ELECTROMAGNETIC

THEORY

ELECTROMAGNETIC THEORYA CRITICAL EXAMINATION OF FUNDAMENTALS(formerlytitled:

Electromagnetics)

BY

ALFRED O'RAHILLYPROFESSOR OF MATHEMATICAL PHYSICS UNIVERSITY COLLEGE, CORK

With a Foreword by Professor A. W. CONWAY, F.R.S..'..

in

two volumes

VOLUME I"'"\

DOVER PUBLICATIONS,

INC.,

NEW YORK

Published

in Ltd.,

Canada30

by

Company,

Lesmill

General Road,

Publishing

Don

Mills,

Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, LondonW.C.2.

This Dover edition, first published in 1965, is an unabridged and oorrected republication of the work first published by Longmans, Green & Company, Inc.

and Cork University Presstitle:

in 1938 under the former Electromagnetics This edition is published by

arrangement with Longmans, Green & Company, Inc. This work was originally published in one volume, but is now published in two volumes.special

Library of Congress Catalog Card

Number

65-15S1}

Manufactured in the United States of America

Dover Publications,180 Varick Street

Inc.

New

York

14,

N.Y.

.5-37/

0G3Kt

FOREWORDTheclash of ideologies in world affairs has its counterpart in

domains of scientific thought. The two main theories of light-emission and wave-theory have fought their battles throughout many generations of scientists with varying success, new experimental facts turning the scale now in one direction and now in another and at the present moment, one does not expect a decision on the issue but rather hopes for some new point of view which will reconcile the amazing and ever increasing number of new observations. The mathematician is apt to regard the matter as a conflict between ordinary and partialseveral;

Hamilton succeeding Lagrange, Schrodinger succeeding Bohr. In electrodynamics the action-at-a-distance formulae (which after all is an emission idea) of Ampere and his followers were succeeded by Maxwell's mathematical formulation of Faraday's views of the aether. The coming of the electron and the LienardSchwartzschild force-function bring us back again to action-at-adifferential equations,

what the author calls propagated far-action. How of Maxwell remains ? His displacement-current in free space is now well known to be only a means of transforming an equation of Poisson's type to the general equation of wavepropagation. His electric stresses in free space have disappeared. Not everyone will agree with the author's estimate of Ritz ; but everyone will agree with the elegance of his electrodynamic formula. Everything that Ritz has written in his short life is worth study, and the author deserves credit for bringing to light again his formulae. Ritz and Einstein are direct opposites. 0. The theory of special relativity, by its very nature, can add ^ nothing to our electrodynamical knowledge. A neglect of this -s fact has led to much loose reasoning and the author has unsucceeded in finding many unjustified deductions. Cj doubtedly v' especially in attempted extrapolation to electronic phenomena.distance, to

much

;

Ti

FOREWORD

In a world which depends so much on electricity, it is rather not yet agreement on electric units and on the basic relationship of quantities such as permeability to mass, length and time. The author goes very thoroughly into this matter and his substitution of measure-ratios ' instead of ' dimensions clears up a lot of difficulties. The classical theory of electricity still remains the inspiration, The appearif only by its symbols, of much modern speculation. ance of this book, which shows what the classical theory is and what it rests on, is opportune. But it is more than a review of existing knowledge, it is a courageous attempt at reconstruction ; and if we do not always agree with the writer, he certainly makes us reflect. I recommend this book to every serious student of Electromagnetics.startling to find that there is' '

Arthur W. Conway.

PREFACEThis bookis intended to be an essay in constructive criticism. Existing expositions are freely criticised without regard for authority, which should find no place in science. The arguments here urged are not put forward dogmatically, but rather for the

purpose of awakening teachers from their dogmatic slumber.' Counter-arguments will be welcomed by the author, but he proposes to pay no attention to mere contradiction on the part of the orthodox.' As originally planned, the work included a detailed criticism of the theory known as Relativity. But the material became so bulky that publication of this latter portion has been deferred. In order to forewarn the reader and to facilitate the critic, the' '

main theses are summarised as follows

:

.

(1)

The

history of electromagnetic theory

is

rewritten.

It is

maintained that Maxwell's views, which were logically stated by Helmholtz and Duhem, are really off the main line of development. The ideas of Gauss and Weber are vindicated ; the proposals of Lorenz and Biemann are claimed to supersede Maxwell'sdisplacement-current.

In particular, the synthetic statement of the accepted ' electromagnetic theory is shown to be the forceformula published in 1898 by Lienard, who is still alive and active. (3) But it is also shown that the universally ignored alternative formula proposed by Bitz in 1908 is equally, and even more, successful. The interest of this formula is that it is really and radically relativist in the proper acceptation of that muchabused word. Even if Bitz's theory is not accepted, it has at least the merit of proving the unsoundness of most of the arguments adduced in favour of the prevalent view.(2)'

classical

(4) Einstein's

use of Voigt's transformation, generally

known

as the special theory of relativity, is subjected to fundamental criticisms that is, as regards electromagnetics, for it is proposed;

-

;

'

viii

PREFACE'

to treat optics subsequently. In particular, Lorentz 's local time and Minkowski's ' space-time ' are rejected. ' aether ' is declared to be (5) Contemporary discussion of theis'

a mere logomachy, a waste of time. An important distinction urged between the quantitative equations of physics and thediscourse(6)'

of physicists.

elementary but radical exposition of the meaning of the symbols of physics is worked out. This is shown to have many practical and even philosophical consequences. It implies the rejection of Bergson's view of duration, of Bridgman's operational theory, and of Eddington's bundles of pointer-readings.' In fact, an attempt is made to sweep idealism and pseudomysticism completely out of physics ; it is held that physical'

An

any philosophical problem. and dimensions,' still the subject ofcontroversy in scientific periodicals and of votes at International Congresses, is treated in a simple but revolutionary, manner. It is claimed that thereby an end is put to barren discussions which have now lasted over fifty years, and that there is no further excuse for electrotechnologists to continue talkingscience(7)is

incapable of solvingunits

-.

:.

The question of

'

nonsense.I wish to

;

.'

thank the National University of Ireland and the Cork University Press for making liberal grants towards the cost >'.,.. of publication..-i-.

For criticisms, suggestions, and help in proof-reading I wish to thank the following: my old teacher Prof. A. W. Conway, F.R.S. of University College, Dublin former pupil and present colleague Mr. M. D. McCarthy M.A.;

Acknowledgments.

my

Rev. Prof.

M.F.

Egan,

S.J.,

of University College, Dublin;

Prof. A. J. McConnell of Trinity College,

Dublin

;

Prof.

McKenna

Owen

of University College, Galway. For any views here expressed I am, of course, solely responsible.

REOISTRilt's HODSS,,

Alfred O'Rahilly..i

UNXTEaSITY COIXEQX, COEK.

CONTENTS TO VOLUME ONECHAPTERI

MathematicalFAGS1.

Vectors

-

-

-

1

2.

Stokes

-

-

-. -

-

-

-

-

-

.

-.

3. 4.5. 6. 7. 8.

Green

-.

-.

-

-

\-

...*

* 719 21

Vectors varying with the Time.

-

-

.

-

A Differential EquationThe Rateof

Change-

of

an Integral

23*," "

Linear Circuits

-.-.-

Some

Integrals

.:..CHAPTERPOISSON

-2528

II

1.

Polarisation

-

3*

2. 3.

Scalar and Vector Potentials

.."."''-

A Doublet Shell

-

-

'..'..'''-

'-

37 41 44 50 56 60 6S

4.5. 6.

Free Energy of a Doublet System Free Energy of a System of Singlets and Doublets

-

-

-

A Polarised MediumTheUnits

-

-

-

-

*

*

7.8.

Localisation of Energy

CHAPTER

III

Maxwell1.

The Faraday-Mossotti HypothesisMaxwell's ' Displacement ' The Displacement Current *

2.3.

" .84 95*"

77

x

CONTENTS

CHAPTER IV

1.

2. 3. 4.

Equivalence The Electrodynamic Potential Induction in Linear Circuitsin the

--.Medium-

Ampere

Neumann

---.-...

FAOB

-102 -114.

123

The Energy of Currents and Magnets-

6. Stresses 6.

-

Point-Charges

... . *

120

-

136 151

CHAPTER VHELMHOLTZ1.

DUHEM"

The Derivation

of Maxwell's Equations-

2. 3.

Helmholtz's Constant

--

"

-

161

-

.

The Status

of Maxwell's Equations

-

-

-

169 176

CHAPTER VILOBENZ1.

The PropagatedFar-Actions

Potentials

.......RlEMANNVII, ,

.

\g\

2.

190

CHAPTER1.

LrforABD2.3.4.

Atomism in Electricity . The Potentials for Point-ChargesTheForoe-FormulaConclusions.

..

. . .

... .

..

_

203

.

. !

.

;

.

._.

212 215

w

22

CHAPTERPOYNTING1.

VIII

AbBAHAM LOKENTZ'

Electromagnetic Mass

2. 3.4. 5. 6.

Page-SchottPoynting

..." --....,'.'.".'.. .

'

,

Localised Energy

.. .

Electromagnetic

Momentum.

Mass and. Energy "'

-

.

304

CONTENTS

ii

CHAPTER IXVOIOT1.

An Algebraic FormulaThe Doppler Effect A Moving Reflector The Force-FormulaMaxwell's Equations Bate or DurationSubrelative Systems-

Jag*

2.

3. 4.5. 6. 7. 8. 9.

-....... .

324 327 33a

342 3473513 6g

The Mass-FormulaApplications to Electromagnetics

374 3g3 404 41Q

10. 11. 12.

The Metaphor of Four DimensionsConclusion-

...

.

.

.

,

.

Some

Objections

427

CHAPTER

I'

Mathematical1.

Vectors.

.

A general elementary knowledge of vector analysis is presupposed. The object of this chapter is to explain the notation employed, to collect some of the more important formulae which are required, and to prove some purely mathematical theorems beforehand so that the ensuing physical arguments may be clarified by being separated from difficulties which are merely',

analytical.

A vector is denoted by a in Clarendon type, its rectangular components by ax av as The scalar product of two vectors Z&j&j. is denoted by ab or, where ambiguity might arise, by (ab) or c = b(ac)

c(ab).

.

I

2

MATHEMATICALis

The vector operator V or (3/3*, 3j3y, d/3z) scalar operator SV is called div, so thatdiv a

called grad.

The

= "Zdajlx.

The vector operator FV

is

called curl, 1 so that

curl

a

-

Mi

j

kJdz

3x

dy

axIf div

ay

o2it

a=0, a

is

said to be a circuital vector;

can be ex-

curl b (see p. 16). If curl a pressed as a a irrotational vector and can be expressed as

=

= 0, a is said to be an a = grad (see p. 15)..

(1.7a)

The vector produot of V^ and a may be thus written or it may be denoted by F(V# a), since there is no danger of misinterpreting the round brackets to represent a scalar product. Likewise the vector produot of a and b + c may be written F(a, b + c).brackets are inserted merely for convenience.;

a The

MATHEMATICALIn the present casedr/dt or

=

dr/dt

+

Voir(1.7b)

v=Theaccelerationfis

f+JW.

given

by

= dv/dt = {d/8t + Voj) (f + Fwr) = f + Fcbr + 2Fcof + FoFtorbecomes

(1.7o)

If o> is constant, this

f=f + 2Fu)f + io(ior)-to s r

(1.7d)

There is another distinction in the meaning of differentiation which will also be required. Let / be any function pertaining to any particular moving particle. Let dfjdt denote its rate of change as the particle moves about whereas df/dt will mean the local rate of change with the time. That is, the former denotes the time-rate of change of /, following the individual history of the particle, while the latter denotes the time-rate of change at a fixed point of space (at a point given relatively to the reference-frame and not participating in the motion). We have;

d l.dtdt.

dx dt'

Or,

more

succinctly,

djdt=djdt+(\V).2. Stokes.

(1.7e)

Consider the line-integral

./=jVds)=j;

Fig. 2.

between the fixed terminal points P and Q (Fig. 2), which in general depends on the form of the path PQ. Here s = dx/ds.x

STOKESa unit vector along ds. As we pass from one path to a neighbouring one, the variation in the integral isetc.,is

so that s

=

\dsZAJSx,fixed),

on integrating by parts (the terminals beingAZilx

where

du

du,

d

Now

i-.-*4?Hence

A or

9

(^>

3

M

(

du*

du'\

A=

Vs

curl u.

Therefore

BJ

= Jfc(SrTscurlu)= |(8rFdscurlu) = |(curludS),,

.

.

where dSintegralis

= FSrds is the directed element of area, and the taken over the strip between the two paths.

Hence

=

(curl

udS) over the whole

area.

But the left-hand side is the line-integral over the closed path forming the contour of S, the direction being given by the arrows in the figure and the direction of dS being related positively (in a right-handed manner) to this direction of circulation. That is,(j)(uds)

=:

f

(curl

udS)

(1.8)

or, as

we may

express

it

the circulation

(line-integral) of

a

G

MATHEMATICALis

vector

the flux (surface integral) of

its curl.

This

is

Stokes's

theorem. 8If

u is an

irrotational vector (curl

u

= 0), we have m(uds) = 0.

is, u = V^ fS(

-

(9.22,

become

w -iS^A^^j^*^ TT** ^ M G ***7 3 ^? 7T = = w f-d fTL/ - *X"i ffSLStof the force, whichrealJv

F -4- r IIit

we

T-

5. T el,

? A* n ard formula

th6

Xr

we

negiecwthk

the

rti0n

so that the totaI f is f the force inve S ti-

The

(7.14) put v' *- component of the f

u v

- (1 - w /c)g ujc r

(1

_ M, /Cr

) (1

_ Mr/e)?j

with a corresponding expression for

Cf_

^'^

= (1 -V/c^/V( i _ lv/c)(1 _ mJc2)31(1

[(1

^^(9.22b)

- v*J#lZgjcJc - v/c

.

gx (l

- Xutwjc%

THE FORCE-FORMULAWe can thus find G'JGX and similarly G"jGy G'JO,. v}2 0), the transformation is along x (m>,,

345

=

=

When w(9.23)

is

G'=(1.YS /P.Y2 /P)G.Henceif

wx = and u> = wt =G'

0,(l, p,

=

P)G.

(9.24)

We

can also verify that in general

.

'0'JG

= G'JG = y*l?,z.

(9.25)

exactly as in the formula (9.17). in general G'JGX is not equal to1

But it must be pointed out that F'JFX For

- (1 -vw IcWlF =

^W'-^A/'I

Whereas

l-(l-vwx lci)G' IGx =, x v/c. \gr (l - Zuxwx /c*) -

(1

- u /c)Zgxwx lc - g (l - w [c)}r

t

r

x ;[wj-jyjtf) r i(i7 J)s?a/c.

,

-

(

J

- v>TfcJg,ux lc -

(

1

- /c)( 1 - tojcjfc Jr

.

,

Ife'/e

we

re-insert the

= ei/i = ?,as follows:

constant x and take the charge-ratio instead of unity, these formulae become modi-

fied

*','"''

'

'G'

= ? /x2

2

.(

'

)G.

(9.25a)

formulae follow from the application of Voigt's transformation to the Lienard-Schwarzschild force-formula, quite independently of any ulterior physical interpretation which may be given subsequently to these purely algebraic results. We are not at the moment criticising the relativist interpretation, we are prescinding from it. There are two advantages in ourAll these

separate the algebra from irrelevant glosses. on the fundamental force-formula instead of dealing with Maxwell's equations. 'We are thus enabled to see at once that the transformation (1, (3, p), commonly given in relativity text-books, is not general. can also see that the:

treatment(2)

(1)

We

We operate

directly

We

transformation depends on the particular form of the forceformula and would be different for a force which" was a differentfunction of the velocities

and

acceleration.

It

is

not usually

1

H346realised

VOIGTwhat a hazardous hypothesis:

is

quotations as the following

contained "rained

in L m such typj ca

,

extend to molecular actions the result found ft, ... uncl for the forces. Lorentz, viii. 202. electric We must now obtain the transformation of the f For this purpose we shall consider the particular^T**?****force. f eIeot Though established in a particular case ,.

We

?,

act Tolman,!

JJe/o&V%, 1934,

3-1

.

^^-2S3&3SS^ -Mrconationof the

p 47

System3 on which they

raay

that

correcLss

&'*!%%*>L,,*t,0*&** letters sh iJut th difference to the transformation. " makes no

where the event (XTZT), represents (X'r^TO, is chosen arbitrar^r It

n,

/,'

Wens

?**

Z

T* **

in-

b0 ,n *

yVPKui' 3 /c)-,2 )-2

Now if a =so that

(1

- >7CT* and a' = (1 = (1 - ^/c )(l - iot/c 2/