the current line. There-fore, from Eq. (III.198) it follows that after one complete passage alongthis line the potential at the initial point acquires a new value. In otherwords, it becomes a many-valued function that of course contradicts thedefinition of the potential of the Coulomb field. This result clearly indi-cates that something is wrong with Eq. (III.197), since it is not prudent toassume that the fundamental physical principle

divj = 0

is incorrect.Now we will study this problem from a slightly different point of view

and demonstrate that a nonzero electric field cannot simultaneously satisfyEqs. (111.196) if Ohm's law, written in the form j = 'Y E, is everywhere valid.In fact, from the equation curl E = 0 it follows that the electric field doesnot have closed vector lines. Then, in accordance with Eq. (IIU97) wehave to conclude that the current density field j also does not have closedvector lines. On the other hand, from the equation divj = 0 it follows thatthe field j cannot have open lines and, again in accordance with Ohm's


law, Eq. (lII.197), this is also true for vector lines of the electric field.Presenting the results of this analysis in the form of a table; we have

Open ClosedField line line

E NO NO

NO NO

We have to come to the conclusion that only a zero field can satisfyEqs. (111.176), since the absence of both types of vector lines simply meansthat fields E and j vanish. Taking into account the fact that both equationsof the system (III.l96) have been derived from physical laws, we are leftwith one choice only, namely, that it is necessary to modify Eq. OII.197).From the physical point of view this result leads us to the conclusion thatthe electric field caused by charges only cannot maintain a constantmovement of charges in a conducting medium.

In essence we have already observed this phenomenon in studyingelectrostatic induction. As we know by placing charges either inside of theconductor or on its surface, their electric field does not remain constant,but instead changes with time until it disappears altogether with thecurrent density field. To perform modifications to Eq. (III.197) we willimagine the following experiment. Suppose that there is a tank, made froma nonconducting material with two parallel metallic plates P + and P -,placed at some distance from each other and bearing charges e+ and e",respectively (Fig. 111.11 c). It is clear that these charges create an electricfield directed from plate p+ to P-. Inasmuch as the medium betweenthem is nonconducting, both charges and their electric field remain con-stant.

But now let us fill the tank with an ionic conductor so that part of theplates are located above the solution, and the conductivity of the ionicconductor 'Ye is much smaller than that of metallic plates. Due to theelectric field E, current arises in the solution, and correspondingly positiveions move to the plate with the negative charge, while negative ionsapproach the plate P+, which has the positive charge. This movement ofions results in a decrease of plate charges until both the electric andcurrent density fields vanish. Assuming that this process is sufficiently slowit can be described in a very simple manner. In accordance withEq. (II1.13), the electric field between plates at some instant is

(III.199)


and it is perpendicular to the plates, since the influence of their edges isneglected. Here l is the charge density at this moment. The field Ecreates a current density j, which equals

(III.200)

and correspondingly the current Ie is

(III.20l)

where Se is the area of each plate that is immersed into the solution, whileIe is the amount of positive and negative charge arriving every second atplates p+ and P-, respectively. It is obvious that this process leads to adischarge of the plates. To maintain a constant current, we will supposethat above the solution the plates are connected with each other through aconducting medium C j , which has a conductivity 'Yj' Thus our model,along with two plates, consists of the solution C, and the conductor C j • Itis clear that in both conductors Coulomb's electric field E C has the samedirection, namely, from the plate P+ to P-.

Now we assume that within the conductor C j along with the Coulombfield there is another field, E?", which has an opposite direction, but it isrelated in the same manner with the current density, and its magnitude isgreater than that of the Coulomb field; that is,

(III.202)

Correspondingly, within the conductor C j there are two current densityfields, which are opposite to each other,

and in accordance with Eq. (III.199) the total current density in C j is

(III.203)

This new field, Eext, is called an extraneous field to emphasize that itsgenerators can be of any origin (thermal, chemical, mechanical) exceptcharges.

Due to this field the positive charges move within the conductor C j

against the electric field E C, and this allows us to compensate for discharge

of the plate P + through the solution. At the same time negative chargesmove in the opposite direction in C j and increase the charge e- on the

(III.204)


plate P-; but we will pay further attention to positive charges only. It isclear that the current inside of the conductor Cj is

t,= l'i ( E~xt - ~) s,

where S, is the area of each plate above the solution.Thus, it is natural to distinguish two parts of this model of the current

flux. The first one is called external, and it corresponds to the solution C~,

where only a Coulombic electric field exists. This field is directed from theplate P" to P-, creating the current, equal, by Eq. (III.20l), to

II~ = 'Y~-S~

6 0

which is trying to discharge plates. In the second or internal part, conduc-tor Cj , both the Coulomb and the extraneous forces act on charges, and inaccordance with Eq, (III.202) the current in Ci is

t, = 'Yi( E~xt - ~ )Si

which tends to restore the charges on the plates. It is obvious that if thecurrents I~and I, are equal to each other, the charges on the plates willnot vary with time and therefore the electric and current density fields willremain constant. In other words, we have demonstrated that the presenceof the extraneous field is necessary to create a time-variant current.

Let us illustrate the relationship between this field and the current.Suppose the field E~xt becomes greater. Then, in accordance withEq. (III.204) the current I, also increases, and this results in an increase inthe amount of charge arriving at the area Si of the plate P+, which isspread over the whole plate. The same is true for negative charges on theplate P-. Correspondingly, the electric and current density fields increasein the solution too, until the current I~ becomes equal to Ii' A similarprocess is observed if the extraneous force decreases.

Now we will make several comments.

1. Several features distinguish Coulomb's and the extraneous fieldsfrom each other, and they are

(a) The voltage of the extraneous field E~xt, unlike the Coulomb field, isin general path dependent; that is, the voltage of this field

tE~xt. d/a

is a function of the contour t along which the integration is performed.


(b) The extraneous field cannot be caused by time-invariant electriccharges, and correspondingly it does not obey Coulomb's law. At the sametime, as was already mentioned, generators of this field can have verydifferent natures such as thermal, mechanical, chemical, or electromag-netic.

(c) The extraneous force, unlike the Coulomb force, is usually equal tozero in those places where its generators are absent. There are exceptions,however, and one is the induction force.

(d) Extraneous and Coulomb fields can create a current in a conductingmedium, while it is impossible to maintain a constant movement ofcharges with only a Coulomb field.

3. In general every circuit consists of internal and external parts. In theexternal part only the Coulomb field E C acts on charges, while in theinternal part both the Coulomb and extraneous fields define a movementof charges.

4. There are cases when both forces, E C and E ext, have the same

direction. This occurs, for instance, when an accumulator is being charged.5. The Coulomb and extraneous forces have only one common feature;

namely, they are related to the current density vector in the same manner,and therefore we have

(III.20S)

The above equation is Ohm's law in differential form, which is valid withinthe internal and external parts of any electric circuit. It is obvious that forthe external part, where Eext

:; 0, we again have

j=yE

In most practical cases of electrical prospecting we are interested in thestudy of the behavior of the field in those places of a conducting mediumwhere extraneous forces are absent, and correspondingly we can still makeuse of Eq. (HU97).

6. The voltage of the extraneous force Eext between terminal points ofthe internal part of a circuit

taken along a current line, is called the electromotive force :ff.

(HI.206)


It is clear that it' characterizes the work performed by the extraneousforce in moving a unit positive charge against the Coulomb field.

Now let us consider several examples.

Example 1 Behavior of the Field and Its Potentialalong a Quasi-linear Circuit

Suppose that the external part of the circuit, shown in Fig. lII.lld, is auniform conductor with a constant cross section S, but within the internalpart there is an electromotive force equal to it'. First consider the externalpart, where the current density field is uniform, since

I = constant and S = constant

In accordance with Ohm's law,

j = '}'E or E=pj

we have to conclude that the electric field inside of the conductor alsodoes not vary.

E = constant

This is the Coulomb field, caused by charges, and it is natural todetermine the charges' location. Certainly they are not located within theconductor, since it is uniform, Eq. (I1I.152), and the sum of charges thatconstitute the current equals zero. Also they cannot be concentrated onlynear the terminal points of the internal part, because a field caused by twosuch charges of opposite sign by all means is nonuniform. It is impossiblethat the sources of this field are located outside the circuit, inasmuch asdue to the electrostatic induction they do not have any influence on thefield inside the conductor.

Thus, there is only one place where sources of the uniform electric fieldE can be located, and this place is the conductor's surface. As soon asthere is a current I in the circuit these charges are distributed in such a"clever" way that the electric field becomes uniform everywhere within theexternal part of the circuit. Schematically their distribution is shown inFigure III.lld.

As usual, let us make some comments.

1. Changing the position of the external part of the circuit in space, thecurrent I does not of course vary. This happens because the distributionof the surface charges also changes in such a manner that they maintainthe same electric field in the conductor.


2. If the cross section or a conductivity of the external part of thecircuit changes, the electric field becomes in general nonuniform, and itsbehavior is again governed by charges. In particular, as conductivitychanges both the surface and volume charges are present, and they aredistributed in such a way that their electric field provides the same currentthrough every cross section of the conductor.

3. In general there are two types of charges on a conductor surface.One of them creates the electric field inside, while another plays acompletely different but also important role, since due to this charge thecurrent in the circuit depends on the electromotive force and its resis-tance, only. In fact, suppose we have a charge near the circuit(Fig. III. l ld). In accordance with Coulomb's law it creates inside of theconductor a field, which can be very strong, while the current 1 does notchange at all. This happens due to the charges that appear on theconductor surface, and their field along with that of the external charge,creates a zero field inside of the conductor (the electrostatic induction).

As we already know, in the internal part of the circuit the currentdensity is defined by both the extraneous and the Coulomb fields. Thelatter, however, is caused by all charges that arise on the whole surface ofthe conductor as well as inside, if it is not uniform.

Now we will study the voltage between two arbitrary points a and b ofthe circuit (Fig. Ill.I l d). If within this interval extraneous forces areabsent, then we have

U(a)-U(b)= jbE.dl'= fbpj.dl'a a

or

bpj'dl'U(a) - U(b) =1f -- =1R ab

a 1(III.207)

where R ab is the resistance of the circuit between the points a and b. Inparticular, when these points coincide with terminal points of the internalpart (current source) we obtain

U+- U_=1R e (111.208)

where R; is the resistance of the external part of the circuit.Next suppose that the internal part with electromotive force it' is

located somewhere between points a and b. Then applying Ohm's law,

or


we have

fb fbPj· dl' f+U(a) - U(b) = E· dl'=I -- - E ext

• dl'a a I -

whence

U( a) - U( b) = IR ab - 15' (III.209)

where R ab consists of the resistance of the internal part R, and theresistance of that section of the external part located between points aand b. In considering the whole circuit when a = b, we obtain

(III.21O)

(III.211)

where R is the total resistance of the circuit.From Eqs. (111.208) and (III.21O) we obtain the relationship between

the potential difference at the terminal points and the electromotive force.

su, 1U -U = =15'----

+ - R, + R e 1 + (RjR e)

Thus this potential difference is practically equal to the electromotiveforce if the resistance of the external part is much greater than that of theinternal one.

(III.2I2)

As follows from Eqs. (II1.207) and (III.209), the potential decreasesalong the current line within the external part while it increases in theinternal part; this behavior is shown in Fig. III.12a.

Example 2 The Current Line of an ElectricalProspecting Array

Now we will consider a different contour, which includes the followingelements:

(a) An internal part (current source) with electromotive force 15';(b) A linear conductor (wire) with resistance R w ;

(c) Two electrodes with grounding resistances R A and R B , whichprovide the contact with the volume conductor; and

(d) An arbitrary medium with resistivity p, which in general can changefrom point to point.


+

_.a+a

i:i-.r.J&Y·· k 2II.

bu

'----------_1

a

c

Fig. 1l1.12 (a) Potential distribution along a circuit; (b) the two-electrode array; and(c) double layer in a conducting medium.

Then, in accordance with Ohm's law, the current in such a circuit(Fig. III.12b) is

(111.213)

where R, is the resistance of the current source. As we know, thegrounding resistances depend on the distribution of the medium resistivityeverywhere regardless of distances from the electrodes. In other words, achange of the resistivity somewhere beneath the earth's surface has tocause a change of the current I and, respectively, measurements of thecurrent with this simple two-electrode array can in principle give informa-tion about the electric properties of the medium. However, this is trueonly in theory, since the grounding resistance is mainly defined by theresistivity of that part of the medium located very close to the electrode.The size of this area seldom exceeds 1 m. Correspondingly, the current Iis practically insensitive to a change of the resistivity beyond this range.For this reason the two-electrode array is not used in electric methods


except for some applications in logging; instead, a standard array consistsof two different parts, namely,

(a) A current line AB, which includes the electromotive force. The salefunction of this part is to create the electric field in a conducting medium.

(b) A measuring line MN, which contains a voltmeter, aIIowing us tomeasure the voltage of the field between receiver electrodes caused by allcharges arising in the medium. This separation of the current and receiverlines of the circuit is the distinguishing feature of most electrical prospect-ing arrays.

Example 3 The Extraneous Force and Electric Charges

In Section 111.5 we studied the distribution of charges in a conductingmedium when the extraneous force is absent. Now we will consider a moregeneral case, proceeding from the equations

or

and Ohm's law,

sdivE= -,

8 0

divj = 0

'(2) _ '(1)in -in

j = y(E + E eXI)

Repeating the same procedures as in Section 111.5, we have

divj = div y(E + EeXI) = div yE + div yE ext = 0

or

'Y divE + E· grad y + div yEext = 0

or

sy- + E· grad y + div yE eXI = 0

8 0

Therefore the volume charges related to the extraneous force are

div y Eexl

{)ext = -80----

Y(111.214)


They are situated within the internal part of a circuit, and as soon as theelectromotive force if; and the internal resistance R; are given, thedistribution of these charges has hardly any interest for us. It is moreuseful to consider surface charges related to the extraneous field.

Generalizing Eq. (III.153) we have

or

Inasmuch as

we have

Here

Div yEext

!ext = -Eo yaY

av Yl + yzY =

2

(111.215)

Div yEext = Y2E~2)ext - YIE~I)ext

and E~2)ext and E~l)ext are normal components of the extraneous force atthe front and back sides of the surface, where its normal indicates thedirection of the current density vector.

In particular, near terminal points of the internal part we have

"IC _ Yi E ext..:.,-- -EO--;W 0-

Y(III .216)

since the field Eext vanishes in the external part of a circuit.Of course, these charges have an influence on the total electric field

both inside and outside of a conductor. If both components E~~ and E~~

are equal, then the charges have the same magnitude but opposite signs.

where

! = ~Eext+ EO av 0 ,

Y"IC_= Yi E extc; -EO--;W 0

Y(III.217)


It is also clear that in the case when near the terminal points theconductor is uniform, Yi = Ye , we have

(III.218)

Example 4 The Contact Electromotive Force

Suppose that the extraneous force E ext is distributed within a very thinlayer with thickness h, and that 5 is its mean surface area, as is shown inFig. II1.12c. The boundary surfaces of this layer are 51 and 52' and thenormal n is directed from 51 to 52. In general, the conductivities of themedia inside and outside the layer are different, and the thickness h canvary. Also we will assume that the normal component of the extraneousforce is positive if it has the same direction as the normal n, but itbecomes negative if the normal component of the extraneous force andthe normal n are opposite to each other.

It is convenient to represent the field of the current density as a systemof current tubes with sufficiently small cross sections. Then, every elementof the layer Ci plays the role of an internal part of a corresponding circuit.It is obvious that the charges arising on surfaces 51 and 52 [Eq. (III.216)are maintained with the help of the extraneous force, while the Coulomb'sfield tends to discharge them.

Taking into account the fact that the layer thickness is very small, it isnatural to assume that within every element of the layer Ci the force Eext

does not change, and therefore

fti = f.2Eext . do = hE~xt1

(III.219)

Correspondingly the distribution of charges on terminal sections is definedby Eq. (III.217); that is, elementary surfaces d5 1 and d52 bear chargeswith the same magnitude but opposite signs.

Now let us make one more simplification based on the assumption thatthe layer thickness is very small. In this case the resistance R, of everyelement of the layer is negligible with respect to the external resistance R;of the corresponding tube current.

Then, in accordance with Eq. (III.212) we have

IlLS The Work of Coulomb and Extraneous Forces, Joule's Law 299

That is, the difference of potentials at boundary surfaces equals theelectromotive force acting near the same point.

It is clear that charges situated on surfaces Sl and Sz create the electricfield in a conducting medium, and if the observation point is located atdistances from the layer significantly exceeding h, this system of chargescan be considered as a double layer. It is very easy to find a relationbetween the electromotive force and the density of dipole moments 1). Infact, from comparison of Eqs. 011.72) and (III.212) we have

(III.220)

Inasmuch as the field of the double layer is defined by the distributionof the function 1), let us consider the limiting case,

but the dipole moment 1) remains the same. This means that we havearrived at the mathematical model of the double layer, when the positiveand negative charges are situated on the front and back sides of thesurface S, respectively. Then, making use of the results derived in SectionIII.2, we have

and

au+ au a?;c-----

at at at (111.221)

eu; auan an

where z"= ?;c = 1) / Co is called the contact electromotive force.Also, applying the concept of the solid angle it is not difficult to

calculate the field of the double layer at any point. This approach is widelyused in the theory of methods, based on a study of natural electric fields.

III.S The Work of Coulomb and Extraneous Forces,Joule's Law

As is well known the movement of charges in a conducting medium(current) is always accompanied by heat production. The relationshipbetween this heat and the fields E C

, Eext,j is discussed in this section. First,


we will suppose that the conductor is a quasi-linear circuit surrounded byfree space, as shown in Fig. III. 11d; later, a more general case will bestudied.

In the previous section we demonstrated that a constant current canexist in a conducting medium only if there is an extraneous force, whichcan have any origin except Coulombic. Induction or electromagnetic fields,for instance, are a very important class of extraneous forces widely used togenerate constant currents. Extraneous forces also arise due to chemicalor physical inhomogeneities in conductors. Such forces appear near con-tacts having different chemical contents (galvanic element, accumulator),or different temperatures (thermoelement), or in the presence of a con-centration gradient of electrolyte ions (concentration galvanic element).Mechanical forces can also provide transportation of charges (electrostaticgenerators). Inasmuch as all of these systems are vital for the existence ofconstant current, they are usually called current sources.

Now we will consider the work produced by a current source and thedistribution of its energy along a circuit. In general, a moving charge issubjected to the influence of both the Coulomb and extraneous forces(Ohm's law), and correspondingly they perform work, which can be easilycalculated. In fact, during the time interval dt the amount of chargepassing through every cross section of the circuit element ClZ is the same,and it is equal to

de = I dt (III.222)

where I is the current.In particular, the amount of charge that enters and leaves the element

C\2 is the same. It may be proper to notice that opposite charges with anequal magnitude pass through various sections, and their velocity isextremely small. Thus the elementary work of forces along the displace-ment dl' can be written

dA = I dt(E + E ext) • dl' (III.223)

because E = I dt(E + E ext) is the force acting on charge de. Respectively,

the work related to the translation of the charge between section S1 andSz during one second is

A= If\E + Eext) • dl'

1

where dl' is directed along the current line.

(III.224)

111.8 The Work of Coulomb and Extraneous Forces, Joule's Law 301

It is clear that A has the dimension of power, and in SI units it ismeasured in watts.

1 watt = 1 volt X 1 ampere

In accordance with the principle of conservation of energy this work has tobe transformed into some form of energy W, and the amount of thisenergy is the same as A.

W=Ij,\E+E eXI) 'd/

1(III.225)

At the same time experiments demonstrate that this energy appears asheat Q, within the element C12 , per unit time.

Q =1j,\E + E eXI) . d/

1(III.226)

In other words, the total work of Coulomb and extraneous forces, per-formed within some element of the conductor, is converted into heatwithin the same element. Equation (III.226) is in essence a formulation ofJoule's law, and it is valid provided that neither chemical reactions norconductor motion consume the energy of a current source.

In particular, the amount of heat that appears in the whole circuit is

since the circulation of the electric field vanishes, but

P=If?

(III.227)

(III.228)

is the work performed by the current source during one second.Thus, all energy of the source of the constant current in a conductor is

transformed into heat.In addition, let us make several comments.

1. The equality ¢E' d/= 0 means that the work done by the Coulombforce along the whole circuit is zero. This result can be expected, since inthe opposite case, due to a decrease of the energy of this field, it wouldnot be possible to preserve the time-invariant current.

2. In the external part of the circuit the extraneous force is absent andin accordance with Eq. (III.226) we have

(!II.229)


This shows that every second the work of the electric field is convertedinto heat, and this fact can serve as another argument illustrating thenecessity of a continuous extraneous force.

3. Joule's law, Eq, (111.226), can be described in terms of the current I,resistance R, and voltages of the electric and extraneous forces. In fact,making use of Ohm's law we have

(III.230)

where R 12 is the resistance of the conductor element en'If the extraneous force is absent within this element we have

(111.231)

That is, the amount of heat that appears every second is directly propor-tional to the resistance and square of the current magnitude. In particular,for the external part with resistance R; we obtain

Q =/(U -U )=J2Re + - e (111.232)

4. In the internal part of the circuit the amount of heat that appearsevery second is

or

Qi =J{+ (Eext - E) dt= te- I(U_- U+) (III.233)

since the fields Eext and E have opposite directions, but E'" > E.Equation (111.233) can be written as

(111.234)

It is clear that the first term on the right-hand side of the equationdescribes the work of the Coulomb electric field E related to the move-ment of charge Jdt per unit time along the external part of the circuit.

Ae=J(U+-U_) =IJ-E·d/+

In other words, this term characterizes the amount of current sourceenergy required to maintain the constant electric field E. Certainly, thecurrent source must perform at least this work, A e , but in reality it alsoproduces additional work, Ai' For instance, in an electrostatic generatorof the current a significant part of the work is related to heat losses due to

111.8 The Work of Coulomb and Extraneous Forces, Joule's Law 303

friction between its elements. This additional work A j can be described byintroducing the internal resistance R, of the current source, which absorbsthe heat energy so that

Also we can imagine a current source that in fact has an internalresistance; and then, in accordance with Joule's law, the amount of heatthat appears within the source is

5. Considering alternating fields we will demonstrate that some part ofthe current source energy travels through the surrounding medium aselectromagnetic energy, and within the external part of the circuit it isconverted into energy of the electric field. Correspondingly the transfor-mation of the source energy into heat can be approximately presented inthe following way:

Current sourceenergy, P

additionalwork: Ai = Qi

electromagneticenergy

1electric field

energy

6. In general, the total amount of heat that appears due to the constantcurrent does not coincide with Joule's heat Q. This is related with the factthat near a contact between different conductors Peltier heat arises; inaddition, Thomson heat appears in the presence of a temperature gradi-ent. However, unlike Joule's heat, these two forms of heat are linearfunctions of the current and usually their contribution is very small.


7. This analysis remains valid if instead of a quasi-linear conductor acurrent tube is considered. Let us take an elementary volume dV =(dS . dl), where dS is a section of the current tube.

Then, the amount of charge passing through this surface during thetime interval dt, is

de = (j . dS) dt

and the work performed by the Coulomb and extraneous forces can bepresented as

(j . dS){(E + E eX1) • dl'} dt

Therefore, the amount of heat that appears every second is

dQ = {j .(E + E eX1) } ( dS . dl')

= {j '(E+EeX1)}dV

At the same time, the work done by the extraneous force is

Thus, in general in a unit volume during one second two processesoccur, namely,

(a) The extraneous force performs work.(b) Heat appears, and its amount is given by

(III.235)

where p is the resistivity of the medium.

In particular, for an elementary volume of the external part of a currenttube we have

dP-=0dV '

dQ £2_ =j .E= _ =pj2dV p

(III.236)

1II.9 Determination of the Electric Fieldin a Conducting Medium

In the previous sections we have attempted to describe the most essentialfeatures of the electric and current density fields in a conducting medium,and in particular the following questions have been studied:

(a) The behavior of the electric field at regular points and near inter-faces of media with different resistivities.

IlI.9 Determination of the Electric Field in a Conducting Medium 305

(b) The distribution of charges in a conducting medium.(c) The current density field and its relation to the electric field, Ohm's

law.(d) The role of extraneous forces.(e) Electromotive force and resistance, and so on.

Now we will discuss basic questions related to solving the forwardproblem, namely how to determine the electric field in a conductingmedium. The importance of this subject is obvious since the calculation ofthe electric field is a cornerstone of the quantitative interpretation of allelectrical methods.

As was shown in Chapter II the solution of the forward problem of thegravitational field is based on direct use of Newton's law, Eq. (I1.5),

and therefore it consists only of numerical integration. The density of mass(5(q) is a physical parameter of the medium, and of course it is indepen-dent of the field g and the geometry of the model. In other words, as soonas a model of the medium is chosen, the sources of the gravitational fieldare simultaneously specified; that is, we know the sources before their fieldis calculated. This fact allows us to make use of Newton's law and todetermine the field g in the most simple way. In contrast, Coulomb's law

1 (5(q) 1 l(q)E(p) = --f --3- L q p dV + --I. -3-L q p dS

47Tco v L q p 47Tco S L q p

cannot be used to determine the electric field in a conducting mediumbecause the density of electric charges, unlike the density of mass, is not aphysical parameter of the medium that can be specified. Instead, itdepends on such factors as resistivity and electric field.

In fact, in accordance with Eqs. (III.152) and (III.154),

1(5 = -copE' grad -,

p

and in particular, this means that we can expect an infinite number ofcharge distributions given a single model of a conducting medium.

Thus, to calculate the electric field E, we have to know the density ofcharges (5 and l. These quantities, however, can be defined only if thefield is already known. So again, as in the cases of electrostatic inductionand the polarization of dielectrics, we are faced with "the closed circle"


problem, which does not allow us to use Coulomb's law for the fieldcalculation. This fact alone requires us to formulate boundary-value prob-lems and explains the fundamental difference in how we approach solu-tions of forward problems in the theory of the gravitational and electricalmethods. In electrical methods we are forced to solve boundary-valueproblems, and certainly this approach is much more complicated than itwould be if we knew the charge distribution and could directly applyCoulomb's law.

Before we formulate the boundary-value problems it is appropriate topresent the system of field equations for both fields E and j. Summarizingthe results derived in previous sections, we have [Eq, (III.237)]

Coulomb'slaw

Ohm'slaw


curlE = 0curl] = 'VI' X E

(A)

(B)E~2) =E[1)

j?) - j~l) = (1'2 - I'1)Et

sdivE = -

EO

divj = 0

(III.237)Let us make several comments concerning this system.

1. Since the dielectric constant E does not have any influence on theelectric field in a conducting medium, we assume that all parts of themedium have the same dielectric constant, equal to that of free space, andtherefore 0 and k are volume and surface densities of free charges only.

2. The system (III.237) describes both fields E and j at the external partof the current tubes, where extraneous forces E ex! are absent. This ishardly a shortcoming, since in electrical prospecting practice the internalpart of a circuit is not usually considered.

3. The system is written for two fields E and j, but the latter can bedetermined, provided that the electric field is already known. Moreover,

III.9 Determination of the Electric Field in a Conducting Medium 307

taking into account the fact that only the voltage of the electric field ismeasured in electrical methods, we will concentrate our attention entirelyon the field E. As follows from Eq. (III.237) and as was demonstratedearlier, the system of equations for the electric field is

curlE = 0s

divE =-EO

(III.238)

It is clear that this system is the same as that in free space. This is notsurprising since the origin of the constant electric field remains the sameregardless of the type of medium. However, there is one fundamentaldifference. Unlike free space, the charge density 0 and 1, cannot bespecified prior to calculation. Since 0 and 1, are unknown in the aboveequations, it is natural to replace them by two equivalent equations thatdescribe the principle of charge conservation, Eqs. (III.142), (III. 143).These equations also contain information about charges, though they arenot present in explicit form.

Indeed, the equation divj = 0 can be rewritten as

div j = div y E = y div E + E . grad y = 0

or

Egrad 'YdivE = ----

y

That is, we again derive the second equation of the system (III.238). Alsothe continuity of the normal component of the current density,

holds only if a certain amount of charge appears near this point; that is, itestablishes a relation between the field and charges, as well as theequation

1,E(2) - E(l) = -

n nEO

Then, after this replacement we obtain the system of field equations of theconstant electric field in a conducting medium, which is used to solve the


forward problem in the theory of electrical methods [Eq. (III.239)].

Coulomb's Ohm's Principle oflaw law charge conservation

t t (III.239)

I curlE = 0 II div yE = 0

£(2) _ £(1) = 0 It t

Let us make several comments here.

'V £(2) = 'V £(1) I12 n 11 n

1. This system describes the behavior of the electric field at the regularpoints of a conducting medium as well as at interfaces of media withdifferent resistivities.

2. The field equations (III.239) have been derived from three physicallaws, namely Coulomb's law, Ohm's law, and the principle of chargeconservation. For this reason they contain exactly the same information asthese laws.

If we imagine, for instance, that the first equation is invalid,

curlE "1= 0 or

then this would mean that the field E does not obey Coulomb's law, andthat the field is caused by vortices as well as sources.

If the field does not satisfy the second equation,

divj "1= 0,

this means that charges continuously accumulate and therefore the fieldcannot be time-invariant.

3. This system does not describe the field behavior due to a doublelayer. This special case was considered in detail earlier.

4. It is clear that any constant electric field, regardless of the distribu-tion of resistivity of the medium and the position of primary sources, is asolution of the system (III.239). This happens because every time-invariantelectric field must obey Coulomb's law, Ohm's law, and the principle of

111.9 Determination of the Electric Field in a Conducting Medium 309

charge conservation. Respectively, we can say that the origin of the systemis such that it has an unlimited number of solutions.

Now we will use the analogy between the field in free space and in thepresence of dielectrics, and introduce a system of equations describing thepotential U in a conducting medium.

First of all, from the first equation curl E = 0, it follows that

E = -gradU

Substituting this into the second equation divj = 0, we obtain for regularpoints of a medium

V(yVU)=O

Then making use of the equality

auE ---t- at

(III.240)

conditions for the electric field at interfaces can be represented as

and

where U1 and Uz are values of the potential at the back and front sides ofthe interface, respectively.

Inasmuch as both components of the electric field have finite values, thepotential U is a continuous function at an interface with some chargedensity 2,. In fact, from the equality

it follows that as the points p] and F:» which are located on oppositesides of the interface, approach each other, the potentials at these pointsbecome equal.

U] = Uz

It is clear that if the potential does not change through an interfacethen its tangential derivatives at both sides are equal to each other andtherefore the conditions

are equivalent.

andau] auz-=-at at

310 TIl Electric Fields

Summarizing these results we arrive at the following system of equa-tions describing the behavior of the potential in a conducting medium:

Coulomb'slaw

Ohm'slaw


(III.241)

aUI auz1'1 an = I'z an

As usual, let us make several comments.

1. Since the potential U is a scalar function, very often it is moreconvenient to solve the forward problem making use of Eq. (111.241) ratherthan Eq. (111.239).

2. In essence Eq, (111.240) is Poisson's equation. In fact, differentiatingwe obtain

or

and as was shown earlier, the right-hand side characterizes the density ofcharges at regular points.

3. In most practical cases in electrical prospecting and electric loggingwe can assume that the conducting medium is a piecewise uniform one,and that it is composed of different media each having a constant resistiv-ity. Because of these assumptions, the equation for the potential isdrastically simplified and we again arrive at Laplace's equation.

(III.242)

From the physical point this result is obvious since, in a piecewise uniformmedium, volume charges are absent.

4. As in the case of the system for the electric field, Eqs. (11.241) do nottake into account the behavior of the potential caused by a double layer.


Fig. 111.13 (a) Illustration for theorem of uniqueness; (b) the first type of medium model;(c) the second type of medium model; and (d) illustration in deriving the integral equation.

5. The potential U of any constant electric field is a solution of thesystem (III.241), and correspondingly the latter does not uniquely defineU. Moreover the equation E = - grad U determines the potential up to aconstant. Thus, the system (III.241) has an infinite number of solutions.

Our next step is to outline an approach that will allow us, in principle,to determine the constant electric field in any conducting medium. SinceCoulomb's law cannot be used for this purpose, it is natural to refer eitherto the system of field equations or that for the potential U. However, thesesystems alone do not uniquely define the field and therefore it is necessaryto bring additional information. As is known, this task is solved with thehelp of the theorem of uniqueness; the importance of this theorem isdiscussed in detail in Chapter I.

With some insignificant modifications that take into account an arbi-trary change of the medium conductivity, we will again describe thistheorem and formulate boundary-value problems. Suppose that the poten-tial U is considered in a volume V, which is surrounded by boundarysurfaces 51 and 52' Also there is an interface 512, where the conductivitycan have different values at the front and back sides (Fig. III.l3a).

Of course, the potential U within this volume is a solution of the system(III.24l). It is appropriate to notice that the surfaces 51 and 52 are usually


different surfaces. They can be a surface of the current electrode, or theearth's surface, or a spherical surface of infinitely large radius, etc.

To determine conditions at boundary surfaces that uniquely define thefield, let us assume that there are two different solutions of the system(III.24l): U(1) and U(2); that is,

and

and

V('Y VU(I)) = 0,

U( I ) = U(1)I 2'

U (2) = U(2)I 2'

V( 'Y VU(2») =°aup) aup)

'Y -- -'Y --I an - 2 an

auF) aUF)'Y I ---a;;- = 'Y2---a;;-

(111.243)

where 'YI' UI and 'Y2'U2 are the conductivity and potential at the back andfront sides of the interface S12' respectively.

We will consider the difference of these solutions.

U(3) = U(2) - U(1)

which in accordance with Eqs. (III.243) satisfies the following conditions:

and

U (3) - U(3)1-2

aup) aUp)'Y - - - 'Y - -

I an - 2 an (III.244)

Then we will introduce the vector X

and make use of Gauss' theorem

1 divX dV = ~ X . dS + ~ X . dS + ~ X . dSv* Sj Sz So

(III.245)

(111.246)

where So is a "safety" surface surrounding the interface S12' where thevector X is a discontinuous function, because the conductivities 'YI and 'Y2are not equal to each other. Correspondingly Gauss' theorem is applied tothe volume V*, which is confined by the surfaces Sl' S2' and So'

111.9 Determination of the Electric Field in a Condncting Medium 313

Substituting Eq. (III.245) into Eq, (I1I.246) and differentiating, weobtain

As So approaches S12 the integration over So is reduced to that over bothsides of the interface S12' Taking into account the fact that at the backand front sides of this surface

respectively, we have

and "2 = -no

and V* ~ V.Now making use of Eqs. (III.244), the equality (111.247) is strongly

simplified and we obtain

The latter relates values of the potential and the field on the boundarysurfaces S1 and 52 with those inside of the volume. Suppose that thesurface integrals in Eq. 011.249) vanish. Then the right-hand side is alsozero.

Inasmuch as the integrand y(VV(3))Z is positive (y > 0), we have toconclude that

grad V(3) = 0 (111.250)

That is, the function V3 does not change within the volume V.Bearing in mind the fact that V3 is the difference of two solutions of the

system (111.241) we can say that if the surface integrals of Eq, (III.249) areequal to zero, then two arbitrary solutions of the system (111.241), V1 and


U2 , can differ from each other by a constant. However, the field E isuniquely defined since

E = ~ grad U = - grade U + C)

Now we will formulate boundary conditions, such that the surfaceintegrals in Eq. (I1I.249) become equal to zero. At this point it is naturalto make use of results derived in Chapter I. Then we obtain three forms ofthe boundary conditions on the surfaces 51 and 52'

1. U= cp( q)

au2. an = t/J(q) (I1I.251)

au3. ¢.Y- as =1

5 an

and 5 is an equipotential surface.

Here cp(q), t/J(q), and 1 are given functions. It is essential that everyboundary condition of (III.251) together with the system (I1I.241) uniquelydefines the electric field in the volume V. This is the main result derivedfrom the theorem of uniqueness.

Correspondingly, we can formulate three boundary-value problems.

Dirichlet's Problem1.

( a) \7(y\7U) =0 at usual points

(b)aUI aU2

YI an = Y2 an on -812

(c)

Neumann's Problem2.

(a) \7(y\7U) =0 at usual points

(b) 1= U2 ,

aUI aU2

YI an = Y2 an

(c) on 52

3.


The Third Boundary Problem

(a) 'V( y'VU) = 0 at regular points

(b)au! aU2

1'1 an = 1'2 an

(c)au

~ y-d5 =/,51 an

au~ l'-d5=-/

52 an

where 51 and 52 are equipotential surfaces.

Now we will make several comments that perhaps will help to clarifythe role of the theorem of uniqueness.

1. The theorem of uniqueness does not provide an algorithm fordetermining the field, but instead it formulates conditions that uniquelydefine the field.

2. Every boundary problem consists of two parts, namely,(a) The system of equations for the potential (111.241).(b) One of the boundary conditions.3. All three boundary conditions always have clear physical meaning,

and it is usually very simple to formulate them.4. It is essential to understand that types of boundary conditions can

vary from point to point, and of course they can be different at differentboundary surfaces. At the same time it is necessary to note that at allpoints of boundary surfaces without exception one of these boundaryconditions must be specified.

5. In deriving the system of equations for the potential (11.241) ·andformulating the boundary conditions, we have assumed that the model ofthe conducting medium has one interface. However, it is obvious that ourresults remain valid in the general case, when there are several interfacesbetween media with different resistivities. In accordance with Eq. (III.24l),at every interface both the potential and the normal component of thecurrent density have to be continuous.

6. In spite of the fact that the theorem of uniqueness does not suggestan algorithm for calculating the field, it formulates the main steps ofsolution that have to be accomplished in order to find the electric field.

As examples of the process of formulating boundary-value problems, wewill consider two models of a conducting medium often used in the theoryof logging and electrical prospecting.


Example 1 The First Model (Fig. IlI.13b)

Suppose that a current / goes into a medium through the electrode A,located near the point O. The medium is everywhere uniform except atthe interface S12' Now let us start to extract information about thebehavior of the field that is essential to formulating a boundary-valueproblem. First, at regular points the potential U satisfies Laplace's equa-tion

since y is constant.Then it is obvious that in accordance with Coulomb's law the potential

caused by all surface charges tends to zero as the distance Lop from theelectrode increases unlimitedly.

U--70

In other words, we can say that on a spherical surface S2 with infinitelylarge radius, the potential U is equal to zero. In other words, we haveformulated the boundary condition at infinity. Concerning the behavior ofthe potential near the interface S12' it is clear that both the potential andthe normal component of the current density have to be continuous.

au! aU2

Yl an =Y2 an

The latter can be considered to be the surface analogy of Eq. (111.240).We have described the behavior of the potential everywhere except at

the electrode surface SI' which along with S2 confines the volume V.Since as the electrode conductivity is many orders of magnitude greater

than that of the surrounding medium, it is possible to treat S1 as anequipotential surface.

U( SI) = constant

Also it is natural to assume that the current I through the electrodesurface is known, and in accordance with Ohm's law it is related to thepotential U by

au~ y-dS =/

Sl an

where n is normal directed into the electrode.


Thus, the boundary-value problem in this case can be described by thepotential satisfying the following conditions:

1. At the equipotential boundary surface of the electrode

au~"h-dS=!

51 an (111.252)

2. At regular points of the medium the potential is a solution ofLaplace's equation

3. The potential and the normal component of the current density ateither side of the interface are related by

aUI aU2

'YI an = 'Yl an

4. At the boundary surface 82 , which has an infinitely large radius, thepotential tends to zero.

U-+O

As follows from the theorem of uniqueness, these four conditionsuniquely define the electric field, as well as the potential, and therefore wecan say that the boundary-value problem is well formulated.

At the same time, it is proper to notice that the boundary conditionnear the source can very often be simplified. Suppose that the currentelectrode A is a small sphere with radius a; then we can express thepotential U as a sum.

where Uo is the potential due to the charge on the electrode surface, whileUl is the potential caused by charges that appear at the interface Sl2'

Then making use of Eq. (1II.163) we have

PI!U= +Ul41TLop

where Pl is the resistivity of the medium that directly surrounds theelectrode. If we assume that the electrode radius is sufficiently small, thenin approaching its surface the potential Ul , caused by remote charges,tends to some finite value, while the potential Uo becomes very large.


Therefore, the boundary condition near the source can be rewritten as

PI!V -7 Va = --- if Lop> a

41T'Lop

Perhaps it is appropriate to make two comments.

(a) In the limit we can replace the surface charge by a point charge.This replacement does not change the field outside the electrode.

(b) From the physical point of view it is clear that this simplification isvalid even in cases where instead of a spherical electrode there is a smallelectrode with an arbitrary shape, and the surface SI is located at somedistance from this electrode.

Thus the boundary-value problem can be represented as

PI!1. VI -7 Va = --- as Lop -7 0

41T'Lop

2.

3.

4.

V2V = O

au; aU;+1Y;a;; = Yi+1 ----a;;- on S;

(III.253)

Here we have made one obvious generalization, by assuming that amedium contains several interfaces S;.

Certainly the boundary condition V -7 Va is much simpler thanEq, (III.252), but the latter is more general, and in particular it has to beused if the electrode is located sufficiently close to charges arising atinterfaces.

Example 2 The Second Model (Fig. III.13d

This model consists of an upper nonconducting half space and a piecewiseuniform conducting medium located beneath the earth's surface. Now wewill formulate the boundary problem for the conducting half space,surrounded by the electrode surface SI' and the boundary surface S2'which includes itself, the earth's surface So, and a half-spherical surface ofinfinitely large radius Ss'

S2 = So + S,

Taking into account the fact that the normal component of the currentdensity equals zero at the boundary with the nonconducting medium, it is

I1I.9 Determination of the Electric Field in a Conducting Medium 319

very simple to formulate the boundary condition at the earth's surface. Infact, according to Ohm's law, the normal component of the electric fieldalso vanishes at the conducting side of this surface.

Thus, the boundary-value problem is formulated in the following way:

1.

2.

\l 2 u = 0

au; aU;+1Yia;; = v. + I ----a;;- on s,

3. (III.254)

4.

5.

au-.-70an

U.-70

on So

Let us make two comments.

1. Because the normal component of the electric field is known at allpoints of the earth's surface (En == 0), we are able to determine the field ina conducting medium without considering free space.

2. In accordance with Eq. (III.174) the potential due to the charge ofthe current electrode located at the earth's surface is

PI!Uo = - - -

2'77" Lop

and correspondingly the boundary condition on the surface S I is

PI!U.-7 Uo= ---

2'77"Lop

After we have formulated boundary-value problems for the electricfield in a conducting medium, it is natural to take the next step andconsider methods to solve these problems. There are at least three suchmethods.

1. The method of separation of variables2. The method of finite differences3. The method of integral equations

320 m ElectrieFltl1d$

But their. study is thesUbjeet of applied mathematics, and far beyond thescope of tIns .monogra,t)h.

Neve~less It1t~(} next section we will demonstrate several times theappn~tlon ·.iii' thtm~thoo of separation of variables. The use of thismethod .. ,~er. is .limited to models of the medium with relativelysimple ~~ nfooterfaces between medi? w~th different con?uctivities.~*. tonslderationof the field beha.vlOr In such models IS of greatl?~ 'itrt~rest, and in fact these studle.s helped to develop the basic~,~f electrical methods. At the same time more complex models of a~tbJotlng medium require that we apply much more complicated ap-:proaches than the method of variable separation, such as the methods oflntegra!equation and finite differences. Both of these methods vividlydemonstrate the degree to which solution of the forward problem inelectrical methods is more complicated than in gravity methods.

For illustration we will derive here the integral equation with respect tothe potential of the electric field, and with this purpose in mind, we willconsider a two-layered medium with an arbitrary inhomogeneity within theupper layer, as is shown in Fig. III.13d. We will use the followingnotations: 11 and 12 are the conductivities of the upper and second layer,respectively; 'Yi and Sj are the conductivities and the surface of inhomo-geneity. So and SI are the earth's surface and the interface between layers,respectively.

To derive the integral equation we will proceed from Green's formula.

(au aG)f (G'V 2u - UV 2G) dV =,{,. G- - U- dS

v ~ an an(III.255)

where n is the normal directed outward from the surface S, whichsurrounds the conducting medium, and G is an arbitrary Green's function.At the same time U is the potential of the electric field and, in accordancewith Eqs. (111.254), satisfies the following conditions:

1. At regular points it is a solution of Laplace's equation,

2. Near the current electrode it tends to Uo,

where the electrode is located in the vicinity of the point O.

and


3. At infinity the potential tends to zero as if the source were a pointcharge,

cV~--~O

Lop

where C is some unknown constant.4. At the interface between the upper layer and the formation

eo, avzV, = Vz, /" an = /,z an

5. At the surface of the inhomogeneity Sj

eu, eu,Vj=Vj , /"an =/'ia;;

6. At the earth's surface the normal component of the current densityis zero; that is,

eo,-=0an

Now we will choose a function G such that the volume integral inEq. (111.255) vanishes, but the surface integral is reduced to that over thesurface inhomogeneity only. With this purpose in mind, let us suppose thatthe Green function G, which depends on two points q and p, is, up to aconstant, the potential at the point q, caused by a unit charge situated atthe point p in a two-layered medium when the inhomogeneity is absent.Correspondingly, the function G satisfies the following conditions:

1. Within the upper layer and the formation, except at the point p,

VZG =0

2. Near the point p, which is an observation point,

1G(q,p)~-

i:3. At the interface between the upper layer and the formation

aG j aGz/' 1 ---;;;; = /'z---;;;;

4. At the inhomogeneity surface

andaG j eo,--=-an an


5. At the earth's surface

aG j-=0an

6. At infinity function G decreases as

L ~ooqp

(111.256)

In applying Eq. 011.255) we have to take into account the fact that it isonly valid provided that the first and second derivatives of functions V andG exist. For this reason we will surround the current electrode and thepoint p by "safety" surfaces SA and Sp, respectively, and apply Green'sformula to each uniform part of the conducting medium.

Within the second layer we have

f [ avz aGz]G z- -Vz- dS=O

51 sn: an_

since VZG z = VZV = 0 and both functions G z and Vz decrease at least asfast as 1/L qp at infinity. Here n., is directed into the upper layer.

Applying Green's formula to the volume occupied by the inhomogene-ity, we have

(III.257)

because VZGi = VZVj = 0, and 11+ is directed outward from the volume.Finally, outside the inhomogeneity and within the upper layer we obtain

(III .258)

The first three integrals can be drastically simplified. We will consider thefirst integral where the integration is performed over the sphere aroundthe current electrode, and in the limit its radius tends to zero. Inapproaching the current electrode V, ~ p,I/41TR, where R is the radius

(III.261)

(III.260)

11l.9 Determination of the Electric Field in a Conducting Medium 323

of the spherical surface, we have

au] PII au] PIIaR - 41TR2 or an 41TR 2

since the directions of Rand n are opposite to each other.Taking into account the fact that the Green function and its derivative

have finite values in the vicinity of the electrode, and making use of themean value theorem, we obtain

~ [au ] aGI]

G --u-_· dSSA l an I an

= [G ~ - !!L aG ]41TR2I 47TR2 41TR an

=PIIGI(O,p) as R-40 (III.259)

It is easy to see that the right-hand side of Eq. (III.259) equals 41TU*,where U* is the potential at the point p in the horizontal layered mediumwhen the inhomogeneity is absent; that is,

~ [aUI aGI]

G I - - U]- dS = 47TU*(p)SA an an

The second integral equals zero, because both derivativesaGjan vanish at the earth's surface; that is,

~ [aUI aGI]

GI--UI- dS=OSo an an

Finally, we consider the third integral around the observation point p,where function G tends to infinity at a rate proportional to IjR inapproaching point p; that is,

1 aGI 1G -4 - and -- = -

I R an R 2

since nand R have opposite directions. Applying again the mean valuetheorem, we obtain

~ [aUI aG]]

G --U- dSS I an 1 an

p

[1 aUI 1 ]

= -- - U - 41TR 2R an I R 2

as R -4 ° (III.262)


where UI(p) is the potential of the electric field in the presence of theinhomogeneity.

Thus, instead of Eq. (III.258) we can write

47TB* (p) - 47TUI( p) +~ [GI

aUI _ UIaG I ] d5

~i im : n :

+f [G I aUt - UI aGj

] d5 = 0 (III.263)51 Bn; Bn;

Now multiplying Eqs. (III.256) and (III.263) by 'Yz and 1'1' respectively,and adding them we have

(III.264)

Taking into account the fact that at the interface 51

eo, eu,1'1 an = 'Yz an

and

aG I aG z1'1--;;;; = l'z--;;;;

but since n l and n z have opposite directions, the sum of the last twointegrals in Eq. 011.264) equals zero, and therefore

47T'YP*(p) - 47T'YPt(p)

Suppose the point p approaches the surface 5 i . Then applying themethod used for the study of the double-layer field we have


Whence

(III.265)

Next, considering the point p inside the inhomogeneity we can obtain

In the limit, when the point p approaches Sj we have

(III.266)

Adding Eqs. (III.265), (III.266) we obtain

47TYP*(p) - 27T{YP\(P) + YP2(P)}

Since

andau! aU2

Y\ an =Yi an

while

we finally obtain

u(p) = ~u*(p)Yj +Y1

(111.267)

Equation 011.267) is an integral equation with respect to the potential U,since both points p and q are located on the surface Sj'



1. The potential U*(p) in a layered medium is assumed to be known,and it can easily be derived by applying the method of separation ofvariables.

2. The integral equation (111.267) remains valid if the inhomogeneity issituated in a horizontally layered medium with n layers. In this case, thefunction G has to describe the potential in this layered medium.

3. In particular, an inhomogeneity can be some structure at one of theinterfaces of the layered medium.

4. Considering electrostatic induction, we demonstrated that the inte-gral equation with respect to charges can be treated as a system of linearequations. Of course, this is also true for Eq. (111.267).

5. As soon as the potential U is known at the surface Sj, it can becalculated at any point outside the inhomogeneity.

6. The integration on the right-hand side of Eq. (III.267) is performedat the surface Sj, except the point p where the potential U(p) is calcu-lated.

III.10 Behavior of the Electric Field in aConducting Medium

Now we will consider several examples that illustrate field behavior indifferent models of a conducting medium. They are chosen in a way todemonstrate the application of electrical methods in different areasof geophysics. Also this study will include a discussion of the resolution ofelectrical methods, their depth of investigation, the influence of geologicalnoise, etc.

Example 1 Influence of an Inhomogeneity on theElectric Field

Suppose that an inhomogeneity with conductivity Yj is surrounded by auniform medium having conductivity Ye (Fig. III.l4a), and Eo describes thefield behavior in the absence of the inhomogeneity. This field is oftencalled the primary field. Due to this field, charges arise at the surface ofthe inhomogeneity, and in accordance with Eq. (III.154) their density isdefined by

(III.268)

III.10 Behavior of the Electric Field in a Conducting Medium 327

Fig. III.14 (a) Behavior of the field if Yi> Ye; (b) behavior of the field if Yi < Ye; (c) electricand current density lines at an interface; and (d) field behavior in the presence of a resistiveinhomogeneity.

where Pi and Pe are the resistivities of the inhomogeneity and thesurrounding medium, respectively, while E~v is the average value ofthe normal component, and the normal n is directed toward the inside ofthe inhomogeneity.

Proceeding from the principle of charge conservation, we have toconclude that the magnitude and sign of ~ vary in such a way that thetotal amount of surface charge equals zero. In most practical cases thismeans that the positive charges arise at one side of the inhomogeneitysurface, while negative ones appear at the opposite side. Therefore, thereis a closed line of points along which the density ~ equals zero. Of coursein general the distribution of charges is much more complicated. Thesecharges are sources of the secondary field Es ' and, correspondingly, thetotal field outside and inside of the inhomogeneity is

(III.269)


Here, it is proper to make several comments.

1. In accordance with Coulomb's law, the secondary field E, is relatedto the charge density 4 by

Es{p) = _1_~ 4{q~Lqp dS47TEO s L qp

where 4(q) is unknown. At the same time this equation allows us to derivesome useful features of the behavior of the field.

2. In general both fields, Eo and E, have different magnitudes anddirections.

3. It is obvious that only the secondary field contains information aboutthe resistivity, shape, dimensions, and location of the inhomogeneity. Allof these quantities, as well as Pe' are usually called geoelectric parameters.

4. The potential and current density fields can be represented as thesum of two fields,

while

for points inside and outside the inhomogeneity, respectively.

Let us assume that a body with a relatively simple shape is moreconductive than the surrounding medium ('Yi > 'Ye), and the primary elec-tric field is directed from the back to the front side of the inhomogeneity(Fig. III. 14a). Then in accordance with Eq, (IIL268) negative chargesappear at the back side, while positive ones arise at the front side. In fact,the normal component E~v is positive at the back side, but the contrastcoefficient

Pi -Pe

Pi +Pe

is negative, and correspondingly 4 < O. In contrast, at the front sidepositive charges arise, since E~v < O.

Now we will consider the behavior of the component of the total field,which is directed along the primary field. It is convenient to present thesecondary field as a sum,

E, = E:+ E;

where E: and E; are the fields caused by the positive and negativecharges, respectively. First we will take some point PI' located outside andin front of the inhomogeneity. In this case the field E: produces a positive

111.10 Behavior of the Electric Field in a Conducting Medium 329

component along the field Eo, while the field of the negative charges, E;,has the opposite direction. Taking into account the fact that the amount ofpositive and negative charge is the same, but the latter is located atgreater distances, we can say that along the primary field

IE,"o1 > IE;:oIand therefore, due to the presence of the inhomogeneity, the field in-creases at the point PI; that is,

(1II.270)

where E,o is the vector component of the secondary field in the directionof the primary field.

Next consider the field behavior at the point Pz, also located outside ofthe body and opposite the point PI' In this case the field of negativecharges, E;, is directed toward the body, opposite to the field of positivecharges. Since the negative charges are located closer, we can againconclude that in this area the field increases in the direction of theprimary field.

Now we will take the point P3' also located in the surrounding mediumbut near the equatorial plane of this model. As is seen from Fig. III.14aboth fields E: and E; produce vector components that are opposite tothe primary field; that is

(1II.271)

Summarizing these results we can distinguish three areas, namely, in frontof and behind the conductive inhomogeneity where the field increases inthe direction of the primary field, and the area around the equatorialplane where the field becomes smaller. At these boundaries the compo-nent of the secondary field along the primary field equals zero.

Now we will consider the behavior of the field inside the inhomogeneityif 'Yi > 'Ye. Inasmuch as the positive and negative charges are located at thefront and back sides of the inhomogeneity, their field produces a compo-nent opposite to the primary field. Correspondingly, the total field in thedirection of the primary field decreases. With an increase in the ratio ofconductivities 'YJ'Ye' the secondary field inside the inhomogeneity alsoincreases and in the limit becomes equal in magnitude to the primary field.Therefore, in this case of the ideal conductor the field within the inhomo-geneity vanishes. As follows from this analysis, with an increase of conduc-tivity 'Yi' the voltage between two arbitrary points of the inhomogeneitydecreases. In other words the better the conductor the smaller the electricfield is inside it, and to some extent this linkage can serve as a characteris-tic of a conductor.


Next we will study the behavior of the potential of the secondary field~. From the equation

1 ,(..~(q)dSUs(p) = -'Y"

4r.Eo s L qp

it follows that the potential is positive if the point p is located near thefront side of the inhomogeneity, and it is negative if the observation pointsare near the back side. .

Correspondingly, there is a surface passing through the inhomogeneitywhere the potential Us equals zero. In the case of the ideal conductor, thepotential of the total field remains constant at all points of this body, evenif the potential of the primary field changes.

Finally, let us discuss the current density field. Outside the inhomo-geneity the behavior of the current field is similar to that of the electricfield, since j = 'YeE. Inside the body the current density increases in spiteof a decrease of the component of the electric field in the direction of Eo.Since the amount of charge passing through any elementary surfacecannot be infinitely large, with an increase of the conductivity 'Yi themagnitude of the current density tends to some finite limit that dependson the shape and size of the inhomogeneity. The vector j is directed alongthe primary field Eo.

Until now we have studied the behavior of the field in the presence of amore conductive inhomogeneity. Next suppose that the body is moreresistive, 'Yi < 'Ye • As follows from Eq, (III.268), positive and negativecharges arise at the back and front sides of the body, respectively(Fig. III.14b), and therefore the distribution of the field is opposite to thatconsidered in the previous case. For instance, the field increases inside theinhomogeneity as well as in the area around the equatorial plane, while itdecreases in front of and behind the body. With an increase of theresistivity, Pi' the electric field inside gradually increases and in thelimiting case of an insulator it reaches some finite value, which depends onthe shape and size of the inhomogeneity. As concerns the potential Us, itsbehavior is similar to that in the previous case, but the positions of zoneswith the positive and negative values U, change. It is obvious that thebehavior of the current density outside the inhomogeneity is similar to thatof the electric field, but inside the current density tends to zero as theresistivity increases.

In addition let us consider the behavior of the vector lines of both fieldsE and j near the inhomogeneity surface. As follows from Eqs, OII.237),

and j~ =j~ (III.272)

III.lO Behavior of the Electric Field in a Conducting Medium 331

where the indices "i" and "e" correspond to points of the inhomogeneityand the surrounding medium, respectively. Then making use of Ohm's lawwe also have

and (III.273)

Taking into account the continuity of the tangential component of theelectric field and the normal component of the current density,Eq. (III.272), we obtain

-j .eit it

Yij~ Yej~and (III.274)

As is seen from Fig. III.l4c the direction of the field near the interface canbe characterized by the angle Q', formed by the field E with the normal n.It is clear that

Then, Eqs. 011.274) can be rewritten as

'Y; 'Ye

tan Q'; tan Q'eor

tan Q'j 'Y;

tan Q'e 'Ye

(III.275)

Of course, the same result is obtained if we proceed from the equation

In accordance with Eqs. (III.275) the vector lines of the fields E and jrefract near the boundary, and this occurs in such a way that at either sidethe value of tan Q' is proportional to the conductivity of the medium.These vector lines approach the normal in the medium with higherresistivity as if they were trying to reduce their path length in this medium.

Also it is clear that if one of the angles equals 0 or 1T/2, then the otherangle has the same value, provided that both conductivities have finitevalues. In other words, in these cases vector lines are not refracted,although one of the field components has a discontinuity.

This study shows that the vector lines of the field j concentrate inside ofthe inhomogeneity as well as in front and behind it, if 'Yi > 'Ye . This is

332 ID Electric Fields

accompanied by rarefaction of these lines near the lateral part of the bodysurface. In the opposite case of a more resistive inhomogeneity, the vectorlines of j are more rarefied in front of and behind the body as well asinside it. At the same time they are concentrated outside near the lateralsurface. This behavior of current lines creates the impression that currenttends to concentrate in the more conductive medium.

If the inhomogeneity is an ideal conductor, then its surface becomes anequipotential surface and therefore the tangential component of the fieldE" equals zero. This means that the vector lines of both fields E and j areperpendicular to the interface. As concerns the behavior of the field insidethe body the electric field vanishes, since Pi = 0, but the current densitylines pass through continuously. In the opposite case, when the body is aninsulator, 'Yi = 0, the normal component of the current density equals zero,

j~ = 0, near the interface and therefore E~ also vanishes. Correspondinglythe vector lines of both fields are tangential to the interface of theinsulator.

Let us consider one more feature of the behavior of the field related tothe influence of an arbitrary inhomogeneity. Inasmuch as the sum ofinduced charges on the inhomogeneity surface equals zero, their field atdistances essentially exceeding dimensions of the body approaches that ofan electric dipole and, in accordance with Eq, (111.59), we have for thepotential U.

(III.276)L ---+00qp

1 M' L q pU ---+ ------;;-~

5 47TE O L~p

where L q p is the distance between the observation point and any pointwithin the inhomogeneity; M is the dipole moment, which is proportionalto the amount of charges having one sign, and it depends on the primaryfield Eo as well as the shape and dimensions of the inhomogeneity. Ingeneral the field inside the body is not uniform, but there is an exception:When the inhomogeneity has a relatively simple shape and the primaryfield Eo is uniform within its vicinity. In this case the dipole moment M isproportional to the field Eo, and it has either the same direction asr. > 'Ye , or the opposite one if 'Yi < 'Ye .

It is also obvious that Eq. 011.276) can be useful in evaluating thesecondary field when observation points are located at distances suffi-ciently exceeding inhomogeneity dimensions. In contrast, often lateralchanges of resistivity occur near the earth's surface, and in such casesmeasurements are usually performed outside and inside the inhomogene-ity. For illustration the behavior of the potential and the tangentialcomponent of the electric field, caused by surface charges, is shown inFig. III. 14d.

III.IO Behavior of the Electric Field in a Conducting Medium 333

Unlike the potential, the normal component of the electric field is adiscontinuous function at the surface of the inhomogeneity, and thisdiscontinuity is defined from Eq. OII.272).

j~ = j~ or

Such inhomogeneities can be objects of investigation when electric meth-ods are used for mapping; but at the same time they can present geologi-cal noise, which is often a very serious obstacle for the application ofelectrical prospecting.

Until now we have considered the influence of an arbitrary inhomo-geneity on the electric and current density fields. Now let us briefly discussa case when the electric field is not disturbed by the presence of lateralresistivity changes.

As follows from Eq. (111.268), charges are absent on the inhomogeneitysurface if at every point the normal component of the primary electricfield EnO equals zero. In fact, in such cases, continuity of the normalcomponent of the current density is automatically satisfied without accu-mulation of surface charges. For instance, suppose that the primary fieldEo is directed along the strike of a two-dimensional dike, shown inFig. III.15a. Then the density of surface charges I is equal to zero, andtherefore, the secondary field E, is absent. This means that everywherethe electric field coincides with the primary one and in this case it isimpossible to discover the dike. At the same time the current density canbe extremely large within the conducting body, since

if r.> Ye

Of course, if there is a component of the primary field perpendicular tothe dike surface, then charges appear and they create a secondary field,which contains information about the geoelectric parameters of the body.

Example 2 Distribution of Charges in aLayered Medium

Suppose that a medium is a system of uniform regions with differentresistivities and the current electrode A is located in its internal partenclosed by surfaces S;, as is shown in Fig. III.15b. Let us calculate theamount of charge arising at each interface. First of all, the electrode


Fig. 111.15 (a) Electric field along a two-dimensional model; (b) the current electrode insideof a piecewise uniform medium; (c) model of a layered medium; and (d) a sphere in auniform field.

charge eA equals

(III.277)

where PI is the resistivity of the medium that surrounds the electrode,whose resistivity is neglected.

In accordance with Eq, (111.19), the density of charge at the interface S,between media with resistivities Pi and Pi+ I is

I =E (E i + l -E i )o n n

or making use of Ohm's law and the principle of charge conservation wehave

(111.278)

where q is an arbitrary point of the surface Si'Taking into account the fact that the surfaces S, are closed around the

electrode, the same amount of charge I passes through every one of them.


Correspondingly, the total charge on the surface S, can be calculated as

or

(III.279)

Certainly, this is a very simple expression that shows that the amount ofcharge distributed on the interface S, is directly proportional to thedifference of resistivities and the current.

Before we continue, it is appropriate to make several comments.

1. Equations (111.278), (111.279) do not apply at the interface with aninsulator since Pi+I ~ 00.

2. The sign of the charge e i is defined by the resistivity difference. It ispositive if Pi+ I > Pi' but negative if Pi+ I < Pi' At the same time the sign ofthe density I can vary from point to point.

3. Equation 011.279) determines the total amount of charge, but itsdistribution still remains unknown.

Now we will write down expressions for the charges arising at eachsurface Si' including the current electrode. Then we have

eA = lOoPI]

e l = loo(pz - PI)!

ez = lOO(P3 - pz)!

e3 = lOO(P4 - P3)]

eN = lOO(PN - PN-I)]

where PN is the resistivity of the external region.Performing a summation over all the charges that appear in the

medium, we obtainN

e = Lei = lOOPN]i~ 1

(111.280)

This is an interesting result which demonstrates that the total charge

(III.281)


arising at all interfaces coincides with the charge on the electrode surface,as it if were located in the external uniform medium with resistivity PN' Asin the case of gravitational masses, any distribution of the volume orsurface charges confined within some volume V creates practically thesame field as that of a point charge, if the observation points are locatedfar away and e * O. Therefore, the asymptotic behavior of the potential ofthe total field in the medium, shown in Fig. IIUSb, is

PN1U(p)~-

41T"Lqp

This result is of great practical importance since it shows that if thecurrent and receiver electrodes are located on the surface of the layeredmedium (Fig. III.ISc), and the distance between electrodes increases, thedepth of investigation of such an array also increases. In other words, thevoltage, measured between receiver electrodes located in the medium withresistivity PI' approaches that of a uniform half-space with resistivity PN'In essence, Eq. (III.281) explains one of the most important features ofgeometrical soundings used in electrical prospecting as well as in logging.

In addition let us notice the following:

1. Strictly speaking each region of the layered medium is not enclosedby its two interfaces (Fig. III.1Sc). However, our results remain valid, sincewe can mentally imagine two surfaces at infinity, which make every layerclosed, but the amount of charge passing through them equals zero.

2. Charges induced at each interface of the layered medium are notconfined with an area of finite dimensions, but are spread over the entireinterface. However, as was shown in Section III.S, their density ~ de-creases relatively quickly, and due to this fact the field of these chargestends to that of a point source as the distance from the current electrodebecomes sufficiently great.

3. Let us note that the amount of charge in the medium does notchange if there is some region surrounded by a surface S* (Fig. III.1Sb)and located outside of the current electrode. In fact, the total chargearising on this surface is

and in accordance with the principle of charge conservation, the integralon the right-hand side equals zero. Here Po is the resistivity of this region.This means that an equal amount of positive and negative charge appearson the surface S*, and correspondingly, with an increase of distance their


field tends to that of an electric dipole. In other words, in the presence ofsuch an inhomogeneity the asymptotic behavior of the field is still de-scribed by Eq. (III.28l).

Example 3 A Conducting Sphere in a UniformElectric Field (Fig. m.isn

Suppose a sphere with radius a and conductivity 1'z is situated in auniform electric field Eo. The surrounding medium has conductivity 1'1'Before we discuss the solution of the boundary-value problem, let usdescribe several obvious features of the primary and secondary fields. Firstof all we have assumed that the primary field is uniform. Certainly such afield cannot exist everywhere, since that would require an infinite powersource. Nevertheless this approximation is very useful, if we restrictourselves to a study of the field within some finite region around thesphere and assume that sources of the primary field are located at greatdistances from observation points.

Inasmuch as the field Eo intersects the surface of the sphere, bothpositive and negative charges arise and their sum equals zero. Thesecharges are sources of the secondary field and due to their presence weare able in principle to detect a conductor. It is natural to assume adistribution of charges possessing axial symmetry with respect to the axis,which is directed along the primary field and passes through the spherecenter. For this reason we can expect that the potential, as well as boththe fields E and j, have the same symmetry. Of course it is an assumptiononly, but very soon with the help of the theorem of uniqueness we willprove that this assumption, as well as others, is correct.

Assuming the axial symmetry of the field, we will introduce a sphericalsystem of coordinates R, 8, ep with its origin located at the center of thesphere and z-axis directed along the primary field. Then in accordancewith Eqs. (III.253) the boundary-value problem is formulated in thefollowing way:

1. at regular points

2.aU1 auz

1'1 aR = 'Yz aR if R=a (III.282)

3. as R -) 00

where Uo is the potential of the primary field.

338 m Electric Fields

(III.283)

The boundary condition at infinity is obvious since the secondary fieldof induced charges decreases with an increase of the distance from thesphere.

Taking into account the relative simplicity of the problem let us attemptto find a solution proceeding from our understanding of the behavior ofthe field. The potential of the primary field can be determined easily sincethe field Eo has only a component along the z-axis and therefore

su;E =--

o azWhence

Inasmuch as the constant is not essential for determination of the field, wewill put it equal zero. In other words, it is assumed that the potential ofthe primary field vanishes in the plane z = O. Then we have

(III.284)

Now let us consider the secondary field caused by surface charges. Aswe know the total charge equals zero and correspondingly far away fromthe sphere their field is equivalent to that of an electric dipole. Now wewill assume that this behavior is observed everywhere outside of thesphere, regardless of the distance from the origin. Also let us suppose thatan unknown dipole moment M is directed along the field Eo. Then thepotential of the secondary field outside of the conductor is

M·RU = ------.,,-

s 47TSoR 3 (111.285)

(III.286)if n».«

Thus, the potential outside of the sphere is

McoseUI = ~EoR cos e+ 2

47TSoR

Finally, we will assume that charges are distributed in such a way thatinside the sphere they create a uniform field, directed along the z-axis.Then the total field within the conductor can be represented as

and

if n s. a (III.287)

where C is also an unknown constant.

rn.lO Behavior of the Electric Field in a Conducting Medium 339

With the help of several assumptions we have described a potential byEqs. (III.286) and (III.287). Now it is time to check whether all of theseassumptions are correct or wrong and, if they are valid, to determine theunknowns M and C. Of course, this task will be performed with the helpof the theorem of uniqueness, and this means that using Eqs. (II1.286),(III.287) we will attempt to satisfy conditions of the boundary-valueproblem, Eq. (111.282). First, it is clear that the potentials of all electricfields caused by electric charges and considered outside of these charges,satisfy Laplace's equation; and, in particular, the potential of the uniformfield and that of the electric dipole are its solutions.

This fact can also be proved by substituting these functions intoLaplace's equation. As follows from Eq. 011.286), the condition at infinityis satisfied too. Now let us find out whether we can provide continuity ofthe potential and the radial component of the current density at thesphere surface, using Eqs. (III.286), (III.287). Then we have for thepotential,

and for the normal component of the current density,

Yl{ - Eo - 2M 3 }cos 8 = -YzEoC cos 847TEoa

Thus, we have obtained two equations with two unknowns.

(III.288)

(111.289)

As is well known, a linear system of two equations with two unknownsdoes not always have a solution, and if the system (111.288) cannot besolved, this means that our assumptions or part of them were incorrect.

However, Eqs. 011.288) have a solution, and we obtain

Yz - Yl 3M=47TEo a Eo

Yz + 2Yl

3YlC=---

Yz + 2Yl


Thus instead of Eqs. (III.286), 011.287) we have

and

if R ~a

(III.290)

if R < a

Then in accordance with the theorem of uniqueness we can conclude thatEqs, (111.290) describe the potential of the electric field caused by chargeson the surface of the sphere, if the primary field is uniform. In fact, thefunctions UI and U2 satisfy all three conditions of the boundary-valueproblem, Eq. 011.282). Moreover, as follows from this theorem, there is noother solution except that given by Eqs. (111.290).

This example vividly illustrates that regardless of the approach used tofind a field, as well as the assumptions which have been made, only thetheorem of uniqueness decides whether this function is a solution of thegiven problem.

Now we will describe some features of the secondary field inside andoutside of the sphere. First, consider the distribution of surface charges.Making use of Eq. (III.19) we have

(III.291)

since nand R are directed inside and outside of the sphere, respectively.Taking derivatives of the potential with respect to R we obtain

aUI Y2 - YI a3

aR= -Eocos8-2 -3Eo cos 8

Y2+ 2YI R

and

Since

auE =--

R aR


we have

or

( III.292)

Thus, the density of charges I is distributed symmetrically with respect tothe z-axis, and it decreases as cos 8 with an increase of the angle 8. Themaximal magnitude of charge is observed at () = 0, where the primary fieldis perpendicular to the surface, and it is equal to zero in the equatorialplane, 8 = 'TT12. The dependence of the charge density on conductivities isdefined by the coefficient K iz-

(III.293)

(III.294)

which varies between -1/2 and 1, as the ratio YZ/'Yl varies from °to 00.

lt is clear that the same relatively weak dependence on conductivity isinherent for the electric field, too. This means that the resolving capabili-ties of electrical methods, applied in mining prospecting, are usually verypoor. As follows from Eq. (III.293), even for relatively small values ofYz/Yl the coefficient K 12 is almost equal to unity. That is, conductorshaving completely different conductivities can create practically the samefield. In other words, the high sensitivity of electrical methods allows us onthe one hand to detect relatively small changes of resistivity, and on theother hand it does not permit us to determine the resistivity of theseconductors.

Now let us consider the behavior of both fields E and j inside of thesphere. In accordance with Eq. (III.290), the electric field E z is uniformand directed, as is the primary one, along the z-axis.

auzE = -- orz aR

3Yl 3Ez = Eo = Eo

Yz+2YI 2+(Yz/Yl)

Thus, with an increase of the conductivity of the sphere (yz > Y1)' the fielddecreases and in the case of the ideal conductor tends to zero, while the


current density

increases and approaches its limit, which is independent of the conductiv-ity of the sphere and is equal to

jz = 3y1Eo

In the opposite case of a more resistive sphere, the electric field Ezbecomes greater with an increase of resistivity pz; but this change isrelatively small. In fact, for an insulating sphere we have

£z = I.5Eo

In conclusion it is appropriate to notice that this analysis of the fieldbehavior is useful for understanding different aspects of electrical methodsin mining and engineering geophysics and also for studying the influenceof geological noise in electromagnetic methods such as magnetotelluricsoundings.

Example 4 Elliptical Cylinder in a UniformElectric Field

Now we will study a field in a slightly more complicated case. Consider anelliptical cylinder located in a uniform medium and a primary electric fieldEo, which is uniform and perpendicular to the cylinder axis (Fig. III.16a).In solving the boundary-value problem, Eq. (III.253), we have to providecontinuity of the potential and the normal component of the currentdensity at the cylinder surface. To simplify this procedure we will intro-duce an elliptical system of coordinates g, TJ.

y = a cosh gcos TJ, Z = a sinh gsin TJ (III.295)

where a = {a Z - b Z) l / Z is the eccentricity of the cylinder, and a and barethe major and minor semiaxes of the cylinder, respectively. This system isdefined by two families of elliptical and hyperbolic cylinders that areorthogonal to each other and have the same focus at the points

y=±a, z=O

Both coordinates change in the following way:

os g< 00, 0 ~ TJ < 27T

and the coordinate TJ is measured from the y-axis.

III.I0 Behavior of the Electric Field in a Conducting Medium 343

Fig. III.16 (a) Elliptical cylinder in a uniform field; (b) current electrode at the boreholeaxis; (c) behavior of Bessel functions; and (d) behavior of the integrand in Eq. 011.336).

It is essential to note that the cylinder surface coincides with one of thecoordinate surfaces, g= go. In this system the metric coefficients are

2 1/2hl=h2=a(cosh g-COS2 1) )

and correspondingly Laplace's equation is written in the very simple form

(III.296)

In the same manner as in the previous case the secondary field arisesdue to induced charges that appear on the cylinder surface, and corre-spondingly we represent the potential of the total field in the form

(III.297)

Inasmuch as the primary electric field is directed along axis y, its potential


Ua is

Ua= -Eay = -Eaa cosh gcos 7]

or

(III.298)

Substituting Eq. (III.298) into Eq. (III.296) it is easy to see that functionsof the type

or

satisfy Laplace's equation.Now we will make two assumptions about the field behavior, namely,

(a) Inside the cylinder the field remains uniform, and it is still directedalong the y-axis,

(b) Outside the cylinder the secondary field decays with an increase ofthe distance from the cylinder as a function e-e and depends on the angle7] in the same manner as the primary field.

Correspondingly, we can write

(III.299)

Uz = -aEaB cosh gcos 7]

Here A and B are unknowns that are independent of the coordinates gand 7].

It is obvious that both functions U1 and Uz satisfy Laplace's equation,and U1 obeys the boundary condition at infinity.

as

From the two conditions at the interface where g= ga,

aU1 eo,1'1 ag = I'z ag

we obtain a system of equations determining the unknown coefficients A

lII.lO Behavior of the Electric Field in a Conducting Medium 345

and B. Solving this system we have

A=

(1 - ~ ) sinh goe~o'Y2

1 + - tanh go'YI (III.300)

1 + tanh goB = -~'Y""'2--

1 + - tanh go'YI

Therefore, the functions UI and U2 , given by Eqs. (III.299), (111.300),satisfy all the requirements of the boundary-value problem, and corre-spondingly they describe the potential of the electric field.

For the components of the field E we have

1 auE =---

'7 h2 aTJ

or

(III.301)

To describe some features of the behavior of the field, we will take intoaccount the fact that

btanh go =-,

a

1 + b/a

1 -b/a

First let us consider the distribution of charge as a function of theparameters of the elliptical cylinder and conductivity of the surrounding


medium. As was shown earlier, the surface density I is defined by thediscontinuity of the normal component of the electric field.

and in accordance with Eqs. (111.301) we have

( ~: - 1) ego tanh go cos 7]

(III.302)

The latter shows that the charge density has finite values at all pointson the cylinder surface and at 7] = ±1T/2 it is zero; but it increases towardpoints 7] = 0, 1T, where under the conditions

1'2 b b--«1'-«11'1 a a

it is

Earlier it was shown that the relationship between surface chargedensity and field strength can be written as

where E~ and E~-q are the normal components of the primary field andthe field of the surface charges, respectively, except for the charge locatedat the point q. As follows from Eq. (111.301), the components E~ andE~-q have the same direction if 1'2> 1'1' and they are opposite to eachother if 1'2 < 1'1' For this reason the charge density and the secondary fieldin the presence of a highly conductive cylinder, if (a> b), are greater thanin the case of the highly resistive inhomogeneity.

Proceeding from Eqs. (III.301) we will briefly describe the field behav-ior outside the cylinder. First of all it is clear that if the cylinder is moreresistive than the surrounding medium 1'2 < 1'1' the electric field dependsweakly on the conductivities and is mainly defined by the value of b/ a.With an increase of b/ a the electric field ETJ increases as well, becausethe observation point becomes closer to the charges. When the cylinderis characterized by higher conductivity than the surrounding medium

1II.10 Behavior of the Electric Field in a Conducting Medium 347

(1'2/1'1» 1), and the ratio of axes is not small, or more precisely when

1'2 b--» 11'1 a

the secondary field is controlled by the geometric parameters only. How-ever, with a sufficiently elongated cylinder in the direction of the field Eo,

1'2 b-- «11'1 a '

b-« 1a

(III.303)

the field of charges is relatively small, E~ «Eo, but it depends on theconductivity of the cylinder in both cases, whether 1'2/1'1 > 1 or 1'2/1'1 < 1.

In essence, inequalities (111.303) are conditions, as the influence of anelongated cylinder on the electric field practically vanishes, and it behavesas an infinite layer with thickness 2b.

It is interesting also to notice that the ratio of the field magnitudescontributed by charges on the surface of an ideally conducting cylinderand those on an insulating one is equal to the ratio of semiaxes of thecylinder, alb.

Finally, consider the field inside of the elliptical cylinder, which can berewritten as

1 +blaE - E

2 - 1 + ('Y2bl'Y1 a ) °or

where

b roL=---

1 +bla

if a rb > 1 (III.304)

is often called the depolarization factor, since it characterizes the fielddecrease inside of the conductor as 1'2/1'1 > 1.

Let us point out the following features of the field E 2 :

1. Surface charges are distributed in such a way that both the sec-ondary field inside of the elliptical cylinder and the primary field areuniform and have the same direction.


2. In the case of the circular cylinder we have

1L=-

2and

2Eo£2=-----

1 + (Y2IYl)

3. With an increase of the ratio alb the influence of charges decreasesand we have

ifY2 b--~O

Yl a

if a ~ b

This shows that with an increase of conductivity the cylinder has to bemore elongated in order to us to neglect the field due to surface charges.

4. With a decrease of the conductivity of the surrounding medium Y 1 ,

the field inside of the cylinder decreases as well. As YI approaches zerothe surrounding medium becomes an insulator, and the surface chargesare so strong that their field completely compensates the primary one.Thus, in this case the elliptical cylinder behaves as an ideal conductorregardless of its resistivity. Of course, this conclusion is valid for anarbitrary conductor and any type of primary electric field (electrostaticinduction).

5. If the elliptical cylinder is an insulator we have

EoE =--

2 l-L'

and correspondingly the maximal increase of the field E 2 is observed forthe circular cylinder when L = 1/2.

E 2 = 2Eo

6. In accordance with Ohm's law the current density in the conductingcylinder is

Y2Eo

1+(~~-I)L

Here it is appropriate to distinguish two cases.

(a) A very elongated cylinder when the field E 2 is practically equal toEo- Then the current density becomes directly proportional to the cylinderconductivity.

III.lO Behavior of the Electric Field in a Conducting Medium 349

(b) A very conductive cylinder for which the product

is much greater than unity, even for a relatively elongated conductor. Insuch cases the current density is independent of the cylinder conductivity.

but it can essentially exceed the value of the normal field jo as a » b.

We have considered two examples, when a sphere or an ellipticalcylinder is placed in a uniform field. In both cases the field inside ofconductors also remains uniform. Such behavior is not a simple coinci-dence. In fact, we can prove that the field inside an ellipsoid with anyratios of axes a, b, c and arbitrary orientation with respect to the primaryfield Eo also remains uniform; but in general it does not have the samedirection as Eo.

In particular, if the primary field is directed along the major axis of thespheroid then the field inside is

and

1 - e2

( 1 + e )L=-- In---2e2e 3 1- e

Here

and b = c < a

For instance, for elongated spheroids the depolarization factor L is

350 m Electric Fields

simplified and we have

b? (2a )Lz- In--1 «1aZ b

if a/b»1

In the next example we will consider a problem that plays a fundamen-tal role in electrical logging.

Example 5 The Electric Field of the Point Source atthe Borehole Axis

Suppose that a very small current electrode A is placed at the boreholeaxis (Fig. III.16b). The borehole radius is a, and the conductivities of theborehole and surrounding media are 'Yl and 'Yz, respectively. ! is thecurrent through the electrode. Our goal is to determine the electric fieldwithin the borehole. Before we formulate the boundary value problem, letus make use of the results derived in Example 2 and discuss the distribu-tion of charges and some general features of the behavior of the field.First of all the charge arising on the electrode surface is

and it creates a primary electric field

PI!EO=-4L3 Lop

7T" Op

(III.305)

(111.306)

where Lop is the distance from the electrode to an observation point p, Inthe presence of this field electric charges also appear at the interfacebetween the borehole and the formation, and as is well known theirdensity is

where E~v is the normal component of the field, caused by charge eA andsurface charges except in the vicinity of the point q. From this equation wecan conclude that the surface density I decreases very rapidly with anincrease of the distance from the electrode. In accordance withEq. (111.278) we can represent the density I as

1= co(pz - Pl)jn


since the normal component of the current density is a continuous func-tion.

Then, integrating over the entire borehole surface we obtain

Therefore, there are two charges. One of them is located on the surface ofthe small current electrode, the other is the charge distributed with adifferent density on the borehole surface, and their sum is

e = eA + es = EOPII + EO(P2 - PI)I = EOP21

That is, the total charge coincides with the charge on the current electrodeas if it were located in a uniform medium with resistivity P2'

The potential of the primary field is

PIIUo(P)=-4L

1T Op

while the potential caused by surface charges is defined by

1 f :2:(q) dSUs(p) =-

41TEO S L q p

(III.30?)

(III.308)

It is obvious that the latter cannot be used for calculation of the field sincei; is not known, but it is useful for our study.

Let us consider the influence of the resistivity of the borehole and theformation near and far away from the electrode. As follows fromEqs. (III.30?), (III.308), in approaching the electrode the primary fieldbecomes very large if the electrode radius is sufficiently small, while thesecondary field tends to some constant, since the distance Lop is alwaysgreater or equal to the borehole radius. Thus near the electrode theprimary field prevails, and therefore only information about the boreholeresistivity can be obtained.

In the opposite case, when the observation point is very far away fromthe electrode, charges spread over the borehole surface create practicallythe same field as that of an elementary charge es ' located at any pointnear the current electrode. As was mentioned earlier, the same equiva-lence is observed for the gravitational field, since as soon as the field isconsidered at sufficiently great distances from either masses or charges


their distribution inside some volume is not important, and we can placeall the mass or charge at any point of this volume. In our case, consideringthe electric field far away, one can mentally replace the surface distribu-tion of charges by the elementary charge es' situated at the electrode,along with the charge eA. Thus the potential and the electric field at largedistances from the current electrode can be represented as

as Lop» a

(III.309)

These equations show that as the distance between the current elec-trode and an observation point p increases, the field approaches that of auniform medium with resistivity P2' in spite of the fact that the point pcan be located within the borehole having resistivity Pl. As follows fromthe analysis performed in Example 2, the same asymptotical behavior isobserved in the case in which there is an invaded zone with resistivity PI:>.between the borehole and the formation.

After this qualitative analysis of the behavior of the field we will takethe next step and derive exact formulas. These will establish relationsbetween the field and its potential on the one hand and medium resistivi-ties, the borehole radius, and the distance from the observation point tothe current electrode on the other hand.

First we will choose a cylindrical system of coordinates r, ip, z with itsz-axis directed along the borehole axis and with its origin located at thecenter of the current electrode (Fig. III.16b). Then due to the symmetry ofthe model with respect to the z-axis and the plane z = 0, where thecurrent electrode is located, it is natural to assume that the potential isindependent of the coordinate cp and that it is an even function of z;that is,

U=U(r,z)=U(r,-z) (III.31O)

Now we are ready to formulate the boundary-value problem, which inaccordance with Eq. (III.253), describes the behavior of the potential inthe following way:

1. Inside of the borehole and within the formation, the potential Usatisfies Laplace's equation.


2. Near the current electrode, potential UI tends to that of the elec-trode charge.

as R~O

where

3. With an increase of the distance R from the current electrode thepotential in the borehole, UI , as well as in the formation, U2 , approachzero.

as R ~ 00

4. At the borehole surface both the potential and the normal compo-nent of the current density are continuous functions.

au! aU21'!a; = 1'2a; if r =a

We will look for the potential as a function satisfying all four require-ments. First of all we will find a solution to Laplace's equation. Takinginto account the axial symmetry, it is proper to present this equation in thecylindrical system of coordinates.

since

a2u 1 au a2u-+--+-=0ar2 r ar az2

au-=0alp

(HI.311 )

This is a differential equation of second order with partial derivatives,since the potential U depends on two coordinates, rand z. To solve thisequation we will suppose that its solution can be represented as theproduct of two functions, so that each function depends on one argumentonly, and consequently we have

U=T(r)cP(z)

Substituting Eq. (111.312) into Eq. (111.311) we obtain

d2T cP dT d2cPcP dr2 + -; dr + T dz 2 = 0

(III.312)

(III.3l3)


and dividing both sides of the equation by 1>T,

1 d 2T 1 dT 1 d 21>

--+---+--=0T dr 2 rT dr 1> dz?

On the left-hand side of Eq. (III.313) it is natural to distinguish twoterms.

1 d 2T 1 dTTerm 1 = - -- +--

T dr 2 rT dr

and

At first glance it seems that they depend on the arguments rand z,respectively, and Eq. (III.313) can be represented as

Term 1( r) + Term 2( z) = aHowever, such equality is impossible, since changing one of the argu-ments, for example r, the first term varies while the second one remainsthe same, and correspondingly the sum of these terms cannot be equal tozero for arbitrary values of rand z.

Therefore, we have to conclude that every term does not depend on thecoordinates and is constant. This fact constitutes the key point of themethod of separation of variables, allowing us to describe the potential asa product of two functions. For convenience let us represent this constantin the form ±m2

, where m is called a constant of separation.Thus instead of Laplace's equation we obtain two ordinary differential

equations of the second order.

1 d 2T 1 dT---+--=+m2

T dr? rT dr -(III.314)

Let us emphasize that replacement of the differential equation with partialderivatives by two ordinary differential equations is the main purpose ofthe method of separation of variables, since the solution of the latterequations is known.

To choose the proper sign on the right-hand side of Eq, (III.314) wewill take into account the fact that the potential U in the borehole and in


the formation is a symmetrical function with respect to the coordinate z,Eq, (III.31O). For this reason we will choose the minus sign on theright-hand side of the equation for 4>, and correspondingly we have

(III.315)

where

d 24>4>"(z) =-

dz?

As is well known the latter has two independent solutions, sin mz andcos mz; but we will use only cos mz, since it is an even function of thez-coordinate.

Thus, the function 4> can be written as

cfJ(z,m) = Cm cos mz (III.316)

where Cm is an arbitrary constant independent of z.As follows from Eqs. (III.314), on the right-hand side of the equation

for function T(r) we have to take the sign" +" and therefore,

1T" ( r) + - T' ( r) - m 2T = 0

r

where

( III.317)

dTT'=-

dr '

Introducing the variable x = mr we have

dT dT dx dT-=--=m-dr dx dr dx

and

Substituting these equalities into Eq, OII.317) we obtain

d 2T 1 dT-+---T=Odx? x dx

(III.318)

This equation is also very well known and is often used in variousboundary-value problems with cylindrical interfaces. Its solution is modi-fied Bessel functions of the first and second type but zero order, [o(x) andKo(x), respectively. Their behavior is shown in Fig. III.16c, and they havebeen studied in detail along with other modified Bessel functions. Also wewill use modified Bessel functions of the first order, [I(X) and KI(x),which describe derivatives of functions [o(x) and Ko(x); the relations


between them are

(111.319)

Graphs of these functions are also given in Fig. 1II.16c. It is useful toshow the asymptotic behavior of these functions.

fo(x)~l Ko(x) ~ -In x

x 1 (111.320)fj(x) ~ 2" Kj(x)~- as x~o

x

and

1Ko(x) ~e-xv 7Tfo(x)~exJ

27TX 2x

1Kj(x) ~e-xv 7Tf j ( x) ~ e' fj;;X as x~oo

27TX 2x

Let us notice that modified Bessel functions are described in numerousmonographs, there are many tables of their values, different representa-tions of these functions, relations between them, polynomial approxima-tions, etc. Certainly, application of these functions is as convenient as thatof elementary functions.

Thus, a solution of Eq. (111.318) can be represented as

T(x) = Dfo(x) + FKo(x)

or

(1II.321)

where Dm and Fm are arbitrary constants that are independent of r.Now making use of Eq. (III.312), for each value of m we have

(1II.322)

where Am and Em are unknown coefficients that depend on m.It is clear that the function VCr, Z, m) satisfies Laplace's equation and

we might think that the first step of solving the boundary-value problem isaccomplished. However, this assumption is incorrect, since the functionVCr, Z, m) depends on m, which appears as a result of the transformationof Laplace's equation into two ordinary differential equations, while thepotential V describing the electric field in the medium is independent of


m, Inasmuch as the function Ut;r, z, m) given by Eq, (111.322) satisfiesLaplace's equation for any m, we will represent U as the definite integral.

which is independent of m,Thus we have arrived at the general solution of Laplace's equation,

which includes an infinite number of solutions corresponding to differentcoefficients Am and Bm. Now we are ready to perform the second step insolving the boundary-value problem: to choose among the functions Amand Bm solutions, which obey the boundary conditions near the currentelectrode and at infinity. With this purpose in mind, we will take intoaccount the asymptotic behavior of the functions Io(mr) and Ko(mr). Aswas shown earlier, Ko(mr) tends to infinity as its argument approacheszero, and therefore this function cannot describe the potential of thesecondary field within the borehole. At the same time the function IaCmr)increases unlimitedly with an increase of r and correspondingly it cannotdescribe the field outside of the borehole. Thus, instead of Eq. (111.323),we can write

if r -s a

(111.324)

if r ~ a

It is clear that these functions satisfy both Laplace's equation and theboundary conditions. In fact, in approaching the current electrode thefunction U1 tends to the potential caused by the charge on its surface,while with an increase of r the function Uz, due to the presence ofKo(mr), decreases. Also both integrands in Eqs. 011.324) contain theoscillating factor cos mz, and therefore the functions U1 and Uz tend ~o

zero as the distance along the z-axis increases. This asymptotic behaviorwill be considered later in detail.

Next we will satisfy the last requirement of the boundary-value problemand find coefficients Am and Bm such that they provide continuity of thepotential and the normal component of the current density on the bore-hole surface. To simplify our transformations, it is important to representthe potential of the primary field in terms of the same functions as the

(II1.325)


secondary field. Such a representation is very well known, and it is calledthe Sommerfeld integral.

1 2 00

- = -1 Ko(mr) cos mzdmR 17" 0

where

and r =1= 0

Then an expression for the potential inside the borehole is

if r'*O (II1.326)

where

PI1C=-

217"2

Now the conditions at the interface r = a can be written as

(III.327)

and

= 1'21 00

BI1IK~(ma) cos mzdmo

where

100x) = dl~X) ,dK (x)Ko(x) = __0_

dx

(III.328)

Both equations contain an infinite number of unknowns Am and Bm ,

and they can be considered as integral equations with respect to Am andBm • Fortunately, there is one remarkable feature of these integrals thatallows us to drastically simplify them. In fact, they are Fourier cosinetransforms, and from the theory of these integrals it follows that theirequality results in the equality of their integrands. Therefore we have

CKo(ma) +Amlo(ma) =BmKo(ma)

1'1{ -CK1(ma) +AmI1(ma)} = -1'2 Bm K l( ma )


since

This is a dramatic simplification, inasmuch as instead of integral equationswe obtain for every value of m a system of two linear equations with twounknowns, Am and Em' whose solution is

and

Let us notice that in deriving the expression for Em the equality

has been used.Thus the functions VJ(r, z ) and Vir, z ), given by Eqs. (III.324) and

(111.329), satisfy all requirements of the theorem of uniqueness; corre-spondingly we can say that these functions describe the potential of theelectric field caused by the electrode charge and charges distributed on theborehole surface.

Inasmuch as measurements of the potential difference as well as theelectric field are of a great practical importance in electrical logging, let uswrite their expressions in the borehole.

PJI[ 1 2V(r,z)=- +-(YJ-Y2)

417 /r 2+z 2 17

00 Ko(ma)KJ(ma)lo(mr) cos mz dm ]

X fa Y2 Io(ma)K J(ma) + y11](ma)Ko(ma)

PJI[ z 2E, = -4 2 2)3/2 + -( Y] - Y2)

17 (r +z 17

00 mKo(ma)KJ(ma)lo(mr)sinmz ]xl dmo Y21o( ma) K]( ma) + Y]1]( ma) Ko(ma)

(III.330)


sinceau

E =--Z az

In particular, if measurements are performed at the borehole axis r = 0,these expressions are slightly simplified, since r = °and we have

PI I [ 1 2U(O,L) = 47T L + 7T(Y]-Y2)

00 Ko(ma)K](ma) cos mLdm 1X fa Y210( ma)K I( ma) + y]I](ma)Ko(ma)

(III .331)

and

PI! [1 2EAO,L)= 47T L2 + 7T(Y]-Y2)

00 mKo(ma)K](ma) sin mLdm 1X fa yJo(ma)KJCma) + y]IJ(ma)Ko(ma)

where L is the distance between the current and receiver electrodes of thenormal and lateral probes, which, as was described earlier, measure thepotential and the electric field, respectively. In essence in both cases adifference of potentials or voltage is measured. However, with the normalprobe the second receiver electrode is located far away and its potential ispractically zero, while with the lateral probe both receiver electrodes areclose to each other, and we can say that the voltage is equal to the productof the electric field and the distance between these electrodes.

Here it is appropriate to make to comments.

1. Algorithms for integration and calculation of modified Bessel func-tions are very well elaborated, and for this reason determination of thepotential and the electric field by Eqs. (III.33l) is a relatively simple task.

2. Applying Eqs, (III.33l) and making use of the principle of superposi-tion, we can solve the forward problem for more complicated probesformed by several current and receiver electrodes, such as theseven-electrode laterolog.

Now let us consider Eqs. OII.33l) in detail. For simplification let usintroduce new variables

x=ma,L

a=-,a

YI P2s=-=-

Y2 PI

(III.332)


and introducing them into Eq. OII.33!) we obtain

U(O, L) = Ua[1 + (5 - l)a ~ {.oA~ cos axdx]

Ez(O, L) = Eaz[1 + (s -1)a2 ~ fa"'xA: sin axdx]

where Ull and Eaz are the potential and electric field at the z-axis, causedby the charge eA if it were located in a uniform medium with resistivity Pl'

and

PI!U,=--

a 4rrL '

(III.333)

Thus the electric field and its potential can be represented as theproduct

(III.334)

and the functions FE and Fu depend on two parameters, namely, theprobe length, expressed in units of the borehole radius a, and the ratio ofconductivities s.

It is obvious that these functions characterize the influence of themedium and the probe length, since they show how the field and itspotential at the borehole axis differ from the corresponding functions in auniform medium with the borehole resistivity PI' Let us rewriteEqs. (III.334) as

andU

F =-u Ua

Very often these ratios are presented in the following form:

andp~ U-=-=FPI Ua u

(III.335)

where p; and p~ are called the apparent resistivity of the lateral andnormal probes, respectively. Of course, we can introduce similar expres-sions for the apparent resistivity of more complicated probes.

As an example, we will consider the relationship between the potentialU and the geoelectric parameters of the medium. With this purpose in


mind, we will study the behavior of the function Fu ' In accordance withEqs. (III.332),

2a 00

Fu = 1 + (s - 1)-1 A~ cos a x dx (III.336)7T 0

where A~ is given by Eq. OII.333) and it is independent of the para-meter a.

With a decrease of the probe length, the ratio a tends to zero andaccording to Eq. (III.336) we obtain

2a 00

Fu~l+(s-l)-lA~dx~l if a~O7T 0

That is, this potential approaches the potential caused by the charge eA

only, which is located on the surface of the current electrode. Certainly,this is a known result, but it is derived in a different way. Now we willinvestigate the opposite case in which the probe length increases, andcorrespondingly the parameter a tends to infinity.

To explain the asymptotic behavior of the function Fu ' let us payattention to the integrand in Eq. (III.336). This is the product of twofunctions.

A~ cos ax

One of these functions, A~(x), gradually decreases without a change ofsign, while cos aX is an oscillating function. The interval ~x, within whichit does not change sign, is defined by the condition

7TLix= -

a

With an increase of the parameter a this interval decreases and,correspondingly, A~ becomes practically constant within every interval~x. Taking into account the fact that A~ is a continuous function of x, wecan say that with a decrease of ~x the integrals over neighboring intervalsare almost equal in magnitude, but have opposite sign. In other words,they cancel each other, and with an increase of a this behavior manifestsitself for smaller x. This means that in the limit as a tends to infinity theintegral in Eq. (III.336) is defined by very small values of x, and this isillustrated in Fig. III. 16d. Taking this fact into account, we will simplifythe expression for A~(x), replacing the functions Io(x) and I/x) by theirasymptotical formulas, Eq. (III.320). Then, instead of Eq. (III.333) weobtain

if x~O


Correspondingly, the asymptotic expression for the potential is

[2a 00 ]

V(O, L) = o; 1 + (s - 1) --;;:- fa Ka(x) cos axdx

and in accordance with Eq. (111.325) we have

[a] pz!

V(O, L) = Va 1 + (s - 1) ~ = Vas=-vl +az 47TL

if a ~ 00

as a ~ 00

Again we have demonstrated that with an increase in probe length thepotential in the borehole approaches that of a uniform medium with theresistivity of the formation. Let us note that by applying the same ap-proach we can derive similar expression for the electric field.

Now we will discuss the results of calculation of the ratio

o, V

Pi ti;presented in log-log scale in Fig. III.17a. The index of the curves iss = Pz!Pi' and the parameter a is plotted along the horizontal axis. Thefunction V is given by Eq. (III.336).

We will distinguish several features of these curves, which reflect thebehavior of the potential at the borehole axis.

1. All curves have left and right asymptotes, corresponding to theresistivity of the borehole and formation, respectively.

2. With an increase of the formation resistivity, or more precisely s, theapproach to the right asymptote takes place at greater distances fromthe current electrode. Similar behavior is observed in the case when theformation is more conductive.

3. With an increase of the separation all curves intersect the rightasymptote, have a maximum, and then decrease, gradually approachingtheir asymptotes. This indicates that there is a range of separations wherethe potential exceeds that in a uniform media with resistivity P:» if s> 1.

4. In the case of a more conductive formation, apparent resistivitycurves also intersect the right asymptote.

5. The more the parameter s differs from unity, the smaller thedistance should be between the current electrode and the observationpoint in order to neglect the influence of charges at the borehole axis.

6. If the formation is relatively resistive, s » 1, there is an intermediatezone where the apparent resistivity curve has a slope that is approximatelyequal to 45°. This zone becomes wider with an increase of s. Suchbehavior of the apparent resistivity means that the potential remains


practically constant. In fact, from Eqs. (III.307) and (111.335) it follows that

Pa = 47TUL or log Pa = log 47TU + log L

(III.337)

and this equation describes a straight line with a slope of 4SO if U =constant.

Now we will study the behavior of the field within this zone. In thesimplest case, when the formation is an insulator, P2 = 00, the distributionof currents in the borehole can be represented in the following way. Verynear the electrode A the current lines are almost radial in direction. Withan increase in distance they tend to be parallel to the borehole axis.Correspondingly, at sufficiently large distances from the electrode A wecan expect a uniform distribution of current density, which has only az-component, equal to

Ijz= -227Ta

Here I is the current arriving at the electrode from the wire, while a is theborehole radius. The coefficient 1/2 is introduced since the current I issymmetrically distributed with respect to the electrode A. Applying Ohm'slaw,

. EzJ-= -" PI

we find that the electric field inside of a borehole with resistivity PI is

PII IE =--=-

z 27Ta 2 25

where 5 is the borehole conductance

(III.338)

Equation (111.337) describes an electric field, which is uniform within across section of the borehole, and it does not change along the z-axis.Therefore, the voltage V, measured between two arbitrary points Mand N, -

V= U(M) - U(N) =EzMN (III.339)

remains constant.It is obvious that the current is accompanied by the appearance of

surface charges that create completely different fields inside and outsideof the borehole. In fact, inside the borehole the electric field is uniform if


a »1 and it is directed along the z-axis, Eq. (111.337), while outside,r > a, charges create only a radial field.

A different behavior of the field and currents occurs when the sur-rounding medium is conductive (pz =1= (0) since the radial component of thecurrent density outside the borehole is not equal to zero. In other words, aleakage of current from the borehole into the formation occurs. Corre-spondingly, with an increase in the distance from the electrode A thecurrent through the borehole cross section decreases. It is obvious thatthis behavior of the current along the borehole is observed because bothvertical components of the current density and the electric field decreasewith the distance z. It is clear that with an increase of the formationconductivity, the electric field E; would decrease more rapidly.

This description allows us to assume that within the intermediate zonethe borehole behaves as a transmission line, and now we will demonstratethat this assumption is correct. In accordance with Ohm's law the changeof the potential dU along an arbitrary element of the borehole, dz(Fig. m.rn», is

I( z) dzdU= ----

S(III.340)

where dzjS is the resistance of the borehole element dz, U(z) is thepotential at the point p, and It z) is the current through the boreholecross section at point p. The negative sign in Eq. (III.340) is introducedsince

dU= U(z + .lz) - U(z) and U(z) > U(z + .lz)

A change in current along the borehole occurs due to leakage into theformation. In other words, we can consider that the surrounding mediumwith resistivity pz is connected in parallel with the borehole. Conse-quently, within the interval dz this leakage current dl, is equal to <dlt z),and it is related to the potential U( z ) by

T TU( z) = - dI = - - dI( z)

dz r dz(111.341)

where T is the resistance per unit length of the formation to radialcurrent. Thus, Eqs. (1ll.340), 011.341) can be represented as

dI 1-=--U(z)dz T

(III.342)


After differentiation of this system we obtain

and (111.343)

where

1n=--

ISf(III.344)

Thus, the distribution of potential and current is defined by the param-eter n, provided that our assumption about the field behavior within theintermediate zone is correct. Let us note that U(z) is the potential at somepoint of the borehole with respect to infinity, where U equals zero. Inparticular, any cylindrical surface with a sufficiently large radius that iscoaxial with the borehole has practically zero potential. As is well known,any solution of Eqs. (III.343) has the form

where A and B are unknown constants. Due to leakage, the current in theborehole tends to zero as z ~ ± 00, and we have

I(z) =Be-n z if Z> 0 (III.345)

Because of the symmetry with respect to the electrode A, half of thecurrent goes in one direction, while the other half goes in the oppositedirection. Therefore, the initial condition is

tI( z) = "2 as z = 0

since the effect of the leakage is negligible near the source. Correspond-ingly, we have

ffez) = <e ?"

2(III.346)

where I is the current near the electrode.Making use of Ohm's law we have, for the electric field at any point of

the borehole,

. I(z) I(z)E; = Pj}z = PI lra 2 = -S-

or

IE = -e-z / P

z 2S(111.347)

III.I0 Behavior of the Electric Field in a Conducting Medium 367

Taking into account the fact that

auE =--

Z az

we obtain for the potential

IffU(z)=- -e-z / P2 S

(111.348)

(III.349)

To compare this result with the curves given in Fig. III.17a we willagain introduce the apparent resistivity, corresponding to Eq. (III.348).

Then making use of Eq. 011.335) we have

Pa = 27T"L !!.e-L / PPI PI VS

where L is the probe length. .It is easy to see that the results of calculations by Eq, (111.349) coincide

with the data given in Fig. III.17a, provided that the transverse resistanceT equals P2

T=P2 (IIl.350)

and the formation is much more resistive than the borehole.This fact strongly confirms that within the intermediate zone, where

P2»> Pl' the behavior of the current and the potential is governed by thetransmission line equation when its conductance is equal to that of theborehole. The transverse resistance is equal to the formation resistivity ifthe surrounding medium is uniform. Let us write expressions for thepotential, the electric field, and the second derivative of the potential.

Ilt2 .~U(z)=- _e-Z/ySP2

2 S

IE(z) = _e-z/YS;;;

25(111.351)

As follows from their comparison, the second derivative of the potentialis more sensitive to the formation resistivity. In fact, in accordance with


Fig. III.I7 (a) Apparent resistivity curves; (b) electric field in a borehole surrounded by aninsulator; (c) two-layered medium; and (d) Bessel functions Jo, Jo.

IIl".lO Behavier of the Electric Field in a Conducting Medium 369

Eqs. (III.343) we can conclude that

S d 2U(z)

12 = U(z) dz 2 (III.352)

and since these equations are differential equations, the latter can also beused in the case when the formation is a horizontally layered medium.

In conclusion we will make two comments.

1. Equation (III.352) shows that in principle we can measure theformation conductivity through the casing as well as during drilling.

2. Comparison of the results of calculations by Eqs. (III.349), (III.350)with the apparent resistivity curves presented in Fig. III.17a also demon-strates that even for a relatively conductive formation, 10 < s < 1000, theleakage effect described by Eq. (111.343) plays a very important role.

Example 6 The Electric Field on the Surface of aMedium with Two Horizontal Interfaces(Fig. III.17c)

Suppose that a current electrode A is placed on the surface of a two-layeredmedium. The upper layer has thickness h, and 11 and 12 are conductivi-ties of the layer and the basement, respectively.

First let us consider the distribution of charges, which appear atinterfaces only. In accordance with Eq. (III.173) the charge on the elec-trode surface is

(III.353)

and correspondingly the potential and the electric field of the primary fieldare

and (III.354)

Due to the presence of the primary electric field Eo, surface chargesarise at the interface between the layer and the basement; and in accor-dance with Eq, (III.178) their density is

Taking into account the fact that the current through this interface equals


!, we obtain for the total charge at the surface

or (111.355)

es = EO(PZ - PI)!

The charges distributed over the interface then induce charges at theearth's surface. As was demonstrated earlier, Eq. (111.172), each elemen-tary charge located in a conducting medium creates at the boundary withan insulator a surface charge with the same sign and magnitude. For thisreason, the induced charge on the earth's surface, eo, coincides with thatat the interface, es , but it is distributed in a different manner; that is,

(III.356)

Thus the total surface charge in a medium is

(111.357)

(III.358)

This means that the charge, which arises at all surfaces, coincides withthe electrode charge, as if the electrode were located at the surface of auniform half space with the resistivity of the basement, pz. As in theprevious example, this analysis allows us to establish the asymptoticbehavior of the field as a function of the distance between the current andreceiver electrodes. In fact, let us represent the potential as the sum

PI!U=-+U

21fT s

where U. is the potential caused by charges at the bottom of the layer andat the earth's surface. Inasmuch as the potential U. has a finite valueeverywhere, in approaching the current electrode both the potential andelectric field are mainly defined by the charge at the electrode surface;that is,

andPI!

Er(r) ~ -2z1fr

(111.359)

In other words, with a decrease of the separation between the currentand receiver electrodes, r, the depth of investigation also decreases, sincethe field is practically defined by the resistivity of the upper layer, PI' only.In the opposite case, when the observation point is far away from thecurrent electrode, the influence of surface charges, as in the previousexample, is the same as if the total charge were placed at the current

ill.to Behavior of the Electric Field in a Conducting Medium 371

electrode. Then, in accordance with Eq. CIII.357) we obtain

pz! pz!U(r)--7

21T r,and Er--721Trz (1II.360)

This means that with an increase of the separation r, the depth ofinvestigation increases as well, in spite of the fact that the current andreceiver electrodes are placed in the upper layer with resistivity Pl' Thishappens because the electric field and its potential become functions ofthe basement resistivity pz only, and this result does not depend on thethickness of the upper layer. Moreover, as follows from the study of thecharge distribution in a layered medium, this asymptotic behavior remainsvalid, regardless of the number of layers and the presence of structureshaving finite dimensions.

In essence we have explained the main concept of geometric soundingsbased on measuring the voltage at different separations from the currentelectrode.

Now we will derive equations for the potential and the electric field atany separation from the current electrode, and with this purpose in mind,we will solve the boundary-value problem.

First of all taking into account the axial symmetry with respect to thevertical axis, perpendicular to interfaces and passing through the currentelectrode, we will choose a cylindrical system of coordinates r, q;, Z asshown in Fig. III.17c.

In accordance with Eqs, (111.254) the boundary-value problem is formu-lated in the following way:

1. Within the layer and the basement, the potential satisfies Laplace'sequation,

if 0 ~z ~ h

and

if z:2:: h

2. In approaching the current electrode, the potential UI tends to thatcaused by the charge at the current electrode.

PI!U --7 -- if R --7 0

I 21TR

where R = Vr z+ z z .3. At the earth's surface the normal component of the current density

is zero, and therefore

aUI-=0az as z = 0


4. With an increase of the distance R from the current electrode, thefield and its potential tend to zero.

as R -+ 00

5. At the interface z == h the potential and the vertical component ofthe current density are continuous functions.

aU1 aU2

1'1 az =1'2 az if z = h

Let us notice that, since the conducting medium is surrounded by surfacesat every point of which boundary conditions are defined, we do not needto consider the field above the earth's surface.

Now we will determine the potential that satisfies all these conditions,beginning with Laplace's equation. In accordance with Eq. CIII.31l) wehave

a2u 1 au a2u

-+--+-=0ar2 r ar az 2

since, due to the axial symmetry, the potential U is independent of thecoordinate cpo Applying the method of separation of variables and repre-senting the potential as

U(r,z) == T(r)q,(cp)

we again obtain two ordinary differential equations.

1 d 2T 1 dT 2---+--=+mT dr? rT dr -

If we choose the positive sign on the right-hand side of the equation forTCr), as was done in the previous example, then the solutions would bethe modified Bessel functions IoCmr) and KoCmr), which have singularitieseither at infinity or at points of the z-axis, respectively. Inasmuch as allthese points are located within the upper layer and in the basementfunctions, IoCmr) and Kimr) cannot describe the potential, which every-where has a finite value except at the origin of coordinates. For this reasonwe will take the negative sign on the right-hand side of the equation for T


and then obtain

(III .361)

(111.362)

The solution of the second equation consists of exponential functions.

(III.363)

where em and Dm are unknown coefficients, which are independent of z.Introducing a new variable x = mr we can represent the first equation inthe form

d 2T 1 dT-+--+T=Odx? x dx

(111.364)

The solutions of this equation are Bessel functions of the first andsecond type but zero order, Jo(x) and Yo(x), respectively, which arethoroughly studied and widely used in numerous theoretical and engineer-ing problems. Inasmuch as the function Yo(mr) has a logarithmic singular-ity at points on the z-axis, it cannot be used to describe the field. Thebehavior of the function Jo(x) is shown in Fig. III.l7d. In particular, wehave

if x« 1

and

if x» 1

Thus for every value of the separation constant m we obtain

(III.365)

and correspondingly the general solution of Laplace's equation, whichdoes not depend on m, is

(111.366)

Having accomplished the first step in solving the boundary-value problem,let us satisfy the other conditions.


Representing the function Uir, z ) in the upper layer by

we see that Uj(r, z ) satisfies the boundary condition near the electrode aswell as Laplace's equation.

To satisfy the condition at the earth's surface, auj/az = 0, we will takethe first derivative of the potential with respect to z. Then we obtain

eu, Pj[zaz 21T(r 2 +z 2 ) 3/ 2

(III.368)

Letting z = 0 we have

(111.369)

This is a very complicated integral equation with respect to unknownsem and Dm , but fortunately for us integrals of the type

have one remarkable feature similar to that of Fourier integrals; namely,from the equality

or

(111.370)

it follows that

(111.371 )

Thus, instead of Eq. (111.369) we have

Ill.I0 Behavior of the Electric Field in a Conducting Medium 375

and correspondingly the expression for the function U, is slightly simpli-fied.

(I1I.372)

(I1I.373)

In the basement, where z increases unlimitedly, we will represent thesolution to Laplace's equation in the form

Uz(r,z) = ['Bme-mZJa(mr)dmo

which of course satisfies the condition at infinity, as z ~ 00. Also, due tothe oscillating character of Ja(mr), both functions U1 and Uz tend to zeroas the coordinate r increases unlimitedly.

To satisfy the conditions at the bottom of the upper layer it is neces-sary, as in Example 5, to represent the potential of the primary field in theform corresponding to that of the secondary field. With this purpose inmind, we will make use of the Lipschitz integral.

or (111.374)

if z> 0

Substituting Eq. (111.374) into Eq, (I1I.372) we have

Ul ( r, z) = {Or Ce- mz + Cm(emz + e-mz)] Ja(mr) dma

where

pJC=-

27T

Respectively, continuity of the potential and the normal component of thecurrent density occurs at the interface z = h, if

(111.375)

(III.376)

(III.377)


and

= -"/2[OBme-mhmJ

o(mr) dmo

Now, again making use of Eqs, (III.370), (III.371), a drastic simplifica-tion occurs and we obtain for every m two equations with two unknowns.

Ce-mh + Cm(emh+ e-mh) = Bme-mh

"/1[ _Ce- mh + Cm(emh - e-mh)1= -"/2 Bme-mh

Let us note here that in both the case of the borehole and that of thehorizontally layered medium we, in essence, observe one of the mostimportant features of the so-called special functions, namely, their orthog-onality. This feature makes them extremely useful for solving numerousboundary-value problems.

Solving the system (III.376) we have

K e-2mh P 1C = 12 1

m 1 - Kl2e-2mh 2'77"

where

K_ P2 -P1

12 -P2+PI

Therefore, the functions UI and U2 , given by Eqs. (III.372), (III.373),satisfy all the conditions of the boundary-value problem, provided that thecoefficients Cm and B m are defined by Eq, (IIL377). Correspondingly, U1and U2 describe the potential of the electric field when the currentelectrode is located at the origin of coordinates. In particular, if theobservation points are located at the earth's surface we have

PII[l 00 e-2mh

]UI(r) = - - + 2K121 _2mhJO(mr) dm

2'77" r 0 l-K 12e

and (III.378)


where E, = -aular is the radial component of the electric field andJ1(x) = -dJO<x)/dx is the Bessel function of first order (Fig. III.l7d).

We have derived formulas (IIl.378) for a two-electrode array, but byapplying the principle of superposition it is a simple matter to obtain anexpression for the voltage of an arbitrary system of electrodes. Also, bymaking use of the same approach in solving the boundary-value problem itis easy to generalize Eq. OII.378) for an n-layered medium. Algorithms forcalculating integrals that describe the field and its potential at the surfaceof a horizontally layered medium are very well developed and havestandard procedures.

Introducing a new variable, x = mr , we will write Eq. (III.378) for theelectric field as

(III.379)

or

where EOr is the primary electric field and FE is a function that dependson two parameters only.

Pzs=-

PIand

ha=-

r

(1II.380)

Before we demonstrate the results of calculation of the electric field, letus study its asymptotic behavior.

As follows from Eq, (III.378), when approaching the current electrodethe integral tends to zero, since J1(x) ~ 0 as x ~ 0, and correspondinglythe field is defined by the electrode charge.

PIIEr~EOr= --2

27rr

In the opposite case, when the distance r increases and r/h >>>- 1, we willtake into account that, due to the oscillating character of the Besselfunction J1(x), the integral is mainly defined by values of the integrand atthe initial part of the integration. Then, letting the exponents approach avalue of one, we have

as m~O


and therefore,

PJ [ 12K12 100

]E; = - 2" + mJ 1( mr ) dm27T r l-KI2 a

Inasmuch as

amJ 1( mr ) = - -Ja(mr)ar

then, in accordance with Eq. (111.374),

1 00

- = 1Ja(mr) dmr a

and we have

if r ~ 00

or

since

K_ P2-Pl

12 -P2 +Pl

if r ~ 00

if r ~ 00 (111.381 )

Thus, regardless of the ratio of resistivities, with an increase of theseparation r, the electric field approaches the value of the field in auniform half space with the resistivity of the basement. Of course, bothasymptotics, as either r ~ 0 or r ~ 00, have been derived earlier proceed-ing from the charge distribution.

Now we will consider the special case when the basement is aninsulator and the separation r is much greater than the upper layerthickness h. Then, letting K 12 = 1 and assuming that m ~ 0, we have

me- 2m h m 1-----;;:---;- ~ -- = -1 - e- 2m h 2mh 2h

since e- 2m h :::: 1 - 2mh.

as m ~O


Therefore we obtain

PI I [ 1 1 a100

]= - - - - - Jo( mr) dm27T r 2 h ar a

if r ~ 00

Finally, by neglecting the first term we have

IE=--

r 27TSr

rif -» 1

h(III.382)

where S = Y1h is the conductance of the upper layer.Equation 011.382) shows that with an increase of the distance the

electric field becomes inversely proportional to the conductance S, and itdoes not depend separately on the thickness and resistivity of the upperlayer. In other words, by performing measurements of the field far awayfrom the current electrode we can determine the conductance S of thelayer only, if P2 = 00.

It is helpful to arrive at Eq. (III.382) in a different way. As is seen fromFig. III.18a, with an increase of r the vector of the current densitybecomes practically horizontal and independent of both coordinates randz. Then the total current through any lateral surface of the cylinder withradius r and height h is

and taking into account Ohm's law, j, = YIE" we again obtainEq. (111.382).

It is always useful to derive the same equation from the mathematicaland physical points of view. For instance in our case this approach allowedus to understand that the asymptotic behavior of the field described byEq. (111.382) corresponds to a uniform distribution of the current densityin the upper layer. Moreover, now we are able to generalize Eq. (111.382)for an n-layered medium, provided that basement is an insulator(Fig. III.18b).

Since the electric field is horizontal far away from the current electrode,the equipotential surfaces become themselves lateral surfaces of cylinderswith axis z. We will consider two arbitrary equipotential surfaces located


Fig. HI.1S (a) Current density distribution in a conductive layer; (b) horizontally layeredmedium and distribution of currents in the far zone; (c) apparent resistivity curves; and (d)model of a medium with a vertical contact.

at distance /).r. The voltage between them can be written as

(1II.383)

where I, is the current in ith layer, R, is the resistance of the cylindricallayer with thickness Sr and height hi. The latter is the thickness of ithlayer. It is obvious that

v = Er/).r andPi/).r /).r

R·=--=--I 27Trh i 27TrS j

where S, is the conductance of the ith layer. Then instead of Eq. (111.383)we obtain

t,=-=

Sj(111.384)

III. to Behavior of the Electric Field in a Conducting Medium 381

Also taking into account the fact that far away from the electrode wecan assume that the layers are connected in parallel, we have

(III.385)

or

S2 S3 SnI=I1+-I1+-I + ... +-1

S S i S 11 1 1

Thus

(III.386)

wheren

is called the total conductance of a system of layers, and this parameteroften plays an important role in the interpretation of an electrical sound-ing.

Finally, from Eqs. (III.384) and OII.386) we have

II I rE = -- = -- if -» 1, Pn+1 = 00 (III.387)

r 27TrSI 27TrS h

that, in fact, is the general version of Eq. (III.382).Let us note that the range of separations, where Eq. OII.387) describes

the field, is usually called the S-zone.To complete the study of the asymptotic behavior of the field, it would

be natural to investigate the case when the basement is an ideal conduc-tor. Such analysis, however, would require an appendix about someremarkable features of functions of a complex variables. For this reasonlet us restrict ourselves to only one comment, namely, that the field faraway from the current electrode decays exponentially, if the basement isan ideal conductor, and the power of the exponent is proportional to theratio r/h,

Now we will consider the apparent resistivity curves, Pa/Pl> calculatedby Eq. (III.379), provided that

and presented on a log-log scale in Fig. III. 18c. The index of the curves isthe parameter s, equal to PZ/PI'


In accordance with the behavior of the electric field, the left asymptoteof all curves is equal to unity; that is,

asr

- --7 0h

Then, with an increase in separation the influence of the basementgradually increases and in the limit the curves approach their rightasymptote, equal to Pzlpl; that is,

rif - --700

h

As follows from a study of the distribution of surface charges, it is seenfrom the curves that with an increase in resistivity difference, approach tothe right asymptote takes place at greater separations. In the case of amore resistive basement, S » 1, there is an intermediate range of separa-tions, when all curves approach that which corresponds to a nonconduct-ing basement. Therefore, within this range, the apparent resistivity P«depends on only one parameter of the medium, namely the conductanceof the upper layer S.

The behavior of the curves of PalPI' given in Fig. III. 18c, clearlydemonstrates the main concept of the geometrical or Schlumberger sound-ings, which are mainly performed with the symmetrical four-electrodearray, shown also in Fig. III.l8c. This method is widely used in ground-water and engineering geophysics, and the results of measurement areusually presented in the form of the apparent resistivity P« as a function ofthe distance between the middle point 0 and the current electrode, AB12.

Concluding this example, let us make several comments related to thesolution of the inverse problem, that is, to interpretation of Schlumbergersoundings.

1. In accordance with Eq. (III.379), the apparent resistivity Pa can berepresented as

Applying the same approach for a solution of the forward problem in ann-layered medium, we can show that the apparent resistivity has a similarexpression.

(III. 389)


where Pi and hi are the resistivity and thickness of the ith layer, respec-tively.

2. Presenting the apparent resistivity curves on a log-log scale, we have

log o,= log PI + log F( A,log :1 )where A is the set of parameters of the medium pJpl' hJh1•

Equation OII.389) shows that a change of the resistivity of the firstlayer, Pi> as well as its thickness hI' do not alter the shape of the curve Pabut results in a parallel shift only. This fact essentially simplifies theinterpretation.

3. In the theory of inverse problems of geophysics it has been provedthat the inverse problem for geometric soundings performed at the surfaceof a horizontally layered medium is unique. In other words, only one set ofgeoelectric parameters generates a given curve of the apparent resistivityor the field. Certainly, this is a very important result, representing inessence the theoretical foundation of the interpretation of geometricalsoundings. However, uniqueness of this inverse problem only holds pro-vided that the field is measured with absolute accuracy, which of coursedoes not correspond to real conditions. In fact, several factors are alwayspresent and introduce some error into values of the apparent resistivity.These factors are

(a) Errors in measuring the voltage between receiver electrodes.(b) Errors in determining distances between electrodes.(c) The presence of a geological noise, which includes lateral changes

of resistivity, topography effect, etc.; in other words, everything thatproduces deviation of the real model of the medium from a horizontallylayered one.

Thus, there is always a difference between the measured curve of theapparent resistivity and that calculated for a horizontally layered medium.Correspondingly we can say that the interpretation of geometrical sound-ings is an ill-posed problem.

This means that by matching the measured and theoretical curves weare only able to establish limited ranges within which every parameter canvary, instead of determining their exact values. As a rule, the width ofthese ranges is different for different parameters, and this can be ex-plained in the following way. The electric field measured at the earth'ssurface is caused by all charges distributed at interfaces, and it is obviousthat the relative contribution of charges at the top and bottom of somelayer strongly depends on its resistivity, thickness, and position. For thisreason it is natural to expect that some parameters of a given medium can


be determined with a great error only, while others are defined with anaccuracy sufficient for practical applications. Correspondingly, we can saythat interpretation of sounding data consists of determining parametersthat characterize with relatively high accuracy certain features of a hori-zontally layered medium. In particular, these parameters can be either thelongitudinal conductance of some layers, S; = h;/p;, or the transverseresistance, 1'; = p;h;, of others.

This consideration clearly shows some similarity between the interpre-tation of gravitational data and electric soundings.

Example 7 The Electric Field at the Surface of aMedium with a Vertical Contact

We will consider the behavior of the electric field in the presence of avertical contact, shown in Fig. III.18d. Let us introduce a Cartesian systemof coordinates x, y, Z, with its origin at the center of the current electrodeA and x-axis directed perpendicular to the contact between media withconductivities 'Yl and 'Yz. Suppose that the electrode A is located in themedium with conductivity 'Yl at a distance d from the contact.

It is clear that the primary electric field, caused by the electrode charge,gives rise to an appearance of charges at the contact. In turn these chargescreate an electric field that generates charges at the earth's surface.Certainly, the distribution of charges at both surfaces is established as theresult of their interaction. Therefore, the potential at every point of amedium is a sum of the potentials, caused by the electrode charge Uo andsurface charges u..

U= Uo+ U.By analogy with previous examples, it is convenient to write the potentialin the conducting medium as

x~d

x~d(III.390)

Let us note that the potential Uz is the potential of the total field, causedby all charges.

Now in accordance with Eq. (111.254) we will formulate the boundary-value problem for the points of a conducting medium in the following way:

1. At regular points the potential satisfies Laplace's equation.


2. At the earth's surface, Z = 0,

au-=0az

since the normal component of the current density vanishes.3. The potential and the normal component of the current density are

continuous functions near the vertical contact.

aUI aU2

Yj ax = Y2 ax

4. Near the current electrode the potential tends to that of the primaryfield.

as R ----) 0 (111.391 )

In particular, if the current electrode is located at the earth's surface,

pjIU----)Uo = - -

2-rrR

5. With an increase of the distance from the current electrode thepotential decreases.

where

U----)O as R ----) 00

(2 2 2)

1/ 2R= x +y +z

All these conditions uniquely define the potential in the conductingmedium, and it is not necessary to look for a solution above the earth'ssurface.

Unlike Example 6, application of the method of separation of variablesfor a solution of this boundary-value problem does not allow us to derivesimple expressions for the potential. This is related to the fact that thevertical contact does not coincide with any coordinate surface of theCartesian system. However, there is an elegant approach allowing us toreduce this problem to another one whose solution is much simpler.

With this purpose in mind, we will mentally transform the mirror imageof the conducting medium with respect to the earth's surface to the upperhalf space. After this transformation we obtain a new model of a conduct-ing medium with only one planar surface x = d and two current electrodeshaving equal charges, p j Is o/ 4-rr , and symmetrically situated with respectto the plane corresponding to the earth's surface (Fig. III.l9a). Next


a b

A "(, "(2"(, "(2

2A d

0

A "(, "(2 "(1 "(2

c d

Pa A...--.1.-. M Pa A"L

P," p;- MNP2 P2

X X

P1<P2P1 < P2

Fig. III.19 (a) Equivalent model with two current electrodes; (b) equivalent model whencurrent electrode is on the earth's surface; (c) apparent resistivity curve for a two-electrodearray; and (d) apparent resistivity curve for a three-electrode array.

suppose that both electrodes approach each other; then, in the limit, asthe separation distance approaches zero, we have one current electrodewith the charge eAPtlEo/21T placed at the plane z = 0 (Fig. III.19b).

Now let us show that the potential in the lower part of this model,z > 0, coincides with that of the original one. To prove this fact, we willformulate the boundary-value problem for the potential for the newmodel, provided that z > O.

1. At regular points the potential satisfies Laplace's equation.

2. In the plane z = 0, due to symmetry with respect to the z-axis thenormal component of the electric field caused by the surface chargesequals zero; that is,

au-=0az


3. At the interface x = d

aVI aV2

11 ax = 12 ax

where VI and V2 are potentials in a media with conductivities 11 and 12'respectively.

4. In approaching the current electrode,

(111.392)

5. At infinity the potential tends to zero.

as R ~ 00

Comparing Eqs. (111.391), (111.392) shows that they completely coincide.Therefore, in accordance with the theorem of uniqueness, the potentialsin both models are also the same. For this reason it is sufficient to solvethe boundary problem for the new model, Fig. III.l9b, which is muchsimpler. Indeed, taking into account the axial symmetry of the model andthe potential with respect to the x-axis, it is convenient to use a cylindricalsystem of coordinates. Then making use of the results obtained in Exam-ple 6, expressions for the potential can be written in the form

(III.393)

x~d

x~dV2 = 100

Bme-mxJo( mr) dm,o

where Am and Bm are unknown coefficients and r = Vy2 + Z2.As was demonstrated earlier, both functions VI and V 2 satisfy Laplace's

equation and the boundary conditions. To provide continuity of thepotential and the normal component of the current density at the interfacex = d, we will make use of Eq. (111.374) and then obtain the followingsystem of equations for determination of Am and Bm:

(111.394)

where C = ptI/27T.


Solving this system we have

A =K Ce- 2mdm 12

B = 2P2 Cm PI + P2

Substituting Eqs. (III.395) we obtain

U = PI! [2. + K (X>e- m (2d - X)j (mr) dm]I 217' R 12Jo 0

and

(III.395)

x ~ d (111.396)

x~dPI! 2P2 100

U2 = - e-mXjo(mr) dm217' PI + P2 0

Then taking into account Eq, (III.374) we arrive at extremely simpleexpressions for the potential.

x~d

since

and

UI

= PI! (2. + K 12)

21T R R I

PI! (1 + Kn)U2 = 217' R x ~ d

2P21 +K12 = - - -

PI + P2

(III.397)

Thus we have solved the boundary-value problem and found the potentialin the presence of a vertical contact. It is appropriate to emphasize againthat Eqs, (111.397) describe correctly the field beneath the earth's surfaceonly.

As follows from these equations the field in the medium with conductiv-ity 'YI is equivalent to that caused by two elementary charges, one of them,eA , located at the origin 0, and the other K 12 eA , placed at the mirrorimage of the origin with respect to the contact. At the same time the fieldin the medium with conductivity 'Y2 coincides with that of the elementarycharge (l + K 12 )eA , located on the surface of the current electrode. Inother words, the presence of the vertical contact does not change the

(III.398)

1II.10 Behavior of the Electric Field in a Conducting Medium 389

geometry of the field in this part of the medium. Thus we have demon-strated that the field of surface charges, distributed over the contact, isequivalent to that of an elementary charge equal to the total surfacecharge K I2eA.

For illustration we will consider the behavior of the potential and thefield at the earth's surface along the x-axis. In accordance withEq. (III.397) we have

UI

= PI I (~+ K 12 )

27T L 2d - L

L~d

where L is the separation between the current and receiver electrodes.Correspondingly, for the electric field we have

L~d

PI / [ 1 K 12 1E 1X = 27T L2-(2d-L)2

pJ (1 + K 12 )

E2x = 27T L2

i.-:«(III.399)

where L is the distance from the current electrode to the middle pointbetween the receiver electrodes M and N, located very close to eachother.

Suppose that P2 > PI and consider the behavior of the potential at thepoint M as a two-electrode array AM with the constant separation Lmoves along the x-axis. If the array is located in the medium withresistivity PI' far away from the contact, the influence of surface charges isnegligible and

d»L

d=L

Approaching the contact, the contribution of positive charges becomesgreater. In particular, when the receiver electrode is located at the contactwe have

PII(I + K 12 )U=-----

27TL

As follows from the second equation in (HI.398), the potential remainsconstant if the current and receiver electrodes are located at each side of


the contact, and it is equal to

but the width of this zone coincides with the array length L. Finally, whenthe whole array AM is located in the medium with resistivity pz theexpression for the potential follows directly from Eqs. (III.398) afterreplacement of PI' K l z, 2d - L by Pz, K z!' 2d + L, respectively, and thenwe have

pz! ( 1 K zl ) pz! ( 1 K IZ )

U= 27T L + 2d+L = 27T L - 2d+L

Inasmuch as negative charges arise at the contact, with an increase ofthe distance from this interface the potential gradually increases andapproaches that of a uniform medium with resistivity pz.

pz!U~Uo=-

27TLd»L

The curve of the apparent resistivity for a two-electrode array

is shown in Fig. III.19c.Now we will study the behavior of the electric field component E l x

along the line x, when pz > PI. Far away from the contact, as in the caseof the potential, its influence is small, and we have

if d» L

Taking into account the fact that the charges, located at the currentelectrode and at the contact surface, are positive and the electric field ismeasured between them, the total field E l x decreases in approaching thecontact. In particular, when the point 0 is located in the vicinity of thecontact we have

d~L

As soon as the measuring point 0 intersects the contact, the electric field


increases and becomes equal to

d~L

This increase is natural since the electric fields, caused by the charges atthe current electrode and at the contact, have the same direction. Thus, adiscontinuity of the field Ex occurs at the contact and we have

Ezx l+KJZ Pz

e., 1 - K JZ PI

This equation can, of course, be derived from the continuity of the normalcomponent of the current density, and it shows that electrical methods canbe useful in detecting lateral changes of resistivity near the earth's surface.

In accordance with Eq. (III.399), the electric field does not changewhen the current electrode and the measuring point are located atdifferent sides of the contact, and it is equal to

This behavior of the electric field can be explained in the following way.As the current electrode approaches the contact, the surface charge anddensity of charges near the x-axis increase, and correspondingly theelectric field should increase too. However, the distance to the measure-ment point 0 simultaneously increases, so that results in a decrease of thefield; and as Eqs. (III.399) show, these two effects compensate each other.The width of the zone where the electric field is constant, as in the case ofa two-electrode array, equals L.

As soon as the current electrode intersects the contact, negative surfacecharges arise. Correspondingly the secondary field has a direction oppositeto that of the primary one. When both electrodes are located in themedium with the resistivity Pz, the expression for the electric field is

and with an increase in distance from the contact, the field graduallyincreases and approaches the primary field, Eo)Pz).

p zlEx ~ EoAPz) = 2rrLz


The apparent resistivity curve

e, E

PI Eo/PI)

is given in Fig. III.19d.

Example 8 Self Potential at the Borehole Axis in aUniform Medium

Until now we have considered examples of the behavior of the field whenthe current was introduced into a medium with the help of a man-made source. Several phenomena occur, however, that result in theappearance of a constant natural field in the earth. For instance, extrane-ous forces arise in areas where filtration of water through permeablerocks, such as sand and sandstone, takes place. In this case the extraneousforce of an electrokinetic origin can be represented as

E, = -Kf grad p

where p is the pressure and K, is a coefficient that depends on theproperties of water and rocks, and which is usually positive. Also, extrane-ous fields arise due to diffusion of a solution through rocks, when anionsand cations move with different velocities. Then this force is written as

Ed = K d grad log c

where c is the solution concentration filling rock pores, while K d is amultiplier that depends on the solution and rock structure and is positiveif the average velocity of anions is greater than that of cations. Anotherexample of a current source of a diffusive nature is the contact electromo-tive force,

which arises at interfaces of solutions with different concentration C I andC2 ·

Self-potential methods based on measuring natural electric fields areapplied in various problems of engineering, ground water, and mininggeophysics, as well as for detecting permeable zones crossed by wells.

Now we will investigate the behavior of the spontaneous potential atthe borehole axis, assuming that the conducting medium is uniform. Thismeans that the influence of charges at the borehole surface and at the top


z

a

Q

p

c

8 03.., <, 823

802 h

821

d

b ++

:+- +

Q : +- ~ - - ~ - -- +

h

M

z

++ -+ -+ :+ -+ : - - - - --

-+++++r

+++++

index of curves

iiFig. 111.20 (a) Model of a borehole intersecting a layer; (b) distribution of double layers; and(c) distribution of potentials on the borehole axis.

and bottom of the layer is neglected. At the same time, due to electro-chemical processes, double layers appear at these interfaces and they giverise to an electric field. The system of double layers consists of two planedouble layers S21 and S23 of infinite extent, located at the bottom and topof the layer, and cylindrical double layers SOl' S02' S03' located at theborehole surface against the medium beneath the layer, the layer itselfand the medium above the layer, respectively (Fig. III.20a).

The potential at the observation point M is equal to the sum ofpotentials caused by each double layer.

(III.400)

(III.401)

In accordance with Eqs. (111.69) and (III.22l) the potential of a uniformdouble layer is

wU= -15'

47T

where w is the magnitude of the solid angle, subtended by the double-layer


surface, as viewed from the observation point M; and it' is the voltagebetween both sides of the double layer.

To describe the behavior of the potential, we will introduce a cylindricalsystem of coordinates r, ip, Z with origin 0, where the horizontal plane atthe middle of the layer intersects the borehole axis z. Also we will use thefollowing notations: h, d, and z are the layer thickness, the boreholediameter, and the distance from the observation point to the origin,respectively; it'ZI' it'Z3, it'OI' it'oz, it'03 are the electromotive forces of thecorresponding double layers or the potential differences between surfacesof a double layer; that is

it'ik = U(i) - U(k)

As was demonstrated in Chapter I, the solid angles, subtended by aninfinite plane and a closed surface, are equal to ± 27T and either 47T or 0,respectively, depending on the position of the observation point withrespect to this surface.

Then the solid angles for each double layer can be represented as

(Ill A02)

where wp and wq are the solid angles, subtended by the borehole cross-sections with the coordinates z = =+= h/2, respectively. SubstitutingEqs. (IlIA01), (IlIA02) into Eq. (IlIAOO) we obtain

(Ill A03)

lIUO Behavior of the Electric Field in a Conducting Medium 395

In accordance with Eq. (1.48),

hz--

2

J(z- %r+ :Zh

z+-2

J(z + ~ r+ :Z

(111.404)

22

where f3p and f3q are the angles shown in Fig. 1II.20b.Correspondingly, instead of Eq, (I1I.403) we have

2z + h g"23 - g"03 + g"ozU = ----;:=====:==-----

V(2z+h)z+dz 2

2z - h g"Zl - g"01 + g"oz g"03+ g"Ol----;:====;:==-----+---V(2z-h)z+d z

If the layer is surrounded by a uniform medium and the mineralizationof water near the bottom and top of the layer is practically the same, theng"Zl = g"Z3' g"OI = g"03' and therefore

or (111.405)

wherez

z* =-d'

hh* =-

d


As follows from this equation, with an increase of the distance from thelayer, the potential U tends to a constant equal to ,wOl; that is,

U(oo)~,w01

Since the potential at the observation point M is measured with respect tothat at infinity we have

where

liU= U(z*,h*) - U(oo) = -iF(z*,h*),ws (111.406)

2z* + h*F(z* h*) = ---;======

, V(2z*+h*)2+ 1

2z* - h*(IIIAO?)

is a function characterizing a change of liU along the borehole axis, and

(IIIAOS)

is the algebraic sum of electromotive forces at interfaces of the layersurrounding medium and the borehole.

As is seen from Fig. III.20c, the curves illustrating the dependence ofthe function liU /,ws on the coordinate z* are symmetrical with respect tothe origin. The index of curves is h*.

In conclusion let us emphasize that in spite of the fact that this solutionis an approximate one, since it does not take into account the change ofresistivity, its analysis is very useful for understanding the behavior of thespontaneous potential at the borehole axis.

References

Alpin, L.M. (1966). "The Theory of Field." Nedra, Moscow.Alpin, L.M., Sheiman, S.M. (1936). Calculations of self-potential oil-gas report. ONTI.Bursian, V.P. (1972). "The Theory of Electromagnetic Fields Applied in Electrospecting."

Nedra, Moscow.Dachnov, V.N. (1967). "Electric and Magnetic Methods of Logging." Nedra, Moscow.Kaufman, A.A. (1990). The electric field in a borehole with a casing. Geophysics, 55:1.Smythe, W.R. (1968). "Static and Dynamic Electricity." 3d, ed. McGraw-Hill, New York.Tamrn, I.E. (1946). "Theory of Electricity." GITTL, Moscow.Wait, J.R. (1982), "Geo-electromagnetism." Academic Press.

Chapter IV Magnetic Fields

IV.l Interaction of Currents, Biot-Savart's Law, the Magnetic FieldIV.2 The Vector Potential of the Magnetic FieldIV.3 The System of Equations of the Magnetic Field B Caused by Conduction

CurrentsIVA Determination of the Magnetic Field B Caused by Conduction CurrentsIV.S Behavior of the Magnetic Field Caused by Conduction CurrentsIY.6 Magnetization and Molecular Currents: The Field H and Its Relation to the

Magnetic Field BIV.7 Systems of Equations for the Magnetic Field B and the Field HIV.8 Behavior of the Magnetic Field Caused by Currents in the Earth

The External and Internal Components of the Normal Field BN

Behavior of the Secondary Magnetic Field Due to Induced MagnetizationThe Secondary Magnetic Field When Interaction between Molecular Currents

Is NegligibleThe Magnetic Field Due to the Remanent Magnetization

References

In this chapter we will discuss the theory of the time-invariant magneticfield and its application in geophysics.

As is well known, magnetic methods are used to solve various problemssuch as

1. Mapping the basement surface and sediments in oil exploration.2. Detecting different types of ore bodies in mining prospecting.3. Detecting metal objects in engineering geophysics.4. Mapping basement faults and fracture zones.5. Determining zones with different mineralization in logging, as well

as inspecting casing parameters.6. Studying the magnetic field of the earth and its generators.

In addition it is appropriate to notice that often the behavior ofalternating magnetic fields practically coincides with that of the time-invariant magnetic field. Therefore, some results derived in this chapterremain valid for alternating fields, as well.

397

398 IV Magnetic Fields

IV.I Interaction of Currents, Biot-Savart's Law,the Magnetic Field

Earlier we introduced gravitational and electric fields by considering theinteraction of masses and charges, respectively. Following this pattern wewill introduce the magnetic field by studying the interaction of constantcurrents. Also, in the same manner as in the case of the electric field, wewill first develop the theory of magnetic fields in free space, and then theeffect of media (magnetic materials) will be taken into account.

Numerous experiments performed in the last century demonstratedthat currents interact with each other; that is, mechanical forces act atevery element of a current circuit. It turns out that this force depends onthe magnitude of the current, the direction of charge movement, the shapeand dimensions of the current circuit, as well as the distance and mutualorientation of the circuits with respect to each other.

This list of factors clearly shows that the mathematical formulation ofthis interaction should be a much more complicated task than that forgravitational and electric fields. In spite of this fact, Ampere was able toformulate an expression for the interaction of currents in a relativelysimple manner.

(IV.I)

where II and 12 are magnitudes of the currents in the linear elements dl;and d/2 , respectively, and their direction coincides with that of thecurrent density; L qp is the distance between these elements and L qp isdirected from point q to the point p, which is located at the center ofcurrent elements; and finally f.Lo is a constant equal to

f.Lo = 41T' 10-7 Him

f.Lo is usually called the magnetic permeability of free space.In applying Ampere's law it is essential to note that the separation

between current elements must be much greater than their length; that is,

Let us illustrate Eq. (IV.1) by three examples shown in Fig. IV.1.

(a) Suppose that elements d/I and dt; are in parallel with each other.Then as follows from definition of the cross product, the force dF(p) isdirected toward element dt;, and the two current elements attract eachother (Fig. rv.io.

IV.I Interaction of Currents, Biot-Savart's Law, the Magnetic Field 399

a b

q

c

11d/ 1

..-- - - -----1:.,.,-.,.r -dF(q)=o

Fig. IV.I (a) Interaction of currents having the same direction; (b) interaction of currentshaving opposite direction; and (c) interaction of current elements perpendicular to eachother.

(b) If two current elements have opposite directions, the force dF(p)tries to increase the distance between elements, and therefore they repeleach other (Fig. I'V.La),

(c) If the elements dt; and dt; are perpendicular to each other, as isshown in Fig. IV.lb, then in accordance with Eq. (IV.I) the magnitude ofthe force acting at the element dtl equals

while the force dF(q) at the point q is equal to zero. In other words,Newton's third law becomes invalid. This contradiction results from thefact that Eq. (Iv.I) describes interactions between current elements in-stead of closed current circuits.

By applying the principle of superposition the force of interactionbetween two arbitrary currents is defined as

(IV.2)


where .2"1 and .2"2 are the current lines along which integration isperformed, and p ¢ q.

Due to these forces actin~ at different points of the contour, varioustypes of movement can occur. In the same manner, masses and chargesmove under action of the gravitation and electric fields, respectively. It isalso appropriate to notice that in SI units F is measured in newtons.

Inasmuch as there is interaction between currents, it is natural byanalogy with the gravitational and electric fields to assume that currentscreate a field, and due to the existence of this field other current elementsexperience the action of the force F. It is natural to call such a field themagnetic field, and it can be introduced from Ampere's law as

Here

and

dF( p) = I) dl'( p) X dB( p)

/La dl'( q) X L qpdB(p) = -I 3

47T L qp

(IV.3)

(IVA)

dl'zCq) = d/(q),

(IV.5)

Equation (IlIA) is called the Biot-Savart law, and it describes therelationship between the elementary linear current and the magnetic fielddB. The vector dB is often called the vector of magnetic induction and itcharacterizes the magnetic field in the same way that vectors q and Edescribe the gravitational and electric fields, respectively. In accordancewith Eq. (IVA) the magnitude of the magnetic field dB is

/La dt.dB = -4I(q)-2- sm(Lqp,dl')

7T L qp

where (L q p , de) is the angle between the vectors Lq p and d/, and thevector dB is perpendicular to these vectors as in shown in Fig. IV.2a. It isobvious that the unit vector ba , characterizing the direction of the field, isdefined by

In SI units the vector of the magnetic induction is measured in teslas, andit is related to other units, such as gauss and gamma, in the following way:

1 tesla = 109 nT = 104 gauss = 109 gamma

IV.I Interaction of Currents, Biot-Savarl's Law, the Magnetic Field 401

Fig. 1V.2 (a) Magnetic field of a current element and (b) magnetic field of a surface current.

Now we will generalize Eq. (IVA) assuming that along with linear currentsthere are also volume and surface currents.

First let us represent the product I dl" as

I dl" = j dS dl" j dSdt=j dV ( IV.6)

where dS is the cross section of the current element, and j is the volumecurrent density.

If the current is concentrated in a relatively thin layer with thicknessdh, which is small with respect to the distance to the observation points, itis convenient to replace this layer by a current surface. As is seen fromFig. IV.2b the product I dl" can be modified in the following way:

I dl" = j dV = j dh dS = i dS

Here dS is the surface element, and

i =j dh

(IV.7)

is the surface density of the current.Now applying the principle of superposition for all three types of

currents and making use of Eqs. (IVA), (IV.6), (IV.7) we obtain thegeneralized form of the Biot-Savart law.

/-La [f j X L qp .:L q p ;., rf.. dl" X L q p ]B(p) = - 3 dV+ 3 dS + LJ Ii'f' 3 (IV.S)4rr v L qp S L q p ;=1 L q p


1. Equation (IV.S) allows us to calculate the vector of magnetic induc-tion everywhere including areas inside of volume currents.

2. Unlike volume distribution of currents, the linear and surface analo-gies are only mathematical models of real distribution of currents, which


are usually introduced to simplify calculations of the field and study itsbehavior. For this reason the equation

.( ) XLB( p) = ~ f J q 3 qp dV

47T V L q p

(IV.9)

in essence comprises all possible cases of the current distribution.3. In accordance with the Biot-Savart law the current is the sole

generator of the magnetic field, and the distribution of this generator ischaracterized by the magnitude and direction of the current density vectorj. And as was shown in Chapter III the vector lines of j are always closed.This means that the magnetic field is caused by generators of the vortextype and correspondingly we are dealing with a vortex field, unlike thegravitational and electric fields.

4. All experiments that allowed Ampere to derive Eq. (IV.!) werecarried out with closed circuits. At the same time Eq. (IV.1), as well asEq. (IVA), is written for the element d/, where the current cannot exist ifthis element does not constitute a part of the closed circuit. In otherwords, Eqs. (IV.!) and (IVA) cannot be proved by experiment, but theinteraction between closed current circuits takes place as if the magneticfield B, caused by the current element I d/, is described by Eq. (IVA).

Let us illustrate this ambiguity in the following way. Suppose that themagnetic field dB, due to the current element I d/, is

fLo dl'( q) X L q pdB(p) = - 3 + I grad ep dE

47T L q p

where ep is an arbitrary continuous function. Then, the magnetic fieldcaused by the current in the closed circuit is

fLOI~ dl'x L q p ¢B( p) = - 3 + I grad ep dE47T:.:z L q p

or, making use of results described in Chapter I,

However, it is important to emphasize that this ambiguity vanishes whenthe interaction or the magnetic field of closed current circuits is consid-ered. In other words, Eqs. (IV.8), (IV.9) uniquely define the magneticfield B.

IV.! Interaction of Currents, Biol-Savart's Law, the Magnetic Field 403

5. In accordance with Eq, (IV.S), the magnetic field caused by a givendistribution of currents depends on the coordinates of the observationpoint p only, and it is independent of the presence of other currents. Theright-hand side of Eq. (IV.S) does not contain any terms that characterizephysical properties of the medium, and therefore the field B at point p,generated by the given distribution of currents, remains the same if freespace is replaced by a nonuniform conducting as well as polarizablemedium. For instance, if the given current circuit is placed inside of amagnetic material, the field B, caused by this current, is the same as if itwere in free space.

Of course, as is well known, the presence of such a medium results in achange of the magnetic field B, but this means that inside of the magneticmaterial along with the given current there are other currents, which alsoproduce magnetic fields. This conclusion directly follows from Eq. (IV.S),which states that any change of the magnetic field B can happen only dueto a change of the current distribution.

Later we will take into account the influence of currents in magneticmedia, but now it is assumed that such media are absent and onlyconduction currents are considered.

6. As follows from Eqs. (IV.3) and (IV.6), (IV.?) elements of linear aswell as surface and volume currents, placed in a magnetic field Baresubject to the action of a force, which is

F=Id/X B, F = (i x B) dS, F = (j x B) dV (IV.lO)

At the same time, forces acting on elementary charges and masses due tothe electric and gravitational fields are

and

dF=AdtE,

dF=Adtq,

dF=IEdS,

dF=IqdS,

dF=oEdV

dF = oqdV

(lV.ll)

From comparison of Eqs. (IV.lO) and rrv.in we can conclude thatthere is an analogy between vectors B, E, and g. In fact, these threevectors determine the force acting on the corresponding generator of thefield. In this sense the vector B, describing the magnetic field, is similar tothe vector E, which characterizes the electric field. There is anothercommon feature of these fields, namely, they are caused by generators ofone type only, which have an obvious physical meaning: charges andcurrents.


In this context let us describe the forces acting on an electron. It isobvious that the force of the electric field is

(IV.12)

where e is the charge of the electron.Next we will consider an elementary value dV and suppose that the

current is formed by motion of electrons. Then the current density j canbe presented as

j=nev

where n and v are the number of electrons in this volume and theirvelocity, respectively.

Therefore, the force of the magnetic field B acting on all electrons is

and correspondingly, every electron is subjected to a force equal to

where

FB =evX B (IV.13)

e<OEquations (IV.12), (IV.l3) again indicate the similarity between the fieldsE and B, which define the force acting on the electron due to the electricand magnetic fields. Let us notice that Eq. (IV.l3) was derived by Lorentz,and correspondingly FB is called the Lorentz force.

Until now we have discussed the analogy between the vectors E and B,but it is also proper to emphasize their difference. In fact, the electric fieldis a source field, caused by charges, while the magnetic field B is a vortexfield, generated by time-invariant currents. Therefore, it is natural toexpect that the behavior of these fields differs essentially from each other.For instance, an electric field E forces an electron to move along its vectorline, while a magnetic field B creates the force FB , which is perpendicularto this field.

7. In the next volume Part B we will demonstrate that the Biot-Savartlaw remains valid even for a certain type of alternating electromagneticfield. This is also true for Coulomb's law.

8. Although calculation of the magnetic field, making use of theBiot-Savart law, is not a very complicated procedure, it is still reasonableto find a simpler way of determining the field. With this purpose in mind,by analogy with the scalar potential of the gravitational and electric fields,we will introduce a new function. Moreover, there is another reason toconsider this function, and it is related with the fact that the Biot-Savart

!V.2 The Vector Potential of the Magnetic Field 405

law allows us to determine the magnetic field B, provided that thedistribution of currents is given. However, we will study cases when thecurrents can be known only if the magnetic field is already determined.We faced a similar problem of "the closed circle" when the electric fieldwas investigated in the presence of dielectrics and conductors. Then wederived the system of field equations and formulated the boundary-valueproblems, allowing us to find the electric field. The same approach will bedeveloped for the field B, and it turns out that this task is essentiallysimplified if first we introduce a new function called the vector potential ofthe magnetic field.

IV.2 The Vector Potential of the Magnetic Field

We will proceed from Bior-Savart's law.

.( ) XLB(p) = ~ f J q 3 qp dV

4'1T V L q p

As was shown in Chapter I,

L q I p I--.!!.!!...=\!--=-\!-L~p i: i;

(IV.14)

(IV.IS)

Substituting Eq, (IY.IS) into Eq. OY.14) we have

B(p) = ~ f j(q) XV_1_ dV= ~ f (V _1_ Xj) dV (IV.16)4'1T v L q p 4'1T V L q p

since the relative position of vectors forming the cross product is changed.Now we will make use of the equality

p

p j PI. v Xj\! X - = \!- XJ +--

-: -; -;

which follows from the vector identity (1.169).

\! X ( epa) = \!ep X a + ep\! X a

Applying Eq. (IV.l?) we can rewrite Eq. (IV.16) as

p

/-La f p j /-La f v X jB(p) = - \! X -dV- - --dV4'1T v L q p 4'1T V L q p

(IV.l7)

(IV.18)


The current density j is a function of the point q and does not depend onthe location of the observation point p. Therefore, the integrand of thesecond integral is zero and

f.L0j p j(q)B(p) = - curl--dV

4'7T v L qp

(IV.19)

Inasmuch as the integration and differentiation indicated in Eq. (IV.19)are carried out with respect to the two independent points q and p, wecan interchange the order of operations and obtain

or

where

p f.L0jj(q)B(p) = curl- - dV

4'7T V L qp

B(p) = curl A

A(p) = ~ j j(q) dV4'7T V L qp

(IV.20)

(IV.21)

(IV.22)

Thus, the magnetic field B, caused by constant currents, can be ex-pressed through the vector potential A defined by Eq, (IV.22). ComparingEqs. (IV.14) and (IV.22) we see that the function A is related to thedistribution of currents in a much simpler way than the magnetic field is,and therefore one reason for introducing this function is already demon-strated. In accordance with Eq. (IY.22), A is a vector, unlike the scalarpotential of the gravitational and electric fields, and its magnitude anddirection at every point p depends essentially on the current distribution.Now let us derive expressions for the vector potential A, caused by surfaceand linear currents.

Making use of the equalities

j dV= i dS

it follows from Eq, (IV.22) that

A = f.Lo f idS4'7T s L q p

and

and

j dV=Idl'

(IV.23)

Applying the principle of superposition we obtain an expression for the

IV.2 The Vector Potential of the Magnetic Field 407

vector potential caused by volume, surface, and linear currents.

(IV.24)

The components of the vector potential can be derived directly from thisequation. For instance, in Cartesian coordinates we have

/-La [f i, dV f iy dS N rr. dey]A =- ---+ --+ '[J'Y-y 47T v L q p S L q p ;=1 I L q p

(IV.25)

Similar expressions can be written for the vector potential components inother systems of coordinates.

As is seen from Eqs. (IV.25), if a current flows along a single straightline, the vector potential has only one component, which is parallel to thisline. It is also obvious that if currents are situated in a single plane, thenthe vector potential A at every point is parallel to this plane. Later we willconsider several examples illustrating the behavior of the vector potentialand the magnetic field B, but now let us derive two useful relations for thefunction A, which simplify to a great extent the task of deriving the systemof the magnetic field equations.

First, we will determine the divergence of the vector potential A. Asfollows from Eq. (IV.22), we have

P p /-La j(q)divA = div-f --dV

47T v L q p

Since differentiation and integration in this expression are performed withrespect to different points, we can change the order of operations andthen obtain

P P "( )

divA = ~ f div~dV47T v L q p

(IV.26)

The volume over which the integration is carried out includes allcurrents, and therefore it can be enclosed by a surface S such that outside


of it currents are absent. Correspondingly, the normal component of thecurrent density at this surface equals zero.

jn = 0 on S

The integrand in Eq, (IV.26) can be represented as

(IV.27)

(IV.29)

(IV.28)

p

p j Vj. pl. p 1V-=-+J'V-=J'V-

i-; -; L q p -:

because the current density does not depend on the observation point. andp

div j(q) = 0

Thus, we haveq

p 1 q 1 q j Vjj,V-=-j'V-=-V-+-

i: L q p i: i;q

q j div j= - div-- +--

i: i;In accordance with the principle of charge conservation,

q

div j( q) = 0

and thereforep 1 q'

• t'7 di JJ' v-= - lV--Lq p i:

Correspondingly, Eq, OV.26) can be written asq •

divA = _!!:!!-1div _J_. dV41T v L q p

On the right-hand side of this equation both integration and differentia-tion are performed with respect to the same point q so that we can applyGauss' theorem. Then we have

ILo 1. q j ILo (j. dSdivA = -- div-dV= --j,--

47T v L q p 47T S L q p

ILo f i; dS=-- --

47T S L q p


Taking into account the fact that the normal component of the currentdensity t; vanishes at the surface S, which surrounds all currents,Eq. (IV.27), we obtain

divA= 0 (IV.30)

This is the first relation that is useful for deriving the system of fieldequations. Let us note that in accordance with Eq. (IV.30) the vector linesof the field A are always closed.

In Chapter III it was shown that the potential of the electric field Usatisfies Poisson's equation.

which has a solution of the form

U=_1_j8dV41Teo v L q p

As follows from Eq. (IV.25) every Cartesian component of the vectorpotential has the same form as the potential U, and therefore by analogy italso satisfies Poisson's equation; that is,

Multiplying each of these equations by the corresponding unit vector i,j, kand summing, we obtain the equation for the vector potential A.

(IV.32)

Now we are ready to derive the system of field equations of themagnetic field, but first let us consider several examples illustrating thebehavior of the field and its vector potential.

Example 1 The Magnetic Field of the CurrentFilament (Fig. IV.3a)

Taking into account the axial symmetry of the problem we will choose acylindrical system of coordinates r, ip, Z, with its origin situated on thecurrent-carrying line. Starting from the Biot-Savart law we can say thatthe magnetic field has only the component Btp' which is independent ofthe coordinate rp. From the principle of superposition it follows that the


a b

c

dl

zp

x

d

Fig. IV.3 (a) Magnetic field of a current line; (b) magnetic field at the axis of a current loop;(c) magnetic field of current loop at an arbitrary point; and (d) magnetic field of a magneticdipole.

total field is the sum of fields contributed by the current elements I dz.Then we have

(IV.33)

where L qp = (r 2 + Z2)3I2 and z is the coordinate of the element dz. Thecoordinates of the observation point are rand z = 0, and z1 and z2 arecoordinates of terminal points of the current line.

It is clear that the absolute value of the cross product is

IdzXLqpl =dzLqpsin(dz,Lqp)

= dz L qpsin f3 = dz L qpcos a

Thus,

/La! j Z 2 dzf3 = -- -- cos a'I' 47T ZI L~p

(IV.34)


Inasmuch as z = r tan a we have

dz = r sec? a da and

(IV.35)

(IV.36)

Substituting these expressions into Eq. (IV.34) we obtain

/-La l f"'2B", = -- cos a da4'ITf "'I

Thus, the final expression for the magnetic field caused by the currentflowing along a straight line has the form

Bcp( p) =/-La! (sin a 2 - sin a 1 )

4'ITr

where az and a 1 are the angles subtended by the radii from the point p tothe ends of the line.

Next suppose that the current-carrying line is infinitely long so that thetwo angles a z and al have values 'IT12 and - 'IT/2, respectively. Then

/-Lo!Bcp(p) = 2'ITr

In the case of a line that is only semi-infinite, a j = 0 and a z = 'IT12, wehave

/-Lo!B (p) =-

'" 4'7Tr(IV.37)

(IV.38)

Now we will assume that a z = a and a 1 = -a. Then in accordance withEq. (IV.35) we have

/-Lo! • /-La l tB = --SIn a = -- ----:-=

'" 2'ITr 2'ITr (r 2 +( 2 ) 1/ 2

where 2t is the length of the current-carrying line. If t is significantlygreater than the distance r, the right-hand side of Eq. (IV.38) can beexpanded in a series in terms of (rIt )2. Then we obtain

412 IV MngneticFields

We see that if the length of the current line 2t is four or five times largerthan the separation r, the resulting field is practically the same as that dueto an infinitely long current-carrying line. It is proper to notice thatEq, (IV.35) is often used in electromagnetic methods for calculating theprimary magnetic field caused by the current in an arbitrary circuit.

Example 2 The Vector Potential A and the MagneticField B of a Current Flowingin a Circular Loop

First assume that the observation point is situated on the axis of a loopwith radius a, as is shown in Fig. IV.3b. Then in accordance withEq. (IV.22),

Inasmuch as the distance L qp is the same for all points on the loop, wehave

By definition the sum of the elementary vectors dl' along any closed pathis zero. Therefore, the vector potential A at the z-axis of a circular currentloop vanishes.

Now we will calculate the magnetic field on the z-axis. From theBiot-Savart law, Eq. (IV.8), it can be seen that in a cylindrical system ofcoordinates each current element I dt creates two field components dBzand dBr • However, it is always possible to find two current elements I d/that contribute the same horizontal component at any point of the z-axisbut with opposite signs. Therefore, the magnetic field has only a verticalcomponent along the z-axis.

As can be seen from Fig. Iv.Jb, we have

since Id/XLqpl = Ldt.


Having integrated along the closed path of the loop, we finally obtain

(IV.39)

where

M = I7ra2 =]S

with S being the area enclosed by the loop.When the distance z is much greater than the radius of the loop a, we

arrive at an expression for the magnetic field, which plays a very importantrole in the study of the magnetic and electromagnetic fields. Neglecting ain comparison with z we have

/-LaMB =--

z 21TZ 3 if z» a (IVAO)

When the intensity of the field does not separately depend on the currentor the loop radius, but is defined by the product M = IS, we call this thefield of the magnetic dipole. Thus, a relatively small current-carrying loopwith radius a creates the same magnetic field as a magnetic dipole havingthe moment M = 1Ta 2

] oriented along the z-axis. It can also be seen fromEq. (IY.39) that when the distance z is at least five times greater than theradius a, the treatment of the loop as the magnetic dipole situated at thecenter of the loop results in an error of no more than 5%.

So far we have considered the vector potential and the magnetic fieldonly along the z-axis. Now we will investigate a general case, and first ofall calculate the vector potential at any point p. Due to symmetry thevector potential does not depend on the coordinate 'P. For simplicity wecan then choose the point p in the x - z plane, where 'P = O. As can beseen from Fig. IY.3c every pair of current-carrying elements, equallydistant from point p and having coordinates 'P and - 'P, create a vectorpotential dA located in a plane parallel to the x - y plane. Inasmuch asthe whole loop can be represented as the sum of such pairs, we concludethat the vector potential A caused by the current-carrying loop has only

(IVAI)


the component A<p. Therefore, from Eq. (IV.22) it follows that

A = /-La l rf, dt'<p'I' 417 'Y R

/-L aI 7T a cos r.p d r.p

= 217 fa (a 2+ r2+ Z2 - 2ar cos r.p )1/2

where de<p is the component of dl' along the coordinate line r.p, and

de<p =acos r.pdr.p, R = (a 2+ r 2+Z2 - 2arcos r.p)1/2

Letting r.p = 17 + 2a we have

dip = 2 do:

and therefore

and cos r.p = 2 sin2 a-I

al/-La j7T/2 (2sin 2 a-I) daA<p = -- [2 ] 1/2

17 0 (a +r) +z2-4arsin2a

Introducing a new parameter

2 4ark =--------;;---

(a+r)2+ z 2

and carrying out some fairly simple algebraic operations we obtain

(IVA2)

where K and E are complete elliptical integrals of the first and secondkind.

(IVA3)

j 7T/2 2 . 2 1/2E ( k) = (1 - k sin a) da

o


These functions have been studied in detail and there are standardalgorithms for their calculation.

Using the relationship between the vector potential and the magneticfield, as given in Eq. (IV.2l), we have in a cylindrical coordinate system.

aA<pB=--

r az '1 a

Bz=--(rA<p),r ar

As is known for elliptical integrals,

and

aK

akE K

k'

aEak

E K---

k k

az 4ar'

Bk k k 3 k 3

-=-----ar 2r 4r 4a

Therefore after differentiation we have

(IV,44)

Thus, in general the magnetic field caused by the current flowing in acircular loop can be expressed in terms of elliptical integrals.

Example 3 The Magnetic Field and Its Potentialfor a Relatively Small Current Loop(Fig. IV.3d; the Magnetic Dipole andIts Moment (Fig. IV.3d)

Suppose that the distance from the center of the current-carrying loopto the observation point Ro is considerably greater than the loop radius;that is,

cos ep dep


Then, Eq. (IVAl) can be simplified so that we have

Il-ola TT cos ep de:A<p ::::: -2-1 ( 2 1/2

1T 0 R o - 2 ar cos ep )

laMo JTT= 21TRoo -[1-_-(-=-~-a_~-r -)-co-S-ep-j-:-l/",",,"2

::::: lall-o 1'"(1 + a: cos ep)cos ep dip

21TR o 0 R o

lall-o JTT= --- cos ep dip

21TRo 0

where the relation

1------:::-n::::: I-nx(1 +x)

(IVA5)

has been used assuming that nx« 1. The first integral in Eq. (IVA5)vanishes so that we obtain

or

(IVA6)

where S is the area of the loop and index "0" is omitted, that is Ro = R.Now we consider a spherical system of coordinates, R, 8, ep with its

origin at the point 0 and with the same z-axis; that is, from this z-axis asz > 0, the direction of the current is seen counterclockwise. ThenEq. (IVA6) can be rewritten as

ll-olS. .A = --2Slll 81<p

41TR(IVA7)


Next we will introduce the moment of the loop as a vector directedalong the z-axis, whose magnitude is equal to the product of the current inthe loop and its area; that is,

M = ISzo= Mzo (IV.48)

where M=IS.It is essential to note that the moment M and the direction of the

current form a right-handed system.Thus, instead of Eq. (IV.47) we can write

A MoM. fJ"=--sm I47T"R2 'P

or

(IV.49)

since

M X R = MR sin fJi<p

Equation IV.49 will be used to account for the influence of molecularcurrents in magnetic materials.

Now proceeding from Eqs. (IV.2l), (IV.49) and taking into account thefact that

AR=Ae=O

we obtain the following expressions for the magnetic field in a sphericalsystem of coordinates:

Mo a(sinfJA<p)BR = RsinfJ afJ '

Whence

v« a( RAcp)B = -----'--

e R aR ' B<p=O

2MoMBR = --3 cos fJ,

47T"R

MoM.Be = --3 sin fJ,

47T"R

These equations describe the behavior of the magnetic field of arelatively small current loop; that is, its radius is much smaller than thedistance from the loop center to the observation point. This is the mostimportant condition to apply to Eqs. (IV.50), while the values of the loopradius and the distance R themselves are not essential. We will call themagnetic field, described by Eqs, (IV.50), that of a magnetic dipole withmoment M. Here it is appropriate to make two comments.

(IV.51)


1. In the case of the electric field the "dipole" means a combination ofequal charges having opposite signs, when the field is determined atdistances essentially exceeding the separation between these charges. Atthe same time the notion of a "magnetic dipole" does not imply theexistence of magnetic charges, but it simply describes the behavior of themagnetic field due to the current in a relatively small loop.

2. The magnetic field of any current loop, regardless of its shape, isequivalent to that of a magnetic dipole when the field is defined atdistances much greater than loop dimensions. In other words, any currentcircuit creates a magnetic field such that far away from currents itcoincides with the field of a magnetic dipole.

The main features of the field of the magnetic dipole directly followfrom Eqs. (IY.50), and they are

(a) At points of the dipole axis z the field has only one component B,directed along this axis, and it decreases inversely proportional to Z3.

/-LaMB =--

z 277" Z3

(b) At the equatorial plane () = 77"/2, the radial component BR van-ishes, and the field has the direction opposite to that of the magneticdipole.

(IV.52)

(IV.53)

(c) Along any radius e= constant, both components of the field, BR

and Be' decrease inversely proportional to R 3• At the same time the ratio

of these components, as well as the orientation of the total vector withrespect to the radius R, does not change. In fact, according to Eq. (IV.50)we have

Be 1- = - tan eBR 2

(d) It is interesting to notice that a very simple dipole field describesthe main part of the magnetic field of the earth. This fact is also useful inpaleomagnetism studies.

Now let us suppose that there are several relatively small loops withdifferent moments Mi' Then applying the principle of superposition wehave for the total moment of this system,

M=~Mi

Thus we have replaced a system of small current loops by one small loop

(IV.54)

IV.2 The Vettor Potential of the Magnetic Field 419

with the moment M. If there is a continuous distribution of such currentloops, then for the total moment we have

M= l p( q ) dVv

where q is an arbitrary point of the volume and P characterizes thedensity of moments

dMp= - (IV.55)

dV

In accordance with Eqs, (IV,49) and (IV.55) the vector potential dA,caused by current loops in an elementary volume dV, is

(IV.56)

(IV.57)

where L qp is the distance between an elementary volume dV and theobservation point p.

Now applying again the principle of superposition, we obtain for thevector potential A, caused by a volume distribution of current loops, thefollowing expression:

A(p) = ~ 1P(q) ; L q p dV41T V L qp

which plays a fundamental role in the development of the theory of themagnetic field B in the presence of magnetic materials.

Example 4 Mechanical Force and the Rotation Moment

In accordance with Eq. (IV.2) the mechanical force acting on the contour2' placed in the magnetic field B can be represented as

where dl'is the contour element, directed along the current.In particular, if the magnetic field B is uniform in the vicinity of the

contour 2' we have


Since the integral is a sum of vectors d/, which form a closed polygon ~,

we have

~d/=O:5&'

Correspondingly, the total force F acting on the current contour in auniform magnetic field B equals zero.

As is well known, the movement of rotation with respect to an arbitrarypoint 0 is defined as

M r = ~Oq X dF =/¢Loq X (d/X B)

Making use of the identity

a X (b X c) = (a . c)b - c(a . b)

we obtain

(IV.58)

(IV.59)

If we assume again that the magnetic field is uniform in the vicinity of thecontour ~, then the second integral in Eq. (IV.59) can be represented as

Applying Stokes' theorem we have

~ L Oq • dl'= fcurl L Oq ' dS:5&' s

where S is the area surrounded by the contour 2'. Performing thecalculation of curl L Oq in a spherical system of coordinates it is easy to seethat

curlLoq == 0

and therefore instead of Eq, (IV.59) we have

(IV.6D)

Considering the two-dimensional gradient in Chapter I we derived theequality


which can be presented as

(IV.6l)

Then making use of Eq. (Iv.Sl) we can rewrite Eq. nV.6D) in the form

M r = If dS X V(L Oq • B)s

Here dS = dS nand n is the unit vector normal to the surface S, whichforms together with the direction of the current a right-handed system.

Taking into account the fact that the magnetic field is uniform, we have

V(L o q ' B) = (BV)L Oq = B

and thus

M r = If dS X B = - IB X f dSs s

or

Mr=MXB

where M is the magnetic moment of the current contour, and

M = If dS = If n dSs s

(IV.62)

(IV.63)

If the current contour is plane, then Eq, IV.62 is simplified and we obtain

Mr=IS X B (IV.64)

where S = Sn and S is the total area of the contour.As follows from Eq. (IV.64) the moment of rotation is located in the

plane of the current contour.

Example 5 Behavior of the Tangential Componentof Field B near Surface Currents

First suppose that the current is uniformly distributed at the plane surfaceSand i is the current density (Fig. IV.4a). Then in accordance with theBiot-Savart law, the magnetic field caused by surface currents is

~O f i(q) X L q pB(p) = - dS

4rr s L~p(IV.6S)


Fig. IVA (a) Magnetic field of a uniform distribution of currents on a plane; (b) magneticfield due to an arbitrary distribution of surface currents; (c) normal component of themagnetic field near surface currents; and (d) circulation of the magnetic field.

To find the tangential component of the field we will multiply bothsides of Eq. (IY.65) by the unit vector t, which is parallel to the surface S,and obtain

J.L (i XL) . tBt ( p) = B . t = _0 f 3

q pdS

47T S L q p

or

J.Lo (tXi)'LBt(p) = -f ' 3 qp dS

47T S L q p

(IV.66)

Inasmuch as both vectors t and i are tangential to the surface S, the crossproduct in Eq, (IV.66) can be written as

t X i = in sin( t, i)

where i is the magnitude of the current density and n is the unit vectorperpendicular to S. Correspondingly, for the tangential component of the


magnetic field we have

JLo f L q p • dSBt(p) = -isin(t,i) 3

4rr s L q p

or

JLo f L p q • dSBt(p) = - -isin(t,i) 3

4rr s L q p

(IV.67)

where dS = d.S n. As was shown in Chapter I the integral is equal to thesolid angle w(p) subtended by the surface 5 as viewed from point p.

Finally, we have

JLoiBt(p) = - -sin(t,i)w(p)

4rr(IV.68)

For instance in the direction perpendicular to the current, we obtain thetotal tangential component

(IV.69)

since sinfi, t) = 1.As was shown in Chapter I the magnitude of the solid angle increases

as p approaches the surface 5 from both the front and back sides.

and

respectively. Therefore, the tangential component of the field in thevicinity of the plane surface 5 is

(IV.70)

Here Bt(p) and Bt-(p) are the total tangential components of themagnetic field at the front and back sides of 51' respectively.

From Eqs. (IV.70) it follows that in general the tangential componentB, is a discontinuous function at any point of the surface 5, and thisdiscontinuity is caused by the current at this point.

(IV.7!)

Now suppose that the surface 5 is an infinite plane. Then in accordancewith (IV.69) the tangential component B, from both sides of the plane


does not change and equals

J-LoiB =+-

t - 2 (IV.72)

regardless of the position of the observation point.At the same time the normal component Bn vanishes, due to symmetry.Now we will study the behavior of the tangential component B, near an

arbitrary surface S, when the current density i is some function of thepoint q (Fig. IV.4b). It is clear that the field Bt(p) can be presented as thesum of two fields,

(IV.73)

where B((p) and B(-q(p) are tangential components of the field, gener-ated by the current element i dS(q) and the remainder of the currents.

Considering the behavior of the field near the point q, we can say thatthe field Bi :" is a continuous function, since its generators are located atsome distance from this point. At the same time, when p approaches thesurface, p ~ q, the solid angle subtended by the element dS( q), tends to± 27T. Therefore, we can write

Bt(p) = J-LOi~P) +B(-q(p)

J-Loi(p)Bt-(p) = - 2 +Bt-q(p)

(IV.74)

The latter shows that the discontinuity of the tangential component at anypoint of the current surface is always defined by the current density at thispoint only, and is equal to

(IV.75)

This equation is often called the surface analogy of the first field equation,and it can be written as

orCurlB = J-Loi (IV.76)

where B+ and B- are the magnetic fields at the front and back sides ofthe current surface, respectively.

IV.3 The System of Equations of the Magnetic Field B 425

It is appropriate to notice that Eq. (lV.76) also remains valid for a widerange of electromagnetic fields applied in geophysical methods.

IV.3 The System of Equations of the Magnetic Field BCaused by Conduction Currents

In principle, the Biot-Savart law allows us to determine the magnetic fieldif the currents are known. In many cases of current distribution in anonuniform conducting medium, however, such an approach becomesextremely cumbersome and hardly practical. Moreover, as will be shown inthe next sections, in the presence of magnetic materials it is impossible tospecify some of the currents if the field B is unknown. Again we are facedwith the problem of "the closed circle." Therefore, as in the case of theelectric field, it is natural to formulate a system of field equations andboundary-value problems.

First, making use of Eq. uv.zi: we discover that divergence of the fieldB vanishes. In fact we have

div B = divcurl A (lV.77)

As is well known, the right-hand side of Eq. (lV.77) is identically zero.Therefore,

divB = 0 (IV.78)

This means that the magnetic field does not have sources and, correspond-ingly, the vector lines of the magnetic field B are closed. Next, applyingGauss' theorem we obtain the integral form of this equation.

~B. dS = 0s

(IV.79)

That is, the total flux of the field B through any closed surface is alwaysequal to zero. Now we will derive the surface analogy of Eq. (lY.78) andwith this purpose in mind consider a very thin layer with current density j.Calculating the flux of the field through an elementary cylindrical surface,as is shown in Fig. IV.4c, we have

B(2l. dS2+ B(l) . dS l + B . dS * = 0

where

(IV.80)

dS 2 =dS n , dS\ = -dS n

and dS * is the lateral surface of the cylinder.


Then reducing the layer thickness t so that the total current remainsthe same, we obtain in the limit as t --+ 0 a surface current with density i,and Eq, (IY.SO) is simplified.

B(2) dS - BO) dS = 0n n

or

(IV.S!)

Thus, the normal component of the magnetic field B is always acontinuous function of the spatial variables. Such behavior is in contrast tothat of the normal component of the electric field, and it points out thatsurface magnetic charges, as well as volume charges, cannot exist. We nowhave three forms of the first equation that describe the magnetic fieldcaused by constant currents.

~B' dS = 0s

div B = 0 (IV.S2)

Each of them expresses the same fact, namely, absence of magneticcharges.

Let us make two comments.

1. Equations (IV.S2) have been derived assuming that the field B iscaused by conduction currents. However, they remain valid in the pres-ence of magnetic materials, when the field is also generated by molecularcurrents.

2. These equations were obtained from the Biot-Savart law for directcurrents, but in actuality they are still valid for alternating magnetic fieldsand in effect represent Maxwell's fourth equation.

At this point we will develop a second equation for the magnetic field.Making use of Eq. (IV.2l)

B = curIA

and the identity

curlcurlM = graddivM - V2M

we have

curl B = grad divA - V2A

Considering the fact that

divA = 0


and taking into account Eq. (IV.32), we obtain

curl B = - V'zA = J.Loj

Thus, the second equation for the magnetic field is

curl B = J.Loj

Consequently outside of currents we have

curlB = 0

(IV.S3)

(IV.S4)

It is clear that Eq. (IV.S3) expresses the fact that currents are generatorsof the vortex type and these are what create the magnetic field.

Applying Stokes' theorem we obtain the integral form of the secondequation.

¢c B· d/= f curlB' dS = J.LoD· dS~ s s

or

(IV.S5)

where I is the current flowing through the surface S, bounded by the path2' (Fig. IVAd). It is proper to notice that the mutual orientation of vectorsdl' and dS is not arbitrary but is defined by the right-hand rule. Thus, thecirculation of the magnetic field is defined by the value of current Ipiercing the surface surrounded by the contour 2', and it does not dependon currents located outside of the perimeter of this area.

It should be obvious from the fact that the circulation is zero, it doesnot follow that the magnetic field is also zero at every point along 2'. Ofcourse, this path 2' can pass through media with different physicalproperties. Next let us consider a conducting layer (Fig. IV.5a). Then,applying Eq. (IY.S5) as well as the approach used in deriving Eq. (IY.Sl),we have

¢B' d/=Bf) dt -BP) dt + {Bn(qz) -Bn(ql)}~h

=jM~h

where qz and ql are points located at elements of the path .2' normal tothe layer.

In the limit as thickness approaches zero, and taking into account thatin reality the volume density J has a finite value, we obtain

B(2) - B(1) = 0I I as ~h ~ 0 (IV.S6)


Fig. IV.5 (a) Circulation inside a conducting layer; (b) tangential component B, near surfacecurrents; (c) magnetic field of a cylindrical solenoid; and (d) magnetic field of a toroidalsolenoid.

In accordance with Eq. (IV.86) the tangential component of the magneticfield is a continuous function of position.

Thus we have derived three forms of the second equation for themagnetic field caused by direct currents, showing that the circulation ofthe magnetic field is defined by the current flux through any surfacebounded by a path of integration, and currents are vortices of themagnetic field. These forms are

curl B = ,uoj, (IV.87)

It is interesting to notice that the last of these equations is valid for anyalternating field, and it is usually regarded as the surface analogy ofMaxwell's second equation. On occasion it is convenient to assume thatthere is a surface current with density i at some interface (Fig. IV.5b).Then, repeating the operations carried out above, we find that the tangen-tial component of the magnetic field is discontinuous at the surface and

(IV.88)


where t and t reptesent two mutually perpendicular directions, bothtangent to the surface.

Although the first two equations in Eqs. (IV.87),

and curl B = /Loj

were derived from expressions for the magnetic field caused by constantcurrents, they remain valid for quasi-stationary fields, which are widelyused in electromagnetic methods.

Now let us summarize these results and present the system of equationsof the magnetic field caused by conduction currents in differential form.

Biot-Savart's law

I curlB = /Loj

B(2j - B(1) = II i It t ,....0 I In

In diVB=oj

(IV.89)

It is proper here to make several comments concerning Eqs, (IV.89).

1. This system has been derived from the Biot-Savart law in the sameway that the systems of equations for gravitational and electric fields wereobtained from Newton's and Coulomb's laws, respectively.

2. The Biot-Savart law and the system (IV.89), together with boundaryconditions, contain the same information about the magnetic field. Themagnetic field described by Eqs. (IV.89) is the classical example of thevortex field. Its generators are currents characterized by the currentdensity field j.

3. At surfaces where the current density i equals zero, both the normaland tangential components of the magnetic field are continuous functions.

4. The system (IV.89) characterizes the behavior of the field in freespace as well as in a conducting medium. Moreover, Eqs. (IV.89) are evenvalid in the presence of a medium that has an influence on the field(magnetic material), provided that the right-hand side of the first equation,

curlB = /Loj

also includes the molecular currents inside of magnetic materials.


5. It is also proper to notice that the system (IV.89) coincides with thatfor alternating magnetic fields caused by conduction currents.

Earlier we showed that as the distance from currents increases themagnetic field behaves like that of a magnetic dipole and, correspondingly,it tends to zero at infinity.

(IV.90)

In accordance with results obtained in Chapter I we can say thatEqs. (IV.89), (IY.90) uniquely define the magnetic field. This conclusion isnot surprising. In fact, as follows from the Biot-Savart law, as soon as thecurrents are known the magnetic field is uniquely defined by their distribu-tion. In other words, there is only one magnetic field, which correspondsto a given current density field j. In this relation it is natural to raise againthe following question: Why do we need to derive and study the system offield equations if the field B can be calculated directly from the Biot-Savartlaw? In essence we have already answered this question, but let us makeadditional remarks to clarify this point.

(a) Taking into account the fact that currents appear everywhere in aconducting medium, and that they can be specified, we do not need toconsider the boundary-value problems for volumes of finite dimensions. Inother words, we will investigate the magnetic field caused by all currents ina conducting medium.

(b) There is a certain similarity between calculations of the gravita-tional and magnetic fields caused by conduction currents, in spite of theirdifferent origin. In both cases the distribution of generators can bespecified prior to the field determination, and correspondingly they can becalculated by making use of either Newton's or the Biot-Savart law.

(c) At the same time it is relevant to point out the fundamentaldifference between calculations of the magnetic and electric fields, whenthe latter are considered in the presence of conductors. This is related tothe appearance of unknown induced charges on conductor surfaces. How-ever, as will be shown later, this difference completely vanishes when themagnetic field is studied in the presence of magnetic materials.

(d) In spite of the relative simplicity with which we calculate thegravitational field using Newton's law, several methods have been devel-oped that drastically simplify this procedure. Some of them are describedin Chapter 1. Therefore, it is even more useful to study similar approachesfor the magnetic field, especially taking into account the relatively compli-cated form of the Biot-Savart law and the necessity of performing integra-tion over the whole conducting space. This is why in the next section wewill use the vector and scalar potentials, as well as some transformations,


which will allow us to facilitate determination of the magnetic field. Indescribing these approaches it is more convenient to proceed fromEqs, (IV.89) rather than using the Biot-Savart law, though of course thesystem of field equations was derived from this law.

Before we continue to study this subject, let us show one application ofthe system (IV.89) determining the field caused by a solenoid. We willconsider a cylindrical current surface with a constant cross section, shownin Fig. IV.5c. The current density has the component ilp only and itsmagnitude does not change.

and ': = constant

where 'Po is the unit vector tangential to the surface. The solenoid has aninfinite length along the z-axis,

Let us assume that the field B. inside of the solenoid is uniform,directed along the z-axis, and is equal to the current density,

(IV.91)

while outside of the solenoid the field vanishes.

(IV.92)

It is easy to see that these functions, given by Eqs. (IV.9l), (IV.92), satisfythe system of field equations (IV.89) as well as the condition at infinity.Therefore, in accordance with the theorem of uniqueness, they describethe magnetic field of the solenoid.

Certainly, it is a very simple behavior, but this result is hardly obvious.First of all it is difficult to predict that the field inside, B~ , is uniform overthe cross section, since the field due to a single current loop varies greatly.Also it is not easy to predict before calculation that the field outside of thesolenoid is zero. From this study we can conclude that if the cylindricalsolenoid has a finite extension along axis z, the field is practically uniformwithin its central part.

Let us note that a similar application of the theorem of uniqueness fordetermination of the field was demonstrated several times in Chapter III,when we investigated the electric field in the presence of conductors. Forinstance electrostatic screening, as well as the uniform distribution ofcurrents over the cross section of a cylindrical conductor, were establishedwith the help of the theorem of uniqueness.

Let us also consider a toroidal solenoid, shown in Fig. IV.5d. Thecurrent density magnitude is


Making use of the integral form of the first equation (IV.85),

and taking into account the axial symmetry, we have

(IV.93)

where Y is a circular path of radius r, located in the horizontal planewith its center situated at the toroid axis; and I is the current passingthrough a surface surrounded by this path Y. Due to the axial symmetrythe magnetic field has the component Be only and, in accordance withEq. (IV.93), we can conclude that

(a) The magnetic field outside of the solenoid equals zero.

Be = 0

(b) Inside of the solenoid the field B} is not uniform and it equals

(IV.94)

That is, it becomes bigger at points located closer to the toroid z-axis. Thelatter shows that component i", varies on the toroid surface.

(c) With an increase of the ratio of the toroid radius R to that of itscross section ro' the field B} becomes more uniform.

(d) If the toroid has an arbitrary but constant cross section and

we can still apply Eq. (IV.93). This means that outside of the toroid thefield Be equals zero, while inside it is easily calculated from Eq, (IY.94).

IVA Determination of the Magnetic Field B Causedby Conduction Currents

Now we will describe methods that allow us to simplify calculations of themagnetic field. We will begin with the vector potential A. In the previoussection this function was introduced as

B = curl A (IV.95)

IVA Determination of the Magnetic Field B Caused by Conduction Currents 433

and

(IV.96)

divA=O

It is essential to note that Eqs. (IV.95), (IV.96) are valid everywhere,including volumes occupied by currents.

As was mentioned earlier the relationship between the current andvector potential is much simpler than that for the magnetic field. There-fore, it is usually easier to find the vector potential and then, making useof Eq. (IY.95), to determine the magnetic field. Also, it is sometimespossible to describe the magnetic field with the help of two or even onecomponent of the vector potential, and this simplifies calculation of thefield B, too.

In accordance with Eqs. (IV.89) at the surface current, we have

or, taking into account Eq. (IV.95),

and n . B(2) = n . WI)

Inasmuch as the normal component ev X A), includes only derivatives indirections tangential to the surface, the last equality remains valid, if werequire continuity of the vector potential.

Correspondingly, the system of equations for the vector potential is

Biot-Savart's law (IV.97)

IV'A~j-~~l'n-x-e-V-X-A:-(2-))---n-x-e-v-"-X-A:-(1-))-=-j.L-O-iI IA(1) = A(2) I


As follows from the behavior of the magnetic field at infinity, the vectorpotential also tends to zero like the field of a magnetic dipole, Eq. (IV.57).

(IV.98)

In accordance with results obtained in Chapter I, Eqs. (IV.97), (IV.98)after some modifications together with the boundary condition at infinity,constitute a boundary-value problem that uniquely defines the field B.Thus, proceeding from the vector potential we can determine the magneticfield in two ways, demonstrated below.

Determination of themagnetic field

Boundary-valueproblem for A

Until now we have considered the vector potential A, defined byEq. (IY.96), when its divergence equals zero.

div A = 0

However, we can imagine an infinite number of vector potentials describ-ing the same magnetic field B. In fact, let us introduce a new vectorpotential A* related to A by

A * = A + grad cp (IV.99)

where cp is a scalar function that is continuous, along with its firstderivative.

Substituting Eq, (IV.99) into Eq. (IV.95), we have

B = curl A = curl A * - curl grad cp

or

B = curl A*

IV.4 Determination of the Magnetic Field B Caused by Conduction Currents 435

since

curl grad 'P = 0

It is essential to note that by changing function 'P in Eq. (IV.99) the field Bremains the same, and this flexibility in choosing the vector potential turnsout to be a very useful factor in solving some boundary-value problems.

In accordance with Eqs. (IY.96) and (IY.99) let us write down anexpression for A* as

P-o f j( q)A * = - -- dV + grad 'P

47T v L q p

Taking divergence from both sides of Eq. (IV.99) we obtain

(IV.lOO)

because div A = O. Thus, in general, the divergence of the vector A* is notequal to zero. At the same time, if the function 'P(q) is harmonic, "V 2'P = 0,then we obtain again

div A* = 0

The vector potential A satisfies the vector analogy of Poisson's equation,

while the potential A * is usually a solution of a different equation. Indeedmaking use of Eq. (IV.99) we have

where

L = grad 'P

or

(IV.101)

Let us note that by properly choosing the function 'P(q), it is possible totransform this equation into a homogeneous one.

Now we will simplify the system of field equations, assuming that theobservation points are located outside of the volume where conductioncurrents are present. This is a case of great practical interest in geophysi-cal methods based on measuring the magnetic field.


In accordance with Eqs. (IV.89) the system of field equations outside ofthe generators is essentially simplified, and we have

curlB = 0 divB = 0Bf) - B?) = 0 B~2) - B~1) = 0

or

curlB = 0 divB = 0 (IV.102)

since the magnetic field is a continuous function in a volume where surfacecurrents are absent.

As we know (Chapter I), this field is harmonic, and it can be expressedthrough a scalar potential. In fact, from the first equation

curlB = 0it follows that

B = - grad U (IV. 103)

where U is a potential describing the magnetic field in the vicinity of anypoint where the current density equals zero. Substituting Eq. (IV.103) intothe second equation of the system (IV.102) we obtain

(IV.l04)

divB = 0

on So

That is, the potential U is a harmonic function.It is natural that the system of equations (IV.102) is given in a certain

part of the space outside of the currents. For instance, it can be a halfspace above the earth's surface. It means that if Eqs. (IV. 102) are the solesource of information about the magnetic field, we do not know currentdistribution and therefore the Biot-Savart law cannot be applied. At thesame time, the field B can be found if along with the system (IV.I02)either the normal or tangential components are given at the boundarysurface. In other words, the magnetic field can be determined by solving aboundary-value problem. In particular, for the upper half-space above theearth's surface So, we can formulate two problems.

1. The first problem

curlB = 0

Bt='Pl(q)

andat infinity

or2. The second problem

curlB = 0

Bn = 'P2(q)

divB = 0

on So

IVA Determination or the Magnetic Field B Caused by Conduction Corrects 437

and

at infinity

Here CPt and CP2 are given functions that can be obtained by measuring themagnetic field at the earth's surface. Of course, these problems can thenbe solved by making use of Laplace's equation and formulating corre-sponding conditions at the boundary surfaces. This study demonstratesthat, as in the case of the electric and gravitational fields, the boundaryconditions uniquely define the harmonic magnetic field. From a practicalpoint of view, this means that knowing the field at the earth's surface wecan calculate this field at any point of the upper space. This is calledupward continuation. In this relation it is appropriate to notice that abovethe earth's surface the gravitational, magnetic, and electric fields can beexpressed through a scalar potential, and this similarity is explained by thefact that fields are considered outside of their generators.

As an example of an application of the scalar potential, we will find itsexpression for the field caused by the current in a relatively small loopwith the dipole moment M. In accordance with eq. (IV,49) we have

(IV.lOS)

To express the magnetic field as the gradient of some function, we willperform a transformation of the right-hand side of Eq. (IV. IDS).

Taking into account the equality

v x (cpa) = cp curIa + Vcp x a

we have

/La (II}B(p) = - -3- curl(M X Lqp ) - (M X Lq p ) X V-3-47T L q p L qp

Inasmuch as

(IV.I06)


the second term in Eq. (IY.106) can be presented as

1 3- (M X L qp ) X \7-3- = -5- (M X L q p ) X L q p )

L q p L qp

Here the equality

a X (b X c) = b(a· c) - c(a· b)

has been used.Next we will consider the term

which is easy to find if we calculate its Cartesian components. By defini-tion for the x-component we have

a- - (M x - zM ) = 2Maz z x x

since the dipole moment M does not depend on coordinates of theobservation point.

By analogy,

and

and therefore

curl(M X L q p ) = 2M

Thus, instead of Eq, (IV.106) we obtain

(IV.107)

IVA Determination of the Magnetic Field B Caused by Conduction Corrects 439

A comparison of terms on the right-hand side of this equation shows thatthe magnetic field can be represented as

where

B = -grad U (IV.IOS)

(IV.109)

In fact, taking the gradient of the potential U and making use of theequality

grade cp]cpz) = cp] grad cpz + cpz grad cp]

we arrive back at Eq. (IY.IO?).Now suppose that a distribution of dipole moments of small current

loops in volume V is characterized by the vector P.

dMP= dV

Then, by applying the principle of superposition, the potential U, causedby such a system of loops, is

JL p. LU(p) = _0 f 3 qp dV

4'7T V L q p

(IV.lIO)

Let us note that this equation plays a very important role in the theory ofmagnetic methods, which will be discussed in detail later.

Next we will consider a general approach to calculation of the magneticfield due to currents in a nonuniform conducting medium. A model of apiecewise uniform medium is shown in Fig. IY.6a. The determination ofthe magnetic field B in such a model usually includes three steps, namely,

1. Solution of the forward problem for the electric field, that is,determination of the field E at every point of the medium.

2. Making use of Eq. (IV.96), determination of the vector potential A;and finally

3. Calculation of the magnetic field from the equation

B = curIA

The second step of this procedure implies a volume integration overwhole space, and taking into account the fact that the magnitude anddirection of the current density vector j varies from point to point,determination of the vector potential is usually very cumbersome. This is


Fig. IV.6 Models of piecewise uniform medium.

especially noticeable in relatively simple models, when the electric fieldcan be expressed in an explicit form. To simplify calculations of themagnetic field we will describe an approach that allows us to replace thevolume integration in the equation (IV.96),

A(p) = ~ f j(q) dV41T V L qp

by a surface integral.First, we will consider a model of the medium when an inhomogeneity

with conductivity 'Yi is located beneath the earth's surface, and theconductivity of the surrounding medium is 'Ye (Fig. IV.6b). The part of thevector potential caused by currents inside of the inhomogeneity is

(lV.ll!)

IVA Determination orthe Magnetic Field B Caused by Conduction Corrects 441

since j(q) = YiE(q) = -YiVU(q}, and Vi is the volume occupied by theinhomogeneity.

Proceeding from the equality

q U q 1 1 q

V-=UV-+-VU-: i; -:

we have

/-LoY; 1 q U /-LO'Yi 1 q 1Ai(p)=-- V-.-dV+- U(q)V-dV (IV.112)

477" V; L q p 477" V; L q p

where the index "q" means that a variable is considered at an arbitrarypoint q of the volume V. Inasmuch as

q 1 p 1'V-= -V-

L qp i;

we can change the order of integration and differential in the secondintegral of Eq. (IV.l12), and then we have

/-LOYi 1U(q) V_1_ dV= - ~'Yi ~ 1U(q) dV477" V; L qp 477" V; L qp

In other words, this integral is the gradient of some function and thereforeit does not have an influence on the magnetic field. Correspondingly wewill consider the vector potential Ai in the form

(IV.113)

Applying the equality

f. grad TdV= fTdSv s

we obtain

(IV.1l4)

where Sj is the surface surrounding the volume ~; and dS = dS n, where nis directed outside of the inhomogeneity.


By analogy, the vector potential caused by currents in the surroundingmedium can be represented as

(IV.lIS)

where dS * is the element of the surface Sj, provided that the normal isdirected inside of the inhomogeneity, that is, dS = -dS n. At the sametime So is the earth's surface with its normal directed outward.

Performing a summation of Eqs. (IV.1I4), (IV.lIS) we obtain

A(p) =Aj(p) +Ae(p) = /LoCYe-yJ f U(q) dS+ /LoYef U(q) dS41T S; L q p 41T So L q p

(IV.1I6)

Thus we have replaced the volume integration by a surface integral,which is usually much simpler. Moreover, instead of the electric field, theintegrands of these surface integrals contain the potential U, which alsofacilitates calculations. By applying the same approach it is very easy togeneralize Eq. (IV.116) for more complicated models. For instance, if thesurrounding medium consists of two uniform parts with conductivities Yleand Y2e' respectively (Fig. IV.6c), we obtain

A(p) = /Lo(Yle-yJ f U(q) dS- /LOYle f U(q) dS41T Si L q p 41T SOl L q p

where S12 is the interface between media with conductivities Yle and Y2e'the and direction of dS 12 is shown in Fig. IV.6c.

Let us emphasize again that the vector potential A(p) in this equationdiffers from that given by Eq. (IV.96) by the gradient of some scalarfunction, but this does not have any effect on the magnetic field. Also it isappropriate to add the following:

(a) Equations (IV.1I6), (IV.1I7) do not take into account the vectorpotential A, caused by currents in wires, but this part of the magnetic fieldcan be easily calculated.

IVA Determination of the Magnetic Field B Caused by Conduction Corrects 443

(b) Applying the equality

f grad" TdS = ¢c Tv dts Sf

we can derive the surface analogy of Eq. (IV.117).

Finally, we will consider one more approach based on simultaneousdetermination of the electric and magnetic fields. Proceeding from resultsderived in the previous chapters as well as this chapter, we can representthe system of equations of the electric and magnetic fields as

I

III

curlE = 0

curl B = /Loj

II

IV

div yE = 0

divB = 0(IV.lI8)

First let us introduce the scalar and vector potentials.As follows from Eqs. I and IV of system (IV.lI8),

E = -gradU and B = curlA (IV.lI9)

Then, substituting Eqs. (IV.lI9) into Eq. III of the system (IV.lI8) andtaking into account Ohm's law, we obtain

curl curIA = -/LoY grad U

Inasmuch as

curl curl A = grad divA - V2A

we have

grad divA - V2A = - /LoY grad U (IV.120)

Now we will make use of the fact that an infinite number of potentialsdescribe the same field and choose a pair of A and U, which simplifiesEq, (IV.120). For instance, suppose that

divA = - /LoYU (IV.121)

Then, instead of Eq. (IV.120) we arrive at the vector form of Laplace'sequation,

and in accordance with Eq. (IV.119)

IE = -- grad divA,

/LoYB = curIA

(IV.122)

(IV.123)


Thus, we have expressed both fields through the vector potential only andderived Eq. (IV.I22), which describes the behavior of function A at regularpoints.

To formulate boundary-value problems we have to include also bound-ary conditions near the current electrodes and current lines, connectingthem, and at infinity. In addition it is necessary to specify the behavior ofthe vector potential at interfaces between media with different resistivities.This behavior of the vector potential follows from the continuity oftangential components of the electric and magnetic fields, if i = 0, as wellas the normal component of the current density and that of the magneticfield.

Now we are ready to consider several examples illustrating the fieldbehavior in a conducting medium.

IV.S Behavior of the Magnetic Field Caused by Conduction Currents

In this section we will consider several examples illustrating the behaviorof the magnetic field in different models of a conducting medium.This analysis is interesting from two points of view. First of all, it has prac-tical meaning since some geophysical methods-for instance, themise-a-la-masse-t-ete based on a study of the field B, caused by a constantcurrent, although in practice low frequency fields are measured. Second,in many cases the constant magnetic field describes the asymptotic behav-ior of alternating fields widely used in electromagnetic methods.

Example 1 A Current Electrode in a Uniform Medium

Suppose that a current electrode is located in a uniform medium withresistivity p (Fig. IV.7a). Then, as was shown in Chapter III, the electricfield as well as the current density has a radial component only, and

Ij = 41TR 2 Ro

where I is the current and Ro is the unit vector.In accordance with the Biot-Savart law every current element creates a

magnetic field, and our goal is to find the field at any point p due to allcurrents in the conducting medium.

IV.S Behavior of the Magnetic Field Caused by Conduction Currents 445

Fig. IV.7 (a) Current electrode in a uniform medium; (b) current electrode and current linein a uniform medium; (c) magnetic field beneath the earth's surface; and (d) magnetic fieldon the earth's surface of horizontally layered medium.

To determine the field B we will consider a plane passing through theelectrode and the observation point p. As is seen from Fig. IY.7a it isalways possible to find two current elements located symmetrically withrespect to the plane. Applying the Biot-Savart law we can easily see thattheir magnetic field equals zero at any point of this plane. Then, takinginto account the arbitrary orientation of the plane, we conclude that themagnetic field caused by all currents in the uniform medium equals zero.

8=0 (IV.124)

This interesting result has been derived by applying a very simpleapproach, but still there is one problem. In fact, from the first fieldequation,

curl B = ,uoj

it follows that up to a constant of proportionality ,uo' curlB is equal to the


current density at the same point. However, in our case, in accordancewith Eq. (IV.124), curl B vanishes despite the presence of currents in thevicinity of every point. To understand this paradox let us recall that theconstant magnetic field is caused by closed currents, but in this examplecharges move in the radial direction, from the current electrode to infinity;that is, current lines are not closed. Correspondingly, the field equationsdescribing the magnetic field cannot be applied in this case.

Now suppose that the current returns to the electrode through the wireas shown in Fig. IV.7b, and therefore the vector lines of the currentdensity field are closed. The magnetic field can be represented as the sumof two fields.

where B] is the field caused by the current in the wire and Bz is the fielddue to the radial distribution of currents in a uniform medium. Inaccordance with Eq. (IV.124), Bz equals zero, and therefore

To determine the field B] we will introduce a cylindrical system ofcoordinates r, cp, z such that its origin 0 coincides with the center of theelectrode. Then, in accordance with Eq. (IV.35) we have

(IV.l25)

Next we will demonstrate that unlike the previous case the field B, givenby Eq. (IV.125), satisfies the first field equation. With this purpose in mindlet us rewrite Eq. (IV.l25) in a spherical system of coordinates with thesame origin.

J-LolB= . (l-cose)'Po

47TRsm e

Taking into account that in spherical coordinates

i R Rio R sin eiep1 a a a

curlB = - -RZ sin e aR ae acp

0 0 R sin es,

(IV.126)

(IV.l27)

curl, B = curl j B = 0

(IV.128)

IV.5 Behavior of the Magnetic Field Caused by Conduction Currents 447

and substituting Eq. (IV.126) into Eq. (IV.127), we have

}.LoIcurlRB = 4rrR2 = }.LOjR'

or

Example 2 A Current Electrode on the Surfaceof a Horizontally Layered Medium

First, we will consider the case when the electrode is located on thesurface of a uniform half space and the current arrives at the electrodethrough a vertical wire, Fig. IV.7c. As was shown in Chapter III theelectric field E at every point of the conducting medium is

pIE = 2rrR 2 R o

and correspondingly the current density vector equals

1. RJ = 2rrR2 a

Thus, the magnetic field, B, is caused by linear and volume currents,and due to the axial symmetry of their distribution we can show that in thecylindrical system of coordinates

and

and

(IV.129)

(IV.130)

Therefore, it is natural to make use of the first field equation in integralform

11B· dl'= }.LoIs

where Is is the current passing through any surface bounded by thecontour 2'. It is appropriate to notice that the system of field equationscan be applied in this example because the current lines are closed. Then,taking into account the axial symmetry of the field, Eq. (IV.130) is


drastically simplified and we have

or

IsB =/La--

'P 27Tr(IV.l3l)

where Sf is the circle with radius r, located in the horizontal plane, andits center is situated on the z-axis.

As follows from this equation at the earth's surface and above, we have

/LaIB =-

'P 27Trif z s; 0 (IV.132)

This expression coincides with that for an infinitely long current-carrying line. According to Eq, (IY.125), if z = 0, the current in a infinitelylong half line placed above the earth's surface creates at this surface amagnetic field.

/LaIB =-

'P 47Tr

Therefore, the currents in the conducting medium that flows from theelectrode also generate the field B. The tangential component of B is

/LaI /LaI /LaIB =---=-

'P 27Tr 47Tr 47Trif z = 0 (IV.133)

It is interesting to notice that these volume currents create the samefield as the half infinitely long current-carrying wire directed verticallydownwards with the current 1. Also it is important to emphasize that themagnetic field at the earth's surface, caused by currents flowing fromthe electrode into a horizontally layered medium, does not depend on thesequence of conductivities of layers. This follows directly from the axialsymmetry, which allows us to apply Eq. (IV.132).

Next, we will study the magnetic field in a uniform half space beneaththe earth's surface, and with this purpose in mind it is appropriate tomake use of Eq. (IV.l31). Since the current density vector has a radialcomponent jR only, the current Is that passes through the sphericalsurface S and is bounded by the circle Sf with radius r (Fig. IV.7c) is


Then, taking into account that

Ii« = yER = 21TR2

we have

and

(IV,l36)

w] = - (IV.l34)

s 27T

where w is the solid angle subtended by the surface S as viewed from theelectrode. In accordance with Eq. (1.48), the solid angle in this case is

w=21T(I-cosa) (IV.13S)where

ra = sin- I -

R

Substituting Eqs. (lV.l34), (lV.l3S) into Eq, (lV.13!), we obtain

B =/-Lo-] (1- Z )'P 21Tr ';r2+z2

This equation describes the magnetic field caused by volume currents aswell as the current in the vertical wire.

Example 3 The Current Flowing in the Wire Groundedat the Surface of a Horizontally LayeredMedium (Fig. IV.7d)

Using the principle of superposition, this pattern of current flow can berepresented as

1. Currents flowing from electrode A into a conducting medium.2. Currents flowing from the conducting medium into electrode B.3. Current] in the wire that connects the electrodes.

In accordance with Eq, (lV.l33) the magnetic field at the earth'ssurface, caused by currents in the conducting medium, is

/-Lo] /-Lo]Bt ( p) = -4--'POI + -4--'P02 (IV,l37)

1TrAp 1TrBp

where rAp and rBp are the distances from electrodes A and B to theobservation point p, respectively, and 'POI and 'P02 are unit vectors asdefined in Fig. IV.7d, It is clear that the field B. is tangent to the earth'ssurface and it is independent of the parameters of the horizontally layered


medium. This is why the magnetic field of constant currents is not used toa study geoelectric parameters of such media.

Next we will consider the third and last element of the system, that is,the current flowing through the wire connecting electrodes and situated atthe earth's surface. According to the Biot-Savart law this current causesonly a vertical component of the magnetic field at points on the earth'ssurface.

!L01 (A dt .B, = -kJ, -Z- sm(dl',Lqp )

47T B L q p

(IV.l38)

where k is the unit vector perpendicular to the earth's surface, dl'is theelement of the current-carrying line, and L q p is the distance from anyelement dt to the observation point.

Equations (lV.13?), (lV.138) completely describe the magnetic field of agrounded current-carrying line located at the earth's surface when theconducting medium is laterally uniform.

Suppose that the field B is observed at distances considerably greaterthan the separation between electrodes A and B, and the current flowpath from A to B is a straight line. Then, the system of charges arising atthe surface of the electrodes, connected by the wire AB, can be consid-ered as an electric dipole with the moment

(IV.l39)

where 1'0 is the unit vector directed along line BA (Fig. IV.8a).To derive approximate expressions for the magnetic field from

Eqs. uv.is», (lV.138) we will use the following notations: r is thedistance from the middle of the dipole to the observation point, and if! isthe angle between the dipole moment and the radius vector r. As may beseen from Fig. IV.8a, the following relations hold:

[

Z ] I/Z

rAp = rZ+(~) -ABrcosif!

[

Z ] I/Z

rBp = r? + ( A:) +ABr cos if! (IV.140)

AB sin a l

2rAp sin if!

AB sin a z

2rBp sin if!

IV.S Behavior of the Magnetic Field Caused hy Conduction Currents 451

Fig. IV.S (a) Electric dipole on the earth's surface; (b) electric dipole in a uniform medium;(c) vertical electric dipole in the presence of a horizontal interface; and (d) horizontal electricdipole in the presence of a horizontal interface.

Considering that r ts- AB we have

AB'Ap:::::'-Tcoscp,

AB'Bp:::::' + T cos cp

and

AB sin cp

2( r _ A: cos tp )

AB sin cp

2( r + A: cos cp )

sin az ::::: ---:---;-;:;----,..-

or

ABsin a l ::::: sin az ::::: -- sin cp

2,


Then, making use of Eq. (IV.137) we obtain expressions for the tangentialcomponents of the field in a cylindrical system of coordinates.

JJ.oI. JJ.oI. JJ. o2I AB .B = - -- SIll a - -- sin a = - -- - SIll if'

r 47Tr 47Tr 47Tr 2r

and

JJ.oI JJ.oI JJ.oI (1 1)B = --- cos a - --- cos a = -- -- - -cp 47TrAp 47Trsp 47T rAp rsp

since cos a = 1.Thus, we have

As follows from Eq. (IV.138) the vertical component of the magnetic fieldof the dipole is

Thus, the magnetic field of the electric dipole at the earth's surface is

JJ.oIAB .B, = - --2- SIll if',

47Tr

JJ.oIABBcp = --2- cos if'

47Tr

(IV.141 )

Equations (IV.14l) vividly illustrate that measurements of the magneticfield of the dipole at the earth's surface do not contain any informationabout the distribution of resistivity in a horizontally layered medium.However, we must recognize that this conclusion does not hold when themagnetic field is measured beneath the earth's surface. Taking measure-ments of the magnetic field at the earth's surface can, however, be usefulin detecting nonhorizontal structures that are frequently of particularinterest in prospecting.


Again making use of the principle of superposition, the magnetic fieldcaused by a current-carrying line with a finite length and arbitrary shapegrounded at the earth's surface can be represented as being the sum offields described by Eqs. (IY.14l).

By investigating alternating magnetic fields caused by groundedcurrent-carrying wires, it can be shown that these equations playa veryimportant role when electromagnetic induction in the field is not particu-larly significant.

Example 4 An Electric Dipole in a Uniform Medium

Suppose that an electric dipole with moment M is located in a uniformmedium with resistivity p (Fig. IV.8b). It is obvious that the vectorpotential of the magnetic field can be represented as the following sum:

(IV.142)

where A+ and A_ are vector potentials, caused by currents in the medium,which flow from the electrode A and into electrode B, respectively, whileAt is the vector potential generated by the current in the wire. Since thecurrent distribution in a uniform medium is a superposition of two systemsof radial currents, their magnetic field equals zero. Correspondingly, thevector potentials A+ and A_ can be expressed as gradients. In particular,they can also be equal to zero. Therefore, the vector potential of theelectric dipole up to the gradient of some function is the same as that ofthe current element dt; that is

(IV.143)

We will use a spherical system of coordinates R, e, 'P with its originlocated at the middle of the dipole and its z-axis directed along the dipolemoment.

Then, Eq. (IV.143) is written as

/-La! dzA=Ak=--k

z 47TR(IV.144)

where k is the unit vector characterizing the direction of the dipole


moment. In accordance with Eq. (IY.144) the vector potential A has onecomponent, A z , and depends on the distance R only. Now it is a simplematter to determine the magnetic field.

Example 5 A Vertical Electric Dipole in a Mediumwith One Horizontal Interface

We will place a vertical electric dipole in a medium with conductivity Yl ata distance h from the interface. The conductivity of the upper space is Yz.Taking into account the axial symmetry of the current distribution, we willintroduce a cylindrical system of coordinates so that the dipole centercoincides with its origin and the z-axis is directed upward, as is shown inFig.IY.8c.

As was shown in Chapter III, the current density is easily calculated atevery point of the conducting medium, and therefore the magnetic fieldcaused by these currents can be determined from the Biot-Savart law.However, this procedure requires a complicated integration over thewhole space. For this reason we will make use of the approach based onsimultaneous determination of the electric and magnetic fields, which wasdescribed in the previous section. In accordance with Eqs. (IV.12l),(IV.123) we have

and

1E = -- grad divA,

/-LoYB = curIA

(IV.145)

div A = - /-LoYU

To facilitate the solution of the boundary-value problem we will assumethat the vector potential of the magnetic field, as well as that of theelectric dipole in the uniform medium, has a vertical component only.Then, taking into account the axial symmetry, Eqs. (IV.145) are simplified,and we have

1 aZAE = z

r /-LoY araz 'E<p=O


and

(IV.I46)B; =Bz = 0,aA zB =--

cp ar

Now we are ready to formulate the boundary value problem for thevector potential A, which is represented in the form

Ar=Acp=O

but

if z s h

if z ~h(IV.147)

and

(IV.148)

where dz is the distance between electrodes, R = "';r2 + Z2, and AL is thepart of the vector potential that arises because the medium is nonuniform.Thus, proceeding from the system of equations for the electric andmagnetic fields, we conclude that the vector potential should satisfy thefollowing conditions:

1. At regular points the vector potential is a solution of Laplace'sequation,

V 2Az = 0

which in a cylindrical system of coordinates is

a2A z 1 aA z a2A z--+--+--=0ar 2 r ar az 2

since A z is independent of the coordinate cp.2. At infinity the potential A z tends to zero.

(IV.149)

A~~O as R ~ 00 (IV.ISO)

3. Near the dipole, the function A z approaches that of the electricdipole in a uniform medium.

(IV.15I)as R ~O/-Lo 1dz

Az~Aoz= 41TR

4. At the interface between media with different conductivities thecomponents i., En and Bcp are continuous and therefore from


Eqs. (IV.146) we have

1 aZAI;:---'YI araz

aZA lZa;z

1 aZA zz---'Yz araz '

aZA zzaz z (IV.I52)

andaA lz aA zz--=--

ar ar

From the first field equation in integral form

(IV.I53)if z =h

11B . dl'= /LoIs

it follows that continuity of the normal component of the current densityresults in continuity of the tangential component of the magnetic field.Correspondingly, the last two equalities of Eqs. (IY.152) can be replacedby one equality, and then the conditions at the interface are

1 aZA l z 1 aZA l z aA lz aA zz---=---'YI araz 'Yz araz ' ar ar

Another simplification is related to the fact that continuity of a functionat some surface provides continuity of its tangential derivatives andtherefore, instead of Eqs. (IV.152) we have

1 aA lz 1 aA zz---=---'Yl az 'Yz az

Thus, we have described the behavior of the vector potential A every-where, and at the same time formulated a boundary-value problem thathas a unique solution.

We will begin by finding a solution of Laplace's equation. Applying themethod of separation of variables and making use of the results derived inChapter III, the vector potential can be represented as

/LoIdz /LuIdz 00

A lz = -4-- + -4--1 (Cme mz +Dme-mZ)Jo(mr) dm1TR 1T 0

md ay.~~

/Lo I dz 1""Azz=~ 0 (Ememz+Fme-mZ)Jo(mr)dm


Taking into account the fact that the vector potential vanishes at infinity,we have to assume that

if z ~ h

Dm =Em =0

Then, Eqs. (IV.154) are slightly simplified and we obtain

/-La! dz /-La! dz 100

A 1z = --- + --- CmemzJaCmr) dm47TR 47T a

(IV.155)

if z ~h

and

/-La l dz 100

A zz = --- Fme-mzJaCmr) dm47T a

These functions satisfy all conditions of the boundary value problemexcept those at the interface when z = h.

To satisfy Eqs. (IV.153) we will follow exactly the same procedure as inthe case of the electric field (Chapter III) and represent the vectorpotential of the electric dipole in a uniform medium as

(IV.156)/-La! dz /-La! dz 100 -mlzlJ C ) d--- = --- e a mr m47TR 47T a

Therefore, the expression for the vector potential in the medium withconductivity 'Y 1 can be written in the form

/-La! dzooA 1z= ~ fa [e-mlzl+Cmemz]JaCmr)dm

Taking into account this representation and substituting Eqs. (IV.155) intoEqs. (IY.153), we obtain two linear equations with two unknowns.

(

e - mh + c emh=F e-mhm m

PI{ _e-mh + Cmemh}= -PZFme-mh

since from the equality of these integrals follows the equality of theintegrands.

Solving this system we have

C K -Zmhm = - 12e ,

2PIF = ----=--

m Pi + pz(IV.157)

here

K = pz - PIIZ

PZ+PI


Thus, expressions for the vector potential A are

fLo! dz fLo! dz 1'"A . = --- - --K e-(Zh-z)mJ (mr) dmL 47T R 47T u 0 a

if z ~ h

if z z. h (IV.158)

(IV.159)if z ~h

and since A lz and A z z satisfy all conditions of the boundary-valueproblem, they uniquely define the magnetic field.

Now applying the equality (IV.156) we express the vector potentialthrough elementary functions

fLo! dz fLo! dz K 12A -------lz 47TR 47T R 1

fLo 1dzA z z = --(1- K 12 )

47TRif z z. h

where

Let us make several comments illuminating the behavior of the field.

1. The potential A 1z in the medium with conductivity {'I is the sum oftwo terms, namely, the potential of the primary field A oz and that of thesecondary field AL. The latter coincides with the vector potential of afictitious vertical dipole, situated at the point with coordinates 0, 2h. Inother words, this point is the mirror reflection of the origin, where the realelectric dipole is located. The current of the fictitious dipole is equal to

(IV.160)

2. The vector potential in the second medium A z z coincides with thatof one electric dipole, located at the origin, and its current is equal to thesum of currents of the real and fictitious dipoles.

3. If the second medium is more resistive, then the moments of the twodipoles have opposite directions; but if the upper space is more resistive,they have the same directions.

4. This study shows that in our example we can apply the method ofmirror reflections, which is also used for determining electric fields. Inother words, the secondary magnetic field is equivalent to that of a


fictitious vertical dipole, located either at the origin where the real dipoleis situated or at its mirror reflection with respect to the interface.

5. In accordance with Eqs. (IV.159) in the upper nonconducting spaceand at the boundary z = h, the vector potential A zz equals zero. Thismeans that the magnetic field caused by the vertical electric dipolevanishes at the earth's surface and above.

B", =- 0 if z ~h and Pz = 00 (IV.161)

This result also follows from the axial symmetry of the field. In fact,applying the first equation of the magnetic field,

~ B· dl'= JLoI~ s

we have for any horizontal circle with the center at the axis z

Brp2Trr = 0 if z ~ h

since current is absent in a nonconducting medium.Due to the axial symmetry, Eq. (IV.16l) remains valid in a laterally

uniform medium. Generalizing this result, we can say that the secondarymagnetic field at the earth's surface and above, caused by the verticalcomponent of all currents in a horizontally layered medium, equals zeroregardless of the current distribution.

6. Beneath the earth's surface the second magnetic field is equivalentto that of a vertical electric dipole located at the point r = 0, z = 21z, andits moment has the same magnitude as that of a real dipole but is oppositein directions. As follows from Eq. (IV.159), both the primary and sec-ondary fields are independent of conductivity and contain informationonly about the position of the dipole.

7. As the electric dipole approaches the earth's surface the magneticfield decreases and at the surface is equal to zero.

8. Let us note that from the gauge condition, Eq, (IV.12l),

divA = - 11-0/'U

it is very simple to determine the electric field. In fact, we have for thepotential UI in a conducting medium

I aA l zUI = - - - - -

11-0/'1 azor

Mz K l zM(2h - z)UI(r,z) = 4 R 3 + 4 R 3

7TBO TrBo I

where M = cOPII dz is the magnitude of the dipole moment.

(IV.162)


In the same manner, we have

(IV.163)

(IV.164)

Example 6 A Horizontal Electric Dipole in a Mediumwith One Horizontal Interface

Next, we will consider a horizontal electric dipole located at a depth hbeneath a horizontal interface. Let us use a Cartesian system of coordi-nates so that the dipole is situated at its origin and the dipole moment Mis oriented along axis x (Fig. IV.8d).

As follows from the previous example, the presence of the horizontalinterface requires us to make use of the vertical component of the vectorpotential A z . Therefore, taking into account the fact that the primaryvector potential has the component along the x-axis,

/-Lo! dxA=--

OX 47TR

we will look for a solution with the help of two components, Ax and A z.In formulating the boundary-value problem for the vector potential A,

A=Axi +Azk

we will proceed from Eqs. (IV.145), and the conditions that constitute thisproblem are .

1. At regular points the vector potential A satisfies Laplace's equation.

or

and (IV.165)

(IV.166)as R~O

2. Near the origin the vector potential tends to that of an electricdipole in the uniform medium with conductivity 1'1'

fJ.-o I dxA~A i=---i

x 47TR

where


3. At infinity the field vanishes and therefore

A ~ 0 as R ~ 00 (IV.167)

(IV.168)

1 aE = --- divA

y fLaY ay

1 aE =--divA

z fLaY az

4. Before we formulate conditions at the interface z = h, let us writedown expressions for the field in terms of the vector potential A. Assumingthat the component A y is absent, A y == 0, and making use ofEqs, (IV.145), we have

1 aEx = --- divA,

fLaY ax

whereaAx aAz

divA=-+-ax az

(IV.169)

andaAx

B = - - (IV.170)z ay

Thus, to provide continuity of the components Ex, E y, jz' Bx' By, andB, the following equalities have to hold at the interface where z = h.

1 1- divA. = - div A,Y] Yz

z = h (IV.171)ifA 1x =A zx

a a- divA1 = - div A zaz az

aA]x aAzxaz az'

Let us note that we have again used the fact that continuity of afunction at the interface results in continuity of its tangential derivatives.Knowing also that continuity of tangential components of the magneticfield provides continuity of the normal component of the current density,we arrive at conditions for the vector potential at the interface.

aA1x aAzx-- --

az azand (IV.I72)

1 1- divA. = - divA zYI Yz

if z = h


This relatively complicated group of equalities has one remarkablefeature. In fact, Eqs. (IY.I72) can be split into two sets. One of themrepresents conditions for the component A x only,

while the other set formulates conditions for the component A z' but alsocontains A x' This splitting of conditions at the interface allows us todrastically simplify the field calculations. This happens because we canseparately formulate boundary-value problems for the components of thevector potential. Indeed, from Eqs. (IY.165)-(IV.172) we obtain

and

f.Lo 1dxA] ~Ao =---

x x 47TR

as R ~ co

BA]xA1x=A zx, --

Bz

as R ~ co

if z = h

(IV.173)

as R ~ a:

1 1- divA. = - divA,1'1 1'z

(IV.174)

It is essential to note that in accordance with the theorem of uniquenessEqs. (IV.173), (IY.174) uniquely define components Ax and A z ' respec-tively.

Thus, the solution of the original boundary-value problem consists oftwo steps, namely,

(a) Determining the horizontal component Ax from Eqs. (IV. 173).(b) Knowing component Ax, determining the vertical component A z

from Eqs. (IV.174).

First of all, proceeding from Eq. (IV.173), we will find the componentA X' As follows from Eqs. (IV. 156) and (IV.166), the vector potential of the

(IV.175)

(IV.177)


primary magnetic field can be presented as

f-Lo 1dx 00

A Ox = --1 e-mIZ!Jo(mr) dm41T 0

Therefore, by analogy with the previous example, solutions of Laplace'sequation VZAx = 0 that satisfy conditions at infinity and near the dipoleare

if z s; h

(IV.176)

if z;:::; h

Here Cm and Dm are unknown coefficients depending on m,Now applying conditions at the interface z =h, Eqs. (IV.173), we have

ie- mil + C emil =D e-mh

m m

_e-mll + C emh = -D e-m hm m

Solving this system we obtain

Cm=O,

In other words, the horizontal component of the vector potential, A x' isnot subjected to the influence of a horizontal interface, and correspond-ingly,

f-Lo1 dxA =A) =Az =--

x x x 41TR (IV.178)

Let us note that this result directly follows from Eqs. (IV.173), sincethey do not contain the conductivity, and therefore we can assume that themedium is uniform. It is easy to prove that in any laterally uniformmedium the horizontal component of the vector potential, A x' of thehorizontal electric dipole is independent of parameters of the medium;that is,

(IV.179)

Correspondingly, taking into account the last equation of (IV.170),

aAxB =--Z ay

we have to conclude that the vertical component of the magnetic field due


to the horizontal dipole is not subjected to the influence of conductivitiesin a horizontally layered medium. Applying the principle of superpositionwe can say that this result remains valid if, instead of the dipole, we have ahorizontal current-carrying wire of arbitrary length.

Now, proceeding from Eqs. (IV.174), we will solve the next boundary-value problem and determine the component A z . To find its expression, itis convenient to make use of the last equality of Eqs. (IY.174), whichfollows from continuity of the tangential components of the electric fields.

1 1- divA. = - divA,1'1 1'2

if z = h

or

~ (aA lx + aA lz) = ~ (aA 2X+ aA 2Z)'Yt ax az 1'2 ax az

(IV.180)

In accordance with Eqs. (IV.175) and (IV.178) we have

aA tx aA 2x aA ox ar--=--=---

ax ax ar ax(IV.181)

Inasmuch as

ar- = cos cpax

andaJo(mr)---= -mJt(mr),

ar

Eq. (IY.18I) can be presented as

aA 1x aA 2x flo! dx 1'"-- = -- = - --- cos cp e-mlzIJI(mr) dmax ax 47T 0

where cp is the angle between the axis x and the radius r, while J,(mr) isthe Bessel function of the first kind.

The equality (IV.180) holds regardless of the values of rand cp, andtherefore the terms containing the component A z have to look similar tothose for aAjax. Correspondingly, we will describe the component A z as

if z 5, h

(IV.182)

if z ~hflo! dx 1'"A 2z = --- cos cp Fme-mZJt(mr) dm

47T 0

Unlike the component Ax' the functions A lz and A 2z depend on the


azimuthal component cp and are a solution of Laplace's equation.

(IV.183)

Substituting Eqs. (IV.182) into Eq. (IV.183) we see that both functionsA I z and A 2z satisfy Laplace's equation. Also it is obvious that they obeythe conditions at infinity.

Finally, substituting Eqs. (IV.182) into the equalities

as z =h

we have

(

E emh =F e-mhm m

(P2 - PI)me-mh = -mP2Fme-mh - mPIEm emh

Whence

E K -2mhm = - 12e , (IV.184)

where

K _ P2 - PI12 -

P2+PI

Therefore

!-tol dx 00

A = - ---K cos cpj e-2mhemzJ (mr) dmlz 47T 12 0 1

if z:s; h

if z ~ h

(IV.185)

Inasmuch as these functions satisfy Eqs. (IV.174) they describe the vectorpotential of the magnetic field caused by the horizontal electric dipolelocated beneath the horizontal interface.

Making use of the equality

we can express the component A z in terms of elementary functions and


obtain

JLoldx V r2+(2h-z)2-(2h-z)A Iz = - -4--K 12 cos cp ~--_ jr=======2=---

rr rvr2+(2h-z)

and (IV.186)

Thus, we have solved the boundary-value problem and determined thevector potential A.

JLoI dxA =---

Ix 47TR

JLoldx Vr 2+(2h-z)2 -(2h-z)A l z = - ---KI2 cos cp----;=======---

4rr ryr2+(2h-z)2

and

JLoI dxA =---

2x 4rrR

if z s; h(IV.187)

(IV.188)

if z ~h

With this information the components of the magnetic field can be easilyfound from Eqs. (IV.170).

As an example, consider the case in which both the dipole and theobservation point are located at the earth's surface: K I2 = 1, z = 0, h = O.Then we have

JLoIdxA =A I =A2 = ---

x x x 4rrr

JLoI dxA = A I = A 2 = - -- cos cp

z z z 4rrr

(IV.189)

The horizontal component of the vector potential A x is caused by thecurrent element I dx, and in accord with the Biot-Savart law it generates


at the earth's surface a vertical component of the magnetic field.

J-toldxB = ---sincp

z 4'7Tr2 (IV.190)

At the same time the vertical component of the vector potential is relatedto the currents in the conducting medium, which produce the horizontalcomponents of the field B. Applying a cylindrical system of coordinateswith axis z we obtain from the equation B = curl A,

or

1 aA zB=--r r acp

andaA zB =-

'I' ar(IV.191)

It is natural that expressions of the field components given byEqs. (IV.190), (IV.19l), and (IV.14l) coincide with each other.

In conclusion let us make several comments.1. Applying the same approach we can arrive at formulas describing

the behavior of the magnetic field in a horizontally layered medium.. 2. Proceeding from Eqs. (IV.187), (IV.188) and the principle of super-position we can derive formulas for the field when instead of the dipole wehave a horizontal current-carrying line of a finite length.

3. Since an arbitrarily oriented dipole can be represented as a sum ofvertical and horizontal dipoles, the results derived in this section can beused to determine the field caused by any distribution of currents.

Example 7 A Horizontal Electric Dipole in a Conducting Half Space neara Vertical Contact

Suppose that a horizontal electric dipole is located in a conductingmedium with a vertical contact dividing two uniform parts with conductivi-ties 'Y1 and 'Y2 (Fig. IY.9a). Let us choose a Cartesian system of coordi-nates so that the dipole is situated at its origin and the x-axis is perpendic-ular to the contact 5 12 and the z-axis is directed upward. First, we willderive equations for the vector potential caused by currents in the con-ducting medium, since the primary vector potential A Ox is known.

J-toldxA =--

Ox 4'7TR

.......•..........".

....


z ,b

X

1, 12 Eo~

,,

M x ,

h512

256

dCz 11

64:'

.' .,/

Eol101

.'~ ..........

1; ,6

10

Fig. IV.9 (a) Horizontal electric dipole near a vertical contact beneath the earth's surface;(b) spheroid in a uniform electric field; (c) spheroidal system of coordinates; (d) curves of theratio B;/Bo'P'

In accordance with Eq. (IV.1l7) we have

A(p) = - I-Lo'YJ f U,(q) as, - I-Lo'Yz f Uz(q) as,47T SOl L q p 47T S02 L q p

(IV.I92)

where L q p is the distance between an arbitrary point q, located at surfacesSo and SIZ' and the observation point p. SOl and Soz are parts of theearth's surface So surrounding media with conductivities 'Y, and 'Yz,respectively, and

So = Sal + Soz

U/q), Uiq), and U(q) are potentials of the electric field caused by


charges that appear at all surfaces, including those of dipole electrodes.Proceeding from results obtained in Chapter III, it is a simple matter tosee that Ut(q) and Uiq) in general are the potential of either four or twohorizontal electric dipoles, which are symmetrical with respect to theearth's surface and the contact.

As follows from Eq. (IY.I92), integration over the earth's surfaceproduces the vertical component of the vector potential, while the sameprocedure for the contact gives rise to the horizontal component. Thus wehave

(IV.I93)

Here it is appropriate to make two comments:

1. The vector potential given by Eq. (IV.I92) can differ from a solutionof Poisson's equation.

A(p) =.!:.:!.- f j(q) dV47T V L q p

by the gradient of some function ep, but this does not have any influenceon the magnetic field since curl grad ep == O.

2. In the previous examples we have demonstrated that singularities ofthe vector potential behavior is caused by only the current in the wireconnecting dipole electrodes. This fact allowed us to present A(p), gener-ated by currents in the conducting medium, through integrals over theearth's surface and the contact only. In other words, we consider thesecondary vector potential while the total potential is

where M = EOPt I dxi is the dipole moment.

In principle, numerical integration enables us to determine the vectorpotential at every point of the conducting medium. However, to find themagnetic field B we also have to carry out calculations of the derivativesthat constitute curl A. For this reason it is convenient, making use ofEq. (IV.I92), to derive an expression for the magnetic field.


Let us consider the integral

U(q)N(p) =f-ndS

s L q p

when n is the unit vector perpendicular to the surface S.Then, taking curl" N on both sides of this equality, we have

P P U( q)curl N = V' X f--n dS

s L qp

or

(IV.194)

(IV.195) .P P n

curl N = f U( q) V' X - dSs L q p

since integration and differentiation are performed with respect to differ-ent points. As we know,

curl cpa = cp curl a + grad cp X a

Therefore

P n pIn X L q pV' X - = -n X grad - = ---;:--

L q p L qp L~p

inasmuch as n is independent of point p and

p 1 L q pgrad- =--

i: L~p

Therefore, in accordance with Eq. OV.192), we have for the magnetic field(B = curl A),

We will introduce the notation

L q p = Ax i + Ay j + Az k

(IV.196)


where

Bearing in mind that

i =j X k,

we have

k= i Xj, j = kx i

and

(IV.197)

(IV.199)

Equations (IV.197) allow us to calculate the components of the magneticfield at every point of the conducting medium, including the earth'ssurface. Now let us demonstrate that the vertical component of themagnetic field can be expressed in terms of elementary functions if boththe dipole and the observation point are located at the earth's surface.

Taking into account Eq. (IV.193), let us consider A .r :

A _- ,uO(-Y2 - 'YI) r U( q) dSJ, (IV.198)

x 47T 512

L qp

Suppose that a horizontal electric dipole is located at a distance t fromthe vertical interface between media with conductivities 'YI and 'Y2. Asfollows from Eq. (IV.117) the expression for the vector potential A~ inthis case is

A*=A*i= ,uO('Y2-'YI) if U*(q) dSx 47T 5 L q p

= ,uO('Y2-'YI) if U*(q) dS27T 5 12 L qp

where U*(q) = tU(q), since the dipole is situated in a uniform mediumwith conductivity 'Y I.

if x:s; t


Comparison of Eqs. (IV.198) and (IV. 199) shows that

Therefore, we can make use of Eq. (IY.159) to calculate the field B, in thepresence of the contact. Changing the directions of the coordinate axis wehave

A = J-Loldx (~_ K l 2 )

Lr 477" R R;

Letting z = aand taking the derivative BA xlay, we arrive at the followingexpression for the vertical component of the magnetic field of a horizontalelectric dipole.

if x ~t

if

(IV.200)

where t is the distance from the dipole to the contact, and x = x p > y = yp

are coordinates of the observation point.As is seen from Eqs. (IV.200) the magnitude of the vertical component

of the field, caused by currents in the medium, gradually increases inapproaching the contact and then begins to decrease if y =1= O.

In conclusion we will make two comments.

1. The representation of the magnetic field in terms of surface integralsallows us to perform calculations of B due to an arbitrarily orientedcurrent-carrying wire, located in a conducting medium, provided that thepotential of the electric field is known at the boundaries with the insulatorand other interfaces.

2. The study of the magnetic field is often used in mapping lateralchanges of resistivity near the earth's surface.

Example 8 A Conducting Spheroid in a UniformElectric Field

Now we will study the influence of a confined inhomogeneity on themagnetic field. With this purpose in mind, consider a spheroid withsemiaxes a and b and conductivity 'Yi' placed in a uniform conducting


medium with conductivity 'Ye . The primary electric field is uniform anddirected along the major axis 2a (Fig. IY.9b). As we know, due toexcitation by the primary field Eo, electric charges arise on the ellipsoidsurface, and they cause a secondary electric field. The charge density at apoint p on the spheroidal surface is described by

"( ) 2 'Yi-'Ye E 3V( )L. P = 00-+-- n PI'i 'Ye

where E~v is the average of values of the normal components of the totalelectric field on the internal and the external sides of the surface near thepoint p. Due to the secondary electric field the distribution of currentdensity changes and, correspondingly, a secondary magnetic field arises.

To understand the behavior of the field B, let us first consider theelectric field. It is almost obvious that with an increase of the semimajoraxis a, the charge density on the lateral surface of the spheroid decreases,and the second field of these charges becomes smaller. In other words, wecan expect that a spheroid markedly elongated in the direction of theprimary field does not significantly distort the electric field. At the sametime, the current density inside and conductor, and hence the magneticfield, increases in proportion to the conductivity 'Yi' In essence such a casecorresponds to an infinitely long cylinder with axis b.

Taking into account the relatively simple shape of the conductor we willapply the method of separation of variables and find the electric field.Then, knowing the current density inside and outside of the spheroid, themagnetic field can be determined. Thus, our first step is to solve theboundary-value problem for the potential of the electric field U. For thispurpose we will introduce a prolate spheroidal system of coordinates ?, YI,'P related to cylindrical coordinates (Fig. IV.9c) by

where c = Va 2- b 2 and

z = c?YI (IV.20l)

-l::;,?<+l,

In particular, the surface of the spheroid with semiaxes a and b is thecoordinate surface Ylo = constant, and

a = CYlo' (IV.202)


The metric coefficients of this system are

_ (YJZ_(Z)I/Zhz - C Z '

YJ - 1(IV.203)

Then, bearing in mind the fact that the field possesses axial symmetry withrespect to the z-axis, Laplace's equation for the potential V is

(IV.204)

It is convenient to represent the potential inside and outside thespheroid as

if YJ::5; YJo

if YJ ~ YJo(IV.20S)

where Vo and Ves are the potentials of the primary and secondary fields,

respectively. As the distance from the spheroid increases, the field ofthe surface charges decreases, and therefore the boundary condition at in-finity is

(IV.206)

At the spheroid surface both the potential and the normal component ofthe current density are continuous functions and hence the conditions atthe interface are

eu, eu,)lia;; =)le a:ry (IV.207)

As we know, Eqs. (IV.204)-(IV.207) uniquely define the electric field.First, applying the method of separation of variables, we will find asolution to Laplace's equation. Representing the potential V as

and substituting it into Eq. (IV.204), we obtain two differential equationsof the second order.

(IV.208)


where n is an integer. These equations are very well known; they arecalled Legendre equations. Their solutions are Legendre functions of thefirst and second kinds, P and Q. Correspondingly, we have

Tn(y/) =AnPn(y/) + BnQn(Y/)

¢(O =cnPn(O + DnQn(Y/)(IV.209)

Legendre functions are another example of orthogonal special functionsand are widely used in mathematics and applied physics.

As an illustration, expressions for the functions Pn(x) and QnCx) for thefirst three values of n are given below.

Po(x)=1

I x + IQ tC x) = -x In -- - I

2 x-I (IV.21O)

I x + I 3xQ (x) = -(3x 2 -I)ln -- - -

2 4 x-I 2

Thus, the general solution of Laplace's equation is a sum.

00

U(~,y/) = L [AnPn(y/) + BnQn(Y/)][cnPnU) +DnQn(O] (IV.211)n~O

Before we continue our search for a solution to the boundary-valueproblem, let us express the potential of the primary field in terms ofLegendre functions. Since Eo is uniform and directed along axis z, wehave

or

Then, making use of Eqs, (IV.201) and (IV.21O) we obtain

(IV.2I2)

That is, the potential of the primary field is expressed with the help ofLegendre functions of the first kind and first order.

Let us note that function PtC?) describes a change of the potential atany coordinate surface where Y/ = constant, and in particular on thespheroid surface Y/ = Y/o' Therefore, it is natural to assume that thepotential of the secondary field depends on coordinate ~ in the same


manner. Bearing in mind that function QI(7])

7] 7]+1QI ( 7]) = - In -- - 1

2 7] - 1

decreases with an increase of the distance, we will present the potentialoutside the spheroid as

if 7]:2:7]0 (IV.213)

Also, we will suppose that the field inside the spheroid remains uniformand directed along axis z; that is,

if 7] < 7]0 (IV.214)

where A and B are unknown coefficients.It is obvious that the functions Vi and U; obey Laplace's equation and

U, satisfies the condition at infinity. Finally, we have to choose coefficientsA and B in such a way that Vi and U; satisfy conditions at the spheroidsurface 7] = 7]0' In accordance with Eqs. (IV.207) this requirement leads toa system of two equations with two unknowns.

PI( 7]0) + AQI( 7]0) = BP I( 7]0)

Ye{p; (7]0) + AQ't( TM} = YiBP; (7]0)(IV.215)

where P;( 7]0) and Q'I(7]0) are first derivatives of Legendre functions withrespect to 7]

and

1 7]+1 7]Q' ( ) = -In _0 0_

I 7]0 2 1 2 17]0 - 7]0-


( ~ -1)7]0(7]~-1)

A= (Yi Y)e . [7]0 7]0+ 1 ]1 + - - 1 (7]6- 1) - In -- - 1

Ye 2 7]0 - 1

1

B = ( Yi) [7]0 7]0 + 1 ]1 + - - 1 (7]6 - 1) - In -- - 1Ye 2 7]0 - 1

(IV.216)


It is clear that the functions U, and Ve , given by Eqs. (IV.213), (IV.2I4),and (IV.2I6), satisfy all the conditions of the boundary-value problem, andtherefore they describe the potential of the electric field when the con-ducting spheroid is placed in the uniform field Eo directed along its majoraxis.

It is convenient to express coefficients A and B in terms of theparameter e.

c

a

As follows from Eqs. (IV.20l),

a = c7Jo,

and

a7Jo+ I=-(I+e),

c

whence

a7Jo- I=-(1-e)

c

and

1B = -------,--

1+ (~: - 1) L

(IV.217)

where L is a geometrical factor, equal to

1 - e2

[ 1 + e ]L =-- In-- -2e2e 3 1- e

(IV.2I8)

In accordance with Eqs. OV.2I7) the uniform electric field inside the


spheroid is

E; = BEo= ( 'Yi )1+ --1 L

Ye

(IV.219)

(IV.220)

and the function L characterizes the effect of surface charges. Forinstance, for a markedly elongated spheroid (e ~ 0,

L~::(ln2: -1)As follows from Eq, (IV.219), as the major axis 2a increases, the

electric field E, approaches the value of the primary field Eo. However, byincreasing the ratio yjYe the spheroid must be increasingly elongated forits field to coincide with that of an infinitely long cylinder. In particular, inthe case of an insulating surrounding medium, the secondary field isalways equal to the primary field inside the spheroid regardless of itslinear dimensions, but it has the opposite direction (the phenomenon ofelectrostatic induction).

From Eq. (IV.219) the current density in the spheroid is

. YiEo

] z = 1+ ( ~~ _ 1) L

For relatively small ratios of conductivities and with L « 1, an increase ofthe spheroid conductivity causes the current density and correspondinglythe magnetic field intensity to also increase in direct proportion to Yi' justas in the case of an infinitely long cylindrical conductor. However, withfurther increase of the conductivity Yi' the secondary electric field in-creases and the total electric field E, becomes smaller. When (yjYe)L » 1the current density and the magnetic field are practically independent ofthe spheroid's conductivity but are proportional to the conductivity of thesurrounding medium,

(IV.221)

Of course, this equation also describes the current density in the case ofan ideal conductor, when the electric field E, equals zero. If the spheroidis more resistive and the ratio alb is relatively small, the current density isproportional to the spheroid conductivity.


Now we will derive an expression for the magnetic field in the planez = O. It is clear that due to axial symmetry the field has only thecomponent B<p' In deriving the secondary magnetic field we will proceedfrom the first equation of the field in the integral form

(IV.222)

First, let us find the secondary current passing through the circle area withradius b. As follows from Eq. (IV.219) we have

(IV.223)

The secondary current through the elementary ring with radius randwidth dr, located at plane z = 0 outside the spheroid, is

(IV.224)

if 11 = 0

where Efz is the difference between the total and primary electric field,which is

1 iJ(Ue-Uo)E e = ------1z h

1a(

Taking into account Eqs. (IV.203) and (IV.213) we have

where

(IV.225)

1 t/27'/ = -(r2+a2 - b2)

c

Substituting Eq. (IV,225) into Eq, (IV.224) and performing integration weobtain

(

7'/2 - 1 7'/ + 1 7'/6 - 1 7'/0 + 1 )I = Tf'V E Ac 2 --.-In -- - on - -- In --+ on

e t e 0 2 7'/ - 1" 2 7'/0 - 1 '10

since rdr = C27'/ d-q.

Correspondingly, the total current is

(

7'/2 - 1 7'/ + 1 7'/6 - 1 7'/0 + 1 )I + I = Tf'V E c2A --In-- - on - -- In -- + on

I e Ie 0 2 7'/ - 1 'I 2 7'/1 - 1 '/0

+ 'YiTfb2BEo - 'YeTf b2EO


Making use of the equality, Eq. (IV.2I5),

ye{1 +AQj(1]o)} =Yi B

and the relation

we obtain

(

1] 2 - 1 1] + 1 )I = I, + I, = 7Tye EOA ( a 2

- b 2) -2- ln 1] _ 1 -1]

Next, applying Eq. (IY.222), we get the following expression for thesecondary magnetic field:

J.Lo(a2

- b2) Eo {1]2- I 1]+1 }

Be = Y A -- In-- - 1]ip 2r e 2 n - 1

or

(IV.226)

whereYer

Bocp = TEo

is the primary magnetic field, provided that it possesses the axial symme-try.

The relationship between the magnetic field and parameters describingthe medium is illustrated by the curves of B;/Bocp in Fig. IV.9d. The leftasymptote of these curves corresponds to the case of the spherical conduc-tor (a = b), as the influence of charges on the electric field is relativelystrong, especially for large values of Y/Ye' As the major axis increases,this influence becomes smaller and the curves approach to their rightasymptote, which characterizes the magnetic field of currents in aninfinitely long cylinder.

Having defined factors that provide equivalence of magnetic fieldscaused by currents in the spheroid and an infinitely long cylinder, let usconsider two-dimensional models in more detail. Suppose a cylindricalconductor with an arbitrary cross section is placed in a uniform electricfield Eo that is directed along the cylinder axis. Since the field Eo istangential to the conductor surface, electric charges do not arise. There-

IV.6 Magnetization and Molecular Currents 481

fore, the secondary electric field is absent regardless of conductivities ofthe medium. In other words, in such models of the field and medium,which are called E-polarization, it is impossible to detect the presence ofinhomogeneities by measuring the electric field. However, the currentdensity differs from j~ inside the cylinder and we have

(IV.227)

where j; is the secondary current density.By mentally dividing the cross section of the inhomogeneity into many

elementary areas, we can say that the secondary magnetic field is a sum offields, caused by current filaments directed along the z-axis. Then, apply-ing the principle of superposition and Eq. (IY.36) we have

(IV.228)

where q is an arbitrary point of the cross section and k is the unit vectoralong axis z, while S is an arbitrary cross section of the two-dimensionalcylinder.

IV.6 Magnetization and Molecular Currents:The Field H and Its Relationto the Magnetic Field B

In Chapter III we considered the behavior of the electric field in thepresence of dielectrics and now, before investigating the influence ofmagnetic materials on the magnetic field, it is useful to summarize themain results of that study. First of all, due to polarization both volume andsurface bounded charges appear in dielectrics. Consequently Coulomb'slaw, which describes the electric field inside and outside dielectrics, iswritten as

where Do and lo are the densities of free charges, while Db and lb arethe densities of bound charges.


It is essential to note that regardless of the dielectric permeability E ofthe medium, the coefficient on the right-hand side of Eq. (IV.229),

1

remains the same.This equation requires some additional comments.

1. The terms dV and dS are an elementary volume and surface,respectively; that is, every one of them is much greater than moleculardimensions. In other words, both dV and dS contain a practically unlim-ited amount of molecules. At the same time the sizes of the elementaryvolume and surface are much smaller than the distance L q p between themand the observation point p.

2. Every atom of a dielectric has either some dipole moment oracquires it in the presence of an electric field. Due to the action of thisfield, dipoles align along the field and their distribution within an elemen-tary volume or a surface is characterized by the dipole moment density P.Such replacement of atomic dipoles by one dipole is a spatial averaging ofmicroscopic quantities (dipole moments of atoms). Therefore, the densityof dipole moments P of elements dV and dS is a macroscopic quantity,which is equal to the mean microscopic density of the atomic momentswithin these elements in the presence of the electric field.

3. Changing the position of the observation point, the microscopic fieldE rnier also varies, but the electric field E, defined by Coulomb's law,represents the macroscopic field, obtained by averaging microscopic fieldswithin elements dV and dS.

4. In our study of the vector polarization Pc we have introduced boundcharges with densities °b and I b and found the relationship betweenthem and the vector P, to be

(IV.230)

Inasmuch as the distribution of bounded charges is unknown, Coulomb'slaw cannot be used to determine the field E, and this remark also appliesto the system of field equations

curlE = 0

CurlE = 0

00 + 0bdivE=---

EO

I o +I bDivE=---EO

(IV.231)


since the right-hand side of the second equation contains the unknownquantities °b and ~ b .

5. To overcome this problem, we performed the following operations:(a) Making use of the relation between bound charges and the polar-

ization vector, Eq. (IV. 230), the system of field equations was modifiedand the new auxiliary vector D was introduced. Then, instead ofEqs. (IV.23l), we obtained

where

curl E = 0

Curl E = 0

°0divD =-

EO(IV.232)

It is essential that the right-hand side of the new system is known.(b) Proceeding from experimental data the relation between the polar-

ization vector P, and the field E was established.

(IV.233)

curl E = 0

where a is the dielectric susceptibility of a medium, which is assumed tobe known.

Having substituted Eq. (IV.233) into Eqs. (IV.232) we arrived at asystem of field equations where only the electric field is unknown.

°0divEE= -

EO

CurIE = 0

(IV.234)

where

is the dielectric permeability.By knowing the parameter of the medium E, we are accounting for the

presence of bound charges without using them to calculate the electricfield.

It is also proper to notice that the system (IV.234) is more complicatedthan the original one, since the dielectric permeability E is, in general, afunction of a position.


A similar approach was used in deriving the system of equations for theelectric field in a conducting medium.

Now we are ready to describe the influence of materials on themagnetic field B. We will follow the same path of study as in the case ofdielectrics. As is well known, some materials-for instance, iron-afterbeing placed in a magnetic field B, produce a noticeable change in thisfield, while other materials have an extremely small influence. This hap-pens due to magnetization, which is displayed in varying degrees by allmaterials. Therefore, from a qualitative point of view we can see asimilarity with the polarization of dielectrics. However, there are manyfundamental differences. In particular, unlike most dielectrics, which aredepolarized when an external field vanishes, there is such a group ofmagnetic materials whose magnetization remains even if the external fieldB disappears. The existence Of these materials, called ferromagnetics, isimportant for magnetic methods in geophysics.

Taking into account our purposes, let us describe the magnetization inthe following way. Suppose for simplicity that a magnetic material is aninsulator and consequently conduction currents are absent. In spite of thisfact, within every molecule different types of motions of electrons occurthat can be approximately visualized as a molecular current. Therefore,every elementary volume contains practically an unlimited number ofmolecular currents. If the permanent magnetization is absent, then thesecurrents are randomly distributed and their magnetic field vanishes. Incontrast, in the case of magnetization molecular currents are mainlyoriented systematically, and consequently they create a magnetic fieldinside and outside of magnetic materials.

If a medium also possesses conductivity, we will distinguish two types ofcurrents, namely conduction currents, which describe a transformation ofcharges through the medium, and molecular ones, which are closed withinmicroscopically small volumes. To calculate the magnetic field B, causedby molecular currents, we will perform their averaging within every ele-mentary volume. In other words, a system of these currents in such avolume is replaced by a distribution of coaxial macroscopic currents withdensity jrn' closed within an elementary volume. Similar averaging is alsoperformed for the conduction and both types of surface currents. Then,for the total density of the volume and surface currents we have

j = J, + jm i = i c + i m

Consequently, the Biot-Savart law is written

}-La [I UC + jm) X L q p 1(ic + i m) X L qp ]B(p)=- 3 dV+ 3 dS (IV.235)477 v L q p S L qp


and it describes the magnetic field B at every point inside and outside themagnetic material. By analogy with the case of a nonmagnetic medium, weobtain the expression for the vector potential A (B = curl A).

(IV.236)

which can easily be derived from Eq. (IV.235).It is appropriate to emphasize that

(a) The Biot-Savart law defines the macroscopic field B, which is themean microscopic field within every elementary volume or surface.

. (b) The coefficient at the right-hand side of Eq. (IV.236)

is independent of magnetic materials. In other words, the Biot-Savart law,as well as Coulomb's law in the presence of dielectrics, correctly describesthe field B in any magnetic material, provided that all currents are takeninto account.

(c) Since the distribution of molecular currents is unknown, theBiot-Savart law cannot be used for field calculation in the presence ofmagnetic materials, and therefore we have to refer to a system of fieldequations. In this sense the analogy with Coulomb's law is obvious.

(d) In spite of the fact that both jm and i m are macroscopic densities ofcurrents, we call them molecular currents to emphasize the differencefrom conduction currents.

The system of field equations in a nonmagnetic medium, Eqs. (IY.89),has been derived from the Biot-Savart law.

. XLB( p) = ~ f Je

3 qp dV417 V L qp

Comparing the latter with Eq, (IY.235) and taking into accountEq. (IY.89) we come to the conclusion that the system of field equations inthe presence of magnetic materials is

curl B = !-LoOe + jm)

Curl B = !-Lo(ie + i m)

divB = 0

DivB = 0(IV.237)

From a theoretical point of view this system does not differ fromEqs. (IV.89). In fact, both of them describe a vortex field. However, there


is one essential difference; namely, the right-hand side of the first equationin the last system contains an unknown density of molecular currents. Toovercome this obstacle we will, first of all, modify the first equation of thissystem in such a way that its right-hand side contains only the density ofconduction currents, which is independent of the field B. In solving thistask we will apply an approach similar to that for the dielectric case. It isconvenient here to proceed from the obvious fact that closed molecularcurrents within every elementary volume create a field B that coincideswith the field of a magnetic dipole with some moment dM. Taking intoaccount Eqs. OV.55)-OV.57) the vector potential, caused by molecularcurrents, can be written as

where

dMP(q) = dV

(IV.238)

(IV.239)

The vector P(q) is called the vector of magnetization, and it character-izes the magnitude and orientation of the dipole moment dM(q). It is alsoclear that the direction of the vector P(q) is perpendicular to the planewhere the currents with density jrn are located, and the vectors P and jrnform a right-handed system (Fig. IV.lOa). As follows from the definition ofthe dipole moment, the unit of measurement for magnetization of thevector is amperes per meter.

A[P] = ~

m

Certainly there is some similarity between the electric polarization vectorP, and that of magnetization P, and this is natural, since both of themcharacterize a distribution of field generators.

In Chapter III we found that

and

and it is also logical to determine similar relations between the vectors Pand jrn'


a

c

I=l

: :


b

d

R

<

Fig. IV.IO (a) Polarization and molecular currents; (b) illustration in deriving vector poten-tial of molecular currents; (c) behavior of the field B; and (d) behavior of the field H.

we represent Eq, (IV.238) as

/La f P 1Am(p) = - P(q) X V' - dV4rr v L q p

Then making use of the equality

curl if!a = if! curl a + grad if! X a

we haveq

/La curl P /La q PAm(p) = -f --dV--f curl-dV

4rr v L q p 4rr v L q p

(IV.240)

(IV.241)

The second volume integral can be replaced by a surface integral since:

f cUrladV=~ n aXad5v So

where 50 and Do are the surface surrounding the volume, and its unitnormal, directed outward.


Therefore we obtain

q

fJ-0 j curl P fJ-0 j no X PA (p)=- --dV-- --d5

rn 47T V L q p 417 So L q p

(IV.242)

The vector potential Arn(p) is caused by all molecular currents includ-ing those located far away from observation points. Consequently we canneglect the surface integral in Eq. (IV.242). In fact, as the distance L fromany system of current loops increases, their magnetic field approaches thatof the magnetic dipole; that is, it begins to decrease as 1/e. Correspond-ingly, the magnetization vector P should also decrease in the same mannerand

"0 X P 1---~k-i; L4

as L ~ 00

where k is a coefficient.Now applying the mean value theorem, we obtain

Thus, instead of (IV.242) we have

. fJ-0jcurIPArn(p) = - --dV

417 v L q p

if L ~ 00

(IV.243)

As follows from Eq. (IV.236) the vector potential can also be written inthe ordinary form

(IV.244)

Comparing these last two equations we arrive at a relationship betweenthe volume density of molecular currents jrn and the vector of magnetiza-tion P.

jrn = curlP (IV.245)

Next we will consider a more complicated model of a magnetic mediumwith some interface 512 , where the vector of magnetization P is a discon-tinuous function (Fig. IV.lOb). In this case we will again perform the sametransformations with Eq. (IY.240) as before, but preliminarily it is neces-sary to enclose the surface 512 by another surface 5 * and then apply


Eq. OV.24l) in the volume V surrounded by surfaces So and S*. Thenecessity of this procedure is related to the fact that curl P does not havemeaning at the surface S12' Thus, instead of Eq. OY.242) we have

f-Lo j curlP /Lo,{.. n , X P flo,{.. no X PA (p) = - --dV- -'Yc dS- -'Yc --dS

m 47T V L q p 47T 5* L q p 47T 50 L q p

(IV.246)

where n * is the unit vector perpendicular to the surface S * and directedoutside of volume V (Fig. IY.lOb).

As S* approaches S12' and neglecting the last integral since So islocated at infinity, we have

f-L0jCUrlP /Lo,{.. H*XPAm(p) = - --dV- -'Yc dS

47T v L q p 47T 5* L q p

Taking into account the fact that integration over surface S * consists overintegration at the back and front sides of the interface S12 ' we obtain

,{.. n * X P j (n * X P) 1 + (n * X Ph'Yc ~-- dS = dS

5* L q p 512 L q p

where the indexes "1" and "2" indicate the back and front sides of theinterface S12 , respectively. As is seen from Fig. IV.lOb,

and

where n is the unit vector, normal to the surface S12' Consequently, wearrive at the following expression for the vector potential caused byvolume and surface molecular currents:

/L curl P /L n X (P - P )A

m( p) = _0 j __ dV + _0 f 2 1 dS (IV.247)

47T v L q p 47T 5 12 L qp

where P2 and PI are magnetization vectors at the front and back sides ofthe interface, respectively. Comparing Eqs. OV.247) and (IV.236) we seethat

(III.248)

That is, the difference of tangential components of the magnetizationvector at the two sides of the interface defines the density of molecularcurrents i m • Thus, we have established relationships between the averagedensity of molecular currents and the vector P, and along with similar

(IV.249)


relations for the electric polarization, they are

curlP=jrn CurlP= i rndivPe = - 8b Div Pe = - ~ b

Now we are prepared to make the first step in deriving the system ofequations of the field B in the presence of magnetic. materials. In fact,making use of Eqs. (IV.249) we can rewrite Eqs. (IV.237) as

curl B = J-LoOe + curl P), divB = 0

Curl B = J-Lo(ie + Curl P), Div B = 0

or

curl(B - J-LoP) = J-Loje'

Curl(B - J-LoP) = J-Loi e,

Introducing the new vector H,

J-LoH = B - J-LoP

we finally obtain

divB = 0

DivB = 0

(IV.250)

(IV.25I)

(IV.252)

curlH = J, divB = 0

CurlH = i c DivB = 0

where the right-hand side of the first equation contains only the density ofconduction currents, which can be specified.

This fact strongly indicates that we have advanced in deriving thesystem of field equations. However, there is still one obstacle to beovercome; namely, it is necessary to establish a relation between vectors Band H, and in this connection let us make several comments.

1. In accordance with Eq. (IV.250) we have

IH=-B-P

J-Lo

That is, H is the difference of two fields with completely different physicalmeanings. Indeed, one of them up to a constant J-Lo describes the magneticfield, while the other characterizes the distribution of molecular currents.Such a combination can hardly be explained from a physical point of view.Later we will demonstrate that, in general, fictitious sources along withconduction currents create this field. This shows once more that H is anauxiliary field, which only allows us to derive the system of field equations.

2. In the vicinity of points, where the magnetization vector P vanishes,the fields Band H differ by the constant J-Lo only.

B = J-LoH


but this does not change the fact that they are fundamentally differentfrom each other.

3. The field H is often called the magnetic field and from an historicalpoint of view such terminology can easily be justified. However, here it willbe called field H.

4. The field H, as well as the magnetization vector P, is measured inamperes per meter; and this unit is related to that in Gauss' system by

A1 - = 4rr' 10-3 oersted

mor

A1 oersted « 79.6 -

m

To use the system of Eqs. (lV.25I) we have to establish a relationbetween the magnetization vector and the field H. Experimental studiesshow that the linkage is much more complicated than that for mostdielectrics, and it can be written in the form

P =X(H)H + Pr (IV.253)

where X(H) is a function that in general depends on the field strength andthe past history of the material; and P, is the remanent magnetization,which remains even when the magnetic field vanishes. However, we willuse the approximate relation

(IV.254)

where X is a constant of the magnetic material, which is independent onthe field, and it is called the magnetic susceptibility. It is clear that theparameter X is dimensionless. Let us make several comments concerningEq. (IV.254).

1. There are three main groups of magnetic materials:(a) diamagnetic(b) paramagnetic(c) ferromagnetic2. In diamagnetic substances X is extremely small (:::::: 10-5) and nega-

tive, so that the magnetization is very weak.3. The susceptibility of paramagnetic materials is positive, and it is

around 10-4 •

4. Remanent magnetization is absent in both these groups of magneticmaterials, and instead of Eq. (IV.254) we have

P=XH


5. Ferromagnetics are usually characterized by large susceptibilities,and they are also able to sustain magnetization in the absence of anexternal magnetic field.

6. The susceptibility of rocks is mainly defined by the presence offerrimagnetics, such as magnetite, titanomagnelite, and ilmenite. Thefollowing table, after Parasnis, demonstrates values of susceptibilities ofrocks.

Susceptibilities SusceptibilitiesMaterial (X 106) Material (x 106)

Graphite -100 Gabbro 3800-90000

Quartz -15.1 Dolomite

Anhydrite -14.1 (impure) 20000

Rock salt -10.3 Pyrite

Marble -9.4 (pure) 35-60

Dolomite Pyrite(pure) -12.5- + 44 (ore) 100-5000

Granite Pyrrhotite 103_105

(without magnetite) 10-65 Haematite (ore) 420-10000

Granite Ilmenite (ore) 3 x 105-4 X 106

(with magnetite) 25-50000 Magnetite (ore) 7 X 104-14 X 106

Basalt 1500-25000 Magnetite (pure) 1.5 X 107

Pegmatite 3000-75000

[After Parasnis (1979)]

7. Equation (IV.254) is applied for one type of ferromagnetics, calledsoft magnetic materials, and with some error it is valid within a certainrange of magnetic field strength.

S. There is a temperature called the Curie point, above which fer-romagnetic properties vanish. Magnetite, for instance, becomes para-magnetic if the temperature is higher than 5S0° C. A decrease in thetemperature below the Curie point results in a restoration of ferromag-netic properties of substances. In this light it is proper to notice that dueto an increase of temperature with the depth at distances exceeding 20 kmfrom the earth's surface, a medium becomes practically nonmagnetic.

9. The remanent magnetization I, can be comparable to or greaterthan the induced magnetization Pind .

P=XH

and in some ferromagnetics can reach 106 Ayrn or greater, while in rocksit can vary from 10 to 100 Ayrn.

IV.7 Systems of Equations for the Magnetic Field B and the Field H 493

10. In general, the induced and remanent magnetization have differentdirections.

11. Magnetization arises due to the action of the magnetic field B onelectrons. Therefore, it would be more natural, instead of Eq. (IV.254), toconsider the equation

P=kB+Pr

However, paying tribute to tradition, we will use Eq. (IV.254).

Now we are ready to establish the relation between the vectors BandH. Substituting Eq. (IV.254) into Eq. (IV.250) we have

B = P-o(H + P) = P-o(H + XH + Pr)or

(IV.255)

where

p- = P-rP-o and P- r = 1 + X (IV.256)

The parameter p- is called the magnetic permeability of the material. Atthe same time P- r is the relative magnetic permeability, and it is obviousthat for diamagnetic and paramagnetic materials, P- r is close to unity,while in ferromagnetic materials it can be very large. For instance inferrites, often used in receiver coils, the relative magnetic permeabilityreaches several thousands. In the practical system of units the parameterp- is measured in henries per meter.

IV.7 Systems of Equations for the Magnetic Field Band the Field H

In the previous section we have derived four equations containing bothfields Band H, Eqs. (IV.25D, and established the relation between them,Eq. (IV.254). These results can be summarized as

Biot-Savart law

I I curl H = j c I

I I Curl H = i c I

I II divB = 0 I

I II DivB = 0 I


and

B = /-LH + /-LoPr (IV.257)

While using this system to determine of the magnetic field, we assume thatthe magnetic permeability /-L, remanent magnetization Pr , and the densityof conduction currents J, and i, are known. For instance, a distribution ofthese currents is defined by the electric field, but it is independent of theconstant magnetic field B.

Taking into account the different nature of the fields Band H and therelatively complicated relationship between them, it is more appropriate toconsider systems of equations for these fields separately. Let us start withthe magnetic field B. As follows from Eqs. (IV.257),

B /-LaH = - - -Pr (IV.258)

/-L /-L

Next, substituting Eq. (IY.258) into the first equation of the system(IV.257) we obtain

(IV.259)

divB = a

DivB = a

B Prcurl - = jc + ILo curl -

/-L IL

B PrCurl - = i , + /-La Curl -

/-L IL

and these form the system of equations of the magnetic field in thepresence of magnetic media. It is obvious that Eqs. (IV.259) are based onthe Biot-Savart law and the relation between fields Band H, both ofwhich were obtained from experimental studies. Of course, from thesystem it follows that sources of the magnetic field are absent, and that theconduction and molecular currents are the sale generators of the field B.

Now we will study the distribution of molecular currents, which, inaccordance with the first equation of the system (IV.259), depend on /-L,and P, and the field B. First, consider their behavior at usual points of amedium. Making use of the equality

curl cpa = cp curl a + (grad cp X a)

we have

B 1 (1)curl - = - curl B + grad - X B/-L /-L /-L

and

Pr 1 (1)curl- = - curiPr + grad - X Pr/-L /-L /-L


Then, the first equation of the field can be rewritten as

CUrlB=,ujc-,u(grad ~ XB) +,uacurlPr+,u,ua(grad ~ XPr)

Since

1 1grad - = - - grad ,u

,u ,u2

we have

1 ,uacurl B = ,ujc + - (grad,u X B) +,uacurl P, -- (grad,u X Pr) (IV.260)

,u ,u

At the same time, the first equation of the system (IY.237) is

curlB = ,ua(jc + jrn)

Comparing the previous two equations we conclude that the volumedensity of molecular currents is

,u - ,ua 1 1jrn = J, + -(grad,u X B) + curl P, - - (grad,u X Pr) (IV.261)

,ua ,u,ua ,u

Thus, in general there are four types of molecular currents. The firsttype,

. ,u - ,ua .J rn = Jc

,ua(IV.262)

(IV.263)

arises in the vicinity of points where the density of conduction currents isnot equal to zero, and both vectors J, and jrn have the same direction if,u > ,ua .

The second type,

1hrn = -(grad,u X B)

,u,ua

appears in parts of a medium where the component of the field perpendic-ular to the direction of the maximal change of magnetic permeability is notequal to zero.

The third type,

j3rn = curl P, (IV.264)

is entirely defined by the behavior of the remanent magnetization, and itarises in the vicinity of points where curl P, =1= O.


Finally, the fourth type of the current density is

1j4m = - -(grad J.L X Pr )

J.L(IV.265)

and this appears in places where the remanent magnetization vector andgrad J.L are not parallel to each other.

Next, let us consider a distribution of surface molecular currents.Substituting Eq. (IY.258) into the equation

CurlH = i e

we obtain

(IV.266)

Pl r + P2 rp ay= _r 2

1 1I:!.{J = - --

J.L2 J.LI

Here B2 , P2r and BI , Pl r are vectors of the field and the remanentmagnetization at the front and back sides of the interface, respectively.

Making use of the equality

we present Eq. (IV.266) as

. 1 I:!.{J I:!.{JCurl B = {Jav i e - {Jav 0 X Bay + J.Lo Curl P, + J.Lo {JaY 0 X Pray

where

{JaY = ~(~ + ~),2 J.L2 J.LI

B I + B2Bay = - - -2

As follows from Eqs. (IV.237),

CurlB = J.Lo(ie + i m)

Therefore, the surface density of molecular currents is

(1 ) K 12

i m= --a-Y - 1 i e+ 2-0 X Bay + Curl P, - 2KI20 X Pray (IV.267)J.Lo{J J.Lo

(IV.268)

(IV.269)


where

K_ J-t2 - J-tl

12 -J-t2 +J-tl

By analogy with the volume density, we distinguish four types of surfacecurrents. The first type,

ilm=~( 2J-t1J-L2 -J-Lo)icJ-to J-t1+ J-tz

occurs in the vicinity of conduction currents at interfaces of media withdifferent magnetic permeabilities.

The current density of the second type is

(IV.270)

and it is directly proportional to the contrast coefficient K 12 and theaverage value of the tangential components of the field. It is appropriateto notice that in accordance with Eq. (IV.74) the function

n X B8V

is the tangential component of the magnetic field at some point q of aninterface, caused by all currents except those in the vicinity of this point.

The third type of the surface density is

i 3m = Curl Pr (IV.271)

and is defined by the difference between tangential components of theremanent magnetization.

Finally, the fourth type of currents arises in places on the interfacewhere the average tangential component of the vector P, differs from zero.

(IV.272)

Next, we will derive the system of equations for the field H. SubstitutingEq. (IV.258) into the second equation of the system (IV,257) we obtain

curlH =jc

CurlH = I,

divJ-tH = - J-to div P,

DivJ-tH = - J-to DivP,(IV.273)

Consequently, the generators of the field H consist of conduction currentsand fictitious sources. To describe the latter we will proceed from the factthat the divergence of any field characterizes the density of its sources.


Therefore, we will introduce the density of fictitious sources of the field Has

and !.m = DivH (IV.274)

First, consider their distribution at regular points of the medium.Inasmuch as

div cpa = cp diva + a . grad cp

we have

or making use of Eq. (IV.274),

H . grad JJ- JJ-o div P,8 = -----

m f.L f.L

Thus, we have two types of sources. One of them,

H· grad JJ-81m =-----

JJ-

(IV.275)

(IV.276)

"arises" in the vicinity of points where there is a component of the fieldalong grad JJ-. Correspondingly, this type of source vanishes if the field Hisperpendicular to the direction of the maximal change of the magneticpermeability. Also, 81m equals zero at places where a medium is uniform.

The second type of sources,

.., JJ-o d'U2m= -- iv P,

JJ-(IV.277)

is related to the behavior of the remanent magnetization only.Next, consider a distribution of fictitious surface sources, introduced by

Eqs. (IV.274).

Since

1V.7 Systems of Equations for the Magnetic Field B and the Field H 499

we have

(IV.278)

Consequently, there are two types of surface fictitious sources, namely,

and (IV.279)

One of them is "located" in the vicinity of points where the averagenormal component of the field H differs from zero. The other is defined bythe behavior of the normal component of the remanent magnetization.

Now we are ready to compare fields Band H, and to accomplish thistask it will be useful to consider them together with the electric field E andthe vector of the electric induction D. Bearing in mind that the behavior ofa field is defined by its generators, let us describe the main features ofthese fields.

1. The magnetic field B is caused by vortices only, and these includeboth conduction and molecular currents.

2. In general, the field H has two different types of generators, theconduction currents and fictitious sources. It is essential to note thatmolecular currents that arise due to magnetization do not have anyinfluence on the field H.

3. The magnetic field B obeys the Biot-Savart law, but this law doesnot describe the behavior of the field H.

4. The force acting on a moving electric charge is defined by themagnetic field B, not the field H.

5. In essence, the field H is an auxiliary field, which was introduced tomodify the system of equations of the magnetic field B.

6. The electric field E is a source field, and it is caused by the free andbound charges only.

7. In general, the field D has two types of generators, namely freecharges and fictitious vortices, but bound charges do not have any effecton the field D.

8. The behavior of the electric field is governed by Coulomb's law, butthe field D does not obey this law.

9. The force acting on the charge is defined by the electric field, but notthe field D.

10. The electric induction vector D was introduced to represent thesystem of equations of the field E in such a way that the right-hand side ofthe second equation contains only free charges, which in the case ofnonconducting dielectrics can be specified.


The summary of this comparison is illustrated by the following table.

PhysicalField Sources Vortices law

B conduction and Biot-Savartmolecular currents law

H fictitious conduction auxiliarysources currents field

E free and Coulomb'sbounded charges law

D free fictitious auxiliarycharges vortexes field

Force actingon charge e

e(v X B)

eE

This analysis clearly demonstrates that there is an obvious analogy be-tween the electric and magnetic fields. In fact, both fields E and B definethe force acting on a charge. Also they are caused by real generators andtheir behavior is governed by physical laws. Of course, there is a differencebetween them, since the electric field has the sources as its origin, whilethe magnetic field is generated by vortices.

In the same manner, we can draw an analogy between the fields D andH. Indeed, both fields are generated by sources and vortices, but one ofthese generators does not have any physical meaning. It is essential to notethat these fields are auxiliary ones and they play exactly the same role,allowing us to derive the systems of equations of the electric and magneticfields.

Now we will consider several examples illustrating the difference in abehavior between fields Band H.

Example 1 Behavior of Fields Band H in the Mediumwith One-Plane Interface

Suppose that there is a planar interface between two media havingmagnetic permeabilities f.LI and f.Lz, respectively (Fig. IY.lOc). It is as-sumed that the uniform magnetic field B is perpendicular to this boundaryand the remanent magnetization and conduction currents are absent.

First of all, it is clear that in such a model vortices of the magnetic fieldare absent. In fact, as follows from Eqs. (IY.261) and (IV.267), the volumeand surface densities of molecular currents vanish. At the same time, the

IV.7 Systems of Equations for the Magnetic Field Band the Field H 501

magnetic field does not have sources,

divB = 0 DivB = 0

(IV.280)

and therefore its vector lines are always closed. In particular, they do notbreak at the interface. Taking into account the fact that the field B isperpendicular to the boundary, we have to conclude that the density of itsvector lines remains the same in both media. From a theoretical point ofview such a field is caused by generators located at infinity.

However, the field H behaves in a different way. In fact, H is related toB by

BH=-

J.L

and therefore it is uniform but has different values in every medium. Forinstance, in a medium with greater magnetic permeability, the field H issmaller. Consequently, the vector lines of this field break off at theinterface (Fig. IV.10d), and fictitious sources arise. In accordance withEq. (IV.279) the density of these is

or

s., = (~- ~)BnJ.L z J.L 1

where the normal component is positive if it is directed from the mediumwith magnetic permeability J.Ll to that with permeability J.Lz and vice versa.For example, if J.Lz > J.Ll ' negative sources with the constant density appearat the interface. At the same time, due to the uniformity of every part ofthe magnetic medium, the volume density of fictitious sources equals zero.

In conclusion we can say that we were forced to introduce fictitioussources of the field H in order to provide continuity of the normalcomponent of the magnetic field at the interface between media withdifferent permeabilities.

Example 2 Behavior of Fields Band H inside the Toroidwith a Gap

Now we will assume that a uniform magnetic medium has the shape of atoroid with a very small gap, and it is surrounded by a nonmagneticmedium (Fig. IV.lla). Also we will suppose that the medium was earlier


c

N

s

d

N

S

N

s

Fig. IV.ll (a) Field B inside a magnetic toroid; (b) field H inside a magnetic toroid; (c) fieldH inside a magnetic toroid with surface conduction currents; and (d) field Hand B inside asolenoid with /.L = /.Lo'

subjected to a magnetic field so that now it possesses permanent (rema-nent) magnetization, and that the vector field P, is uniform and is directedalong the toroid axis.

Since conduction currents are absent and every elementary volume ofthe uniform medium has the same magnetization, the volume density ofmolecular currents equals zero, Eq. (IV.26l). At the lateral surface of thetoroid the density of molecular currents does not vanish. In fact, takinginto account the fact that P2r = 0 and making use of Eqs. (IV.27I),(IV.272) we obtain

and


As is seen from Fig. IV.11a the vectors of the surface current density i m

and the magnetization P; are oriented in agreement with the right-handrule. These currents are a system of current loops with the same density,uniformly distributed on the lateral surface of the toroid. It is obvious thatsuch currents create a practically uniform magnetic field inside the magnetwhich, along with the magnetization vector, is directed along the toroidaxis. At the lateral surface the tangential component of the magnetic fieldchanges from its value inside the magnet to a very small value close tozero; that is,

B21 -B l l = -B

As follows from Eq. (IV.270) this discontinuity means that there is anothertype of surface current density,

which also describes a system of current loops with the same magnitude.Therefore, we can say that the magnetic field due to a permanent magnet,having only surface molecular currents, is equivalent to that of a solenoidwith the same distribution of conduction currents. It is easy to predict thatif the gap width is small with respect to the toroid diameter, then thevector lines of B are almost parallel to each other. This means that insidethe toroid and within the gap, the field B remains the same.

Now we will consider the behavior of the field H. As follows fromEqs. (IY.257).

(IV.281)

That is, H is uniform and directed along the toroid axis. Thereforefictitious sources at the lateral surface of the magnet are absent,Eq. (IV.278).

Also the conduction current density equals zero. However, sources ofthe field H arise at two boundaries between the toroid and its gap. Inaccordance with Eq. (IV.279) we have

I = 2 /-L - /-La Hi"lm + n ,

/-L /-La


B; - /-LaPnH l n = - - - -

/-Land


it is easy to show that positive sources arise at the interface where themagnetic field is directed from the magnet to the gap and vice versa.

The boundary with positive sources is usually called the north pole,while the opposite side of the gap is the south pole. By definition thevector lines of any field start from a positive source and finish at negativesources. Therefore, inside the toroid fields the Band H have oppositedirections, and within the gap they have the same direction and differ fromeach other by the constant /-La (Fig. IV.llb).

B=/-LaH

Now suppose that the toroid does not have a gap, and still the magneticfield and the magnetization vector are tangential to its lateral surface.Since conduction currents are absent, we have to conclude that the field Hwithin the solid toroid, as well as outside, equals zero.

H=O

Consequently, inside this permanent magnet Eq. (IV.28I) is simplified andwe have

but outside the toroid,

B = P, = 0

Example 3 Behavior of Fields Band Hinside the Solenoid

Next we will consider a solenoid that has the same dimensions as thetoroid with the gap, and is shown in Fig. IV.llc. Inside the solenoidmagnetic media are absent, and therefore the field B is caused by theconduction current in the coil only. Since the gap width is small comparedto the solenoid diameter, the magnetic field is practically uniform inside ofthe solenoid and in the gap. The field H is also generated by the coilcurrent only and, in accordance with Eq. (IV.28I),

B = /-LaH

That is, it is uniform and has the same direction as the field B.Suppose that the current density in the coil has a magnitude and

direction, such that the magnetic fields coincide in both gaps of thesolenoid and the toroid. Then, due to uniformity of these fields we canstate that inside the solenoid and the toroid they are also equal to eachother. This happens in spite of the fact that in one case the field is caused


by conduction currents, while in the other, molecular currents are the solegenerators of the magnetic field. Once more this illustrates equivalence ofthese currents as generators of the field B.

Now compare the field H in both models. It is clear that within the gapsof the permanent magnet and the solenoid they are equal to each other,since the magnetic fields coincide. However, inside the toroid and the coilthe behavior of the field H does not have any common features. In fact,inside the solenoid we have

1H=-B

/-La

but inside the toroid, H is caused by fictitious sources in the vicinity of thepoles and is directed opposite to the magnetic field B.

Example 4 Behavior of Fields Band Hinside the Magnetic Solenoid

Suppose that the toroid is wound by a current coil, as is shown inFig. IY.lld, and that both the conduction and molecular surface currentshave the same direction. Consequently, the magnetic field becomesstronger. If the current in the coil is sufficiently large, then the field H ismainly caused by this current, and therefore both fields Band H have thesame direction inside of the toroid.

Now we will return to the system of field equations (IY.259) andconsider several models of a medium, where this system is essentiallysimplified.

Case 1 A Nonmagnetic Medium

In this simplest model conduction currents are the sole generators of thefield B and, from Eqs. (IY.259) we again arrive at the system (IV.89).

curl B = /-Lajc

Curl B = /-Laic

divB = 0

DivB =0(IV.282)


At the same time the system of equations for the field H is

curlH = jc

CurlH = i c

divH = 0

DivH = 0(IV.283)

and the fields differ from each other by the constant fJ.o .

B = fJ.oH

It is obvious that in this case the magnetic field is defined directly from theBiot-Savart law.

Case 2 Uniform Magnetic Medium

Suppose that the medium is everywhere uniform and magnetic permeabil-ity equals u: Then, in accordance with Eqs. (IV.259) and (IV.273) we have

and

curlB = fJ.jc

curlH=jc

divB = 0

divH = 0

(IV.284)

(IV.285)

Comparison with the previous case shows that the magnetic field Bincreases in a uniform magnetic medium, and this increase is directlyproportional to the permeability u: Such behavior is very easy to explain.In fact, since the medium is uniform, the density of molecular currentsdiffers from zero only in the vicinity of conduction currents and, as followsfrom Eq. (IY.262),

. fJ. - fJ.o .JIm = Jc

fJ.o

Consequently, the total current density near conduction currents is

. . fJ. - fJ.o . fJ..J =Jc + ---Jc =-Jc

fJ.o fJ.o

and therefore the magnetic field B also increases in the same manner.However, the field H does not change, and this happens because

(a) It is assumed that the distribution of conduction currents remainsthe same as in the case of a nonmagnetic medium.


(b) Due to uniformity of the medium fictitious sources do not arise.(c) Molecular currents do not generate the field H.

Let us notice that in a uniform magnetic medium both fields Band Hhave a vortex origin.

Case 3 Induced Magnetization Is Absent

Now we will assume that the distribution of molecular currents is definedby the remanent magnetization P, only, and that it is known. At the sametime, conduction currents are absent, and one can neglect the inducedmagnetization Pin; that is,

Pin =XH = 0 (IV.286)

Inasmuch as the vector Pin is also equal to zero in a nonmagnetic medium,as X = 0, we can say that molecular currents are located in a medium withthe magnetic permeability /-L, equal to /-La' Consequently, Eq. (IV.28l) canbe written in the form

(IV.287)

It is essential to note that the magnetic field does not change thedistribution of molecular currents, and in accordance with Eqs. (Iy'26l)and (IY.267) their density is

and (IV.288)

since we can let /-L = /-La .This analysis shows that in this case the field B in a magnetic medium

coincides with that in a nonmagnetic one, provided that in both media thedistribution of currents is the same. Taking into account the fact that allmolecular currents are known, the magnetic field can be determineddirectly from the Biot-Savart law, and consequently there is no need tosolve a boundary-value problem.

By letting /-L = /-La' Eqs. (IV.259) are drastically simplified and we have

curl B = /-La curl P,

Curl B = /-La Curl Pr

divB = 0

DivB = 0

Therefore, at regular points of the medium,


and at an interface,

That is, the discontinuity of the tangential components of both vectors Band P, is the same.

As follows from Eqs. (lV.273), the system of equations for the field H is

curlH = 0 divH = -divPr

(IV.289)n : (Hz - HI) = n : (Pl r - PZr )

and unlike the magnetic field, H has a source origin only.In accordance with Eqs. (lY.275) and (lY.278) these sources are

8 = -divPr (IV.290)

In the next section we will demonstrate the relationship between the fieldH and its sources.

Case 4 Magnetic Field Due to the Remanentand Induced Magnetization, as J, = i, = 0

Consider a more complicated case, where conduction currents are every-where absent, but the magnetic field due to the given distribution of thepermanent magnetization results in the appearance of new molecularcurrents. In other words, both types of magnetization produce generatorsof the field B, and consequently we cannot assume that the magneticpermeability of the medium equals JLa' Therefore, the system of fieldequations is

B PrCurl - = JLa Curl - DivB = 0

JL JL

B Prcurl - = JLa curl-

JL JLdivB = 0

(IV.291)

and, as follows from Eqs. (lV.26l), (lV.267), the density of currentsgenerating this field is

1 1j = - (grad JL X B) + curl P, - - (grad JL X Pr ) (IV .292)

JLJLa JL


and

It is obvious that the first term of the volume and surface density ofcurrents cannot be determined if the field B is unknown. We are againfaced with the problem of "the closed circle." Therefore, the Biot-Savartlaw cannot be used to calculate the magnetic field, and instead of it wehave to formulate a boundary-value problem. In this connection it is usefulto consider the system of equations for the field H. In accordance withEqs. (IV.273) we obtain

curlH = 0

CurlH = 0

div u.H = r u« divP,

Div]LH = -]La Div Pr

(IV.293)

As in the previous case the field H has a source origin only, and thedistribution of its sources is defined by Eqs. (IV.275)-(IV.278). Of course,one type of fictitious sources depends on the field H, and this fact requiresthe formulation of a boundary-value problem too.

Thus, determination of the magnetic field can be, in principle, aCCOm-plished in two ways. One of them is based on the solution of the system(IV.291), while the other allows us to find the field H and then, making useof Eq, (IY.258), to determine B.

Taking into account the fact that H is a source field, the secondapproach in general is more preferable since it permits us to introduce ascalar potential U, which essentially simplifies the calculation of the field.

Case 5 Remanent Magnetization and ConductionCurrents are Absent

Suppose that a magnetic substance is placed in an external magnetic fieldBo ' which is known. Then, due to the induced magnetization, molecularcurrents arise that generate the secondary magnetic field Bs ' Therefore,the total magnetic field B consists of two parts.

Since conduction currents and remanent magnetization are absent, the


system of field equations is markedly simplified and we have

Bcurl- = 0

f.L

n x (BZ -~) = 0f.Lz f.LI

divB = 0

(IV.294)

Consequently, the volume and surface densities of molecular currents are

1j = -(grad f.L X B),

f.Lf.Lo(IV.295)

and they can be determined, provided that the field B is known.This study shows that in general even at regular points the field has a

vortex origin.As follows from Eqs. (IV.273) the system of equations for H is

curlH = 0

and its sources are

H· grad f.L8

m= -----

f.L

div f.LH = 0

0' (f.LzH z - f.LIHI) = 0(IV.296)

(IV.297)

As in Case 4, it is obvious that the field B can be determined by twoways. But taking into account the source origin of the field H it is morenatural to formulate a boundary-value problem with respect to this fieldand then, from the relation B = f.LH, to find the magnetic field. At thesame time, if the medium is piecewise uniform, both approaches areequivalent to each other. In fact, the systems of equations for fields BandHare

curlB = 0 divB = 0

n X (Bz

- ~) = 0 o'(Bz-B1)=0(IV.298)

u.z f.L I

and

curlH=O divH = 0

oX(Hz-H1)=0 0' (f.LzH z - f.LIHI) = 0(IV.299)

Therefore, volume molecular currents are absent, and both fields can be

IV.S Behavior of the Magnetic Field Caused by Currents in the Earth 511

expressed in terms of a scalar potential, in spite of the fact that themagnetic field is caused by vortices located on the surface.

We have considered several cases in which the system of field equationscan be simplified. In general, the magnetic field B can be represented asthe sum of three fields.

B = B(1) + B(2) + B(3)

and each of them satisfies one of the following systems:

B(1)curl - = J, divB(1) = 0

JJ,

B(l)Curl- =ic DivB(1) = 0

JJ,

B(2) Pcurl - = curl ~ divB(2) = 0

JJ, JJ,

B(2) PCurl - = Curl ~ DivB(2)= 0

JJ, JJ,

andB(3)

curl- =0 divB(3) = 0JJ,

W3)

Curl-=O DivB(3) = 0JJ,

It is also obvious that within B(1) and B(2) we can distinguish fieldscaused by given distribution of conduction currents and the remanentmagnetization, and therefore they can be directly calculated from theBiot-Savart law.

IV.S Behavior of the MagneticField Caused by Currents in the Earth

As is well known, the magnetic field measured above and below the earth'ssurface can be presented as the sum of two fields,

(IV.300)


Here BN is the normal magnetic field, caused mainly by conductioncurrents in the earth's core, and Bs is the secondary or anomalous field,generated by molecular currents in the upper part of the earth. Thesearise due to a concentration of ferromagnetic substances in different typesof formations, including basement, intrusive igneous rocks, magnetic orebodies, etc. The main purpose of magnetic methods applied in explorationgeophysics is the study of various geological structures having differentmagnetic susceptibilities. These methods are based on measuring thesecondary magnetic field. In this section we will apply our knowledge ofthe theory of the magnetic field and consider the behavior of this field,caused by the conduction and molecular currents in the earth. Ourapproach will be similar to that used in Chapters II and III.

First of all, let us notice that the normal field BN has a more compli-cated character than the gravitational field gN' Whereas the latter doesnot practically change direction and its variation with time is relativelypredictable, the field B on the earth's surface varies strongly in bothmagnitude and direction. Moreover, it has a relatively large alternatingcomponent that depends on time and is unknown in advance. Here we willconsider only the constant part of this field, which in principle is caused bycurrents inside of the earth as well as by conduction currents in theionosphere. It is obvious that determining the relative contribution ofthese currents to the normal magnetic field BN is vital for understandingits origin. For this reason, I think it is proper to describe here the mainfeatures of the solution to this problem, which was first given by Gauss.

The External and Internal Componentsof the Normal Field BN

Suppose that the normal field BN is known on the earth's surface, and thatsuch information is obtained by measuring BN at various magnetic sta-tions. Also, we assume that conduction currents are absent between theearth's surface and the ionosphere. This assumption is confirmed bynumerous experiments, which are mainly based on the use of the first fieldequation in the integral form

where I is the current that passes through an area surrounded by thecontour 2'. These experiments show that the circulation of the magneticfield along any contour 2', located on the earth's surface and above,equals zero. Therefore, the current between the earth's surface and the

IV.8 Behavior of the Magnetic Field Caused by Currents in the Earth 513

ionosphere can be neglected. Consequently, the system of equations forthe field B above the earth is

curlB = 0 divB = 0

and at its surface both the normal and tangential components of the fieldare continuous functions.

ifILl = J-Lz = ILa

Thus, B is a harmonic field and can be expressed in terms of a scalarpotential U, which satisfies Laplace's equation.

(IV.301)

where

B = -gradU

Next we will choose a spherical system of coordinates R, 0, 'P with itsorigin at the earth's center. Assuming axial symmetry of the field,Eq. (III.30!) can be written as

!- (RZ 8U) + _1_!..- (Sin 08U) = 0 (IV.302)8R 8R sin 0 80 80

Now, applying the method of separation of variables, we will present thepotential U in the form

U(R,O) = T(R)F(O)

Substituting Eq. (IV.303) into Eq. (IV.302) we obtain

F!.- (R Z 8T) + .i.: (Sin 0 8F) = a8R iJR sin 0 iJO iJO

or

1 iJ ( iJT) 1 iJ ( iJF)-- R Z- +--- sinO- =0T 8R 8R F sin 0 se 80

(IV.303)

(IV.304)

It is easy to see that both terms on the left-hand side of this equationare independent of Rand 0, and consequently we arrive at two ordinarydifferential equations.

!!-(RZdT)

=mTdR dR

(IV.305)


and

d ( dF)de sin fJ de + mF sin e = 0

The solution to the first equation is

T (R) =A Rn+ B R-n- In n n

where

(IV.306)

(IV.307)

m = n(n + 1)

It is a simple matter to see that functions Tn(R) satisfy Eq, (IV.305).Correspondingly, Eq. (IV.307) is

d ( dFn )de sine de +n(n+l)Fn=O (IV.308)

Solutions to this equation are Legendre functions of the first andsecond kind, Pn(e) and Qn(O), and some of them are given by Eq, (Iv.210).Inasmuch as the functions Qn(fJ) have singularity along the z-axis, e= 0,they cannot be used to describe the behavior of the field. Therefore,making use of Eqs. (IV.303) and (IV.307), the potential U(R, e) is

U(R,O) = L: (AnR n + BnR-n-I)Pn(cos0)n=l

(IV.309)

It is essential to note that the right-hand side of Eq. (IY.309) consists oftwo terms. One of them decreases with an increase in distance from theearth's center,

L: BnR-n-1pn(cos 0)n=l

while the other00

L: AnWPn(cos 0)n~l

becomes greater as R increases. For this reason it is natural to interpretthe first and second terms as potentials of the magnetic fields caused bycurrents in the earth and ionosphere, respectively. From Eq. (IV.30!),

B = -gradU

IV.S Behavior of the Magnetic Field Caused by Curreuts in the Earth 515

we have

and therefore

auB =--

R eu:1 au

B =---e R ae ' Bcp=O

BR = L [-nAnRn- 1+ (n + l)BnR-n-Z]Pn(cose)n=l

00 aB = - " [A R n - 1 + B R- n

- Z ] -P (cos e)o L.. n n ae nn=l

Introducing notations used in geomagnetism,

and X= -Bo

we obtain for points located at the earth's surface,00

Z= L [nAnR3-1-(n+l)BnRon-z]Pn(cose)n=l

00 aX= " [A R n

-1 +B R-n

-Z ] -P (cos s )L.. nOn 0 ae n '

n~l

(IV.31O)

where R o is the earth's radius and e is the latitude of the observationpoint.

Thus, we have represented the vertical and horizontal components ofthe magnetic field on the earth's surface as a combination of sphericalharmonics, and each of them is a sum of two terms, characterizingcurrents above and beneath the surface. This type of representation is vitalfor separating the total field into two parts, generated by the external andinternal currents.

Suppose that we have performed a spherical harmonic analysis of themeasured values of Z and X. Then we have

Z = L ZnPn(cos e)n=l

and (IV.31l)

(IV.312)


where Z, and X n are the magnitudes of spherical harmonics, which aredetermined from this analysis. It is relevant to notice that a similarapproach is used when any experimental data are approximated by apolynomial or Fourier series.

Now comparing Eqs. OV.31O), OV.31l) and taking into account theorthogonality of the spherical functions, we arrive at two linear equationswith two unknowns, An and Bn.

fz, = nA n R 3- l- (n + 1)BnR-

n-

2

\ z, =AnR3-l + BnR on - 2

Solving this system we obtain for the amplitudes of the spherical harmon-ics, describing the fields of the external and internal currents, the follow-ing expressions:

(n + l)Xn +ZnA =-----.,.--

n (2n + 1)R3- l

nX -ZB = n n Rn+2

n (2n + 1) 0

(IV.3l3)

By performing this analysis Gauss demonstrated that the field caused byexternal currents is negligible and that amplitudes of the first harmonic,Xl and Zl' are at least five times greater than the amplitudes of otherharmonics. Thus, this study allows us to establish the fact that the mainpart of the normal magnetic field B N behaves like that of the magneticdipole.

2B lZ= ---cosO

R3'

or

2J.L oMZ= --cosO

47TR 3 '

since

(IV.314)

n = 1, PI ( cos 0) = cos 0 ,a

-Pl(cosO) = -sinOaoand the moment M is directed from north to south along the rotation axisof the earth, since the influence of longitude was not taken into account.

Vector lines of this field are shown in Fig. IV.12a. I believe that it isonly natural to appreciate the simplicity of Gauss' method, which allows usto establish the fundamental feature of the field BN •


Fig. IV.12 (a) Magnetic field of the earth; (b) components of the magnetic field of the earth;(c) magnetic inhomogeneity in a magnetic field Bo; and (d) magnetic cylinder in a uniformmagnetic field.

As we know, the magnetic field described by Eqs. (IV.3lO) is caused bycurrent loops with diameters much smaller than the distance betweenthem and an observation point. For this reason, it is accepted that thenormal magnetic field BN is caused by conduction currents, located at thecentral part of the earth and oriented in planes that are almost perpendic-ular to the axis of rotation. As follows from Eq. (IV.314) both componentsof the field vary on the earth's surface, and in particular, near the polesthe vertical component reaches 60.103 y. Then letting () = 0, we see thatthe moment M is approximately equal to

M~ 1023 A m2

If we suppose that the radius of this current system is around 1000 km,then the total current is

/=3 X 1010 AThis is really strong current.

The study of the remanent magnetization of rocks is the foundation ofpaleomagnetism, which allows us to reconstruct the history of the mag-


netic field of the earth. It is amazing that the theory of paleomagnetism is,in principle, based on the use of very simple equations (IV.314), whichdescribe the field of the magnetic dipole. In fact, the direction of themagnetic field of the earth that existed at the time when a rock wasformed by cooling, defines the direction of remanent magnetization. Thus,by knowing the orientation of a rock sample, as well as time of formation,and measuring the remanent magnetization, it becomes possible to deter-mine the direction of the magnetic field within a certain range of time inthe past. In particular, these investigations allowed us to discover reversalsin the direction of the field BN • This phenomenon is of great importancein the theory of geomagnetism, as well as for the chronology of geologicalevents.

Moreover, if we assume that throughout the earth's history its magneticfield has behaved almost like that of a magnetic dipole, then it is possibleto determine the apparent position of poles at different times. Indeed, asfollows from Eq. (IV.314), the ratio

X 1- = - tan 8Z 2

defines an angle 8, which is formed by the radius vector R, characterizingthe position of the rock sample and the direction of the dipole moment.Then, drawing a line to the earth's surface along this direction, wedetermine the position of the pole. Having performed such a study basedon Eq. (IV.314), geophysicists discovered that poles wander over theearth's surface, and it turns out that the paths of this movement observedon different continents essentially vary from each other. This fact serves asa strong confirmation of the movement of continents, and one can besurprised that such simple equations are extremely useful in developingmodern concepts of such global problems as plate tectonics.

In conclusion, let us notice that the magnetic field on the earth'ssurface is usually characterized by its magnitude T, inclination I, anddeclination D; and they are related to the field components, as shown inFig. IV.12b, by

T=VH 2 + Z 2 =VX2 + y 2 + Z 2

H= Tcos I,

Ztan 1= H'

Z = Tsin I,

ytan D= H

X=HcosD, Y=Hsin D

In the next two parts of this section we will describe the secondarymagnetic field Bs only, caused by rock susceptibility.


Behavior of the Secondary Magnetic Field Dueto Induced Magnetization

In this part we will study the influence of a piecewise uniform magneticmedium placed in an external field Bo. This field can be, for instance, thenormal field of the earth, BN . An example of such a medium is shown inFig. IY.12c. Due to the field Bo, molecular currents arise in the mediumand they generate a secondary magnetic field Bs. Consequently, the totalfield at every point is

Since the medium is piecewise uniform, the volume density of molecu-lar currents vanishes and the field Bs is generated by surface molecularcurrents only. Of course, the density of these currents is usually unknownprior to the calculation of the field, since their distribution depends on thetotal magnetic field B. In other words, the interaction between currentscan be significant. Consequently, to determine the secondary field Bs' it isnecessary, in general, to solve a boundary-value problem. In formulatingthis problem we will proceed from the system of field equations, which inaccordance with Eqs. (IV.298) is

curl B = 0

B l l Bit---=0f.L 1 f.L I

divB = 0

B l n - BIn = 0(IV.315)

since remanent magnetization and conduction currents are absent.Now we will introduce the scalar potential V in the same manner as

was done in the case of the electric field. In fact, from the first equation,

curlB = 0

it follows that

B = -grad V (IV.316)

Let us note that the choice of sign in this equality is not important, and weselected the negative sign only by analogy with the electric field.

Then, substituting Eq. (IV.316) into the second equation,

divB = 0

we obtain Laplace's equation

(IV.317)

Consequently, the conditions at an interface of media with different


magnetic permeabilities are

1 auz 1 au,-------=0J.Lz at J.L, at '

or

-o. au,---=0an an

and (IV.318)

since from continuity of function UI J.L, the continuity of the derivative inthe tangential direction to the interface follows. Thus, the system ofequations for the scalar potential, describing its behavior in a piecewiseuniform medium, is

vzu=O

u, UI---=0J.L z J.L,

-o. -o.---=0an an(IV.319)

It is obvious that if there are several interfaces, we have to ensurecontinuity of the functions UI J.L and auIan at all of them.

At this point it is useful to write down the system of equations for the.potential of the electric field in a piecewise uniform conducting medium.From Eqs. (111.241) we have

VzU=O

(IV.320)

Comparing Eqs. OV.319), (IV.320) we see that in both cases thepotential is a harmonic function and this fact is obvious, since in thevicinity of regular points of a uniform medium the volume density ofcharges and molecular currents equals zero. At the same time, thebehavior of these potentials at interfaces differs essentially from eachother, and this can be easily explained. In fact, in the case of the electricfield surface charges arise and correspondingly the potential is a continu-ous function, while the normal component of the electric field -auIanhas a discontinuity. In contrast, the presence of surface molecular currentsleads to discontinuity of the tangential component of the magnetic field,and therefore, the potential U has a discontinuity.

Inasmuch as the surface current does not create in its vicinity a normalcomponent En' that is, auIan, it remains everywhere a continuous func-tion. In spite of this difference the system OY.319) can be easily trans-formed into system OY.320). With this purpose in mind, we will introduce

(IV.321)


a function U*, which differs from the potential of the magnetic field U atevery regular point of the medium by a constant multiplier.

1U*=-U

J.L

Then, the system (IV.319) can be written as

VZU* = 0

-o: autUi - ut = 0, J.Lza;; - J.LI a;; = 0

The similarity of Eqs. (IY.320), (IV.32l) is obvious. Correspondingly, inboth cases boundary-value problems are formulated in the same manner.Therefore, determination of the magnetic field caused by surface currentsin a piecewise uniform medium consists of the following steps:

1. Solving Laplace's equation.

AU=O

2. Choosing harmonic functions that satisfy the conditions at an inter-face.

o. UI---=0,J.Lz J.Ll

eu, aUt---=0an an

3. Choosing among these functions those that obey the boundary condi-tions. As in the case of the electric field, the latter requires eitherknowledge of the potential or its normal derivative on surfaces surround-ing a medium where the field B is studied.

Then, from the theorem of uniqueness it follows that the potential U,determined in the above way, uniquely defines the magnetic field.

It is relevant to notice that the function U*, introduced earlier, is thepotential for the field H.

Next, we will consider several examples illustrating the solution of theboundary-value problem as well as the behavior of the magnetic field.

Example 1 A Cylinder in a Uniform Magnetic Field

Suppose that a cylinder with radius a and magnetic permeability J.Li isplaced in a uniform magnetic field Bo' which is perpendicular to thecylinder axis (Fig. IY.12d). The magnetic permeability of the surroundingmedium is J.L e •


Taking into account the fact that the medium is uniform inside andoutside of the cylinder, magnetization does not produce volume molecularcurrents. At the same time, the surface density of these currents is notzero, and in accordance with Eq. (IV.270) we have

where

2Ki = __12 0 X Bav

J.Lo(IV.322)

and n is the unit vector, normal to the cylinder surface and directedoutward.

Inasmuch as the primary field Bo is uniform and has only the compo-nent Box, the current density vector is oriented along the cylinder axis,and it does not change in this direction. In other words, the secondaryfield B, is caused by linear current filaments located on the cylindersurface.

To determine this field we will begin by solving Laplace's equation.First, let us choose a cylindrical system of coordinates r, cp, y, so that they-axis coincides with the cylindrical axis. Then, Laplace's equation can bewritten as

a2u 1 au 1 «u-+--+--=0ar2 r ar r2 acp2

(IV.323)

since the field and its potential are independent of coordinate y. Again,we will apply the method of separation of variables and represent thepotential of the magnetic field in the form

U(r,cp) = T(r)F(cp) (IV.324)

Substituting Eq, (IV.324) into Eq. OV.323) and multiplying all terms bythe function

TF

we obtain

r2 a2T r et 1 a2F----+---+--=0T(r) ar2 T(r) ar F acp2

(IV.325)

(IV.326)


It is easy to see that the sum of the two first terms, as well as the lastone, is independent of coordinates rand cp, and therefore instead ofEq. (IV.325) we arrive at two ordinary differential equations.

1 d 2p---=+n2

F dcp2 -

r 2 d 2T r dT---+--=+n2

T dr? T dr

To choose the proper sign on the right-hand side of these equations, wewiII make use of the fact that the field B is a periodic function of theargument cp with period equal to 21T. Otherwise, it would become amany-valued function. For this reason, we wiII select the negative sign onthe right-hand side of the first equation of Eqs. (IV.326) and then obtain

d 2F

-- +n 2F = Odcp2

This is the well-known equation of the harmonic osciIIator, and its solu-tion is

(IV.327)

where Fn is the particular solution for a given integer value of n.It is appropriate to notice that if we chose the positive sign, then

function F would not be periodic since in this case

This analysis also shows that on the right-hand side of the second equationof (IV.326) we have to select positive sign, and consequently,

d 2T 1 dT n2-+----T=Odr? r dr r 2

The latter is an ordinary differential equation that has also been studied indetail, and its solution is

(IV.328)

Therefore, the general solution to Laplace's equation, represented as asum of partial solutions, is

00

U(r,cp) = L (c~rn+D:r-n)(A~sinncp+B:sinncp) (IV.329)n=O


To satisfy the other conditions of the boundary-value problem, let usintroduce a potential for the primary magnetic field directed along thex-axis. Then we have

Bo = -gradU

whence

oreu;

Bo= --ax

Letting C equal zero we obtain

(IV.330)

It is convenient to represent the potential of the total magnetic field insideand outside of the cylinder as

{u

U( r , cp) = U. = U + Ue a s

if r < a

if r > a(IV.33!)

Here U; consists of potentials for the primary and secondary fields, but U,is the potential of the field outside of the cylinder, caused by surfacecurrents.

Taking into account the fact that the secondary magnetic field haseverywhere a finite value and decreases with an increase of distance fromthe cylinder, the functions U; and U; are

U;( r , cp) = L (A~ sin rup + B~ cos ncp)rnn=O

(IV.332)

Ue(r,cp) = -Borcoscp+ L (A~sinncp+B~cosncp)r-nn~O

It is essential to note that U; and Ue , given by Eqs. (IV.332), satisfyLaplace's equation and the boundary condition at infinity, since

as r -700

Next, we will determine the unknown coefficients An and Bn , and withthis purpose in mind it is natural to apply conditions at the cylindersurface.

and if r = a


From Eqs, (IV.332) it follows that

Boa cos ep 1---- + - L (A~ sin rup +B~ cos nep)a- n

/-L e /-L e n=O

and

1 00

- L (A~ sin rup + B~ cos rup )an

/-Li n=O

-Boa cos ep - L n(A n sin nip + B; cos rup)a-n -1

n=O

L n(A~ sin rup + B~ cos rup)an-

1

n=O

(IV.333)

As is well known, one of the most remarkable features of trigonometricfunctions sin nep and cos rup is their orthogonality, and therefore theequalities

f2IT .. 10 if m =1= no sm mip sm rup dip = 7T if m=n

(IV.334)

f2IT 10 if m =1= no cos mtp cos nip dip = 7T if m=n

and

f2IT .

cos mip sin tup dip = 0o

hold. Here m and n are arbitrary integers.Now, multiplying both Eqs. (IV.333) by sin mip and integrating with

respect to ep from zero to 27T, we obtain

(IV.335)

where m is any positive integer including zero. It is clear that the system(IV.335) has only the zero solution; that is,


In a similar manner we can prove that

Be =B i =0m m

Certainly, this is an important result since we have demonstrated thatthe secondary field inside and outside of the cylinder is described, as wellas the primary one, by the first cylindrical harmonic only. In accordancewith Eqs, (IV.333), the amplitudes of these harmonics, Bf and Bl, aredetermined from the system

1 1-{ -Boa + Bfa-I} = -Bta~e ~1

B B e -2 B i- 0- l a = 1

whence

2 ~i - ~e 2Be =K a B = a B1 21 a + . a

~i ~e

and

2~.Bi- _ I B

1 - a~i +~e

Thus, we have derived the following expressions for the potential:

2~i - ~e a

U; = - Bar cos cp + - Bocos cp~i + ~e r

if r < a

if r > a

(IV.336)

which satisfy all the requirements of the boundary-value problem andtherefore describe the magnetic field for this model.

Since

auB =--

r ar and1 au

B = ---'I' r acp

the secondary field outside of the cylinder is

2e ~i - ~e a

B; = B02 cos cp~i + ~e r

- 11 a 2

B e ~i rOB .'I' = 02 sin cp

~i + ~e r

(IV.337)


Comparison of the potential U; with that of the primary field shows thatthe magnetic field inside the cylinder remains uniform and is directedalong the x-axis. As follows from the second equation of (IY.336), thisfield is

2/1 .E~ = '-1 Eo

f-L e + f-Liif r <a (IV.338)

Now, let us describe some features of the behavior of the field and thedistribution of surface currents. In accordance with Eq. (IV.322) we have

(IV.339)

since

and

if r=a

Here i r , icp' i y are unit vectors along coordinate lines.As follows from Eqs. (IV.336),

E~ = -Eo sin tp +K 21Eo sin cp

. 2f-LjE~ = - Eo sin cp

f-Li + f-L e

and consequently the surface current density is

. 2K12 •1 = - --Eo Sill cp

y f-Lo

or

. 2 f-Li - f-L e .ly = - Eosm cp

f-Lo u, + f-L e

(IV.340)

Therefore, the currents generating the secondary magnetic field are dis-tributed in such a way that in one-half of the surface,

O::::;CP::S7T

they are directed along the y-axis, if f-Li > f-L e , while in the other part,

7T < cp < 27T

the currents have the opposite direction. In particular, the current densityreaches a maximal value along two lines of the plane x = 0, and it vanishesat z = O.


It is natural that the current density is directly proportional to theprimary magnetic field Bo. At the same time its dependence on themagnetic permeability of the medium is defined by the contrast coefficient

K _ /-Li - /-L e12 -

/-Li + /-L e

which varies from - 1 to + 1.Next, consider the behavior of the magnetic field caused by these

currents. In accordance with Eq. (IY.338), the field inside the cylinder isuniform and has the same direction as the primary field. With an increasein magnetic permeability /-Li (/-Li > /-L e)' B~ also increases and, for suffi-ciently large values of the ratio /-LJ/-Le' we have

if/-Li-» 1J-Le

That is, the field of surface currents practically coincides with Bo.In the opposite case, when the surrounding medium has a greater

magnetic permeability, J-L e> J-Li' surface currents have a direction such thatthe primary and secondary fields are opposite to each other inside of thecylinder. Consequently, the total field B~ is smaller than the primary one,and in particular, when (/-LJ/-Le) « 1, it is almost zero.

It is also useful to determine the induced magnetization vector. Bydefinition this is

. Xi. 1 ( /-Lo).P=XiH'= -B'= - 1- - B'/-L i /-L 0 /-L i

since

Taking into account Eq. (IV.338) we have

(IV.341)

That is, the density of dipole moments defined by the primary andsecondary fields is uniformly distributed within the cylinder. Due to thisfact the volume density of molecular currents is equal to zero. It isappropriate to notice that the induced magnetization P has the samedirection as the field Bo' while the orientation of surface currents dependson the ratio /-LJ/-L e.


Now suppose that susceptibility of the medium is much less than unity.

Xe« 1 and Xi« 1

(IV.342)P = [ Xi + Xe]

/I 1+--,-0 2

As seen in the table in Section IY.6, this case is of great practical interestin magnetic prospecting. Substituting X into Eq. (IV.34l) we have

XiBO


Xi +Xe--- «1

2

and expanding the right-hand side of Eq. (IV.342) in a series, we obtain

XiBO Xi(Xi + Xe)P=-- - Bo+ ...

f-Lo 2f-Lo

It is clear that the second term, as well as the following ones, is very smalland therefore can be neglected.

Then,

(IV.343)if X« 1Xi

P=-Bof-Lo

This means that for such an approximation the density of dipole momentsis defined by the primary field only. In other words, we assume thatinteraction between molecular currents is negligible. Correspondingly,their density is

(IV.344)Xi« 1. Bo .ly=Xi-stnqJ

f-Lo

Returning to the general case, let us note that with an increase ofsusceptibility Xi' the relation

is simplified and in the limit we have

1 .P=-B1

f-Loif Xi» 1


It is also interesting to compare the behavior of the magnetic andelectric fields inside the magnetic and conducting cylinder, respectively.Comparison with results obtained in Chapter III shows that parameters 'Yand 1/p- playa similar role. Indeed, with an increase of the conductivity ofthe cylinder with respect to that of the surrounding medium, the fieldinside, E i tends to zero. In the opposite case, 'Yj « 'Ye , it approaches thelimit 2Eo. As follows from Eq. (IV.338), the analogy between 'Y and lip-or p and p- is obvious.

Consider the behavior of the secondary field outside of the cylinder. Inaccordance with Eq. (IV.337) we have

2P-j - P- e aB: = 2" cos cpP-j + P- e r

It is useful to determine the Cartesian components of the field. We have

Bz = Br sin cp + Bep cos tp

B; = B, cos cp - Bep sin cp,

where

Therefore,

Zsin cp = -,

r

xcos cp = -,

r

(IV.345)

B e = 2 P-i - P- e 2 XZz a 2

fLi-P- e (X 2+Z 2)

/I -/I X 2_Z 2

B e _ r-j r-e 2 ----=-x - a 2

P-j+P- e (X 2+Z 2)

As an example, the behavior of these components along the profileZ = Zo is shown in Fig. IV.13a. Features of these curves characterize theposition and parameters of the cylinder. For instance, the observationpoint, where B, = 0 and B; has a maximal magnitude, is located above thecylinder center. At the same time, the x-coordinate of the point where thehorizontal component B~ changes sign equals the distance Zo betweenthe profile line and the cylinder center.

Now we will demonstrate that the field Be, caused by surface molecularcurrents, is equivalent to the field of two linear current filaments with


a b

c d

50

50

10

81 300 ••.•.....•.••.....•..••.•..•.

80--

-d

:::::::::::::::.t--.....;.",-

/

--Fig. IV.13 (a) Secondary magnetic field components Bx and Bz ; (b) elliptical cylinder inuniform magnetic field; (c) receiver coil with magnetic core; and (d) dependence BJBo onratio a/b.

opposite directions located in the vicinity of the cylinder center in theplane x = 0. Suppose that the field is considered at some point alongthe z-axis. As was shown earlier, the field Ex that is generated by the linecurrent passing through point x = 0, Z = Il z/2 and directed along they-axis is

Correspondingly, the field due to the second line current with coordinatesx = 0, Z = -Il z/2 and having the opposite direction is


Therefore, the total field, caused by both currents, is

JLo t::..zI

and assuming that the distance from the observation point to that cylindercenter is much greater than the separation between these currents, wehave

Comparing the latter with the component Bep from Eq. (IV.337), when'P = 1T/2 we see that they coincide if

JLi - JL e 2JLO/t::..Z=21T aBo

JLi + JLe

It is easy to generalize this result and show that such equivalence holdsfor both components of the field at any observation point outside of thecylinder. This current system is often called a linear dipole, and in ourcase it is located in the plane perpendicular to the primary magnetic field.

Unlike the gravitational field, the direction of the normal magnetic fieldof the earth, Bo, varies on its surface. For this reason, the same inhomo-geneity-for instance, the horizontal cylinder, located at the same depthbut at another part of the earth-usually creates a completely differentfield.

For example, suppose that the susceptibilities of both media are small.Then, from (IV.337) we have

x· -x a 2e I e

B; = ---Boz cos 'P2 r

e Xi-Xe a2

.Bep = ---Boz Sill 'P

2 r

(IV.346)

Let us consider one example illustrating the magnitude of the secondaryfield. Assuming that

r- =3,a

and Bo= 50000')'

and making use of Eq, (IV.346), we see that the field magnitude can reach2.5')', but it constitutes only a very small portion of the normal field. In the


practice of magnetic methods, sometimes even smaller fields are mea-sured.

Until now, we have considered only the case where the primary field Bois perpendicular to the cylinder axis. Next, let us suppose that the field Bois oriented along this axis. To determine the influence of such a cylinderon the magnetic field, we will use the following approach.

The normal magnetic field Bo is accomplished by the field H o, which is

The presence of the cylinder does not change this field. In fact, in theprevious section we demonstrated that the field H can be caused only byconduction currents, as well as fictitious sources. Then, taking into accountthe fact that the cylinder is uniform, H o is everywhere tangential to itslateral surface, and making use of Eqs. (IV.276) and (IV.279), we concludethat the volume and surface sources vanish. Also conduction currents areabsent. Therefore, inside and outside the cylinder we have

Consequently, we arrive at the conclusion that outside the cylinder themagnetic field does not change and is equal to the primary field.

However, inside the cylinder, the magnitude of the field B is different,and we have

Thus, the secondary magnetic field B, can be written

if

if

r<a

r>a

(IV.347)

It is obvious that this result is easily generalized to a cylinder with anarbitrary cross section.


As follows from Eq. (IV.322) the density of surface currents generatingthis field is

or

since

(IV.348)

and . ( ILi )B1 + B o= IL

e+ 1 B o

Therefore, the surface molecular currents form a system of circularcurrent loops, located in the planes perpendicular to the y-axis. Theirdensity is everywhere the same. It is essential to note that such a distribu-tion of currents is able to create a very strong magnetic field inside of thecylinder if ILi >>> ILe • The analogy with the solenoid is obvious. In contrast,if the magnetic permeability of the surrounding medium is much greaterthan that of the cylinder, the field of these currents, B~, almost cancels thenormal field Bo. Consequently the total field inside, n', tends to zero ifJL e »> ILi·

Example 2 The Spheroid in a UniformMagnetic Field n,

Suppose that an elongated spheroid with semiaxes a, b (a > b), andmagnetic permeability ILi is placed in a uniform magnetic field Bo directedalong the major axis (Fig. IV.l3b). The magnetic permeability of thesurrounding medium is ILe • As in the previous example, due to inducedmagnetization molecular currents arise on the spheroid surface, and theycreate the secondary magnetic field. To find this field we will againintroduce the potential U

B = -grad U

and formulate a boundary-value problem.Taking into account the shape of the inhomogeneity, it is convenient to

make use of the spheroidal system of coordinates described in Section


IV.5 of this chapter (Example 8). Then, representing the potential as

we see that potential should satisfy the following conditions:

1. At regular points.

LlU=O

2. At the spheroidal surface.

7] = 7]0

J.L e J.L j

3. At infinity.

aUe eo;-=-a7] a7]

if 7] --'> 00

where Uo is the potential of the primary magnetic field Bo.

Applying the approach used in Section IV.5, we assume that thepotential outside of the spheroid is described by the first spherical har-monic. Inside the body the magnetic field is uniform and directed alongthe major axis. In accordance with Eq. (IV.211), the expressions for thepotential are

where the coefficients A and D are unknown, while P/x) and Q\(x) areLegendre functions, given by Eqs. (IV.210), and

c = Va 2 - b?

It is clear that U, and U, satisfy Laplace's equation, and U; tends to Uoas the distance from the spheroid increases. To determine the coefficientsA and D we will make use of conditions at the interface 7] = 7]0 andobtain


Solving this system and making use of Eqs. (IV.2I7), (IY.2I8), we have

(IV.350)

where

I - e2

( I + e )L=-- In---2e2e 3 I - e

andc

e= -a

Let us compare the potentials of the electric and magnetic fields in thepresence of either a conducting or a magnetic spheroid. This comparisonshows that both fields have the same dependence on the geometricparameters and coordinates of the observation point. Moreover, outside ofthe spheroid the electric field and conductivity are related in the samemanner as the magnetic field and magnetic permeability. However, theinfluence of these parameters inside of the spheroid is different. Inaccordance with Eqs. (IV.348), (IV.349), the uniform magnetic field Bi is

f.Li

f.L e

I+ (:~ _ I) L n,

aif - > I

b(IV.35I)

(IV.352)

and it is directed along the major axis.As was demonstrated in Section IV.s, for a markedly elongated spheroid

the function L can be represented in the form

b 2 2aL:::::-ln-

a2 b

and with an increase of the ratio alb, L rapidly tends to zero. Therefore,


in the limit when the spheroid coincides with an infinitely long cylinder,the field inside, Bi, again becomes

At the same time, in accordance with Eq. OY.349), the field outside of thecylinder vanishes.

The fact that the magnetic field inside of a spheroid that is elongatedalong the field can be much stronger than the primary one, plays afundamental role in measurements since it essentially allows us to increasethe moment of receiver coils. Consider a coil with a magnetic core andhaving the shape of the cylinder, as shown in Fig. IV.13c. As is wellknown, such coils are often used for measuring alternating magnetic fields,because the electromotive force induced in the coil is directly proportionalto the rate of a change of this field with time. Therefore the increase offield B inside of the coil due to the presence of the core essentiallyincreases the sensitivity of the receiver.

The behavior of the field B i as a function of the ratio of the semiaxes isshown in Fig. IY.13d. It is clear that the right asymptote of this curvecorresponds to the case of an infinitely long cylinder, when the maximalincrease of the field B i is observed. Cores are usually made from ferriteswith relative magnetic permeabilities reaching several thousands. Forinstance, if we assume that J1-JJ1-e = 5000, then, as is seen fromFig. IV.13d, a maximal increase of the field B i almost takes place,provided that

ab >400

To satisfy this inequality it is usually necessary to use long cores, which areinconvenient for geophysical applications. Correspondingly, shorter coresare applied that still provide a strong increase in the field Bi

. For example,if J1-JJ1- e = 5000 and alb = 20, we have

B.-' = 100Bo

One more remark. It is helpful to notice that the results of the fieldcalculation inside a markedly elongated spheroid can be applied for thecentral part of relatively long cylinders.

Now let us consider the behavior of the field B i when the spheroid istransformed into a sphere with radius a. Taking into account the fact that


parameter e tends to zero as a -7 b, and making use of the series

1 +e 2In-- z2e+ -e3+ ...

1- e 3

Eq. (IV.218) is simplified and we have

1L=-

3

Therefore,

(IV.353)

That is, even in the case of the sphere, the field B i can be almost threetimes greater than the primary field.

Earlier we demonstrated that the potential U is independent of thecoordinate tp, and correspondingly the component of the field B in thisdirection equals zero. This means that surface currents have only anazimuthal component i"" and from Eq. (IV.322) we have

K .• 12. • (Be B')1", = -17) X 1§ § + §

f-Lo

Since

K _ f-L e - f-Li12 - ,

f-L e + f-Li

and

1 auB =---e hI ag ,

the current density magnitude is

(

1) 2 - g2 ) 1/2

h , =c1-g2


Substituting the expressions for the potential we obtain

Thus, the current density reaches its maximal magnitude in the planex = 0 and then gradually decreases in both directions, when g approacheseither + 1 or - 1.

Example 3 The Field of a Vertical Magnetic Dipoleat a Borehole Axis

Suppose that the center of a small horizontal loop with a current I islocated at a borehole axis (Fig. IV.14a). The magnetic permeabilities ofthe borehole and the surrounding medium are J.L i and J.L e' respectively.The borehole radius is a. The influence of the medium on the magneticfield can be described in the following way.

Due to the primary field of the current loop, molecular currents arise inits vicinity, as well as at the borehole surface. Consequently, at every pointthe magnetic field consists of the primary and secondary fields, and thelatter is caused by molecular currents. It is essential to note that thedensity of currents on the borehole surface is defined by the total magneticfield, Eq. (IV.322). Therefore, we have to formulate a boundary-valueproblem to determine the field B. With this purpose in mind let usintroduce a cylindrical system of coordinates r, tp, z and represent thepotential V (B = - grad V) as

r<a

r>a(IV.354)

where Va is the potential of the primary field, caused by the current loopin a uniform medium with magnetic permeability J.Li'

Thus, the potential V has to satisfy following conditions:

1. At regular points of the medium

ilV=O

2. Near the current loop

Vi~Va if R~O


Fig. IV.14 (a) Vertical magnetic dipole at the borehole axis; (b) geometric factor of theborehole; (c) vertical magnetic dipole at the axis of a cylindrical shell; and (d) magneticmedium in the primary field Bo.

3. At the borehole surface

fJ-i

4. At infinity

if r =a

u~O if R~oo

In solving this boundary-value problem we will make use of the resultsderived in Section IIUO of Chapter III (Example 5).

To facilitate the derivations we will take into account the axial symme-try of the field and its potential. In other words, U, as well as the field B, isindependent of coordinate cp, and therefore,

au-=0acp

IV.S Behavior of the Magnetic Field Caused by Currents in the Eatth 541

First, we will find a solution to Laplace's equation, which in cylindricalcoordinates is

a2v 1 eu a2v

-+--+-=0ar 2 r ar az 2

Applying the method of separation of variables and making use ofEqs. (HUIS) and (III.317) we obtain for the general solution,

V(r,z) = ["(Air/o(mr) +BmKa(mr))o

X ( C~ sin mz + D~ cos mz) dm (IV.355)

(IV.356)

(IV.3S7)

Before we proceed, let us represent the potential Va in the same formas the function VCr, z). In accordance with Eq. (IV.I09) we have

JLi M cos 8Vo= 47T R 2

where M = IS, cos 8 = r/R, R = Vr 2 +Z2, and S is the coil area.Representing Va as

JLiM a 1Vo = - 47T az R

and making use of Sommerfeld's integral, Eq. (IV.325),

1 2.00- = -1 Ko(mr)cos mzdmR 7T 0

we obtain

JL.M 1""Vo(r, z) = _I-2 mKo(mr)sin mzdm27T 0

Then, taking into account the behavior of the modified Bessel functions,we arrive at the following expressions for potential V, satisfying theconditions in the vicinity of the current loop and at infinity.

JLoM 1"" [JL' ]VI =--2 m _IKo(mr)+Amlo(mr) sinmzdm27T 0 JLa

JL oM1""U; = --2 mDmKa(mr)sin mzdm

27T 0

where Am and Dm are unknown coefficients.

(IV.358)


Now, applying the conditions at the borehole surface we obtain twoequations in two unknowns.

since

The solution of this system is

A = J.Li (J.Li - J.Le)KOK\

m J.Lo p.,J\KO+ J.LeIOK\

and

Inasmuch as

we finally have

J.Li ( J.Li) rnaKo(rna) K\( rna)

Am

= J.Lo J.L e -1 l+(J.Li -l)rnaKo(rna)I\(rna)J.Le '

(IV.359)

Thus, we have solved the boundary-value problem for the potential.The components of the magnetic field, caused by all currents, are

auB =--

r ar 'au

B =--z az ' B",=O

As follows from Eq. (IV.322) the surface current has only an azimuthal

if r = 0


component i cp' and its density is

since

Then, taking into account Eqs. (IV.357), (IV.358), we obtain

It is clear that the surface currents form a system of circular current loopslocated symmetrically with respect to the plane z = 0, and their densitydepends on the coordinates rand z.

Next, consider the behavior of the field at the borehole axis. Bydefinition,

auB =--

r arand since 1](0) = 0, the field B has only a vertical component Bz . Ofcourse, this fact also follows from the symmetry of the current distribution.

In accordance with Eq. (IV.356), the primary magnetic field is

II.MB O = _'-_1_

Z 2rrz 3

Applying Eqs. (IV.358), (IV.359), the total magnetic field at the boreholeaxis can be represented as

e, = !LiM r~ -(~-1) ~ f' x3Ko(x)K](x)cos

axdx ]

2rr L !Le tt a 0 I+X(:: -1)1](X)Ko(X)(IV.360)

where x = rna, a = Lja, and L = z is the distance between the coil andthe observation point, usually called the probe length. It is convenient to


normalize the total field by the primary one, B~. Consequently, we obtain

The function b, depends only on two parameters, JLJJLe and a.Let us briefly study the asymptotic behavior of the field as a function of

a. As the parameter a decreases the integral on the right-hand side ofEq. OY.36l) tends to some constant, and therefore,

if a --+ 0

In other words, in the near zone the field B~ coincides with the fieldfound in a uniform medium with the magnetic permeability of the bore-hole, JLi' This field is practically caused by conduction currents in the loopand molecular currents arising at its surface. At the same time, theinfluence of currents that appear at the borehole surface is negligible. Tofind an asymptotic expression for the field in the opposite case, when theprobe length is much greater than the borehole radius, we will use theapproach described in Section III.9 of Chapter III (Example 5). It is basedon the fact that the integral in Eq, OV.36l) is mainly defined by smallvalues of the argument x, when parameter a becomes very large. Takinginto account the fact that

if x--+O

we obtain

aZ00

= - -z1 K o( x)cos ax dxaa a

From

00 rr rr

1Ko(x)cosaxdx= ~a 2 1 + a Z 2a

if a» 1


it follows that

if a» 1

Substituting this result into Eq, (IV.36l) we have

(JL - ) JL-bz "" 1 + -' - 1 ",,-'JL e P- e

(IV.362)

Thus, in the far zone, a» 1, the magnetic field is subjected to theinfluence of the magnetic permeability of the surrounding medium.

Suppose that both susceptibilities Xi and Xe are very small. Then, byneglecting the second term in denominator of the integrand inEq. (IV.36l), we obtain

or

(IV.363)

(IV.364)

where Bi = P- oM/21TL3 is the field generated by the current in the loop

only, while B: is the secondary field, caused by surface currents,

P-oMB: = --3 [Xi(l- G f ) +XeGf]

21TLand

a 300

G f = -1 x 3Ko(x)K1(x)cosaxdx

1T 0

Let us rewrite Eq, (IV.364) as

JLoMB: = 21TL3 (XiG b + XeGf )

where

(IV.365)

(IV.366)

Gb + Gf = 1

The functions G b and G, are usually called the geometric factors of theborehole and formation, respectively. In accordance with Eq. (IV.366), thesecondary field consists of two fields, provided that the induced magnetiza-


tion is defined by the primary magnetic field B: only. In other words, theinteraction between molecular currents is neglected, and for this reasoneach term in parentheses in Eq. (IV.366) is the product of the susceptibil-ity and the corresponding geometric factor. The terms containing a prod-uct of susceptibilities, Xi and Xe , are absent.

As follows from Eq. (IV.365), the geometric factor of the boreholedepends on the parameter a only, and its behavior is shown in Fig. IV.14b.Taking into account Eq, (IV.362), the asymptotic behavior of the functionG b is

if a ---) 0if a ---) 00

(IV.367)

Hence, with an increase of the probe length the influence of the boreholedecreases, and the field approaches that corresponding to a uniformmedium with susceptibility Xe .

In conclusion, it is proper to make two comments.

1. Applying the principle of superposition and neglecting the interac-tion of molecular currents, Eq. (IV.366) can easily be generalized to themodel with several coaxial cylindrical interfaces. Then we have

where Xi and Gi are the susceptibility and geometric factor of the ithcylindrical layer, respectively. The function Gi is expressed in terms of thegeometrical factor of the borehole.

2. When using the two-coil probe, the magnetic field is usually gener-ated by an alternating current. However, the frequency is chosen in such away that the influence of electromagnetic induction is very small, and wecan use the theory of the constant magnetic field.

Example 4 The Field of a Vertical Magnetic Dipoleat the Axis of a Thin Cylindrical Surface

We will assume that a vertical magnetic dipole with moment M is locatedat the axis of a thin cylindrical shell with magnetic permeability J-L andthickness I1r. The surrounding medium is nonmagnetic, and it can serve asa model of the borehole and the formation. As is seen from Fig. IV.14c


the internal and external radii of the shell are

arr =r --

I 0 2 and

In this case there are two interfaces, and therefore it is natural toexpect that the solution of the boundary-value problem is more cumber-some than in the previous example. However, taking into account the factthat the shell thickness is much smaller than its radius,

ar-« 1ro

the field determination can be markedly simplified. With this purpose, wewill derive approximate boundary conditions at the shell surfaces, assum-ing that ar tends to zero, but the product u. ar remains constant.

First, let us write down the field equations in the integral form

B'dl't-p.,-=O,

Applying the first equation to the elementary path 2', shown III

Fig. IV.14c, we have

B;az B:az Br(Z+ ~Z)ar-- - -- + --'-----'--

p.,o p.,o p.,

where B; and B: are the fields in the borehole and the external medium,respectively, and B, is the field within the shell.

Since the radial component B, is a continuous function, the latter canbe rewritten as

n: az Be az 1 sn:_z z__ + __raraz=O

p.,o p.,o p., iJz

orp.,o iJBe: _Be = _ ar_r

z z p., iJz

Therefore, in the limit as ar ~ 0, we obtain

if Sr ~ ° (III.368)

That is, the tangential components of the field B, at both sides of such ashell are equal to each other.


By definition from Eq. (IV.368), the continuity of the potential follows:

(IV.369)

(IV.370)

Next, consider the flux of the magnetic field through the closed surfaceS that surrounds the shell element (Fig. IY.14c). Then we have

¢B . dS = B: ( rz) rza tp a z - B:( rI)ria tp az + Bz ( z + a2z)r0a cp a r

-Bz ( z - ~z )ro acpar = 0

or

. aBzB:(rz)rz - B;(rl)rl + -rar = 0az

Letting rl = r2 = "o and taking into account the fact that the tangentialcomponent of the field H is a continuous function at every shell surface,

H = Bz = B~ B:z /-L /-Lo /-Lo

we obtain

. /-L Sr aB:B:-B;= -----

/-Lo az(IV.371)

Therefore, the discontinuity of the radial component of the field is directlyproportional to the parameter

(IV.372)

(IV.373)

and the rate of change of the vertical component Bz along the boreholeaxis.

In terms of the potential, Eq, (IY.370) is

eu; aUj e-u;---= -n--ar ar az z

Both Eqs. (IV.369) and (IV.373) are approximate boundary conditions forthe potential, and it is essential to note that they do not contain thepotential of the field inside of the shell. Consequently, we do not need todetermine field B within the magnetic medium, and this fact drastically

IV.S Behavior of the Magnetic Field Caused hy Currents in the Earth 549

simplifies the solution of the boundary-value problem. In essence, we havereplaced the cylindrical shell with finite thickness, 6.r, by the infinitely thinsurface, having the same values of the radius ro and parameter n.

Now we are ready to formulate the boundary-value problem. It is clearthat the potential U should satisfy Laplace's equation outside of thecylindrical surface, tend to zero with an increase of the distance from thedipole, and obey conditions (IY.369) and (IV.373). Making use ofEqs. (IV.357), (IY.358) the potential inside and outside of the magneticsurface is

r s: ro

r e ro (IV.374)

(IV.375)

Next, applying conditions (IV.369) and (IV.373) we obtain the system ofequations with respect to the unknowns Am and Cm •

Ko(mro) +Am/o(mro) = CmKO(mro)

- CmK I ( mro) + K I ( mro) - A m / I ( mro) = nmCmKO( mro)

Solving this system we obtain for the coefficient Am' characterizing thefield in the borehole,

nrom2KJ(mr

o)A m = - 2

1 +nrom /o(mro)Ko(mro)

Then, making use of Eqs, (IY.374), the vertical component of themagnetic field at the borehole axis is

where

J.L 6.rn =--I ,

J.Lo ro

La=-

ro

and L is the probe length.Thus, measuring the magnetic field at the borehole axis we can in

principle study the change of the casing parameter n l , which is defined byits thickness, radius, and magnetic permeability. Let us also notice that


due to the axial symmetry the molecular currents have an azimuthalcomponent i<p only, and their directions at either side of the surface areopposite to each other.

Until now we have studied the behavior of the field B in relativelysimple models of the medium, and this fact allowed us to apply themethod of separation of variables in solving boundary-value problems.

Next, we will consider a more general case, when a magnetic body ofarbitrary shape and constant magnetic permeability j.L is placed in theprimary magnetic field Bo (Fig. IV.14d). The surrounding medium isnonmagnetic. As before we will assume that conduction currents andremanent magnetization are absent. It is clear that due to the inducedmagnetization molecular currents arise on the body surface, and theygenerate a secondary magnetic field. Let us emphasize again that thedensity of these currents, Eq. (IV.322), is defined by the total magneticfield B,

and therefore, in general, it is necessary to solve a boundary-valueproblem.

To determine the magnetic field we will derive the integral equation forits potential U, applying the same approach as in the case of the electricfield in a conducting medium, Eq. (111.267). Both the potentials of theelectric and magnetic fields are harmonic functions. However, their behav-ior at interfaces is different. For instance, the potential of the electric fieldis a continuous function, while in the case of the magnetic field it has adiscontinuity. At the same time comparison of potentials of fields E and Hshows complete similarity of the equations describing the functions UE

and UH . In fact, from Eqs. (III.24l) and (IV.296) we have

aU1E aU2 EY - - - y - -

1 an - 2 an

This identity allows us to make use of the integral equation derived forthe potential of the electric field. Assuming that the surrounding mediumis uniform and its magnetic permeability is JLo, and replacing y by JL in


Eq. (111.267), we obtain

(IV.376)

where S is the surface surrounding the body; UOH is the potential of thefield H o' which accompanies the primary magnetic field Bo; L q p is thedistance between point p and any point q, located on the surface S; andUH is the potential of the total field H,

H= -grad UH

From the equations

BO = 1L0Ho = - grad U

it follows that

and H, = - grad UOH

Therefore, Eq. OV.376) can be rewritten as

(IV.377)

If the point p is located on the surface S, Eq. OV.377) is an integralequation for the function UH .

Having solved this equation we can determine UH at every point of thesurface S. Then, again applying Eq. OV.377) the potential of the magneticfield U can be found at any observation point p outside or inside of theinhomogeneity. In fact, we have

(lL i- ILO)rr.. a 1

U(p)=Uo(p)+ 4 ~UH---dS1r S an L q p

(IV.378)

Of course, taking derivatives on both sides of Eq. IV.378 with respect tothe coordinates of the observation point, we can calculate the componentsof the magnetic field B( p ).

(IV.379)


The Secondary Magnetic Field When Interactionbetween Molecular Currents Is Negligible

Suppose that the susceptibility of a medium is very small and therefore theinduced magnetization Pin is defined by the primary magnetic field only.

XPin= -Bo

/Lo

In other words, we assume that the primary field is much stronger than thesecondary one:

Bo» a,

As is well known in most practical cases where magnetic methods areused, this inequality is valid.

Taking into account the fact that the vector Pin characterizes the dipolemoment of molecular currents

dM = Pin dV

and making use of Eqs, (IV.109) and (IV.379), we have

au; )= ()Bo(q)'LqpdVp X q 4 L3

7T" qp(IV.380)

(IV.381)

Here dU(p) is the potential of the magnetic field, caused by molecularcurrents in the elementary volume dV.

Applying the principle of superposition, we obtain for the potential ofthe secondary magnetic field B,,

1 B ·LU(p) = -4 f. X(q) °L3 qp dV

7T" V qp

Since the normal magnetic field of the earth, Bo, does not change within amagnetic inhomogeneity, Eq. (IV.38l) can be rewritten as

1 B· LU(p)=-x( ° qp dV

47T" Jv L~p

(IV.382)

Thus, instead of a solution to the boundary-value problem, determinationof the secondary field (B, = - grad U) is reduced to calculation of thevolume integral only, which can be replaced by a surface integral. Such


Fig. IV.IS (a) System of magnetic particles and (b) illustration for deriving the secondarymagnetic field.

drastic simplification of the solution to the forward problem is of greatpractical importance for the application of magnetic methods. In thissense the similarity with the theory of gravitational methods is obvious.

Making use of Eq. (IY.382) we assume that the susceptibility of themagnetic body is constant, although under real conditions this assumptionis usually invalid. In fact, the secondary magnetic field is mainly caused byrocks that contain ferromagnetic particles located inside of an almostnonmagnetic medium. Thus, in performing a field calculation we mentallyreplace the real nonuniform medium by a uniform one that produces thesame secondary field.

To illustrate this procedure let us assume that small spherical particleswith magnetic permeability J-L and radius a are distributed within thenonmagnetic medium (Fig. IV.15a). Also we suppose that the inducedmagnetization in each sphere is not subject to the influence of otherparticles; that is, the interaction between them is negligible. Considerinside the body a spherical volume with radius R o, which contains Nparticles. Then, making use of results obtained in Example 2 of thissection, we can represent the potential caused by the currents in every

(IV.383)

(IV.384)

(IV.385)


particle in the form

fL - fLo a3

Uj(p) = 2 -zBoCOS OJfL + fLo R;

where R, is the distance between the observation point p and the spherecenter and OJ is the angle formed by the primary field and the radiusvector n..

Assuming that the distance R from point p to the center of thespherical volume is much greater than its radius R o,

R»Ro

then the potential due to all N particles located within this volume is

fL - fLo a 3

U(p) = N 2 -zBocos°fL + fLo R

since R, ==: R and OJ ==: 0, but °is the angle between vectors Rand Bo.Now, we will suppose that the spherical volume is filled by a uniform

medium with magnetic permeability fL* such that its molecular currentsgenerate the same magnetic field as those of the original model. It is clearthat the potential of this field is

fL* - fLo R6U(p)= Bo-zcosO

fL* + 2fLo R

By equating the right-hand side of Eqs. (IV.383), (IV.384) we determinethe equivalent magnetic permeability fL* .

fL* - fLo R 3 _ N fL - fLo 30- a

fL* + 2fLo fL + 2fLo

It is convenient to introduce a new parameter

47TNa3 3 a3N

V= - = --;---R 3 47T

a -R33 a

which characterizes the volume of particles per cubic unit of matter.Solving Eq. (IV.385) we obtain

1 + 2VK~z

fL* = fLo 1 - VK*iz(IV.386)


Here

K* _ /L - /La _ _ X_12 - -

/L + 2/La 3 + X

Inasmuch as the term VKi2 is usually very small, we can rewriteEq. (IV.386) as

(IV.387)

or

VxX* ::::--X-

I +-3

For instance, if magnetite occupies one percent of the volume and itssusceptibility equals one, then the equivalent uniform medium has suscep-tibility a little less than 10- 2

. As follows from Eq, (IV.387), parameter X*is often directly proportional to the susceptibility of ferromagnetic parti-cles and the relative volume occupied by them (X < 1).

We have derived Eq. (IV,387), provided that all particles are sphericaland their interaction is absent. At the same time, it is natural to expectthat the susceptibility of the equivalent uniform medium, Xm s depends onthe shape, dimensions, and mutual orientation of particles, as well as ontheir susceptibility. For instance, if elongated particles are not oriented inthe same direction, then the induced magnetization is different for differ-ent particles. Therefore, unlike the case of spheres, the secondary fieldrepresents a vector sum of fields caused by every particle. This shows thatthe susceptibility of an equivalent medium, X*' is a function of theorientation of the primary field Ba.

Next, we will demonstrate that the secondary field and its potential canbe expressed by a surface integral. Inasmuch as

L q p q 1-- = grad-L 3 Lqp

qp

Eq. (IV.382) is presented as

I q IU(p) = -XBa' f grad - dV

47T V L q p

Then, making use of the equality

f grad 'P dV = ~'P dSv s


we finally obtain

1 Bo'dSU(p) = -4X~-L-

1r S qp

or

I r!. BOn dSU(p)=-X~~-

41r S L q p

(IV.388)

where S is the surface of the inhomogeneity and BOn(q) is the normalcomponent of the primary field Bo. The simplicity of Eq. (IY.388) isobvious, and at the same time it plays a fundamental role for solvingforward problems when the susceptibility is sufficiently small.

As follows from Eq. (IV.388) the secondary magnetic field is

or

(IV.389)

since

and BOn is independent of the observation point p.In particular, the horizontal and vertical components of the field B, are

I r!. L q p • ie, = -X'j',Bon- 3- dS

41r S L q p

(IV.390)

Now consider an example where the magnetic body is a rectangularprism formed by the coordinate surfaces of a Cartesian system, and theprimary field has a component Boz only (Fig. IV.I5b). Then, for the


.nponent B, we have

(IV.391 )

(IV.392)

ere Sj and S2 are surfaces located in the xz-plane, dS 2 = dS2k, dS j =

j k, B On(S2) = B oz' and BOn(Sj) = - Boz.The latter can be rewritten as

'e integrals in this equation are well known, and they describe the solidgle w( p), subtended by the corresponding surface. Therefore,

XBz(p) = 47TBoz[w2(p) -wj(p)]

In conclusion, it is appropriate to make one comment. Often, mainlycause of tradition, the field H is used to describe the influence ofiomogeneities on the magnetic field. In accordance with Eq. (IV.389),

X A:. L p q dSHs(p) = -4--'Y,Bon(q) L 3

7Tf-Lo S qp

iere

k(q) =XBOn

(IV.393)

(IV.394)

the density of fictitious sources that create the auxiliary field H.As follows from Eq, (IV.393) the relationship between the electric andtitious magnetic charges and their fields is the same. However, it is clearat the field H and its sources do not have any physical meaning. Thus,e procedure for determining field B, consists of using Coulomb's law forlculation of field H, and then multiplying by f-Lo to obtain Bs .

Ie Magnetic Field Due to the Remanent Magnetization

e will again restrict ourselves to the case when a uniform inhomogeneityth magnetic permeability f-L is surrounded by a nonmagnetic medium.[so, we suppose that the vector P, is known, but conduction currents are


absent. It is natural to start the study of a field behavior from the systemof equations (IV.29l).

since

curl B = fJ.o curl Pr divB = 0

(IV.395)n'(B2-Bd =0

P2 r = 0 fJ.2 = fJ.o '

In accordance with Eq. (IY.192) the volume and surface densities ofmolecular currents are

j = curl P,

and

(IV.396)

where

K_ fJ.o - fJ.

12 -fJ.o + fJ.

Thus, even in a uniform medium the volume density of molecular currentscan differ from zero. This means that in such cases we are not able tointroduce the scalar potential U, since

curlB -=1= 0

Therefore it is more convenient to make use of the source field H. Then,applying the relation

B = fJ.H + fJ.oPr

the magnetic field can be calculated.In general, the vector P, can be an arbitrary function of position, but we

assume that it satisfies the condition

curl P, = 0 (IV.397)

Of course, this equation is valid if the field P, is uniform. Consequently,the volume density of currents vanishes, and the secondary magnetic fieldis only generated by surface currents. As follows from Eq. (IV.396), twotypes of these currents are defined by the remanent magnetization, and


therefore they can be specified. Their total density is

i p = - n X P, - K 12n X P,

or

(IV.398)

The third type of current,

(IV.399)

appears due to induced magnetization, and hence it is unknown prior tofield determination. To find the field B then, we have to solve a boundary-value problem. Making use of Eqs. (IV.395) and the relation B = - grad U,it can be formulated as

1. At regular points

aU=O

2. At the surface of the inhomogeneity

V2 VI 11-0---= --VM ,

11- 11-0 11-

where Vz and VI are potentials outside and inside of the magnetic body,while VM is related to the vector P, as

(IV.400)

The latter follows from Eq, (IV.397).3. With an increase of the distance from the inhomogeneity the poten-

tial U, tends to zero.

To illustrate the behavior of the field caused by permanent magnetiza-tion, let us consider again an elongated spheroid, shown in Fig. IV.13b.Suppose that vector P, is directed along the major axis and it does notchange within the body. Making use of the results obtained in Example 2of this section, we will present potentials VI and Uz , as well as functionVM , in the following way:

V1U, 1/) = DcP t( 1/)PI(nVMU,1/) = -PrcP1(1/)P1( g) (IV.401)


and

where

Pr = IPrlIt is obvious that the functions V2 and VI' given by Eqs. (IV.401), satisfy

Laplace's equation and that U; tends to zero as the distance from thespheroid increases. Taking into account the behavior of potential on thespheroid surface Y'f = Y'fo, we have

AQI( Y'fo) _ DP J( Y'fo) = {Lo p P ( )r 1 Y'fo

{Lo {L {L

AQ;( Y'fo) = DP; ( Y'fo) = D

since


and

Inasmuch as

Y'fo Y'fo + 1QI( Y'fo) = -In-- - 1

2 Y'fo - 1

1 Y'fo + 1 Y'foQ~( Y'fo) = -In -- - -2-

2 Y'fo - 1 Y'fo - 1

and

aY'fo= -,

c

b 2

2 1-Y'fo- -2'c

ce= -

a

the coefficients A and D can be represented in the form

ab 2


and

1 1 + e acb 2 -In -- --

a 2 1 - e b2

D=JL oPr7 (JL )1+ --1 L

JLo

(IV.402)

where L is given by Eq. (IV.21S),As follows from Eq. (IV.40l) the field inside of the spheroid is uniform

and directed, as is the vector Pr , along the x-axis. In fact, we have

and

b2 1+el>-ln--

2ac 1- e 'if a> b

For a markedly elongated spheroid, we have

1

and in the limit

(IV.403)

if

(JL ) b

22a--1 -In-<<1

JLo 2a 2 b

Thus, inside the infinitely long cylinder the magnetic field equals theremanent magnetization P, multiplied by fJ.o' Comparison of Eq. (IV.403)with the relation

shows that the field H equals zero; this result is obvious since conductioncurrents and fictitious sources are absent.


Now let us consider the potential of the magnetic field outside of thespheroid. In accordance with Eqs. (IVA01), (IVA02), we have

TJ TJ+l-In---1

ab? 2 7/ -1Uzeg,TJ)=/La Pr--;;'2 (/L ) g

1+ --1 L/La

Here x = c~TJ, r = c[(l- ex7/2 - 1)]1/2 are the cylindrical coordinates ofan observation point,

With an increase of the distance from the spheroid, coordinate 7] alsoincreases, and in the limit we have

and therefore,

1/2r == C7](l - e) and

R=Vr 2+ x 2 =C7](1+e-e)1 /2=CTJ

x cg7Jcos () = - = -- =~

R cTJ

Then, taking into account that

7] 1) + 1 TJ 1+ l/TJ 1-In--=-ln ::::: 1+-2 1) - 1 2 1- l/TJ 3TJ2

the asymptotic expression for the potential U2 is

ab? cos ()

Uz~U,P'3 (1+ (:, _+)RZ

or

/La M' Ru=---2 47T R3

where

if R -)(X) (IVA04)

is the dipole moment of surface currents.


Next, suppose that the ratio alb increases without limit. Then, inaccordance with Eq. (IVA02), the field B l decreases outside the spheroidand, in the case of an infinitely long cylinder, it vanishes.

ifa- -700

b

In this limit case, surface currents depend on the vector Pr only. In fact,substituting Eq, (IVA03) into Eq. (IV.396) we obtain

i = KilO X P, + Curl P, - KilO X P, = Curl P,


1. We have assumed that the remanent magnetization is uniform withinthe spheroid. It turns out that such simple behavior of the vector P, isobserved when an ellipsoid with constant magnetic permeability /-L andsemiaxes a, b, c is placed in a uniform magnetic field. If the primary fieldis directed along one of these axes, the field B I is also uniform and has thesame direction. In the more general case, when the primary field isarbitrarily oriented with respect to the ellipsoid, the field inside is a sum ofuniform fields directed along corresponding axes.

2. The magnetic field inside the spheroid B] is smaller than that in thecase of an infinitely long cylinder, /-L > /-La. A similar phenomenon wasobserved when we studied the electric field in the presence of dielectricsand conductors. It was found that electric charges arising on an inhomo-geneity surface create a secondary field such that its direction is oppositeto the primary one if CI > Cl or 1'1> 1'1' Because of this the total electricfield inside of a body is always smaller than in the case of an infinitely longcylinder.

However, the molecular currents on the surface of the magnetic spheroidand an infinitely long cylinder have the same direction. At the same time,the current magnitude, as well as the area of current loops, becomessmaller as the spheroid edges are approached. This behavior of surfacecurrents is the sole reason why the field inside an infinitely long cylinder isgreater than that inside of the spheroid.

3. In principle, the study of the magnetic field can be performed withthe help of field H, caused by fictitious sources. They "arise" on thespheroid surface in such a way that H is opposite to the vector Pr . Usuallythe ratio HIPr is called the "demagnetization factor," which indirectlycharacterizes the difference between magnetic fields generated by currentson surfaces of the spheroid and the infinitely long cylinder.

4. The relative simplicity of the spheroid shape allowed us to apply themethod of separation of variables. In the more general case of an arbitrary


body, however, the field is calculated by using either the method ofintegral equations or finite differences.

Next, we suppose that only the remanent magnetization defines thedistribution of molecular currents. In other words, the influence of theinduced magnetization is negligible, and therefore we can say thatthe magnetic permeability of a body equals !-to. Then, by definition thedipole moment of every elementary volume is

dM=PrdV (IVA05)

Consequently, the potential of the magnetic field B, caused by all molecu-lar currents within an arbitrary inhomogeneity, is

II. P'LU(p) = _'-_0 l r

3 qp dV47T V L q p

Applying again the equality

1grad cp dV = ~ cp dSv s

and assuming that P, is constant, we obtain

if X« I (IVA06)

As follows from Eq. (IV.388), the potential of the secondary magneticfield, which arises due to the action of the normal field Bo and theremanent magnetization Pr , is

if x« 1 (IV.407)

if X« 1 (IV.408)

Hence, generalizing Eq. (IV.389), the secondary magnetic field is

__l_rf..XBon + !-tOPnrB( p) - 4 't' L3 L q p dS

7T qp

This equation plays a fundamental role for interpretation of magnetic

References 565

anomalies. Of course, the quality of this interpretation depends, as in thecase of gravity and electric methods, on different factors, such as theaccuracy of measurements, the influence of geological noise, and choice ofa model for the magnetic body. In addition, in most cases we have to havesufficient information about the magnitude and direction of the remanentmagnetization.

References

Bursian, V.P. (1972). "Theory of Electromagnetic Fields Applied in Electric Methods."Nedra, Leningrad.

Ianovski, V.M. (1944). "Earth Magnetism." Leningrad.Parasnis, D.S. (1979). "Principles of Applied Geophysics." Chapman and Hall.Parkinson, W.D. (1982). "Introduction to Geomagnetism." Elsevier.Sabba S. Stefanescu. (1929). Etudes theoriques sur la prospection electrique du sousol.

Premiere serie, Inst. Geol. al Rom. Studii technice si economice.Smythe, W.R. (1968). "Static and dynamic electricity." 3d ed., McGraw-Hill.Tamm, I.E. (1946). "Foundation of Theory of Electricity. GITTL, Moscow.Zilberman, G.£. (1970). Electricity and Magnetism. Nauka, Moscow.


Index

Apparent resistivity, 361, 367-369surface with two horizontal interfaces,

380-383surface with vertical contact, 390

Biot-Savart law, 400-402, 404-405generalized form, 401magnetic fields, 429-431, 484

Borehole axispoint source, electric field, 350-369

apparent resistivity, 361, 367-369asymptotic behavior of functions, 357boundary-value problem, 352-354,

357-359current change along borehole, 365-366current density, 353, 364geoelectric parameters and potential,

361-362modified Bessel functions, 355:"'356potential, 351-355, 358-359surface charge density, 350-351

self potential, 392-396vertical magnetic dipole, 539-546

Borehole conductance, 364Bouguer slab correction, gravitational field,

169Boundary condition, harmonic fields, 95-96Boundary-value problem, 80

conducting sphere, 337-338cylinder in uniform magnetic field,

523-526elliptical cylinder, 344-345magnetic field, 550-553

from conduction currents, 436-437

vertical magnetic dipole at borehole axis,539-542

point source at borehole ax¥' 352-354,357-359

secondary magnetic field of earth, 519-521spheroid in uniform magnetic field,

534-535surface with two horizontal interfaces,

371-377surface with vertical contact, 384-388vector potential, 454-455, 462-464vertical magnetic dipole, axis of thin

cylindrical surface, 547-549

Cartesian system, 11curl,6Odivergence, 46gravitational field, 142Laplace's equation, 87scalar field, 26vector line, 38vector potential, 131

Charge conservation, 123Charge density, conducting spheroid, 473Circular loop, vector potential and magnetic

fields, 412-415Circulation, 53-55

curl and, 58-59electric field, 218-219elementary contour, 62harmonic fields, 82-83

Coefficient of proportionality, 138Conductance, borehole, 364

567

568 Index

Conducting medium, electric field, 122-123Conducting sphere, uniform electric field,

337-342boundary-value problem, 337-338charge distribution, 340-341current density, 339depolarization factor, 349-350potential, 337-340

Conducting spheroid, in uniform electricfield, 472-481

Conduction currents, 484absent, magnetic fields and field H

behavior, 509-511behavior, 444-481

conducting spheroid in uniform electricfield, 472-481

current electrode in uniform medium,444-447

current electrode on surface ofhorizontally layered medium,447-449

electric dipole in uniform medium,453-454

horizontal electric dipole in conductinghalf space near vertical contact,467-472

horizontal electric dipole in mediumwith one horizontal interface,460-467

vertical electric dipole in mediumwith one horizontal interface,454-460

wire grounded at surface of horizontallylayered medium, 449-453

current density near, 506flowing in wire grounded at surface of

horizontally layered medium,449-453

magnetic fieldsequation system, 425-432determination, 432~444

dipole moment distribution, 439surface integration, 439-443vector potential, 432-435volume integration, 435

vector potential, boundary-valueproblem, 436-437

Conductivity, 257Conductor, free space, with electric field,

208-213potential, 233-238

Constant of separation, 354

Contact electromotive force, 298-299, 392Contrast coefficient, 328, 528Copper, electron mobility, 256Coulomb field, 289-292Coulomb forces, work, 301, 304Coulomb's law, 200-213, 305, 481

electric field, 264Cross product, 4-6

definition, 63Curie point, 492Curl, 53, 58-62, 64-66

circulation and, 58-59projection, 59source fields, 101-102

Currentconducting spheroid, 479-480heat production, 299-304

Coulomb and extraneous forces, 301,304

louie's law, 302-303in whole circuit, 301

Current density, 252-254, 404behavior, 259conducting spheroid, 478, 481current electrode, 269dimension, 253electric field and, 74, 255-256extraneous field, 289flux, 258-259

closed cylindrical surface, 261normal component, 262

continuity, 307Current density field, 330-332

as quasi-potential field. 74second equation, 262system of equations, 263

Current density vector, current electrode onsurface of horizontally layered medium,447

Current electrodesurface of horizontally layered medium,

447-449in uniform medium

charge distribution, 267-271magnetic fields behavior, 444-447

Current filament, magnetic fields, 409-412Current line, electrical prospecting array,

294-296Current loop, relatively small, magnetic fields

and potential, 415-419Curvilinear coordinate system, 9-11

field equations, 68-69

Cylinder, uniform magnetic field, 521-534amplitudes of harmonics, 526contrast coefficient, 528dipole moment density, 528-529equation of harmonic oscillator, 523induced magnetization vector, 528-529magnetic permeability, 521normal magnetic field, 532-533potential, 522, 524secondary field behavior, 530-533surface current density, 527

Cylindrical system, 11, 353curl, 60divergence, 46Laplace's equation, 87scalar field, 26vector line, 38

Density of surface masses, 145Depolarization factor, 347, 349-350Dielectric constant, 246Dielectric permeability, 201, 482-483Dielectrics, see Electric fieldDipole moment

electric field, 239magnetic field

every elementary volume, 564molecular currents, 552

Dipole moment density, 482cylinder in uniform magnetic field,

528-529Dirichlet's problem, electric field, 314Dirichlet's boundary-value problem, 95-97,

115source fields, 105vortex fields, 125

Divergence, 43-52definition, 43surface, 48two-dimensional, 51

Dot-cross product, 6Dot product, 4, 6Double cross product, 6

Earthapparent position of poles, 518gravitational field, 158-160

on surface, 159vertical component, 159

Index 569

magnetic field, 516-517Edge lines

density, 56, 63direction, 57normal surfaces of quasi-potential field, 64

Electrical prospecting array, current line,294-296

Electric charges, extraneous force, 296-298Electric current, 252-256

dimensions, 259extraneous field, 289-290

Electric dipoleearth's surface, magnetic fields, 452horizontal, see Horizontal electric dipolepotential and field, 224-226vector potential, 457vertical

in medium with one horizontal interface,454-460

in uniform medium, 453-454Electric dipole moment, wire grounded at

surface of horizontally layered medium,450

Electric field, 200-396behavior in conducting medium, 326-396

charge distribution, layered medium,333-337

conducting sphere, 337-342elliptical cylinder, 342-349inhomogeneity effect, 326-333point source at borehole axis, 350-369self potential at borehole axis, 392-396surface with two horizontal interfaces,

369-384surface with vertical contact, 384~392

charge distribution, layered medium,333-337

electrode charge, 333-334potential behavior, 336total charge on surface, 335

circulation, 218-219conducting medium, 122-123conducting spheroid, 473Coulomb's law, 200-213, 264

conductor in free space with electricfield, 208-213

electric force of interaction, 202-203normal component, planar charge

distribution, 204-208current density and, 74, 255-256determination in conducting medium,

304-326

570 Index

determination... continuedboundary conditions, 314, 317-318boundary-value problems, 306, 314-315,

319closed circle problem, 305-306current density continuity, 307Dirichlet's problem, 314first model, 316-318forward problem, 308Green's formula, 320-322Neumann's problem, 314potential, 310-313second model, 318-319theorem of uniqueness, 311, 315third boundary problem, 315

dielectrics, 121-122difference between total and primary,

conducting spheroid, 479equation system, 307, 482-483extraneous field, 286-299

contact electromotive force, 298-299current density, 289current line of electrical prospecting

array, 294-296electric charges, 296-298electromotive force, 291Ohm's law, 287potential along quasi-linear circuit,

292-294variation with time, 288voltage, 290-291

extraneous forces, 300electrokinetic origin, 392work, 301, 304 •

forces on elementary charges and masses,403

inhomogeneity effect, 326-333charge density, 326current density field, 330-332normal component of field, 333potential, 332primary field, 328secondary field, 327, 330

inside and outside dielectrics, 481in metals, 256potential, 222-224in presence of dielectrics, 238-251

dielectric constant, 246dipole moment, 239polarization, 238-239polarization-induced charges, 243-244

system of field equations, 245, 247vector of electric induction, 246volume distribution of charges, 244

resistance, see Resistancestrength, 203surface with two horizontal interfaces,

369-384apparent resistivity curves, 381-383asymptotic behavior, 377boundary-value problem, 371-377charge distribution, 369-370current density distribution, 379-380Laplace's equation, 373-375potential, 370-375S-zone, 381total surface charge, 370

surface with vertical contact, 384-392apparent resistivity, 390boundary-value problem, 384-388discontinuity at contact, 391at earth's surface, 389equivalent model, 386potential, 384-390

time-invariant, 259-260work produced, 300

Electric field potential, 336along quasi-linear circuit, 292-294asymptotic expression, 363behavior in layered medium, 336conducting sphere in uniform electric field,

337-340continuity, 309current electrode, 269due to polarization, 240elliptical cylinder, 343-345equation system, solutions, 312geoelectric parameters and, 361-362inhomogeneity effect, 332inside borehole, 358-359Laplace's equation, 276, 316-317Ohm's law, 316outside of polarized element, 239-240point source at borehole axis, 351-355Poisson's equation, 409in presence of dielectrics, 240-243

equation system, 250-251secondary field, 330self potential at borehole axis, 392-396surface with two horizontal interfaces,

370-375surface with vertical contact, 384-390

time-invariant, equation system, 213-238behavior near surface charges, 231-232conductor in free space, 233-238differential form, 221-233double layer field, 226-230electric dipole field, 224-226first equation, 215-220integral form, 221outside of conductor, 237as Poisson's equation solution, 235second equation, 213-215theorem of uniqueness, 236voltage along radius-vector, 216-218

Electric force of interaction, 202-203Electric induction vector, 246, 499Electrical permittivity of free space, 201Electromotive force, 291

contact, 298-299Electron mobility, copper, 256Electrostatic induction, 209-213

conducting spheroid, 478distribution of charges, 210-211

Electrostatic shielding, theorem ofuniqueness, 236

Elementary mass, 138-139force acting on, 140gravitational field, 141-145

on surface, 145-146Elevation correction, gravitational field, 168Elliptical cylinder, in uniform electric field,

342-349charge distribution, 345-346current density, 348-349depolarization factor, 347field inside, 347potential, 343-345surface charge density and field strength,

346Elliptical system of coordinates, 342

Ferromagnetics, 484susceptibility, 492

Field equations, 66-81curvilinear orthogonal system of

coordinates, 68-69electric field

conducting medium, 122dielectrics, 121-122

field generators in part of space, 77-78

Index 571

harmonic fields, 81-82homogeneous system, 79

source fields, 105lack of edge lines on normal surfaces, 69in presence of medium, 75-76quasi-potential field, 71-72solution of boundary-value problem, 80source fields, 100-101, 120surface analogies, 67

harmonic fields, 87-88vortex fields, 129-130

vortex fields, 123-124Field generators, 56

only in part of the space, 77-78types, 70-71

Field H, 491-511equation system, see Magnetic fieldmagnetic fields and, 493magnetization vector, 491measurement units, 491

Fields, 22-23Field theory, fundamental relations, 42First equation of electric field, 215-220First equation of gravitational field, 182-186First equation of magnetic field, surface

analogy, 424First Green's formula

source fields, 107vector analog, 131

Flux, 40-47element of normal surface, 41number of vector lines, 41through closed surface, 47through elementary surface, 178-180total, 44

Fredholm integral equation, 212

Gauss' theorem, 49-52, 312electric field potential, 240-241harmonic fields, boundary surface, 92-93source fields, 107vortex fields, 128, 131

Geological noise, 178, 192-193Geomagnetism, 518Geometric model, vector fields, 36-39Gradient lines, 35-36Gradient probe, 286Gradient, see Scalar fieldGravitational constant, 138

572 Index

Gravitational field, 137-199acting on and caused by mass, 139calculation in upper half space, 193-199caused by thin spherical shell, 151-155caused by volume distribution of masses,

148-151change of mass distribution, 163-169components, 142-143determination, 157-178

accuracy factors, 167Bouguer slab correction, 169earth, 158-160elevation correction, 168three-dimensional body, 175-178two-dimensional model, 169-175

elementary mass, 141-145on surface, 145-146

equivalent model of nonuniform medium,161-162

first approximation, 162first equation, 182-186forces on elementary charges and masses,

403forward problem, 305geological noise, 178, 192-193ill-posed problem, 166inside a layer, 149-150inside spherical shell, 151-152interpretation, 162-163inverse problem, 162, 165-167Newton's law of attraction, 137-157nonuniqueness, 164normal component due to plane surface

masses, 145-148outside spherical shell, 152-153parameters describing mass distribution,

165poten tial, 187-188

behavior, 189caused byvolume distribution of masses,

190source fields, 103

principle of superposition, 140-141, 146,155

second equation, 180-181solution of forward problem, 163sphere, 155-157star-shaped bodies, 164upward continuation, equation system,

178-199differential form, 187-199

first equation, 182-186integral form, 186-187second equation, 180-181

Gravitational force, components, 138Green's formula, 83

electric field potential, 320-322first, 107-108

vector analog, 131second, 107-108

vector analog, 132Green's function, 108, 117-119, 197

harmonic, 119singularity near observation point, 116-117vortex fields, 132

Grounding resistancespherical electrode, 279-281voltage measurement and, 281-285

Harmonic fields, 81-100absence of vortices, 84-85boundary condition, 95-96boundary surface, Gauss' theorem, 92-93circulation, 82-83field equations, 81-82potential of the field, 85-91

boundary conditions, 94boundary surface is equipotential, 98-99equations describing behavior, 86field equation surface analogies, 87-88Gauss' theorem, 93known on boundary surface, 94-97Laplace's equation, 87-91normal derivative on surface, 97-98

quasi-potential field, 85voltage, 83-84

Harmonic oscillator, equation, 523Horizontal electric dipole

in conducting half space near verticalcontact, 467-472

medium with one horizontal interface,460-467

Huiler's rule, 21

Ill-posed problem, gravitational field, 166Induced charges, plane interface, charge

distribution, 271-273Induced magnetization

absent, behavior of magnetic fields andfield H, 507-508

cylinder in uniform magnetic field,528-529

magnetic fields behavior, 508-509secondary magnetic field of earth, 519-521very small susceptibility, 552

Inhomogeneity, effect on electric field inconducting medium, 326-333

Inverse problem of the gravitational fieldtheory, 162

Joule's law, current, 302-303

Kirchoff's formula, 118

Laplace's equation, 83,87-91, 188, 191, 193cylindrical coordinates, 353electric field potential, 276, 310, 316-317elliptical cylinder, 343features for harmonic functions, 92normal derivatives on surface, 97-98one-dimensional case, 91potential, 474

on boundary surface, 94-97scalar potential, earth's surface, 513simplest form, 91solution, vector potential, 455-457vector form, 443

Legendre equations, 474-475Lipschitz integral, 375Lorentz force, 404

Magnetic dipolemagnetic fields, 417-418vertical, see Vertical magnetic dipole

Magnetic field, 397-565applications, 397Biot-Savart law, 429-431boundary-value problem, 550-553caused by currents in the earth, 511-565

cylinder in uniform magnetic field,521-534

due to remanent magnetization,557-565

external and internal components,511-518

Index 573

secondary, due to inducedmagnetization, 519-521

secondary, negligible interactionbetween molecular currents,552-557

spheroid in uniform magnetic field,534-539

vertical magnetic dipole at axis of thincylindrical surface, 546-550

vertical magnetic dipole at borehole axis,539-546

conduction currents, see Conductioncurrents

current element, 400-401currents in earth, spherical harmonic

analysis of Z and X, 515-516earth, 516-518equation system

above earth, 513from conduction currents, 425-432determination of solenoid field, 431-432first equation, 425-426, 429magnetic materials, 490second equation, 426-429total flux, 425

equation system with field H, 493-511behavior in magnetic solenoid, 505-511behavior in medium with one-plane

interface, 500-501behavior in solenoid, 504-505behavior in toroid with gap, 501-504first equation, 494-495molecular current distribution, 494-495second equation, 497surface fictitious sources, 498-499surface molecular current distribution,

496field Hand, 493influence of materials, 484-487inhomogeneity effect, 557integral equation for potential, 550-551interaction of currents, 398-401potential, surface integral, 555-556principle of superposition, 418-419secondary

conducting spheroid, 480potential, 550surface integral, 555-556

small spherical particles in nonmagneticmedium, 553-554

spherical coordinates, 417

574 Index

spherical coordinates continuedsurface current, 401vector potential, 405-425

Cartesian coordinates, 407current density, 408current filament, 409-412current flowing in circular loop, 412-415divergence, 407mechanical force and rotation moment,

419-421relatively small current loop, 415-419tangential component behavior near

surface currents, 421-424Magnetic permeability

equivalent, 554free space, 398

Magnetic solenoid, behavior of magneticfields and field H, 505-511

induced magnetization absent, 507-508nonmagnetic medium, 505-506remanent and induced magnetization,

508-509remanent magnetization and conduction

currents absent, 509~511

uniform magnetic medium, 506-507Magnetic susceptibility, 491Magnetization

induced, 492-493unit of measurement, 486

Magnetization vector, 486field H, 491molecular current volume density and, 488

Mechanical force, rotation moment, 419-421Method of separation of variables, 354Molecular currents, 484

density, 489-490dipole moment, 552distribution, 494polarization, 486-487surface density, 496-497, 510, 558types, 495-496vector potential, 486-488volume density, 495, 510, 558

vector of magnetization and, 488

Neumann's boundary-value problem,111-113

electric field, 314source fields, 105vortex fields, 125

Newton's law of attraction, 137-157elementary mass, 138-139, 141-145gravitational field, 137

Ohm's law, 123differential form, 255-256, 291electric field potential, 316extraneous field, 287integral form, 277

Oriented lines, 9~11

Oriented surfaces, 9-11

Paleomagnetism, 517-518Peltier heat, 303Planar charge distribution, electric field,

normal component, 204-208Plane interface, induced charges, charge

distribution, 271-273Plane surface masses, gravitational field,

normal component, 145-148Plate tectonics, 518Point source, borehole axis, see Borehole

axisPoisson's equation, 188, 191, 232

closed circle, 120-121electric field

dielectrics, 122potential, 250, 309-310, 409

fundamental solution, 108-110source fields, 102-104vector analogy, 435vector potential, 131

Polarization, 238-239molecular currents, 486-487potential due, 240

Polarization vector, 482-483Potential, see also Scalar potential; Vector

potentialelectric field, 222-224harmonic fields, 85-91integral equation, 550-551

Potential field, 72Principle of charge conservation, 259-260,

262Principle of superposition

electrostatic induction, 211gravitational field, 140-141, 146, 155magnetic fields, 418-419

Index 575

Quasi-potential field, 61-62, 66edge lines of normal surfaces, 64field equations, 71-72generators within volume, 81harmonic fields, 85sources, 72vortex fields, 124

Radius vector, 2-3Remanent magnetization, 491-493

absent, magnetic fields and field Hbehavior, 509-511

magnetic field due to, 557-565dipole moment, 564elongated spheroid, 561-562molecular current volume and surface

densities, 558potential, 559-560secondary field, 564spheroid, 560-561, 563-564surface current, 558-559

magnetic fields behavior, 508-509rocks, 517-518

Resistance, 274-286theorem of uniqueness, 275-276three-electrode array, 285-286

Resistance groundingspherical electrode, 279-281voltage measurement and, 281-285

Resistivity, 257Right-hand rule, 9-10Rocks

density, 158microcharge movement, 252-253remanent magnetization, 517-518resistivities, 257

Rotation moment, mechanical force,419-421

Scalar fieldclosed path, 28coordinate systems, 26forms of equations, 31gradient, 23-36gradient lines, 35-36integral presentation, 32

surface of discontinuity, 31two dimensional, 32use of level surfaces, 34-35vector flux, 30-31

Scalar functionsdifferentiation, 7observation point position, 1-3

Scalar potentialearth's magnetic field, 520Laplace's equation, earth's surface, 513

Second equation of current density field, 262Second equation of electric field, 213-215Second equation of gravitational field,

180-181Second Green's formula, 194

source fields, 107-108vector analog, 132

Solenoidbehavior of magnetic fields and field H,

504-505magnetic fields determination, 431-432

Solid angle, 12-22an tisymmetric, 18behavior, 16-18calculation, 20-21difference near surface, 20features, 15-19

Source field, 71, 100-123closed circle, 120curl, 101-102Dirichlet's boundary-value problem, 105electric field, conducting medium, 122-123field equations, 100-101, 120first Green's formula, 107fundamental solution of Poisson's

equation, 108-110Gauss' theorem, 107gravitational field potential, 103Kirchoff's formula, 118Neumann's boundary-value problem, 105Poisson's equation, 102-104potential

sources absent in volume, 110-111,114-116

in terms of distribution of sources,111-114

properties, 101-104second Green's formula, 107-108singularities near surface sources, 101tangential components, 101voltage, 102

576 Index

Sources, 70quasi-potential field, 72surface distributions, 72-73

Spheregravitational field, 155-157uniform magnetic field, 537-539

Spherical electrode, grounding resistance,279-281

Spherical system, 11curl, 60divergence, 46Laplace's equation, 87magnetic field, 417scalar field, 26vector line, 38

Spheroidconducting, uniform electric field, 472-481uniform magnetic field, 534-539

boundary-value problem, 534-535Stokes' theorem, 63-66Surface analogy of first field equation, 424Surface current

behavior of magnetic fields tangentialcomponent, 421-424

distribution, 496types, 497

Surface density, 145Susceptibility, magnetic materials, 491-492S-zone, 381

Tesla,400Theorem of uniqueness, 79, 311, 315

electrostatic shielding, 236resistance, 275-276source fields, 106-107

Thin spherical shell, gravitational fieldcaused by, 151-155

Third boundary problem, electric field, 315Thomson heat, 303Three-dimensional body, gravitational field,

175-178Three-electrode array, resistance, 285-286Toroid

with gap, behavior of magnetic fields andfield H, 501-504

magnetic fields determination, 431-432Two-dimensional model, gravitational field,

169-175arbitrary cross section, 175

mass distribution, 170on earth's surface, 172strip, 173-174

Unit vector, 57

Vector fields, geometric model, 36-39Vector functions

scalar and vector components, near surfaceand line, 8-9

scalar function differentiation, 7Vector lines

distribution, 49family of, 39number and flux, 41terminal points, 42-43

distribution, 51vortex fields, 124

Vector of electric induction, 246, 499Vector of magnetization, see Magnetization

vectorVector potential, see also Magnetic fields

current flowing in circular loop, 412-415electric dipole, 457

uniform medium, 453-454ferrornagnetics, 485horizontal electric dipole, 460-466magnetic field

from conduction currents, 432-435currents inside of inhomogeneity,

440-441vertical electric dipole, 454-459

molecular currents, 486-488volume and surface, 489

vortex fields, 126-127Vectors

dot and cross products, 4-6observation point position, 1-3right-hand rule, 9-10scalar and vector components, 3-4

Vertical electric dipolein medium with one horizontal interface,

454-460Vertical magnetic dipole

magnetic field at axis of thin cylindricalsurface, 546-550

magnetic field at borehole axis, 539-546boundary-value problem, 539-542

potential, 539-541primary magnetic field, 543secondary magnetic field, 545-546very small susceptibility, 545

Voltagealong arbitrary contour, 54along arbitrary path, 183-184along closed path, 54-55along radius-vector, 216-218current electrode, 269extraneous field, 290-291harmonic fields, 83-84as integral, 52measurement, grounding resistance and,

281-285number of normal surfaces, 53-54source fields, 102

Volume charges, extraneous force, 296-297Vortex field, 71, 123-136

boundary-value problems, 125distribution of surface vortices, 130

Index 577

field equation surface analogies, 129-130Gauss theorem, 128, 131Green's function, 132normal component at boundary surface,

129quasi-potential field, 124surface integrals, 134-135surface of singularity, 129-130tangential component at boundary surface,

128-129vector lines, 124vector potential, 126-127volume integral, 133

Vortices, 71lack in harmonic fields, 84-85surface, 73

distribution, 72-73, 130

Watt, 301


International Geophysics Series

EDITED BY

RENATA DMOWSKADivision ofApplied Science

Harvard University

JAMES R. HOLTONDepartment ofAtmospheric Sciences

University of WashingtonSeattle, Washington

Volume 1 BENO GUTENBERG. Physics of the Earth's Interior. 1959*

Volume 2 JOSEPH W. CHAMBERLAIN. Physics of the Aurora and Airglow, 1961*

Volume 3 S. K. RUNCORN (ed.). Continental Drift. 1962*

Volume 4 C. E. JUNGE. Air Chemistry and Radioactivity. 1963*

Volume 5 ROBERT G. FLEAGLE AND JOOST A. BUSINGER. An Introduction to AtmosphericPhysics. 1963*

Volume 6 L. DUFOUR AND R. DEFAY. Thermodynamics of Clouds. 1963*

Volume 7 H. U. ROLL. Physics of the Manne Atmosphere. 1965*

Volume 8 RICHARD A. CRAIG. The Upper Atmosphere: Meteorology and Physics. 1965*

Volume 9 WILLIS L. WEBB. Structure of the Stratosphere and Mesosphere. 1966*

Volume 10 MICHELE CAPUTO. The Gravity Field of the Earth from Classical and ModernMethods. 1967*

Volume 11 S. MATSUSHITA AND WALLACE H. CAMPBELL (eds.), Physics of GeomagneticPhenomena. (In two volumes.) 1967*

Volume 12 K. VA. KONDRATYEV. Radiation in the Atmosphere. 1969*

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Volume 13 E. PALMEN AND C. W. NEWTON. Atmospheric Circulation Systems: TheirStructure and Physical Interpretation. 1969

Volume 14 HENRY RISHBETH AND OWEN K. GARRIOTI. Introduction to Ionospheric Physics.1969*

Volume 15 C. S. RAMAGE. Monsoon Meteorology. 1971*

Volume 16 JAMES R. HOLTON. An Introduction to Dynamic Meteorology. 1972*

Volume 17 K. C. YEH AND C. H. LIU. Theory of Ionospheric Waves. 1972*

Volume 18 M. I. BUDYKO. Climate and Life. 1974*

Volume 19 MELVIN E. STERN. Ocean Circulation Physics. 1975

Volume 20 J. A. JACOBS. The Earth's Core. 1975*

Volume 21 DAVID H. MILLER. Water at the Surface of the Earth: An Introduction toEcosystem Hydrodynamics. 1977

Volume 22 JOSEPH W. CHAMBERLAIN. Theory of Planetary Atmospheres: An Introductionto Their Physics and Chemistry. 1978*

Volume 23 JAMES R. HOLTON. An Introduction to Dynamic Meteorology, Second Edition.1979*

Volume 24 ARNETI S. DENNIS. Weather Modification by Cloud Seeding. 1980

Volume 25 ROBERT G. FLEAGLE AND JOOST A. BUSINGER. An Introduction to AtmosphericPhysics, Second Edition. 1980

Volume 26 Kuo-NAN LIOU. An Introduction to Atmospheric Radiation. 1980

Volume 27 DAVID H. MILLER. Energy at the Surface of the Earth: An Introduction to theEnergetics of Ecosystems. 1981

Volume 28 HELMUT E. LANDSBERG. The Urban Climate. 1981

Volume 29 M. I. BUDYKO. The Earth's Climate: Past and Future. 1982

Volume 30 ADRIAN E. GILL. Atmosphere to Ocean Dynamics. 1982

Volume 31 PAOLO LANZANO. Deformations of an Elastic Earth. 1982*

Volume 32 RONALD T. MERRILL AND MICHAEL W. McELHINNY. The Earth's MagneticField: Its History, Origin, and Planetary Perspective. 1983

Volume 33 JOHN S. LEWIS AND RONALD G. PRINN. Planets and Their Atmospheres: Originand Evolution. 1983

Volume 34 ROLF MEISSNER. The Continental Crust: A Geophysical Approach. 1986

International Geophysics Series 581

Volume 35 M. U. SAGITOV, B. BODRI. V. S. NAZARENKO; AND KH. G. TADZHIDINOV. LunarGravimetry. 1986

Volume 36 JOSEPH W. CHAMBERLAIN AND DONALD M. HUNTEN. Theory of PlanetaryAtmospheres: An Introduction to Their Physics and Chemistry, Second Edition.1987

Volume 37 J. A. JACOBS. The Earth's Core, Second Edition. 1987

Volume 38 J. R. MEL. Principles of Ocean Physics. 1987

Volume 39 MARTIN A. UMAN. The Lightning Discharge. 1987

Volume 40 DAVID G. ANDREWS, JAMES R. HOLTON, AND CONWAY B. LEOVY. MiddleAtmosphere Dynamics. 1987

Volume 41 PETER WARNECK. Chemistry of the Natural Atmosphere. 1988

Volume 42 S. PAL ARYA. Introduction to Microrneteorology. 1988

Volume 43 MICHAEL C KELLEY. The Earth's Ionosphere. 1989

Volume 44 WILLIAM R. COTTON AND RICHARD A. ANTHES. Clouds and PrecipitatingStorms. 1989

Volume 45 WILLIAM MENKE. Geophysical Data Analysis: Discrete Inverse Theory, RevisedEdition. 1989

Volume 46 S. GEORGE PHILANDER. El Nino, La Nina, and the Southern Oscillation. 1990.-Volume 47 ROBERT A. BROWN. Fluid Mechanics of the Atmosphere. 1991

Volume 48 JAMES R. HOLTON. An Introduction to Dynamic Meteorology, Third Edition,1992

Volume 49 ALEXANDER A. KAUFMAN. Geophysical Field Theory and Method, Part A.Gravitational, Electric, and Magnetic Fields. 1992


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Geophysical Field Theory and Method: Gravitational, Electric, and Magnetic Fields