Electromagnetic Waves - Delft Academic Press · electromagnetic, acoustic and elastic waves. Hans Blok (1935) was a member of the scientific staff of the Laboratory of Electromagnetic

Electromagnetic Waves— An Introductory Course

Martin Verweij (1961) graduated in 1992 from Delft University ofTechnology, Delft, Netherlands. From 1993-1997, he was a researchfellow of the Royal Netherlands Academy of Arts and Sciences at theLaboratory of Electromagnetic Research of the Delft University ofTechnology, and since 1998 he is associate professor in the samelaboratory. He has written a range of papers on integral transformationmethods for electromagnetic, acoustic and elastic waves. He has beenteaching various classes on basic and advanced electromagnetic wavetheory since 1994 and was chosen Best Teacher of ElectricalEngineering in 2003.

Peter van den Berg (1943) was a member of the scientific staff of theLaboratory of Electromagnetic Research, Delft University ofTechnology, Delft, Netherlands. He graduated in 1971 from the DelftUniversity of Technology, and during the academic year 1973-1974 hewas Visiting Lecturer in the Department of Mathematics, University ofDundee, Scotland. He was appointed as a full professor at the Laboratoryof Electromagnetic Research in 1981. Since 2003 he is a researchprofessor in the Faculty of Applied Sciences of the Delft University ofTechnology. During his career, he has written an impressive amount ofpapers on the numerical analysis of forward and inverse electromagneticwavefield problems, and he has been teaching various classes onelectromagnetic, acoustic and elastic waves.

Hans Blok (1935) was a member of the scientific staff of the Laboratoryof Electromagnetic Research, Delft University of Technology, Delft,Netherlands. He graduated in 1970 from the Delft University ofTechnology, and he was appointed full professor at the Laboratory ofElectromagnetic Research in 1980. He was dean of the Faculty ofElectrical Engineering, Delft University of Technology in the period1980-1982. During the academic year 1983-1984 he was a visitingscientist at Schlumberger-Doll Research, Ridgefield, Connecticut,U.S.A. During his career, he has written a number of papers onresonators and optical waveguides, and he has been teaching variousclasses on electromagnetic waves and signal theory. He has beenemeritus professor since 2000..

Electromagnetic Waves— An Introductory Course

dr. M.D. Verweij

prof.dr. P.M. van den Berg

prof.dr. H. Blok

�

VSSD

© VSSDFirst edition 1999

Second edition 2001-2006, improved 2010

Published by:VSSDLeeghwaterstraat 42, 2628 CA Delft, The Netherlandstel. +31 15 278 2124, telefax +31 15 278 7585, e-mail: [email protected]: http://www.vssd.nl/hlfURL about this book: http://www.vssd.nl/hlf/e016.htm

A collection of digital pictures and an elctronic version can be made available forlecturers who adopt this book. Please send a request by e-mail to [email protected]

All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher.

ISBN 90-407-1836-9 EAN 9789040718366NUR 924

Keywords: electromagnetic waves

Preface

The course ”Electromagnetic Waves” offers an introduction in the the-oretical concepts of electromagnetic waves. This course book contains thebasic material on time-varying wavefields and their applications in electri-cal engineering, e.g., electromagnetic compatibility, communication and re-mote sensing. A prerequisite to this course is a standard course ”Electricityand Magnetism” where, from experimental laws, the Maxwell equations fortime-varying electromagnetic fields are formulated as a system of partialdifferential equations.

Chapter 1 reviews the necessary mathematical background, while Chap-ter 2 introduces the fundamental mathematical equations: the Maxwell equa-tions, the constitutive relations and boundary conditions. The main line ofthe course is the construction of solutions to these equations in some sim-ple configurations. The concept of an electromagnetic wave is introduced inChapter 3, where one-dimensional waves are discussed. A wave phenomenoncan only be understood in connection with an electromagnetic source thatgenerates a wave. For the excitation of one-dimensional waves, the planar-electric-current sheet is chosen. As a simple example of one-dimensional wavepropagation, the parallel-plate waveguide is discussed shortly. In Chapter 4,the two-dimensional waves are studied, in particular specific properties asinterference, Fresnel reflection/transmission factors, Brewster’s angle andtotal reflection are treated. In Chapter 5, the consequences of a weakly in-homogeneous medium are discussed and the theory of electromagnetic raysis introduced. Further, in Chapters 6 and 7, the theory of transmission linesand electromagnetic waveguides is treated. In view of communication ap-plications, the closed parallel-plate waveguide and the open dielectric-slabwaveguide are described in full detail. Finally, Chapter 8 deals with theexcitation of two-dimensional waves and the concept of the far-field approx-imation is introduced.

The student who has successfully completed the present introductorycourse on electromagnetic waves, has learned the basic concepts of electro-magnetic wave propagation. By simplifying the problems in such a way

vi preface

that a description in terms of one-dimensional and two-dimensional wavessuffices, more attention can be given to the physical understanding of thepropagation phenomena. However, it is stressed that in more realistic con-figurations of present-day technology, a full three-dimensional descriptionof electromagnetic wavefields is needed. In this context, it is noted thatthe methodology of handling the radiation and scattering of electromagneticwaves in three-dimensional configurations will be treated in more advancedcourses of the electrical engineering curriculum.

The authors acknowledge Dr. E. C. Slob for compiling the original set ofexercises, problems and answers; Dr. M. D. Verweij for contributing to thematerial of Chapters 4 and 5, revising the exercises, problems and answers,and preparing the final print version; and Mr. K. F. I. Haak for re-checkingthe answers.

Delft, January 1999 H. BlokP.M. van den Berg

Preface to the second edition

This edition is identical to the first edition, except that a number oferrors have been corrected. The authors acknowledge their collegues of theLaboratory of Electromagnetic Research and in particular Dr. D. Quak andMr. P. Jorna for reporting most of these errors.

Delft, September 2001 M.D. VerweijP.M. van den Berg

H. Blok

Preface to the improved second edition

This edition is identical to the second edition, except that some minuteerrors have been corrected. The authors acknowledge Dr. R.F. Remis of theLaboratory of Electromagnetic Research for reporting some of these errors.

Delft, August 2010 M.D. VerweijP.M. van den Berg

H. Blok

Contents

Preface v

1 Introduction 11.1. Cartesian vectors and their properties . . . . . . . . . . . . . 4

1.1.1. Addition, subtraction and multiplication of vectors . . 4

1.1.2. Differentiation with respect to a parameter . . . . . . 6

1.1.3. Differentiation with respect to the spatial coordinates 6

1.2. Exercises and problems . . . . . . . . . . . . . . . . . . . . . 17

2 The Electromagnetic Field Equations 21

2.1. Force exerted on an electric point charge . . . . . . . . . . . . 21

2.2. Maxwell’s equations in vacuum . . . . . . . . . . . . . . . . . 24

2.3. Maxwell’s equations in matter . . . . . . . . . . . . . . . . . . 25

2.4. The constitutive relations . . . . . . . . . . . . . . . . . . . . 282.5. The system of field equations . . . . . . . . . . . . . . . . . . 31

2.6. The boundary conditions . . . . . . . . . . . . . . . . . . . . 32

2.7. Frequency-domain representations . . . . . . . . . . . . . . . 36

2.7.1. The frequency-domain field equations . . . . . . . . . 37

2.8. Polarization state . . . . . . . . . . . . . . . . . . . . . . . . . 392.9. Poynting’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 41


3 One-dimensional Electromagnetic Waves 49

3.1. The planar electric-current sheet as emitter . . . . . . . . . . 50

viii contents

3.2. Steady-state analysis . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.1. Lossless medium . . . . . . . . . . . . . . . . . . . . . 553.2.2. Lossy medium . . . . . . . . . . . . . . . . . . . . . . 57

3.3. Transient emission into a lossless medium . . . . . . . . . . . 593.4. Reflection and transmission problem . . . . . . . . . . . . . . 62

3.4.1. Electric field analysis . . . . . . . . . . . . . . . . . . . 64

3.4.2. Magnetic field analysis . . . . . . . . . . . . . . . . . . 65

3.5. Shielding problem . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5.1. Electric field analysis . . . . . . . . . . . . . . . . . . . 69

3.6. Parallel-plate waveguide . . . . . . . . . . . . . . . . . . . . . 72


4 Two-dimensional Electromagnetic Waves 81

4.1. Plane waves in a homogeneous medium . . . . . . . . . . . . 83

4.1.1. Uniform plane waves . . . . . . . . . . . . . . . . . . . 87

4.2. Interference of two plane waves . . . . . . . . . . . . . . . . . 90

4.2.1. Steady-state analysis: lossless case . . . . . . . . . . . 92

4.3. Reflection of a plane wave by an electrically impenetrable half-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4. Reflection and transmission of a plane wave incident upon aplane interface . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4.1. Uniform plane waves in the frequency domain . . . . . 108


5 Electromagnetic Rays in a Two-dimensional Medium 121

5.1. Homogeneous, lossless medium . . . . . . . . . . . . . . . . . 122

5.2. Parallel polarization . . . . . . . . . . . . . . . . . . . . . . . 124

5.3. Perpendicular polarization . . . . . . . . . . . . . . . . . . . . 128

5.4. Ray trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.4.1. Ray trajectories in a horizontally layered medium . . . 133

5.4.2. Ray trajectories in a radially layered medium . . . . . 138


contents ix

6 Transmission Lines 1456.1. TEM-waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.2. Parallel-plate waveguide . . . . . . . . . . . . . . . . . . . . . 151

6.3. Coaxial line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.4. Propagation properties . . . . . . . . . . . . . . . . . . . . . . 155

6.4.1. Two-conductor transmission line . . . . . . . . . . . . 1566.4.2. Lossless transmission line: steady-state analysis . . . . 158

6.4.3. Transients on lossless transmission lines . . . . . . . . 1606.5. Exercises and problems . . . . . . . . . . . . . . . . . . . . . 164

7 Electromagnetic Waveguides 167

7.1. Parallel-plate waveguide . . . . . . . . . . . . . . . . . . . . . 172

7.2. Propagation properties of modes in a parallel-plate waveguide 178

7.3. Dielectric slab waveguides . . . . . . . . . . . . . . . . . . . . 182

7.4. Propagation properties of guided modes in a dielectric slabwaveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193


8 Excitation of Two-dimensional Electromagnetic Waves 203

8.1. The sheet emitter with a parallel electric current . . . . . . . 204

8.1.1. The far-field approximation . . . . . . . . . . . . . . . 207

8.2. The sheet emitter with a perpendicular electric current . . . . 2138.2.1. The far-field approximation . . . . . . . . . . . . . . . 216


Answers to Exercises 223

Bibliography 241

Index 243

Chapter 1

Introduction

Electromagnetic wavefields have a wide range of applications: from com-munication to medical treatment, from environmental sensing to energy ra-diation.

When following an electromagnetic wave on its course, we start with itsexcitation by an electromagnetic source. Some of them are natural sourcessuch as the sun and stars, others are artificial ones (a transmitting antenna,a laser). Once it has been generated, the wave propagates along a certainpath from the source to the receiver. Depending on the properties of themedium through which the wave passes, this propagation can lead to contin-uous refraction by spatial changes in the medium parameters (for example,the atmosphere), or to discontinuous refraction by an abrupt change in themedium (for example, an interface between two different media). Finally,the wave motion is picked up by an electromagnetic receiver (a receivingantenna, an optical detector).

Each of these aspects is the subject of theoretical and experimental in-vestigation. Usually, when the attention is focussed on a particular detail,the remaining circumstances are chosen as simple as possible. For example,when one wants to investigate the directional characteristics of a transmit-ting antenna, the surrounding medium will be taken of the utmost simplicity,as far as its electromagnetic properties are concerned, and of infinite extent.When studying refraction phenomena during the propagation of an electro-magnetic wave, the source will be taken a simple one, while the influence

2 introduction

of the receiver will be neglected at all. These simplifications are dictatedby the impossibility to take into account the influence of all parameterssimultaneously.

The basic laws of macroscopic electromagnetic theory were formulatedby James Clerk Maxwell and can be found in his famous book (Maxwell

1873). For a survey of the history of the subject the reader is referred toWhittaker (Whittaker 1953). From the theory it follows that there existelectromagnetic waves that travel with a finite speed which in vacuo seemsto be a universal constant, independent of the state of motion in which theobserver carries out his or her experiments. (The latter is not the case forwaves in matter.) Since through a wave motion with constant speed thechanges in position in space and the changes in time are interrelated in arigid manner, electromagnetic waves in vacuum can serve to interconnect thespace-time observations for two observers in relative motion. This concepthas led Einstein (Einstein 1956) to the theory of relativity. We shallconfine our analysis of electromagnetic waves to the case where the sourcesthat generate the wavefield, and the observer are at rest with respect to thematerial media in the configuration.

As in any type of wave motion, the physical quantities that describethe electromagnetic waves, depend on position and time. Their time depen-dence in the domain where the source is acting is impressed by the excitationmechanism of the source. The subsequent dependence on position and timeelsewhere is governed by propagation laws. The physical laws that underlythe properties of waves are induced from a series of basic standard experi-ments. To carry out these experiments, an observer must be able to registerthe position and the instant at which an observation is made. To registerthe position the existence of an isotropic background space is preassumed.In this space, distance can be measured along three mutually perpendicu-lar directions with one and the same position- and orientation-independentstandard measuring rod. To register instants, the existence of a position-and orientation-independent standard clock is preassumed. The standardmeasuring rod is used to define, at a certain position which is denoted asthe origin O, an orthogonal Cartesian reference frame consisting of threebase vectors {i1, i2, i3} that are of unit length each. The orientations ofthese three base vectors form a mutually perpendicular, right-handed triad(Fig. 1.1). (The property that each base vector specifies geometrically a

introduction 3

length and an orientation, makes it a vectorial quantity, or a vector; nota-tionally, vectors will be represented by bold face symbols.) Let {x1, x2, x3}denote the three numbers that are needed to specify the position of an ob-server, then the vectorial position of the observer x is the linear combination(Fig. 1.2)

x = x1i1 + x2i2 + x3i3 . (1.1)

........

........

........

........

........

........

........

........

........

........

........

........

........

...................

................

i1

............................................................................................................................

................i2

............................................................................................................................ ................

i3

O

Figure 1.1. Standard measuring rod and Cartesian reference frame {O, i1, i2, i3}in three-dimensional space.

................................

................................

.....................................

.............................

x

........

........

........

........

........

........

......................

................i1

......................................................................

................i2...................................................................... ............

....

i3O.............................................................................................................................................................................................................................................................................................

...............................................................................................................................................................................................

...............................................................................................................................................................................................................................................................................................

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

........

.....

x1

..............................................................................

x2

..............................................................................

..............................................................................

............. ............. ............. ............. ............. ............. ............. ............. .............

x3

............. ............. ............. ............. ............. ............. ............. ............. .............

............. ............. ............. ............. ............. ............. ............. ............. .............

Figure 1.2. Cartesian coordinates {x1, x2, x3}.

4 introduction

The numbers {x1, x2, x3} are denoted as the orthogonal Cartesian coordi-nates of the point of observation. The time coordinate is denoted by t.

One of the purposes of the basic standard experiments is to define theunits in terms of which the measured physical quantities are expressed. In ac-cordance with international convention, we employ the International Systemof Units (Systeme International d’Unites), abbreviated to SI, for expressingthe measured physical quantities. The mathematical framework by whichthe results from the standard basic experiments are cast into the macroscopicphysical laws that govern the wave motion is furnished by vector calculus.For this reason, the next section summarizes those properties of Cartesianvectors that are needed in our further analysis.

1.1. Cartesian vectors and their properties

The mathematical framework of the theory of electromagnetic waves isfurnished by vector calculus. For this reason we summarize those propertiesof Cartesian vectors that are needed in our further analysis.

1.1.1. Addition, subtraction and multiplication of vectors

Vectors can be subjected to the algebraic operations of addition, sub-traction and multiplication. Let the components of v be given by v1, v2

and v3, and those of w by w1, w2 and w3, then the components of the sum(difference) of v and w is given by

v ± w = (v1 ± w1)i1 + (v2 ± w2)i2 + (v3 ± w3)i3 . (1.2)

The product of the scalar ϕ and the vector v is given by

ϕv = ϕv1i1 + ϕv2i2 + ϕv3i3 . (1.3)

The scalar (dot) product of the vectors v and w is given by

v · w = v1 · w1 + v2 · w2 + v3 · w3 = w · v , (1.4)

cartesian vectors and their properties 5

The length of a vector v is denoted as

|v| = (v · v)12 = (v2

1 + v22 + v2

3)12 . (1.5)

The vector (cross) product of the vectors v and w is given by

v×w = (v2w3−v3w2)i1 +(v3w1−v1w3)i2 +(v1w2−v2w1)i3 = −w×v , (1.6)

or in matrix notation

v × w =

∣∣∣∣∣∣∣i1 i2 i3

v1 v2 v3

w1 w2 w3

∣∣∣∣∣∣∣ . (1.7)

The scalar triple product of three vectors u, v and w is given by

u · (v × w) = u1(v2w3−v3w2) + u2(v3w1−v1w3) + u3(v1w2−v2w1) , (1.8)

or in matrix notation

u · (v × w) =

∣∣∣∣∣∣∣u1 u2 u3

v1 v2 v3

w1 w2 w3

∣∣∣∣∣∣∣ . (1.9)

The scalar triple product has the property

u · (v × w) = v · (w × u) = w · (u × v) (1.10)

= −w · (v × u) = −v · (u × w) = −u · (w × v) .

The vectorial triple product can be written as

u × (v × w) = (u · w)v − (u · v)w . (1.11)

As regards the differentiation of a vector, two cases have to be distin-guished: differentiation with respect to a parameter, and differentiation withrespect to the spatial (Cartesian) coordinates of the space in which the vectorfunction is defined.

6 introduction

1.1.2. Differentiation with respect to a parameter

Let ϕ = ϕ(t) a scalar function and assume that ϕ is a differentiablefunction of the parameter t (in electromagnetics often the time coordinate).Then, the derivative ∂tϕ = ∂ϕ/∂t is also a scalar function. Let v = v(t) bea vector function and assume that v is a differentiable function of the pa-rameter t. Let v1 = v1(t), v2 = v2(t), and v3 = v3(t) denote the componentsof v, then the derivative ∂tv of v is the vector

∂tv = (∂tv1)i1 + (∂tv2)i2 + (∂tv3)i3 . (1.12)

Let ϕ = ϕ(t) be a differentiable scalar function of the parameter t and letv = v(t) and w = w(t) be differentiable vector functions of the parametert, then we have the following differentiation rules:

∂t(ϕv) = (∂tϕ)v + ϕ∂tv , (1.13)

∂t(v × w) = (∂tv) × w + v × ∂tw . (1.14)

1.1.3. Differentiation with respect to the spatial coordinates

Let ϕ be a scalar function and assume that ϕ = ϕ(x) = ϕ(x1, x2, x3) is adifferentiable function of the spatial (Cartesian) coordinates x1, x2 and x3.Then, the derivatives ∂1ϕ = ∂ϕ/∂x1, ∂2ϕ = ∂ϕ/∂x2 and ∂3ϕ = ∂ϕ/∂x3 arealso scalar functions. In this context, the gradient of ϕ = ϕ(x) is introducedas

grad ϕ = ∇ϕ = (∂1ϕ)i1 + (∂2ϕ)i2 + (∂3ϕ)i3 , (1.15)

where∇ = i1∂1 + i2∂2 + i3∂3 (1.16)

is the operator of Hamilton, the so-called nabla operator or del operator.This operator is a vector and acts as a spatial differentiation with respect tothe three spatial coordinates.

Let v be a vector function and assume that v = v(x) = v(x1, x2, x3) is adifferentiable function of the spatial (Cartesian) coordinates x1, x2 and x3.The derivative ∂1v is the vector

∂1v = (∂1v1)i1 + (∂1v2)i2 + (∂1v3)i3 . (1.17)


Similarly, we have

∂2v = (∂2v1)i1 + (∂2v2)i2 + (∂2v3)i3 , (1.18)

∂3v = (∂3v1)i1 + (∂3v2)i2 + (∂3v3)i3 . (1.19)

These three derivatives operating on the vector function v can be combinedin the divergence operator, defined as

div v = ∇ · v = ∂1v1 + ∂2v2 + ∂3v3 , (1.20)

and in the curl operator, defined as

curlv = ∇× v = (∂2v3−∂3v2)i1 + (∂3v1−∂1v3)i2 + (∂1v2−∂2v1)i3 . (1.21)

We note that ∇ is a vector operator satisfying two sets of rules:

• vector rules;

• partial differentiation rules, including differentiation of a product.

We now summarize the rules for the differentiation with respect to thespatial coordinates of the scalar functions ϕ = ϕ(x) and ψ = ψ(x), and ofthe vector functions v = v(x) and w = w(x).

∇(ϕ + ψ) = ∇ϕ + ∇ψ , (1.22)

∇ · (v + w) = ∇ · v + ∇ · w , (1.23)

∇ × (v + w) = ∇ × v + ∇ × w , (1.24)

∇(ϕψ) = (∇ϕ)ψ + ϕ∇ψ , (1.25)

∇ · (ϕv) = (∇ϕ) · v + ϕ∇ · v , (1.26)

∇ × (ϕv) = (∇ϕ) × v + ϕ∇ × v , (1.27)

∇ · (v × w) = (∇ × v) · w − v · (∇ × w) , (1.28)

∇ × (v × w) = (w · ∇)v − w ∇ · v − (v · ∇)w + v ∇ · w , (1.29)

∇(v · w) = w × (∇ × v) + (w · ∇)v + v × (∇ × w) + (v · ∇)w .

(1.30)

We note that we have assumed that the functions ϕ, ψ, v and w are differen-tiable functions of the spatial coordinates. When we assume that ϕ = ϕ(x)

8 introduction

is also a twice differentiable function of x, we have the rules:

∇ · (∇ϕ) = (∇ · ∇)ϕ = (∂21 + ∂2

2 + ∂23)ϕ , (1.31)

∇ × (∇ϕ) = 0 , (1.32)

∇ · (∇ × v) = 0 , (1.33)

∇ × (∇ × v) = ∇(∇ · v) − (∇ · ∇)v . (1.34)

Subsequently, we present the rules for the spatial differentiation of a spatiallydependent function f = f(|x|):

∇|x| =x

|x| , (1.35)

∇|x|n = n|x|n−2x , (1.36)

∇f(|x|) = ∂f(|x|) x

|x| , (1.37)

where ∂f is the derivative of f with respect to its argument. Further, wehave:

∇ · x = 3 , (1.38)

∇ × x = 0 , (1.39)

(∇ · ∇)|x|n = n(n + 1)|x|n−2 , (1.40)

and when a is a constant vector:

∇(a · x) = a , (1.41)

(a · ∇)x = a , (1.42)

(a × ∇) × x = −2a . (1.43)

Interpretation of grad ϕ

We consider a continuously differentiable scalar function ϕ = ϕ(x) andwe take the dot product of its gradient ∇ϕ (del ϕ) and an infinitesimalincrement of length

dx = dx1i1 + dx2i2 + dx3i3 . (1.44)

Thus we obtain

(∇ϕ) · dx = (∂1ϕ)dx1 + (∂2ϕ)dx2 + (∂3ϕ)dx3 = dϕ , (1.45)


ϕ(x) = C�P �

Q

......................................................................................................................................................................................................................................................................................................................

................................

xP

............................................................................................................................................................................................................................................................................................................................................................................

................................

∇ϕ

...................................................................................................................................... .....................

dx................................

..........................................

.................................

.................................

.................................

.........................................

...............................................................................................................................................................................................................................................................................................................................................................................

..........................................................

........................................................

.............................................................................................................................................

........

........

........

........

........

........

......................

................i1

......................................................................

................i2...................................................................... ............

....

i3

O

ϕ(x) = C + ΔC

....................................................................................................................

.......................................

..................................

................................

...................................

...............................................

.............................................................................................................................................................................................................................................................................................................................................................................................

......................................................

..................................................................

................................

Figure 1.3. The gradient vector.

the change in the scalar function ϕ corresponding to a change in positiondx. Now consider P and Q to be two points on a surface ϕ(x1, x2, x3) =constant. These points are chosen so that Q is a distance dx from P. Then,moving from P to Q, the change in ϕ(x1, x2, x3) = constant is given by

dϕ = (∇ϕ) · dx = 0 , (1.46)

since we stay on the surface ϕ(x1, x2, x3) = constant. This shows that ∇ϕ isperpendicular to dx. The vectorial distance dx may have any direction fromP as long as it stays in the surface ϕ = constant, point Q being restricted tothis surface. For vanishing dx, we observe that ∇ϕ is oriented in a directionof the normal to the surface ϕ = constant (see Fig. 1.3).

If we now permit dx to take us from one surface ϕ = C, C being aconstant, to an adjacent surface ϕ = C + dC, then,

dϕ = dC = (∇ϕ) · dx . (1.47)

10 introduction

For a given dϕ, |dx| is a minimum when it is chosen parallel to ∇ϕ, or, fora given |dx|, the change in the scalar function ϕ is maximized by choosingdx parallel to ∇ϕ. This identifies ∇ϕ as a vector having the direction ofthe maximum space rate of change of ϕ.

Very often the notion of directional derivative occurs. When τ is a unitvector, the quantity τ · ∇ϕ is called the directional derivative of ϕ in thedirection of τ , and equals the rate of change of ϕ in the direction of τ , viz.,

τ · ∇ϕ = ∂τϕ = τ1∂1ϕ + τ2∂2ϕ + τ3∂3ϕ . (1.48)

When τ is the tangent along a surface ϕ = constant, we obtain

τ · ∇ϕ = ∂τϕ = 0 , (1.49)

which is consistent with Eq. (1.46).

Interpretation of divv

We consider a continuously differentiable vector function v = v(x). Thedivergence operator ∇ · v (del dot v) results in a scalar quantity indicatingthe outflow of a vector field. It can be obtained from the limiting behaviorof the net outflow integral for a vanishing small enclosed volume. To showthis we first compute the net outflow of a vector field v over the elementarydomain with volume dV = dx1dx2dx3 at the center of the elementary domain(see Fig. 1.4). This latter point is given by xP = {1

2dx1,12dx2,

12dx3}. By

Taylor’s theorem, the field component v1 is

v1(x) = v1(xP) + (∂1v1)(x1− 12dx1) + (∂2v1)(x2− 1

2dx2) + (∂3v1)(x3− 12dx3)

+ higher order terms . (1.50)

The surface integral of the normal component of v (in the direction of theoutward normal) over the top surface {x1 = dx1, 0 < x2 < dx2, 0 < x3 <

dx3} of the volume element, shown in Fig. 1.4, is

∫ dx2

x2=0

∫ dx3

x3=0v1(dx1, x2, x3) dA =[v1(xP) + 1

2(∂1v1) dx1]dx2dx3

+ higher order terms . (1.51)

Documents

Electromagnetic Waves - Delft Academic Press · electromagnetic, acoustic and elastic waves. Hans Blok (1935) was a member of the scientific staff of the Laboratory of Electromagnetic