zotero/storage/FP9NWSJP/.zotero-ft-cache

Oleg D. Jefimenko
WEST VIRGINIA UNIVERSITY
ELECTROMAGNETIC RETARDATION AND
THEORY OF RELATIVITY
NEW CHAPTERS IN THE CLASSICAL THEORY OF FIELDS
SECOND EDITION
Electret Scientific Company Star City

Copyright© 1997, 2004 by Oleg D. Jefimenko
All rights reserved. Reproduction or translation of this book, or any part of it, beyond that permitted by the United States Copyright Act without written permission of the copyright owner is unlawful. Requests for permission or for further information should be addressed to the publisher, Electret Scientific Company, P.O. Box 4132, Star City, West Virginia 26504, USA.
Library of Congress Control Number: 2004092642 ISBN 0-917406-24-9 (paperback) ISBN 0-917406-25-7 (hardcover)
Printed in the United States of America
www.ElectretScientific.com Electret Scientific
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PREFACE
This book is a sequel to my Electricity and Magnetism, 2nd ed., (Electret Scientific, Star City, 1989) and Causality, Electromagnetic Induction, and Gravitation, (Electret Scientific, Star City, 1992). It is a result of a further exploration of the classical theory of fields in search of heretofore overlooked relations between physical quantities and heretofore overlooked applications of the theory. The book is divided into two parts. The first part, Chapters 1 to 5, presents the fundamentals of the theory of electromagnetic retardation with emphasis on recently discovered relations and recently developed mathematical techniques. The second part, Chapters 6 to 11, presents the fundamentals of the theory of relativity based entirely on the theory of electromagnetic retardation developed in the first part.
Electromagnetic· retardation is as yet a fairly obscure concept, and therefore an explanation of what it is and why a book needs to be written about it is in order.
Electric and magnetic fields propagate with finite velocity. Therefore there always is a time delay before a change in electromagnetic conditions initiated at a point of space can produce an effect at any other point of space. This time delay is called electromagnetic retardation. Recent studies have shown that electromagnetic retardation is of overriding importance for the general electromagnetic theory and, by extension, for the entire

vi

PREFACE

classical theory of fields. We now know that electromagnetic retardation manifests itself in many different ways including, but not limited to, electromagnetic cause-and-effect relations, electromagnetic waves generated by oscillating electric charges and currents, electromagnetic fields and potentials of timedependent charge and current distributions, electromagnetic fields of moving charge distributions, mechanical relations between time-dependent or moving charges and currents, dynamics of atomic systems, time relations in moving electromagnetic systems, and the visual appearance of moving bodies. Perhaps the most important recently discovered aspect of the now evolving theory of electromagnetic retardation is that this theory leads to, and duplicates, many electromagnetic relations that are customarily considered to constitute consequences of relativistic electrodynamics. In fact, it is now clear that there exists an intimate relation between the theory of electromagnetic retardation and the theory of relativity. Obviously then, the phenomenon of electromagnetic retardation and its theoretical representation must be thoroughly understood and investigated.
In contrast with the theory of electromagnetic retardation, the theory of relativity is fairly familiar. However, as far as its scientific essence is concerned, the theory of relativity means different things to different people. It is important therefore to give a clear definition of the expression "theory of relativity" as it is used in this book.
In this book, "theory of relativity" (or "relativity theory," or simply "relativity") is used as a collective term for the body of equations, methods, and techniques whereby physical quantities measured in one inertial frame of reference can be correlated with physical quantities measured in any other inertial frame of reference.
As already mentioned, there exists an intimate relation between the theory of electromagnetic retardation and the theory of relativity. On the basis of this relation, all the fundamental equations of the theory of relativity, including equations of relativistic electrodynamics and relativistic mechanics, are derived

PREFACE

vii

in Chapters 6 to 8 in a natural and direct way from equations of the theory of electromagnetic retardation without any postulates, conjectures, or hypotheses. As a result, Maxwellian electromagnetism, electromagnetic retardation, and the theory of relativity are united in this book into one simple, clear, and harmonious theory of electromagnetic phenomena and of mechanical interactions between moving bodies.
An important consequence of the theory of relativity developed in the above manner is the revelation of certain basic errors in the interpretation and use of Einstein's special relativity theory. The nature of these errors and the ways to avoid them are explained in Chapter 9.
One of the most controversial elements of Einstein's special relativity theory is his idea of universal kinematic time dilation, according to which the rate of all moving physical and biological 11clocks II is uniformly dilated in consequence of nothing other than the relative motion of the clocks. As is shown in Chapter 10, moving elementary electromagnetic clocks indeed run slower than the same stationary clocks, but their slower rate is a consequence of dynamic interactions and depends on both the velocity and the construction of the clocks.
An extension of the theory of relativity, as it is developed in this book, leads to a covariant theory of gravitation analogous to relativistic electrodynamics. This extension is presented in Chapter 11, the concluding chapter of the book.
Although the book presents the results of original research, it is written in the style of a textbook and contains numerous illustrative examples demonstrating various applications of the theory developed in the book. Therefore it can be used not only for independent reading, but also as a supplementary textbook in courses on electromagnetic theory and on the theory of relativity.
I am pleased to acknowledge with gratitude a stimulating exchange of correspondence with P. Hillion, J. J. Smulsky, V. N. Strel'tsov, and W. E. V. Rosser on some aspects of the theory of relativity, and with M. A. Heald on the subject of electromagnetic retardation.

viii

PREFACE

I am very grateful to S. W. Durland and D. K. Walker for carefully reading the manuscript and for their most useful suggestions and recommendations.
Special thanks are due to Yu. G. Kosarev who believes that retardation is a universal phenomenon that should be properly treated in a new branch of physics which he proposes to call "retardics." His comments are highly appreciated.
Finally, I am very grateful to my wife Valentina for proofreading the numerous versions of the manuscript and for otherwise helping me to make the book ready for publication.

Oleg D. Jefimenko April 14, 1997
PREFACE TO THE SECOND EDITION

The second edition of this book is intended to update the presentation of the subject matter and to correct the misprints and other errors that appeared in the first edition. Sections 8-2, 9-4, and 11-3 have been rewritten. Two new Appendixes have been added. Particularly important is Appendix 3, containing an analysis of the physical nature of electric and magnetic forces and presenting a novel interpretation of the "near-action" mechanism of electromagnetic interactions.
I am pleased to express my gratitude to my wife Valentina for her assistance in the preparation of this edition of the book.
Oleg D. Jefimenko March 31, 2004

CONTENTS

PREFACE

V

Part I

RETARDATION

1 RETARDED INTEGRALS AND OPERATIONS WITH RETARDED QUANTITIES

1-1 Vector Wave Fields and Retarded Integrals

3

1-2 Mathematical Operations with Retarded Quantities 6

References and Remarks for Chapter 1

14

2 RETARDED INTEGRALS FOR ELECTROMAGNETIC FIELDS AND POTENTIALS

2-1. Maxwell's Equations and the Wave Field

Theorem

15

2-2 Solution of Maxwell's Equations in Terms of

Retarded Integrals

18

2-3. Surface Integrals for Retarded Electric and

Magnetic Fields

28

2-4. Retarded Potentials for Electric and Magnetic

Fields

34

2-5. Electromagnetic Induction

38

References and Remarks for Chapter 2

45

IX

X

CONTENTS

3 RETARDED INTEGRALS FOR ELECTRIC AND MAGNETIC FIELDS AND POTENTIALS OF MOVING CHARGES

3-1. Using Retarded Integrals for Finding Electric and

Magnetic Fields and Potentials of Moving Charge

Distributions

46

3-2. Correlation between the Electric and the Magnetic

Field of a Moving Charge Distribution

58

References and Remarks for Chapter 3

61

4 ELECTRIC AND MAGNETIC FIELDS AND POTENTIALS OF MOVING POINT AND LINE CHARGES

4-1. The Electric Field of a Uniformly Moving

Point Charge

62

4-2. The Magnetic Field of a Uniformly Moving

Point Charge

70

4-3. Electric and Magnetic Fields of a Line Charge

Uniformly Moving Along its Length

73

4-4. The Electric Field of a Point Charge in

Arbitrary Motion

79

4-5. The Magnetic Field of a Point Charge in

Arbitrary Motion

92

4-6. Electric and Magnetic Potentials of a Moving

Point Charge

95

4-7. How Accurate are the Equations for the Fields and

Potentials Obtained in this Chapter?

97

References and Remarks for Chapter 4

100

CONTENTS

xi

5 ELECTRIC AND MAGNETIC FIELDS AND POTENTIALS OF AN ARBITRARY CHARGE DISTRIBUTION MOVING WITH CONSTANT VELOCITY

5-1. Converting Retarded Field Integrals for Uniformly

Moving Charge Distributions into Present-Time

(Present-Position) Integrals

103

5-2. Converting Retarded Potential Integrals for Uniformly

Moving Charge Distributions into Present-Time

(Present-Position) Integrals

115

5-3. Some Peculiarities of the Expressions for the

Fields and Potentials Derived in this Chapter

120

References and Remarks for Chapter 5

126

Part II

RELATIVITY

6 FROM ELECTROMAGNETIC RETARDATION TO RELATIVITY

6-1. Relativistic Electromagnetism, Relativistic

Terminology, the Principle of Relativity,

and Theories of Relativity

129

6-2. Equations for Transforming Electric and Magnetic

Fields of Uniformly Moving Charge Distributions

into Electric and Magnetic Fields of the Same

Stationary Charge Distribution

131

6-3. Inverse Transformations

138

6-4. Equations for Transforming Electric and Magnetic

Potentials of Uniformly Moving Charge Distributions

into Electric and Magnetic Potentials of the Same

Stationary Charge Distributions and Vice Versa 141

References and Remarks for Chapter 6

145

xii

CONTENTS

7 THE ESSENTIALS OF RELATMSTIC ELECTRODYNAMICS

7-1. Basic Relativistic Transformation Equations

148

7-2. Transformation Equations for Velocity and

Acceleration

153

7-3. Transformation Equations for Partial Derivatives

with Respect to Coordinates and Time

156

7-4. The Invariance of the Cartesian Components of

Maxwell's Equations under Relativistic

Transformations

158

7-5. Testing Relativistic Transformations

165

7-6. The Method of Corresponding States

170

References and Remarks for Chapter 7

178

8 FROM RELATIVISTIC ELECTROMAGNETISM TO RELATIVISTIC MECHANICS

8-1. Transformation of the Lorentz Force

181

8-2. Transformation of Electromagnetic Energy and

Momentum of a Parallel-Plate Capacitor

186

8-3. Relativistic Expression for Mechanical

Momentum

190

8-4. Relativistic Mass, Longitudinal Mass, and

Transverse Mass

193

8-5. Transformation Equations for Mechanical Force,

Energy, and Momentum

196

8-6. Transformation of Torque

199

8-7. Rest Energy, Kinetic Energy, and the Relation

between Relativistic and Classical Mechanics

200

References and Remarks for Chapter 8

205

CONTENTS

xiii

9 COMMON MISCONCEPTIONS ABOUT RELATIVITY THEORY

9-1. The Lorentz Length Contraction

207

9-2. The Electric Field of a Moving Parallel-Plate

Capacitor

212

9-3. Using Lorentz Transformations for Finding Electric

and Magnetic Fields of a Moving Parallel-Plate

Capacitor

217

9-4. The Right-Angle Lever Paradox

218

9-5. Is the Magnetic Field due to an Electric Current a

Relativistic Effect?

228

References and Remarks for Chapter 9

231

10 THE RATE OF MOVING CLOCKS

10-1. The Idea of Time Dilation

235

10-2. Clocks Running in Accordance with Einstein's

Special Relativity Theory

237

10-3. Clocks that do not Run in Accordance with

Einstein's Special Relativity Theory

251

10-4. Reconciling the Theory with

Experimental Data

262

References and Remarks for Chapter 10

264

11 GRAVITATION AND COVARIANCE

11-1. Analogy of Electromagnetism with

Gravitation

267

11-2. Relativistic Transformation Equations for

Gravitational and Cogravitational Fields

275

11-3. Relativistic Gravitation and Relativistic

Mechanics

281

xiv

CONTENTS

11-4. Covariant Formulation of the Electromagnetic

and of the Gravitational-Cogravitational Theory 284

References and Remarks for Chapter 11

292

APPENDIXES

Appendix 1. Vector Identities

296

Appendix 2. Transformation Equations for

Momentum and Energy

299

Appendix 3. The Physical Nature of

Electric and Magnetic Forces

302

INDEX

331

I
...____. RETARDATION

1

RETARDED INTEGRALS AND OPERATIONS WITH RETARDED QUANTITIES

The fundamental laws of electromagnetism are represented mathematically by Maxwell's electromagnetic equations. The general solution of these equations for electromagnetic fields in a vacuum is expressed in terms of "retarded" field integrals which constitute the basic mathematical element in the general theory of time-dependent electromagnetic phenomena. A thorough understanding of the properties and use of retarded integrals is therefore indispensable for formulation and application of the theory. In this chapter we shall acquaint ourselves with retarded integrals and with operations involving quantities and expressions appearing in these integrals.

1-1. Vector Wave Fields and Retarded Integrals1

The vector wave field is the field of a vector V which satisfies

the inhomogeneous wave equation (also known as the general

wave equation) V XV xv+ _1 {_ J2V = K(x,y,z,t),
c2 iJt2

(1-1.1)

where K is some vector function of space and time which, for

3

4

CHAPTER 1 RETARDED INTEGRALS

simplicity, will be assumed here to be zero outside a finite region of space (this differential equation constitutes a mathematical expression for a wave-like disturbance that propagates in space with the speed c).
An important property of a vector wave field is that this field can be represented by the retarded field integral and retarded potentials, as explained in the following theorem.

The Wave Field Theorem. A vector field V satisfying Eq. (1-

1.1) and vanishing at infinity can be represented by the retarded

integral

J V = - _1 41r

-[V-'(-V'-r•V-)-dK]V, ,

(1-1.2)

where the brackets are the "retardation symbol," to be explained below, and r is the distance from the source point P'(x', y', z') where the volume element of integration, dV', is located to the field point P(x, y, z) where V is being determined; the primed operator V' operates on the source-point coordinates only. (Note: The integration in the above integral is over all space; except when noted otherwise, the integration in all integrals that follow is also over all space.)
The derivation of Eq. (1-1.2) is mostly of historical interest and will not be presented here.2 In lieu of the derivation we shall show in Example 1-2.3 that Eq. (1-1.1) is satisfied by V given by Eq. (1-1.2).
Corollary I. A vector field V satisfying Eq. (1-1.1), vanishing at infinity, and having zero divergence outside a finite region of space can be represented by the retarded scalar potential cp and the retarded vector potential A as

V = - Vcp +VxA,

(1-1.3)

with cp and A given by

SECTION 1-1 WAVE FIELD THEOREM

5

J cp -_ - 1 41r

[V' •V+Kl] , ----dV
r

+ cp0

(1-1.4)

and (1-1.5)

where K1 and K2 are the ordinary potentials of the function K of Eq. (1-1.1) (so that K = - VK1 + V x Ki), both vanishing at
infinity, and cp0 and Ao are arbitrary constants.
Corollary II. A vector field V satisfying Eq. (1-1.1),
vanishing at infinity, and having zero divergence outside a finite
region of space can be represented by the retarded scalar potential cp and the retarded vector W as

V = - Vcp +W,

(1-1.6)

with

J cp = _1 [V' • V] dV' + 'Po

41r

r

(1-1. 7)

and

(1-1.8)

where cp0 and W0 are arbitrary constants. The proof of these corollaries is presented in Examples 1-2.1 and 1-2.2.

The retardation symbol [ ] indicates a special space and time

dependence of the quantities to which it is applied and is defined

by the identity

[fl = f(x' ,y' ,z' ,t-rlc),

(1-1.9)

where tis the time for which the retarded integrals are evaluated. Thus the value of a function placed between the retardation

6

CHAPTER 1 RETARDED INTEGRALS

symbol [ ] is not that which the function has at the time t for which the integrals are evaluated, but that which it had at some
earlier time t' = t - rlc, or, as one says, the function is retarded.
The integrals of retarded quantities, or retarded integrals, are mathematical expressions reflecting the phenomenon of "final signal speed" - that is, the fact that a certain time rlc must elapse before the results of some event at the point x', y', z' can produce an effect at the point x, y, z separated from the point x', y', z' by a distance r.
Retarded integrals are closely associated with the principle of causality. According to this principle, all present phenomena are exclusively determined by past events. Therefore equations depicting causal relations between physical phenomena must, in general, be equations where a present-time quantity (the effect) relates to one or more quantities (causes) that existed at some previous time. As we shall presently see, in electromagnetic theory retarded integrals are "causal equations" expressing electric and magnetic fields and potentials in terms of their causative sources: the electric charge density p and the electric current density J. 3

1-2. Mathematical Operations with Retarded Quantities

Mathematical manipulations with retarded integrals frequently require applications of the operator V to retarded quantities. When applying V to such functions, one should take into account that they depend on space coordinates not only explicitly, but also implicitly through

r = {(x-x')2 +(y-y')2 +(z-z')2}'12

(1-2.1)

appearing in the retarded time t' - rlc. One also should take into
account that V may operate with respect to x, y, z coordinates as
well as with respect to x', y', z' coordinates. Finally, one should

SECTION 1-2 MATHEMATICAL OPERATIONS

7

take into account that a V operation may be performed upon a
retarded quantity taken at the instant t = constant as well as at the instant t' = t - rlc = constant (the latter operation is identical
with the corresponding operation upon the same unretarded quantity, combined with the subsequent "retardation" of the resulting expression by replacing in this expression t by t - rlc).
Let us designate an unspecified scalar or vector function fix',
y', z', t), together with an appropriate multiplication sign, if
needed, by X. To avoid ambiguities with V operations involving X, we shall employ special notations, as follows. If an operation is to be performed with respect to primed coordinates, we shall use the primed operator V' in writing this operation, and we shall use the ordinary operator V for designating operations with respect to unprimed coordinates. If an operation upon a retarded X is to be performed considering the retarded time t - rlc as constant, we shall denote the operation as [VX] or [V'X], placing both the operator and the function upon which it operates between the retardation brackets, and we shall use the ordinary notations V[X] or V'[X] for operations upon retarded functions when these
operations are to be performed considering the present time t, rather than t - rlc, as constant.
We shall frequently use expressions aIKl operations involving the radius vector connecting a volume element dV' of an electric charge or current (the source point x', y', z') with the point of observation (the field point x, y, z). If this radius vector is directed toward the field point, we shall designate it as r, if it is directed toward the source point, we shall designate it as r' . Likewise, we shall designate the correspoIKling unit vectors as ru
and r 'u· Observe that since r = (x - x ')i + (y - y ')j + (z -
z')k and r' = (x' - x)i + (y' - y)j + (z' - z)k, the vector r'
= - r, so that the result of any operation upon r' or r' with V or
V' is the negative of the result of the same operation upon r or r, and the result of any operation upon r, r', r or r' with V is the negative of the result of the same operation with V'.

8

CHAPTER 1 RETARDED INTEGRALS

We shall now derive several useful operational equations for
I retarded functions. Let us consider the operation o[X]lox' y', z', 1,
where [X] is some retarded scalar or vector function. 4 Taking into
account that retarded functions depend on x', y ', and z' not only directly, but also indirectly through r, we can write

o[X] I =
ox' y',z',1

o[Xl I

+

ox' y',z',1-rlc

o[X] I . o(t-rlc) x',y',z'

o(t - rlc)
ox'

•

( l -2 _2)

We can simplify the last expression by noting that

o[X] o(t-rlc)

I _
x',y',z'-

["o""xa]t I

x',y',z''

(1-2.3)

and that, by Eq. (1-2.1),

O(t-r/c) = X -x I = -co-sa,

ox'

er

C

(1-2.4)

where cos a is the direction cosine of vector r with respect to the x axis (Fig. 1.1). We then obtain

Tt o[X]I =o[X]I

+cosa[oX]I

(1_25)

OX/ Y',z',I OX/ y',z',t-rlc -C-

x',y',z' •

•

Analogous expressions can be obtained also for o[X]loy' I x', z', 1 and for o[X]/oz' I x',y', 1• If we now multiply these expressions by
the unit vectors i, j, and k, respectively, and then add them
together, we obtain the following operational equation

where

V'[X] = [V'X] + r"[oX], C Ot

(1-2.6)

= _r = i(x-x') + j(y-y') + k(z-z')

ru r

r

= icosa + j cos /3 + kcos 'Y

(1-2.7)

is the unit vector directed along r toward the point x, y, z (cos /3

SECTION 1-2 MATHEMATICAL OPERATIONS

9

y

x,y,z

x',y~·

X
z

Fig. 1.1 The direction cosine of r with respect to the x axis is cosa
= (x - x')lr.

and cos 'Y are the direction cosines of r with respect to the y and

z axis, respectively).

In a similar manner we can obtain the corresponding equation

for the unprimed V (assuming that X does not explicitly depend

on x, y, z)

V[X] = _ r"[ax]. C at

(1-2.8)

Combining Eqs. (1-2.8) and (1-2.6), we obtain an equation

correlating one unprimed V operation with two primed V

operations

[V'X] = V[X] + V'[X].

(1-2.9)

Differentiating V{[X]/r} and using Eq. (1-2.9), we obtain the correlation

V [X] = _ rJX] + V[X] = r~[X] + [V'X] _ V'[X], (l-2 _10)

r

r2

r

r2

r

r

and, combining the first and the last term of the last part of Eq. (1-2.10), we obtain a useful equation

[V'X] = V [X] V' [X]

-r-

_r + -r -

(1-2.11)

10

CHAPTER 1 RETARDED INTEGRALS

Another useful equation is obtained by eliminating V[X] from the middle part of Eq. (1-2.10) by means of Eq. (1-2.8):

(1-2.12)

Finally we note that, since
o[Xl _ o[Xl
o(t-rlc) - 7ft'
we have, by Eqs. (1-2.3) and (1-2.13),

(1-2.13)

[aaxt ]

=

o[Xl. ar

(1-2.14)

~
Example 1-2.1 Prove Corollary I to the wave field theorem, assuming that V • V, K1, and K2 are zero outside a finite region of space.
Expressing in Eq. (1-1.2) K as K = - VK1 + V x K 2 and
using Eq. (1-2.11), we have

V __l_f [V'(V' • V)-K] dV'

411"

r

= - - 1-f [V'(V' • V) +V'Kl -V' x Kil dV'

411"

r

(1-2.15)

= - _l_f V [V'. V +Kl] dV' __l_f V' [V'. V +Kl] dV'

_f411"

r

411"

r

+ _ 1 v x [Kil dV' + _l Jv' x [Kzl dV'.

411"

r

411"

r

The second and the fourth integrals of the last expression can be transformed into surface integrals by using vector identities (V20) and (V-21) (see Appendix for a list of vector identities). But since, by supposition V • V, K1, and K2 are zero outside a finite region of space, while the surface integrals are taken over all space, the integrals vanish. We thus have

SECTION 1-2 MATHEMATICAL OPERATIONS

11

V=-_1 Jv[V'•V+K1]dV1 +_1 Jvx [KJdV 1 . (1-2.16)

~

r

~

r

Factoring V out from under the integral signs (we can do so because the integration is with respect to primed coordinates, while V operates upon the unprimed coordinates) and designating the resulting integrals as <P - <Po and A - A0, we obtain Corollary I to the wave field theorem.

Example 1-2.2 Prove Corollary II to the wave field theorem. As in the preceding example, we have

J Jv V = __1 [V'(V' • V)-K] dV' = __1 [V' • V] dV'

J , J 411"

r

411"

- _1 V -[V-' -•Vd] V1 + _1 -[Kd] V, .

411"

r

411" r

r (1-2.17)

The second integral of the last expression is, as in Example 1-2.1, zero. We thus have

V = - _l Jv [V'. V] dV1 + _l J [K] dV1

411"

r

411" r

J J = - v(_l [V' • V] dV') + _1 [K] dV'.

411"

r

411" r

(1-2.18)

Designating the first integral as <P - <Po and the second integral as W - W0, we obtain Corollary II to the wave field theorem.

Example 1-2.3 Show that V given by Eq. (1-1.2) satisfies Eq. (11.1)
Using vector identity (V-16), we can rewrite Eq. (1-1.1) as

(1-2.19)

where we have denoted V(V • V) - K as Z for simplicity.

12

CHAPTER 1 RETARDED INTEGRALS

Let us now divide the volume of integration in Eq. (1-1.2) into two parts: Vol1 and Volz. Let Vol1 be a very small region close to the point of observation, so that within this region the retardation can be neglected. We then have from Eq. (1-1.2)

V = - _!_ J z dV' ,

I

411" Vo/I r

(1-2.20)

where the integral is not retarded. But this integral represents the well-known solution of the Poisson equation5

(1-2.21)

The contribution of Vol1 to V2V in Eq. (1-2.19) is therefore given by Eq. (1-2.21).
Let us now determine the contribution of Volz to V2V in Eq. (12.19). From Eq. (1-1.2) we have

vzv =vz(-1- J [Zldv')=-1- J vz[Z]dV', (1-2.22)

z

411" Vo/Z r

411" Volz

r

where we have placed V2 under the integral sign, because V2
operates upon the unprimed coordinates, while the integration is
with respect to primed coordinates. We can evaluate the last integral in Eq. (1-2.22) by integrating,
in turn, the x, y, and z components of the integrand. Taking into
account that V2 can be expressed as V • V, using Eqs. (1-2.12),
(1-2.8), and (1-2.14), and remembering that V • r = 3, V(l/r") = - (n/r"+ 1)ru, and r • r" = r, we find, after somewhat lengthy but
very simple calculations6

f vzv = - _!_ _lJZ_[Zx]_dV1.

xz

411" VolZ re ZlJt z

(1-2.23)

Since similar equation can be obtained also for the y and z
components of Vz, Eq. (1-2.22) becomes

SECTION 1-2 MATHEMATICAL OPERATIONS

13

V 2 V = - _!_ J iJ2 [Z] dV'.

2

411" Vo/2 TC2fJt2

Factoring out o2!c2of, we have

(1-2.24)

r ' vzv = a2 (- 1 J [Z] dV')

2

c2fJt2

411" Vo/2

(1-2.25)

or, by Eq. (1-1.2), remembering that Z = V(V • V) - K,

(1-2.26)

The contribution of Vol2 to V2V in Eq. (1-2.19) is therefore given by Eq. (1-2.26).
Adding now Eqs. (l-2.21) and (1-2.26), we obtain

a2v
V2 (V + V) = _ _z + Z.

I

2

c2fJt2

(1-2.27)

Since Vol1 can be made as small as we please compared to Vol2, o2V1!c2of can likewise be made as small as we please compared to o2Vzlc2of. Therefore, assuming that Vol1 ~ Vol2, we can add o2Vifc2of to the right side of Eq. (1-2.27) without affecting the
equation. We then have

(1-2.28)
or (1-2.29)

so that V1 + V2, and therefore V given by Eq. (1-1.2) does indeed
satisfy Eq. (1-1.1).

14

CHAPTER 1 RETARDED INTEGRALS

References and Remarks for Chapter 1
1. This section closely parallels a similar section in the author's Electricity and Magnetism, 2nd ed., (Electret Scientific, Star City, 1989) pp. 46-52. 2. The solution of the scalar counterpart of Eq. (1-1.1) was first published by G. Kirchhoff in Ann. der Phys. und Chemie, 18, p. 663 (1883). See also 0. W. Richardson, The Electron Theory of Matter (Cambridge University Press, London, 1914) pp. 189-193 and R. B. McQuistan, Scalar and Vector Fields (Wiley, New York, 1965) pp. 292-305. 3. Causal relations in the domain of electromagnetic phenomena are analyzed in the author's book Causality, Electromagnetic Induction, and Gravitation, 2nd ed., (Electret Scientific, Star City, 2000). 4. The notation ly•.,•.r means "y', z', tare held constant." 5. This integral and the associated Poisson's equation are best known in connection with the magnetic vector potential produced by an electric current. See Ref. 1, pp. 363-364. 6. An alternative method of evaluating the Laplacian of the integral in Eq. (1-1.2) is to use spherical coordinates. See, for example, R. Becker and F. Sauter, Electromagnetic Fields and Interactions (Blaisdell, New York, 1964) pp. 280-281 or M. A. Heald and J. B. Marion, Classical Electromagnetic Radiation, 3rd ed., (Saunders, FortWorth, 1995)pp. 258-260. Observe, however, that this method is based on a presumed spherical symmetry of the integrand in Eq. (1-1.2), and is therefore of limited validity.

2

RETARDED INTEGRALS FOR ELECTROMAGNETIC FIELDS AND POTENTIALS

A basic problem in electromagnetic theory is the obtaining of equations expressing electric and magnetic fields and potentials in terms of their causative sources: electric charges and currents. In the case of time-dependent systems, the most general equations expressing electric and magnetic fields and potentials in terms of charges and currents involve retarded integrals. Electric and magnetic fields and potentials expressed in terms of retarded integrals are called retarded electric and magnetic fields and potentials. In this chapter we shall derive several types of equations for retarded fields and potentials of time-dependent charge and current distributions and shall give examples of the use of these equations.

2-1. Maxwell's Equations and the Wave Field Theorem

The basic electromagnetic field laws are represented by four Maxwell's equations which, in their differential form, are1

V·D = p

(2-1.1)

V·B = 0

(2-1.2)

15

16 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

VxE =

iJB
Tt

(2-1.3)

and

VxH = J + ao, at

(2-1.4)

where E is the electric field vector, D is the electric displacement vector, H is the magnetic field vector, B is the magnetic flux density vector, J is the electric current density vector, and p is the electric charge density. For fields in a vacuum (the only fields with which we shall be concerned in this book), Maxwell's equations are supplemented by the two constitutive equations

(2-1.5) and
(2-1.6)

where e0 is the permittivity of space and µ0 is the permeability of space. (The names and designations of electromagnetic quantities used in this book are the same as those used in Ref. 1.)
In Maxwell's equations electric and magnetic fields are linked together in an intricate manner, and neither field is explicitly represented in terms of its sources. However, with the help of the vector wave field theorem introduced in Section 1.1 we can express each field in terms of its causative sources. To do so, we shall first convert Eqs. (2-1.1) - (2-1.4) into two inhomogeneous wave equations, thereby separating the two fields one from the other.
Taking the curl of Eq. (2-1.3) and using Eq. (2-1.6), we have

VxVxE=-!VxB=-~!vxH. (2-1.7)
Eliminating V x H by means of Eq. (2-1.4) and using Eq. (2-
1.5), we obtain

SECTION 2-1 MAXWELL'S EQUATIONS

17

Rearranging terms and replacing BCJP-o by 1/c2, we finally obtain

a aJ VxVxE

+

1 2E
__
c 2 ot 2

=

-

µ0-ot.

(2-1.9)

Taking now the curl of Eq. (2-1.4) and using Eq. (2-1.5), we have

i i V XV XH

= V XJ

+

V XD at

= VXJ

+ e0 atV X E.

(2-1.10)

Eliminating V x E by means of Eq. (2-1.3) and using Eq. (21.6), we obtain

a a V X V X H = V XJ - 8 0- 2B = V XJ - e0µ0- 2H .

a1 2

a1 2

(2-1.11)

Rearranging terms and replacing e0µ0 by 1/c2, we finally obtain

V X V X H + _!_ 02H = V x J.
c 2 ot 2

(2-1.12)

Equations (2-1. 9) and (2-1.12) are the general electromagnetic wave equations for the electric and magnetic fields, respectively. Applying Eq. (1-1.2) (the vector wave field theorem) to Eqs. (21.9) and (2-1.12), we can write for the electric field

J[V'(V' • E) + µ0 oJ]

E = - _!_

ot dV' ,

411"

r

(2-1.13)

and for the magnetic field

18 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

J] [V' (V' • H) - V' x

H = - _l J-=---------=dV',

41r

r

(2-1.14)

where E and H are determined for the instant t, and the quantities in the brackets are taken at the corresponding retarded time t' t - rlc (c is the velocity of light in a vacuum).

2-2. Solution of Maxwell's Equations in Terms of Retarded Integrals

According to Eqs. (2-1.1) and (2-1.5), V • E = ple0, and

according to Eqs. (2-1.2) and (2-1.6), V • H = 0. Applying these

relations to Eqs (2-1.13) and (2-1.14) and noting thate0/Lo = 1/c2,

we obtain

[V'p + ~ aJ]

E = - _l_J
41re0

c2 ot dV'
r

(2-2.1)

and

H = _1 J[V' x J] dV' .

(2-2.2)

41r

r

Equations (2-2.1) and (2-2.2) constitute solutions of Maxwell's equations for fields in a vacuum and represent the electric and magnetic fields in terms of their causative sources: the electric charge and current distributions. 2 Since the fields in Eqs. (2-2.1) and (2-2.2) are expressed in terms of retarded integrals, these fields are called retarded fields.
There are several special forms into which Eqs. (2-2.1) and (2-2.2) can be transformed. One such special form is obtained from Eqs. (2-2.1) and (2-2.2) by eliminating from them the spatial derivatives. This can be done as follows.
Writing Eq. (2-2.1) in terms of two integrals and using vector identity (V-33) to transform the first integral, we have

SECTION 2-2 SOLUTION OF MAXWELL'S EQUATIONS 19

(2-2.3)

The second integral in the last expression can be transformed into a surface integral by means of vector identity (V-20). But this integral vanishes, because p is confined to a finite region of space, while the surface of integration is at infinity. Transforming the integrand in the first integral by means of vector identity (V-34)
and using ru = r/r, we then obtain for the electric field

J{[p] Jr r3 E = l 41re0

+

r l2c[ T0Pt]}r dV'

-

l
41re0c2

l [ T0Jt]dv 1• (2-2.4)

Similarly, applying vector identities (V-33) and (V-21) to Eq. (2-2.2), taking into account that there are no currents at infinity, and using vector identity (V-34), we obtain for the magnetic field

H = _!_J{[J] + _l raJ]}xrdV'. 411" r 3 r2c at

(2-2.5)

Observe that in Eqs. (2-2.4) and (2-2.5) the vector r is directed toward the point of observation (the field point).
Equation (2-2 .4) represents a generalization of the electrostatic Coulomb's field integral to time-dependent systems and reduces to that integral in the case of time-independent fields in a vacuum. Likewise, Eq. (2-2.5) represents a generalization of the BiotSavart's integral for magnetic fields and reduces to that integral in the case of time-independent systems.3
Another form of the field equation for E can be obtained as follows. According to the conservation of electric charge law (the continuity law),4

20 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

op = - V • J. 01

(2-2.6)

Therefore the contribution that op/at makes to the first integral in Eq. (2-2.4) can be expressed as

f f _l_[ 0P]rdV1 = - [V' • J] rdV'.

,~ fu

,~

(2-2.7)

Using now vector identity (V-30) with ru = r/r for transforming the last integral, and using vector identity (V-8), we obtain

Next, using vector identity (V-23), we transform the first term in the integrand of the last integral, obtaining

f f v ~ V' C

•

[J]dV1
,2

=

T,I;

~([J]
C ,2

• ds')

-

(CJ] • 1)~dV1 . (2-2.9)

,2

C

Since the integration is over all space, and since there is no current at infinity, the surface integral in Eq. (2-2.9) vanishes. Applying vector identity (V-4) to the integrand of the remaining integral in Eq. (2-2.9) and remembering that a V' operation upon r is the negative of the same V operation (see Chapter 1, p. 7), we then have

(2-2.10)

From Eqs. (2-2.7), (2-2.8), (2-2.9), and (2-2.10), we obtain therefore

SECTION 2-2 SOLUTION OF MAXWELL'S EQUATIONS 21

Substituting Eq. (2-2.11) into Eq. (2-2.4) and taking into account
that V'(l/r) = 2r/r', we finally obtain5

J E = _l_ [p] rdV'

41!"t:0 r3

(2-2.12)

[aJ]· [aJ]}dv' _

1 41l"t:0e

J{?[JJ

_

2 r

~ [J]•r

_

r r 3e

7ii

+ 1 r re ai

•

It is important to note that although in Eqs. (2-2.1)-(2-2.12) the charge density, the current density, and their derivatives are retarded, retardation can frequently be neglected, in which case the above equations can be used with ordinary (unretarded) charge density, current density, and their derivatives. Let us define the "characteristic time" of an electromagnetic system as the time T during which the charge density, the current density, or their temporal derivatives experience a significant change. For example, in the case of periodic charge and current variations, T may be assumed to be the period of the oscillations, and in the case of monotonously changing charges and currents, T may be assumed to be the time during which the charge density, the current density, or their temporal derivatives change by a factor of two. Let us now assume that the largest linear dimensions of the system under consideration is L. If T and L satisfy the relation

T ► Lie,

(2-2.13)

then no significant change occurs in the system during the time that the electric or magnetic field signal moves across the system, and therefore the retardation in the propagation of the electric or magnetic fields within the system is negligible. In Section 2.5 we shall discuss in some detail electromagnetic effects in systems to which Eq. (2-2.13) applies.

22 CHAPTER 2 RETARDED FIELDS AND POTENTIALS
T Example 2-2.1 A thin circular ring of radius a and cross-sectional
area s carries a uniformly distributed charge q. At t = 0 the ring
starts to rotate with constant angular acceleration a about its symmetry axis which is also the x axis of rectangular coordinates (Fig. 2.1). Find the electric and magnetic fields at a point x on the
axis for t > 0.

.

Fig. 2.1 Calculation of the electric and ma,gnetic fields

X on the axis of a charged

ring rotating with angular

acceleration a.

The current density J created by the rotating ring is J = pv = pwaOu = pataOu, where p is the charge density in the ring, w is the
angular velocity of the ring, and Ou is a unit vector in the circular direction (right-handed with respect to x). The time derivative of J
is {JJ/<Jt = paaOu. In terms of q, the current density and the derivative are J = (qat/21rs)Ou and {JJ/{Jt = (qa/21rs)Ou.
To find the electric field, we can use Eq. (2-2.4). Since {JJ/<Jt
is in the circular direction, and since r is the same for all points of
the ring, the second integral in Eq. (2-2.4) makes no contribution
to the electric field on the axis (the contributions of any two volume
elements on the opposite ends of a diameter cancel each other). Since the charge density does not depend on time, the contribution
of the first integral is

E = _l_f.!!....rdV', 47re0 r 3

(2-2.14)

which is identical with the expression for the electrostatic field produced by a stationary charge density p. The solution of Eq. (2-

SECTION 2-2 SOLUTION OF MAXWELL'S EQUATIONS 23

2.14) for a charged ring is well known,6 and therefore we shall reproduce it here without calculations. It is

E =

qx

i.

411"eo<a 2 +x 2)312

(2-2.15)

To find the magnetic field, we can use Eq. (2-2.5). Expressing [J] and [iJJ/ot] in Eq. (2-2.5) in terms of q, a, s, and Ou, we have

}x H = _l_ f {qa(t - rlc) (J + _!!!!_o rdV'

411"

211"sr 3 " r 2c211"s "

= _l_J{ qat (J -_!!!!_o +____!f!!_o }xrdV' (2-2.16) 411" 211"sr3 " r 2c211"s " r 2c211"s "
J{ }x = _l qat (J rdV'. 411" 211"Sr3 u

The current formed by the ring is filamentary. Its magnitude is

I = Js = qat/211". Since the current is filamentary, the volume

element dV' in Eq. (2-2.16) can be written as sdl', where di' is a

length element along the circumference of the ring. Furthermore,

we can combine Ou and dl' into the vector dl' = dl'(Ju· We then

have from Eq. (2-2.16)

H = - _l! !..r x dl', 4'71"J r 3

(2-2.17)

which is identical with the expression for the magnetic field produced by a time-independent filamentary current /. The solution of Eq. (2-2.17) for a ring current is well known. 7 It is

H =

1a2 i

2(a2 + x2)312 ,

or, substituting I = qat/211",

H

=

qata 2 •
----,,-...,,.-,,.,,.I.

411"(a2 + x2)312

(2-2.18) (2-2.19)

The surprising result of this example is that neither the electric nor the magnetic field on the axis of the rotating ring is affected by retardation.

24 CHAPTER 2 RETARDED FIELDS AND POTENTIALS
Example 2-2.2 Electromagnetic waves can be generated by a radiating "electric dipole antenna." It consists of a piece of straight open wire which carries a current
(2-2.20)
The current in the wire is produced by cutting the wire in the middle and connecting the two parts to a source of alternating current. If the length l of the antenna is much smaller than the
wavelength of the generated waves, l ~ >.. = 21rclw, the antenna is
called a "Hertzian dipole." In a Hertzian dipole the current is the same along the entire length of the antenna. Find the magnetic and electric fields produced by the Hertzian dipole shown in Fig. 2.2, at a large distance r ► l from the dipole.

r >> I

Fig. 2.2 Calculation of the electric and magnetic fields generated by an electric dipole antenna. (The unit vector <Pu is directed into the page.)

To find the magnetic field, we can use Eq. (2-2.5). Since the current in the antenna is filamentary, we can replace the volume
integral in this equation by a line integral (note that for a
filamentary current JdV' = Jsdl' = ldl', where s is the cross-
section area of the conductor, and di' is a length element vector in the direction of J). Furthermore, since the antenna is along the z axis, we can write Eq. (2-2.5) as

H = _l J{[l] + _ 1 [01]}kxrdl 1 . 41r r3 r 2c ot

(2-2.21)

SECTION 2-2 SOLUTION OF MAXWELL'S EQUATIONS 25

Differentiating Eq. (2-2.20), replacing tin Eq. (2-2.20) and in its derivative by the retarded time t - r!c, and substituting the resulting expressions in Eq. (2-2.21), we then have

J{/ 1
H =-

0 sinw(t - r/c)
-----

+ - 10w-cos-w(t-- r- /c)}k

X

, rd/ .

(2-2.22)

4r

r3

r~

Since, by supposition, r ► A = 2rc/w, the first term in this
integral is much smaller than the second term and can be neglected. Since r ► l, r may be considered the same for all points of the antenna. The integral reduces therefore to the product of the second integrand and the length of the antenna

H = _ / 0 w_cos_ w(t_- r_ /c) kxr/, 4rr2c
or, in terms of the coordinates shown in Fig. 2.2,

(2-2.23)

IaZwcosw(t - rlc)
H = ---,-----sinOcpu. 4rrc

(2-2.24)

To find the electric field, we can use Eq. (2-2.12). Since we are only interested in the electric field at a large distance from the antenna, we can neglect in Eq. (2-2.12) all terms that approach zero at infinity faster than as 1/r. We then have

J{ [aJ]· _r E = 1 4re c2

r r3

at

r

1 [aaJt ]}dv' '

0

which we can write similar to Eq. (2-2.22) as

(2-2.25)

E = _ 1_ J{r(k • r) /0wcosw(t - r!c) - ~/0wcosw(t - r!c)}dt'.

4re c2

r3

r

0

(2-2.26)

Taking into account that k • r = r cos O, and replacing the

integral, as before, by the product of the integrand and the length

of the antenna, we obtain

E = l/0 wcosw(t - r/c)(rcosO _ k).

4re0rc 2

r

(2-2.27)

26 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

Resolving ru and Ou shown in Fig. 2.2 into components along the z and x axes, we can easily find that

rcorsO

-

k

= r cosO u

-

k

= sinOOu.

Therefore we finally have

(2-2.28)

E = l/0 wcosw(t - rlc) sinOO .

47re0rc 2

u

(2-2.29)

An alternative method for obtaining Eq. (2-2.29) is to apply Maxwell's Eq. (2-1.4) to Eq. (2-2.24) and to integrate the result with respect to t. 8

Example 2-2.3 Another system capable of generating electromagnetic waves is the radiating "magnetic dipole antenna," shown in Fig. 2.3. It consists of a circular loop of wire carrying a current
(2-2.30)
Assuming that the radius of the loop is a, find the electric and magnetic fields produced by this antenna at a large distance r ► A
= 271'c/w ► a from it.

Fig. 2.3 Calculation of the electric and magnetic fields generated by a magnetic dipole antenna. (The unit
vector 'Pu is directed into the
page.)
We shall find the electric field produced by the antenna by using Eq. (2-2.4). Assuming that the antenna has no net charge, we only need to consider the second integral in this equation. Since the

SECTION 2-2 SOLUTION OF MAXWELL'S EQUATIONS 27

current in the antenna is filamentary, the volume integral can be

replaced by a line integral (see Example 2-2.2). Differentiating then

Eq. (2-2.30) and replacing tin the derivative by t - rlc, we can

f / write Eq. (2-2.4) as

E

=

-

1
--

_ 0 w_ cos_ w(t_ - r_ lc) dJ1 '

41re0c 2

r

(2-2.31)

where di' has the same direction as the current in the loop. Transforming the integral in Eq. (2-2.31) by means of vector identity (V-18), factoring out the constants, and using vector identity (V-25), we have

J E = _ ~ dS' x V' cosw(t - rlc)

41reoc2

r

(2-2.32)

J{- +

~
41re0c 2

~sinw(t - rlc) + .2..cosw(t - r/c)}r x dS'.

re

r2

"

But w/c = 21r/X and, by the statement of the problem, r ► X.

Therefore the second term in the last integral may be neglected, and

we obtain

J E = _ /ow2 sinw(t - r/c) r X dS'.

41re0c 3

r

"

(2-2.33)

Now, since r ► a, we can replace the integral by the product of the integrand and the surface area of the antenna, so that

E = - -/-ow-2 -s-in-w(-t --rrlc)u X k 1ra2'

41re0c 3

r

(2-2.34)

or

E

=

-/ 0-w2-a-2 s-in-ws(tm - rl

c)
0

.
<

f

,

.

4e0c 3r

"

(2-2.35)

The magnetic field can be determined from Eq. (2-2.5). Since we are only interested in the magnetic field at a large distance from the antenna, we can neglect in Eq. (2-2.5) the first term in the integrand (it is proportional to 1/r2, and for large r is negligible

28 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

compared with the second term, which is proportional to 1/r). We

then have, replacing as before volume integration by line

integration,

f H

=

-

1
_

-I0-w-c-o-sw-(,t---r-lc-) r x di 1 .

411"

r 2c

(2-2.36)

Since r ► a, r may be considered the same at all points of the

antenna, and therefore we may factor out r/r, obtaining

f H = - - -1r X _ 10 w_co_ sw_ (t -_rl_ c) dJ1.

41l"cr

r

(2-2.37)

But the integral in Eq. (2-2.37) is the same as in Eq. (2-2.31) for E. By Eqs. (2-2.37) and (2-2.31)-(2-2.35), we then have

x H

_
-

I0w2a 2 sinw(t - rlc)
-----,,----Si

.
ll

8 r

,1,,
'l'u·

4c 2

r2

(2-2.38)

or

H

(2-2.39)

A

2-3. Surface Integrals for Retarded Electric and Magnetic Fields
A remarkable feature of Eqs. (2-2.1) and (2-2.2) is that they correlate the electric field with the gradient of the charge distribution and correlate the magnetic field with the curl of the current distribution rather than with the charge and current distribution as such. Hence, the equations may be interpreted as indicating that the electric and magnetic fields are associated not with electric charges and currents, but rather with the inhomogeneities in the distribution of charges and currents (a homogeneous, or uniform, charge distribution has zero gradient, and a homogeneous, or uniform, current distribution has zero curl).

SECTION 2-3 RETARDED SURFACE INTEGRALS

29

A frequently encountered charge or current distribution is a

distribution in which the charge or current changes abruptly from

a finite value in the interior of the distribution to zero outside the

distribution. For this type of charge and current distribution, Eqs.

(2-2.1) and (2-2.2) can be transformed into special forms that are

more convenient to use than Eqs. (2-2.1) and (2-2.2) themselves.

Consider first Eq. (2-2.1). In this equation the part of the

integral involving Vp can be separated into two integrals: the

integral over the boundary layer of the charge distribution under

consideration and the integral over the interior of the charge

distribution:

_J -f _f _ 1 [V'p]dV'=-1

[V'p]dV1 +_1 [V'p]dV'. (2-3.1)

41re0 r

41re0 s.1. r

41re0 Inr r

The first integral on the right of Eq. (2-3.1) can be transformed by using vector identity (V-33):

_J -J -J _ 1

[V'p]dV' = -1 V [p]dV' +_1 V' [p]dV'.

41re0 s.1. r

41re0 s.1. r

41re0 s 1. r (2_3_2)

In Eq. (2-3.2), V in the first integral on the right operates upon the field point coordinates only. Therefore it can be factored out from under the integral sign. The integrand in this integral will then be [p]/r. Since both [p] and rare finite, while the integration is over the volume of the boundary layer whose thickness, and therefore volume, can be assumed to be as small as we please, the integral vanishes. The second integral on the right of Eq. (2-3.2) can be transformed into a surface integral by using vector identity (V-20). Equation (2-3.2) can be written therefore as

J _ 1_

[V'p] dV' = _l_J [p] dS', (2-3.3)

r 41re0 B.layer r

41re0 B.layer r

where the surface integral is extended over both surfaces (exterior and interior) of the boundary layer.

30 CHAPTER 2 RETARDED FIELDS AND POTENTIALS
In Eq. (2-3.3), dS' of the exterior surface is directed into the space outside the charge distribution, while dS' of the interior surface is directed into the charge distribution. However, since there is no charge outside the charge distribution, the integral over the exterior surface vanishes. Since the boundary layer can be made as thin as we please, we can make the interior surface of the boundary layer coincide with the surface of the charge distribution. Reversing the sign in front of the surface integral, we can write then Eq. (2-3.3) as

_1_ J [V' p] dV' = - _l_f

[p] dS''

41re0 B.layer f

41re0 Boundary f

(2-3.4)

where the integration is now over the surface of the charge distribution, and where the surface element vector dS' is directed, as usual, from the charge distribution into the surrounding space.
From Eqs. (2-2.1), (2-3.1), and (2-3.4) we obtain

r -J r E =_l_l [p]dS' --1 [V'p]dV' - _1_J_!_[aJLv,.

41re 0

Boundary

f

411"e 0

Int

r

411"e c 2 r at 0

(2-3.5)

This equation becomes especially simple in the case of a constant

(uniform) charge distribution surrounded by a free space. In this

case Vp in the interior of the distribution is zero, and Eq. (2-3. 5)

simplifies to

r -J E = _l_l [p] dS' - - 1 .!.[aJ]dv'.

41re0 Boundary f

41re0c 2 f at

(2-3.6)

Consider now Eq. (2-2.2). Just as in the case of Eq. (2-2.1), we can separate the integral in Eq. (2-2.2) into an integral over the boundary layer of the current distribution and an integral over the interior of the distribution. By the same reasoning as that used to simplify Eq. (2-3.2), we find that the integral over the boundary layer can be written as

SECTION 2-3 RETARDED SURFACE INTEGRALS

31

_If [V' x J] dV' = _If V' x [J] dV'.

411" B.layer r

411" B.layer

r

(2-3.7)

Transforming the integral on the right of Eq. (2-3.7) into a surface integral by means of vector identity (V-21), and taking into account that there is no current in the space outside the current distribution, we obtain, just as we obtained Eq. (2-3.4),

T _If [V' x J] dV' = _I J,

[J] x dS',

411" B. layer r

r 411" Boundary

(2-3.8)

where the integration is over the surface of the current distribution, and the surface element vector dS' is directed from the current distribution into the surrounding space.
Equation (2-2.2) can be written therefore as

r H = _I J,

[J] x dS' + _If [V' x J] dV'. (2-3.9)

r 471" Boundary

411" lntenor r

For the special case of V x J = 0 in the interior of the current
distribution, Eq. (2-3.9) simplifies to

r H = _I J,

[J] x dS'.

r 471" Boundary

(2-3.10)

~
Exam.pie 2-3.1 A thin, uniformly charged disk of charge density p, radius a, and thickness b rotates with constant angular acceleration a about its axis, which is also the x axis of rectangular coordinates. The midplane of the disk coincides with the yz plane of the coordinates, and the rotation of the disk is right-handed relative to thex axis (Fig. 2.4). Using Eqs. (2-3.6) and (2-3.9), find the electric and magnetic fields produced by the disk at a point of
the x axis, if at t = 0 the angular velocity of the disk is w = 0. The disk creates a convection current J = pv = pwR()u =
patROu, where R is the distance from the center of the disk, and (Ju is a unit vector in the circular direction (right-handed with respect
to a). The time derivative of J is {JJ/ot = paR()u· To find V' x J,

32 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

a., 6) E,H X

Fig. 2.4 Calculation of the electric and magnetic fields on the axis of a charged disk rotating with constant angular acceleration a.

we use the relation v = w x Rand vector identity (V-12). Taking into account that w is not a function of coordinates, we then obtain

V' xJ = V' x (pw x R) = p[w(V' • R) -(w • V')R], (2-3.11)

and since R = y'j + z'k, while w • V' = wiJ/iJx', we have

V' x J = 2pw = 2pat = 2pcxti.

(2-3.12)

Examining now Eq. (2-3.6) and taking into account that iJJ/iJt is in the circular direction, we recognize that the second integral in Eq. (2-3.6) vanishes by symmetry (see Example 2-2.1). And since p does not depend on time, we see from Eq. (2-3.6) that the electric field of the disk is the ordinary electrostatic field given by

r E = _l_f

!!..dS' = _P_!

dS'. (2-3.13)

411"Bo r Boundary

47re0 Boundary r

Let us now evaluate the surface integral in Eq. (2-3.13). By the
symmetry of the system, only the two flat surfaces of the disk
contribute to the field on the axis. The back surface is located at x'
= - b/2, the front surface is located at x' = + b/2. The direction of the surface element vector dS' is - i for the back surface and +
i for the front surface. We have therefore

Ia E = _ pi

2'lTRdR

+ pi Ia 2'lTRdR

° ° 47re0 [R 2 +(x +b/2)2] 112 47re0 [R 2 +(x-b/2)2)112

= -~{[a 2 +(x +b/2)2]1!2 -(x +b/2) -[a 2 +(x -b/2)2]112 +(x -b/2)}.

2eo

(2-3.14)

SECTION 2-3 RETARDED SURFACE INTEGRALS

33

Since b <E1 x, we can use the relation
± [a 2+(x b/2)2]112 =[a 2+x 2±xb]112 =(a 2+x 2)112[1 ±xb/2(a2+x 2)].
(2-3.15) Substituting Eq. (2-3.15) into Eq. (2-3.14), we obtain after elementary simplifications

E = pb[l -

x ]•

2eo

(a2+x2)1,2 I.

(2-3.16)

To find the magnetic field, we use Eq. (2-3.9). Substituting [J]
= paR(t- rlc)fJu and [V' x J] = 2pa(t - rlc)i into Eq. (2-3.9), we
have

J H =_1_1

pa.R(t-rlc) fJuxdS' + _!_ 2pa(t-r/c) dV'.

r 471' Boundary

T

471' Int

T

(2-3.17)

By the symmetry of the system, only the curved surface of the disk
contributes to the first integral. At this surface R = a, r = (a2 + x2)112, (Ju x dS' = - i dS', and the surface itself is S' = 21!'ab. In the second integral r is r = (R2 + x2)112 and the volume element is dV' = b271'RdR. The magnetic field is therefore

H = -. pIa-a-[t--(-a:2,+-x 2-)1~12/c=2]2-71-'a-b +ip_a Ja-t---(--R=-2'=+"x"2")"1"1,2-/-c2271'bRdR

471'(a2 +x2)112

271' o (R2 +x2)112

= -i pata 2b +ipaa 2b +ipatb(a2+x2)v2_ipaa2b (2-3.18)

2(a 2+x 2)112 2c

2c

or (2-3.19)

It is interesting to note that neither the electric nor the magnetic field of the rotating disk is retarded, just as was the case with the fields of the rotating ring discussed in Example 2-2.1 (see, however, Example 2-4.2).

34 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

2-4. Retarded Potentials for Electric and Magnetic Fields

The calculation of time-dependent electric and magnetic fields
can sometimes be simplified by using retarded electromo,gnetic
potentials. For the calculation of magnetic fields in a vacuum it is
convenient to use the potentials defined in Corollary I of Section
1-1. Substituting in Eqs. (1-1.3), (1-1.4), and (1-1.5) V = B, V
• V = V • B = 0, K1 = 0, and K2 = µc} [because by Eqs (2-
1.12) and (1-1.1) K = V x Jin the wave equation for H, so that
K = µ0V x J in the wave equations for B = µoli], and leaving out, as usual, cp0 and A0, we have

where

B = VxA,
A = µo J [J] dV' .
41r r

(2-4.1) (2-4.2)

If the current is filamentary, this equation reduces to

A = ~J [J] di',
41rT r

(2-4.3)

where di' is a length element vector in the direction of the current.
For the calculation of electric fields in a vacuum 1t 1s convenient to use the potentials defined in Corollary II of Section
1-1. Substituting in Eqs. (1-1.6), (1-1.7), and (1-1.8) V = E, V • E = V • (D/eo) = p/e0, K = - µofJJ/ot [see Eq. (2-1.9)], and
leaving out cp0 and W0, we have

where while

E = - Vcp + W,
-J cp = - 1 [p] dV' 41re0 r

(2-4.4) (2-4.5)

W = - µo J _!_[oJ]dv'.
411" r ot

(2-4.6)

SECTION 2-4 RETARDED POTENTIALS

35

Using Eq. (1-2.14) and taking into account that the integration in Eq. (2-4.6) involves space coordinates only, we can factor out a/at from under the integral sign, obtaining

- !:!_ J_!_[aJ]dV' = - !._{!:!_ J [J] dV'}. (2-4.7)

41r r at

at 471" r

Therefore, according to Eq. (2-4.2), Eq. (2-4.4) can be written as

E = - V'{) - 00~,

(2-4.8)

where A is the retarded magnetic vector potential given by Eq. (2-

4.2) or Eq. (2-4.3).

The potentials'{) and A given by Eqs. (2-4.5) and (2-4.2) are

the retarded electromagnetic potentials. They represent a

generalization of the ordinary electric and magnetic potentials '{)

and A and reduce to them in the case of time-independent fields

in a vacuum. 9
...

Example 2-4.1 Show that the retarded potentials '{) and A satisfy

Lorenz's condition

V • A

-
-

-

e
0

µ

0O7'f{t) "

(2-4.9)

From Eqs. (2-4.5) and (1-2.14) we have

![ - e µ O'{) = - !:: J!._ [p] dV' = - !:: J 0P]dv'.

0 0 at

411" at r

41r r at

But according to the continuity law, Eq. (2-2.6),

(2-4.10)

so that

- [:] = [V' • J]'

_

O'{)

eoµocii

= µo J [V' • .J]dV'

41r r

•

(2-4.11) (2-4.12)

Transforming the integral in Eq. (2-4.12) by means of vector identity (V-27), we have

36 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

(2-4.13)

The last integral can be transformed into a surface integral by

means of the vector identity (V-19), and since there is no current at

infinity, the surface integral is zero, and so is the last integral. In

the first integral, V can be factored out from under the integral

sign. Therefore we obtain

f -

e_u
ur-o

fJlJctp

= V

• !:5!.. 411"

[Jr] dV'.

(2-4.14)

Eliminating the last integral in Eq. (2-4.14) by means of Eq. (24.2), we obtain Lorenz's condition.

Example 2-4.2 Find the electric and magnetic fields at all points of space far from the rotating ring described in Example 2-2.1 (Fig. 2.5).

cl>u/·E,B
q// r
;o~ ~ a, S' x
r>>a

Fig. 2.5 Calculation of the electric and magnetic fields far from the charged ring rotating with constant angular acceleration. (The unit vector </," is directed into the page.)

At large distances from the ring, the ring constitutes a point charge q, which does not depend on time. Therefore the electric potential of the ring is the ordinary electrostatic potential
(2-4.15)

Since the ring constitutes a convection line current / = qat/21r, the magnetic vector potential of the ring is, by Eq. (2-4.3),

SECTION 2-4 RETARDED POTENTIALS

37

r r r A = ~ ! qa(t-rlc)/21r dl' = qatµo! dl' _ qaµo!dl,. (2_4 _16)

47r

r

81r2

r 81r2c

The last integral on the right of Eq. (2-4.16) is zero. The remaining

integral can be transformed into a surface integral by means of

vector identity (V-18). We then obtain

f f A -_ - qat- µ0 -dJ-.' -_ - qat-µ0 _r:xdS,,

81r2 r

81r2 r2

(2-4.17)

where r'u is a unit vector directed from the point of observation toward the surface element dS'.
Now, since the point of observation is far from the ring, the integral can be replaced by the (vector) product of the integrand and the surface area S' of the ring, so that the vector potential is

A = -qa-tµrou, X S' = - -qa-tµor X S' ,

81r2r2

81r2r2 u

(2-4.18)

where ru is a unit vector directed from the ring toward the point of observation. The magnitude of the vector S' is 1ra2, and the direction is along the x axis. Designating the angle between ru and S' as 8, we then have for the vector potential

A

=

-

-q-a-2a-t,µ,o-

sm.

8,1,.
'I'

,

81rr2

u

(2-4.19)

where cpu is a unit vector in the circular direction left-handed

relative to the x axis.

By Eq. (2-4.1), the magnetic flux density field associated with

this vector potential is

B =V x A

qa 2atµ

81rr3

0 (2cos8r
u

+

sin80 )
u

(2-4.20)

(we do not reproduce the actual calculation of V x A, since it is not important for the purpose of the present example; the calculation is done by using the expressions for the curl of a vector in spherical coordinates'°). It is interesting to note that this field is

38 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

an ordinary (unretarded) field of a current dipole, 11 and that on the
x axis (0 = 0) it reduces to the field found in Example 2-2.1 (for
X ~ a).
Let us now find the electric field of the ring. By Eq. (2-4.8),
(2-4.15), and (2-4.19), we have

E = - -q- r

qa 2cxµ + ___osinOq,

41re ,2 u 81rr2

u

0

(2-4.21)

(2-4.22)

It is interesting to note that although the electric field of the
ring does not depend on t, the presence of the 'Pu term makes the
field different from the electrostatic field of the ring. This term represents the contribution of [oJ/ot] in Eq. (2-2.1) and represents the "electrokinetic field" (see Section 2-5). In the case under consideration, the electrokinetic field is circular and is directed opposite to the current in the ring.
On the x axis, the electric field of the ring reduces to the field found in Example 2-2.1.

2-5. Electromagnetic Induction
Electromagnetic induction is frequently explained as a phenomenon in which a changing magnetic field produces an electric field ("Faraday induction") and a changing electric field produces a magnetic field ("Maxwell induction").
A detailed examination of the causal relations in timedependent electric and magnetic fields shows, however, that neither of the two fields can create the other. 12 The causal equations for electric and magnetic fields in a vacuum are the retarded field equations discussed in Sections 2-2 and 2-3.

SECTION 2-5 ELECTROMAGNETIC INDUCTION

39

According to Eqs. (2-2.1), (2-2.2), (2-2.4), (2-2.5), and (22.12), in time-variable systems electric and magnetic fields are always created simultaneously, because they have a common causative source: the changing electric current {)J/at. Once created, the two fields coexist from then on without any effect upon each other. Therefore electromagnetic induction as a phenomenon in which one of the fields creates the other is an illusion. The illusion of the "mutual creation" arises from the fact that in time-dependent systems the two fields always appear prominently together, while their causative sources (the timevariable current in particular) remain in the background.
As can be seen from Eq. (2-2.1) or from Eq. (2-2.4), a timevariable electric current creates an electric field parallel to that current (parallel to [aJ/at]). This field exerts an electric force on the charges in nearby conductors thereby creating induced electric currents in the conductors. Thus, the term "electromagnetic induction" is actually a misnomer, since no magnetic effect is involved in the phenomenon, and since the induced current is caused solely by the time-variable electric current and by the electric field produced by that current.
The electric field produced by a time-variable current differs in two important respects from the ordinary electric field produced by electric charges at rest: first, the field produced by a current is directed along the current rather than along a radius vector, and second, the field exists only as long as the current is changing in time. Therefore the electric force caused by this field is also different from the ordinary electric (electrostatic) force: it is directed along the current and it lasts only as long as the current is changing. Unlike the electrostatic force, which is always an attraction or repulsion between electric charges, the electric force due to a time-variable current is a dragging force: it causes electric charges to move parallel (or anti-parallel) relative to the direction of the current. If the time-variable current is a convection current, then the force that this current exerts on

40 CHAPTER 2 RETARDED FIELDS AND POTENTIALS
neighboring charges causes them to move parallel to the convection current, rather than toward or away from the charges forming the convection current [the total force is, of course, given by all the terms in Eq. (2-2.1) or Eq. (2-2.4)].
Since the electric field created by time-variable currents is very different from all other fields encountered in electromagnetic phenomena, a special name should be given to it. Taking into account that the cause of this field is a motion of electric charges (current), we may call it the electrokinetic field, and we may call the force which this field exerts on an electric charge the electrokinetic force. 13 Of course, we could simply call this field the "induced field." However, such a name would not reflect the special nature and properties of this field.
Let us designate the electrokinetic field by the vector Ek. From Eq. (2-2.4) we thus have
(2-5.1)
The electrokinetic field provides a precise and clear explanation of one of the most remarkable properties of electromagnetic induction: Lenz's law. Consider a straight current-carrying conductor parallel to another conductor. According to Lenz's law, the current induced in the second conductor is opposite to the inducing current in the first conductor when the inducing current is increasing, and is in the same direction as the inducing current when the inducing current is decreasing. In the past no convincing explanation of this effect was known. But the electrokinetic field provides the definitive explanation of Lenz's law: by Eq. (2-5.1), the sign (direction) of the electrokinetic field is opposite to the sign of the time derivative of the inducing current. When the derivative is positive, the electrokinetic field is opposite to the inducing current; when the derivative is negative, the electrokinetic field is in the same direction as the inducing

SECTION 2-5 ELECTROMAGNETIC INDUCTION

41

current. Since the induced current is caused by the electrokinetic field, the direction of this field determines the direction of the induced current: opposite to the inducing current when that current increases (positive derivative), the same as the inducing current when the inducing current decreases (negative derivative).
Of course, since the direction of the inducing current usually varies from point to point in space, the ultimate direction of the electrokinetic field and of the current that it produces is determined, in general, by the combined effect of all the current elements of the inducing current in the integral of Eq. (2-5.1).
The electrokinetic field also gives a simple explanation of the fact (first noted by Faraday) that the strongest induced current is produced between parallel conductors, whereas no induction takes place between conductors at right angles to each other. This phenomenon is now easily understood from the fact that the electrokinetic field due to a straight conductor carrying an inducing current is always parallel to the conductor.
Although we have been discussing the electrokinetic field as the cause of induced currents in conductors, its significance is much more general. This field can exist anywhere in space and can manifest itself as a pure force field by its action on free electric charges. Of course, because of the c2 in the denominator in Eq. (2-5.1), the electrokinetic field cannot be particularly strong except when the current changes very fast. This is probably the main reason why this field was ignored in the past. Another reason is the temporal (transient) nature of this field.
But even weak electric fields can produce strong currents in conductors, and that is why the current-producing effect of the electrokinetic field is much more prominent than its force effect on electric charges in free space.
If we compare Eq. (2-5.1) with Eq. (2-4.2) for the retarded magnetic vector potential A produced by a current J, we recognize that the electrokinetic field is equal to the negative time derivative of A (observe that µ0 = l/t:02):

42 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

oA Ek = - -ot·

(2-5.2)

However, although Eq. (2-5.2) correlates the electrokinetic field with the magnetic vector potential, there is no causal link between the two: the correlation merely reflects the fact that both the electrokinetic field and the magnetic vector potential are simultaneously caused by the same electric current.
Important as it is, the electrokinetic field has not been studied (or even recognized as a special force field) until very recently, although the fact that the time derivative of the retarded vector potential is associated with an electric field has been known for a long time.
Electromagnetic induction is a phenomenon associated with relatively slow current variations and with electromagnetic fields extending over relatively small regions of space (rapid current variations and time-variable fields extending over long distances are dealt with on the basis of radiation theory; see Examples 2-2.2 and 2-2.3). More specifically, electromagnetic induction applies to systems satisfying Eq. (2-2.13). Therefore, as far as electromagnetic induction is concerned, the retardation in the propagation of the electric field from the inducing current to the conductor in which the induced current is created can be ignored. Removing the retardation symbol [] in Eq. (2-5.1) and factoring out o/ot, we then obtain for the electrokinetic field

(2-5.3)
...
Example 2-5.1 A conducting circular ring of radius R is placed outside a long coaxial solenoid of n turns, radius a and length L, carrying a current / (Fig. 2.6). Using Eq. (2-5.3) find the electrokinetic field and then the voltage induced in the ring when the current in the solenoid is changing. Observe that according to

SECTION 2-5 ELECTROMAGNETIC INDUCTION

43

the conventional explanation of electromagnetic induction, the voltage and the current in the ring is induced by the changing magnetic field at the location of the ring. But this explanation does not work in the present case, because there is no magnetic field at the location of the ring (except for the end-effect field of the solenoid, which is negligible).

tY

L>>R

r

R{\f

•

I I

.

X

Fig. 2.6 Calculation of the voltage induced in a conducting ring placed outside a solenoid carrying a variable current.
Let the axis of the solenoid be the x axis of a rectangular system of coordinates, let the ring be in the yz plane, and let the ends of the solenoid be at x = - L/2 and x = L/2. To find the electrokinetic field induced by the solenoid in the ring, we shall consider a point of the ring located on they axis. We can represent this point by the vector Rj. Consider next a point on the surface of the solenoid at a distance x from the yz plane. Combining cylindrical and rectangular coordinates, we can represent that point
by the vector b = xi + a cos0j + a sin0k. The distance between the two points is then r = Rj - b = - xi + (R - a cos0)j -
a sin0k, so that for r in Eq. (2-5.3) we have, by adding the squares
of the components of rand taking the square root of the sum, r =
(x2 + R2 + a2 - 2Ra cos0) 1'2. The current density in the solenoid
can be written as J = (nl/Lw)Ou = (nl/Lw)(- sin0j + cos0k),

44 CHAPTER 2 RETARDED FIELDS AND POTENTIALS

where I is the current in the solenoid, w is the thickness of the
current sheet, and Ou is a unit vector in the direction of the current. The volume element to be used in Eq. (2-5.3) can be written as dV'
= wad8dx.
By the symmetry of the system, the contribution of the y
component of J to the electrokinetic field is zero. Equation (2-5.3)
becomes therefore (we replace lleoi2 by µ0)

a( Ek= - _ ~ J 2.-Ju2

nI cos Ok

wad8dx), (2-5.4)

at 411" 0 -L/2 Lw(R 2 +a 2 -2Racos8 + x 2) 112

(2-5.5)

Integrating by parts over 8, we obtain

a( Ek= - - k µOn/Ra2f2"fu2

sm• 20

d8dx). (2-5.6)

at 47rL O -u2 (R2 +a 2 -2Racos8 +x 2)3'2

Integrating over x and taking into account that L ;!!> R, a, we

obtain

f !...(k Ek = -

µon/Ra2 2ir

sin20

do). (2-5.7)

at 27rL o (R2 +a 2 -2Racos8)

The integral in Eq. (2-5.7) is just14 7r/R2. The electrokinetic field generated at the point ~ of the ring by the current in the solenoid is therefore (replacing k by Ou)

E = - !...(o µonla2)

8

at u 2RL ,

(2-5.8)

and the voltage induced in the ring is

... (2-5.9)

REFERENCES AND REMARKS FOR CHAPTER 2

45

References and Remarks for Chapter 2

1. See, for example, Oleg D. Jefimenko, Electricity and Magnetism, 2nd ed., (Electret Scientific, Star City, 1989) pp. 500-503. 2. These equations constitute time-dependent counterparts of the corresponding time-independent equations. See Ref. 1, pp. 101-103 and 350-352. 3. Although we have derived Eqs. (2-2.4) and (2-2.5) from equations for electromagnetic waves, they are fundamental electromagnetic equations of general validity. In fact, Maxwell's equations and, hence, the entire Maxwellian electromagnetic field theory can be derived from them (in conjunction with Eq. 2-2.6). See Oleg D. Jefimenko, "Presenting electromagnetic theory in accordance with the principle of causality," Eur. J. Phys. 25, 287-296 (2004). 4. See, for example, Ref. 1, p. 497. 5. A similar equation is derived in W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed., (AddisonWesley, Reading, 1962) p. 248. 6. See, for example, Ref. 1, pp. 99-100. 7. See, for example, Ref. 1, pp. 346-347. 8. For a more detailed solution see Ref. 1, pp. 559-562. 9. Equations (2-4.5) and (2-4.2) were first obtained in 1867 by L. Lorenz. See E. T. Whittaker, A History of the Theories of Aether and Electricity (Thomas Nelson, London, 1953) Vol. I, Chapt. 8 ("Maxwell") pp. 267-268. 10. See, for example, Ref. 1, p. 55. 11. See, for example, Ref. 1, pp. 380-382. 12. See Oleg D. Jefimenko, Causality, Electromagnetic Induction, and Gravitation, 2nd Ed. (Electret Scientific, Star City, 2000) pp. 3-18. 13. The term "electrokinetic" is also used in reference to phenomena associated with the movement of charged particles through a continuous medium or with the movement of a continuous medium over a charged surface. These phenomena have no connection with the electrokinetic field discussed in this book. 14. It is best evaluated by using a computer program for symbolic integration.

3
RETARDED INTEGRALS FOR ELECTRIC AND MAGNETIC FIELDS AND POTENTIALS OF MOVING CHARGES
In this chapter we shall learn how retarded integrals for electric and magnetic fields and potentials can be used for finding electric and magnetic fields and potentials of moving electric charge distributions. We shall also discover important relations between the electric and magnetic fields for two special cases of moving charge distributions: an arbitrary charge distribution moving with constant velocity and a point charge in arbitrary motion.
3-1. Using Retarded Integrals for Finding Electric and Magnetic Fields and Potentials of Moving Charge Distributions
A time-variable electric charge distribution always involves a movement of electric charges. For example, if the density of a charge distribution changes with time, then some electric charges change their location within the charge distribution or move to or from the charge distribution. Conversely, a moving charge distribution is inevitably a time-variable charge distribution because it creates charge density in regions of space which it
46

SECTION 3-1 MOVING CHARGE DISTRIBUTIONS

47

enters and eliminates charge density from the regions of space which it leaves. Consequently, the electric and magnetic fields of a moving charge distribution can be determined from retarded field (or retarded potential) equations derived in Chapter 2 for the general case of time-dependent charge and current distributions.
To use retarded field integrals for finding electric and magnetic fields of moving charge distributions, we need to express the time derivatives ap!iJt and aJ/at in terms of the velocity of the charge distribution under consideration. This can be done as follows. Consider a stationary charge distribution of density p as
a function of x', y', z',

p = p(x',y',z').

(3-1.1)

If this charge distribution moves with velocity v without changing its density, the total time derivative of p is

dp=ap+ ap dx' + ap dy' + ap dz.' =ap+v•V'p.

dt at ax' dt ay' dt az' dt iJt

(3-1.2)

Since p remains the same as the charge moves, dp/dt = 0, so that

ap = - v • V 1p. at

(3-1.3)

A moving charge distribution constitutes a current whose density
is J = pv. Therefore

v. aJ = a(p v) = -(v . V'p) v + p av = -(v • V'p) v + p (3-1.4)

at at

at

Observe that in the retarded field integrals derived in Chapter 2, the denominator r representing the distance between the volume element dV' and the point of observation is not a function of time. Therefore it is not a function of time also in the case of moving charge distributions. A moving charge distribution must be considered as moving past different volume elements of space associated with different but fixed r's. The question arises, if dV'

48

CHAPTER 3 FIELDS OF MOVING CHARGES

is a volume element of space, rather than a volume element of a moving charge distribution, how does one introduce the volume of the charge distribution into the field integrals? To answer this question, let us examine how the electric and magnetic fields of a moving charge distribution are created.
The phenomenon of retardation indicates that time-dependent charge distributions send out electric (and magnetic) field "signals" that propagate in all directions with the velocity of light. The electric or magnetic field created by a time-variable charge distribution at the point of observation is the result of the signals sent out by all the individual charges within the distribution and simultaneously "received" at the point of observation at the instant t. But different charges within the distribution are at different distances from the point of observation, and the times needed for the signals originating from the different charges to arrive at the point of observation are different. Therefore the signals that are received at the point of observation simultaneously at the instant t are sent out from the different charges within the distribution at
different retarded times t' = t - rlc. For a moving charge
distribution these times are different not only because different charges within the distribution are located at different distances from the point of observation, but also because the location of these charges changes as the charge distribution moves. As a result, the region of space from which the field signals responsible for the field at the point of observation are sent is not equal to the region of space, or volume, occupied by the charge distribution when it is at rest.
Consider a charge distribution of length l moving against the x axis with a constant velocity v. The electric field E of the charge is observed at the point O (Fig. 3.1). A field signal is sent from the trailing end of the distribution when this end is at the distance r1 from the point of observation. A field signal is sent from the leading end, when this end is at the distance r2 from the point of observation. Since the leading end is closer to the point

SECTION 3-1 MOVING CHARGE DISTRIBUTIONS

49

of observation than the trailing end, the field signal from the leading end must be sent at a later time, if it is to arrive at the point of observation simultaneously with the signal sent from the trailing end. The difference in the times needed for the two signals to arrive at the point of observation is r1/c - rife. During this time the charge distribution moves a distance (r1/c - rzfc)v. Hence the distance l* between the two points from which the two signals are sent is
(3-1.5)

•
-4--V

I ~ r -V r. - 4 - - / - . .
-" ' -,,

I.. /* --- --~ "1

0.

Fig. 3.1 For the two field signals to arrive simultaneously at 0, the field signal originating from the leading end of the moving charge must be sent later than the field signal originating from the trailing end of the charge.

In this chapter we shall be mainly concerned with the special case of charge distributions for which ri,r2 ► l*. In this case (see
Fig. 3.2), r1 - r2 = l* cos </> = l*(r • v)/rv, where r is the
distance between the midpoint of l* and the point of observation, and </> is the angle between r and v. Substituting this expression for r1 - r2 in Eq. (3-1.5), we have

l • = l * (r • v) /re+ l,

(3-1.6)

or

l* = -e----,--1___,...........
1 -(r • v)/rc

(3-1.7)

Therefore, as already mentioned, the region of space from which

50

CHAPTER 3 FIELDS OF MOVING CHARGES

Fig. 3.2 Geometrical
relations between r, cp,
and l* when r1, r2 ► l*. The significance of
the vector l* will be
explained later.

._v

I*

);',, r1-r2

r2

the moving charge sends out the field signals resulting in the electric and magnetic fields created at the point of observation is not equal to the region of space (volume) actually occupied by the charge. In the case of a charge distribution whose linear dimensions are small compared with the distance from the charge to the point of observation, this region of space, usually called the effective volume, or the retarded volume, AV',et is

A V,eI,

=

AV'
-,..--~-,..- '

1 -(r • v)/rc

(3-1.8)

where AV' is the actual volume of the charge [this equation is obtained from Eq. (3-1. 7) by noting that the volume dimensions perpendicular to the direction of motion are not affected by retardation, and that the dimensions along the direction of motion change in accordance with Eq. (3-1.7)].
Although the distance l* given by Eq. (3-1.5) or Eq. (3-1.7) is a distance between two points in space rather than a length of an object, it is usually called the retarded length of the charge. In fact, it is actually the "visual" length of a rapidly moving body, as the length of the body would appear to a stationary observer. As follows from Eq. (3-1.7), the retarded length of a body moving toward the observer is longer, and the retarded length of a body moving away from the observer is shorter, than the actual length of the body. 1 It should be emphasized that Eqs. (3-1.6)-(31.8) hold only for charges or bodies observed from a distance

SECTION 3-1 MOVING CHARGE DISTRIBUTIONS

51

much greater than the linear dimensions of the charge or body. For a general case, the retarded length or volume of a body cannot be expressed by a simple formula, but can be calculated in terms of the actual length of the body once the position of the body at the time of observation is given (Section 4-3).
Another effect of retardation that needs to be taken into account when applying retarded field equations to moving charge distributions is an apparent distortion of the shape of a moving charge distribution. The distribution appears to change its shape because the retarded times for different points within the distribution are different.
y

• V

0 =-----------------tX►
Fig. 3.3 Geometrical relations between the "present position vector" r0 and the "retarded position vector" r for a charge distribution moving with velocity v in the negative x direction.
Consider a charge distribution moving against the x axis with a velocity v and observed from a point O (Fig. 3.3). The retarded volume element dV' of the charge distribution is at the point P and is represented by the vector r. The present position of the same volume element is at the point P0 and is represented by the vector r0. The distance .:ix' from P to P0 is the distance that the charge travels during the time that it takes the field signal to

52

CHAPTER 3 FIELDS OF MOVING CHARGES

propagate from P to 0, that is, Lix' = v(r/c). We shall now show that, within the charge, any line parallel to they axis when the charge is at rest or at its present position appears to be slanted when the charge is moving and is at a retarded position.
First, let us note that according to Fig. 3.3 the relation between the x component of the present position vector rO and the x and y components of the retarded position vector r is (as usual, we use primes to indicate source-point coordinates)

x' = x~ + vrlc,

(3-1.9)

or (3-1.10)

Differentiating Eq. (3-1.10) while keeping x0' constant, we have

dx' = - -y'(-v/-c)- dy' r[I -(v/c)(x'Ir)] '

(3-1.11)

which can be written as

Y

I..

~-P,., V
-+-,-

r
I

-q-

-

, ,

_

-

\0

'\ 0

rvlc

.. I

I - - -/d~ i -q-~

\rp p

.

Fig. 3.4 A charge at its retarded position appears to be elongated and its vertical lines appear to be slanted.

SECTION 3-1 MOVING CHARGE DISTRIBUTIONS

53

y

I*

-acota
X

Fig. 3.5 Explanation of the vectors l* and a*. The vector l* represents the retarded length of the moving charge, the vector a* represents the "slanted" thickness of the charge.

=

y 1vlc -r[l--(-v/- c)c-os~ cp]

=r[-l--(yr1v-•/cv-)/r-c]

=

(v/c)sincp 1-(r • v)/rc

(3-1.12)

Thus, according to Eq. (3-1.12), a vertical line (x0 ' = constant,
dx0' l<ly0' = 0) within the charge at the present position appears to
be slanted when the charge is viewed at its retarded position (Fig.

3.4), and the angle a of the slant is given by

a = y cot

1vlc
-=-,----,-----,,--~

r[l - (r • v)/rc]

(3-1.13)

In the derivations presented later in Chapter 4, we shall consider a moving charge in the shape of a rectangular prism of length l and thickness a. For determining the magnetic and electric fields of such a charge we shall make use of two special vectors shown in Fig. 3.5: the vector I* representing the retarded length of the charge, given by

I* = - - - - -l - 1 ,• 1-(r • v)/rc

(3-1.14)

54

CHAPTER 3 FIELDS OF MOVING CHARGES

and the vector a* representing the "slanted" thickness of the charge, given by (note that r • v = x'v)

a •

= -

ay 1v/c .
-----,---I

-aJ.

=

-

ay'vlc .
........,------,- 1-

---a-,(-r---,x...1.v..l.c.).,,.J..

r[l-(r·v)/rc]

r[l-(r·v)/rc] r[l-(r•v)/rc]

(3-1.15)

We shall also use the following relation derived in Example
3-1.1 for a charge moving with acceleration v = iJv/iJt'

V['r--(-r 1-• v-)/~c]

= r - rv/c + (r • v)r/c2
r 3[1-(r • v)/rc]2

(3-1.16)

Note that if v = 0 (motion with constant velocity), Eq. (3-1.16)

becomes

V '[-r=- (-r-1· v- -) /-c~]

r - rv/c
= --,------
r3[1-(r • v)/rc]2

(3-1.17)

In dealing with retarded integrals for moving electric charges, we shall frequently use the expression

r - (r • v)/c,

(3-1.18)

where r is the retarded position vector joining a retarded volume element dV' of a moving charge distribution with the point of observation. If the charge distribution moves with a constant velocity v, this expression can be converted to the present position of the charge distribution, that is, to the position occupied by the volume element dV' of the charge distribution at the instant for which the electric and magnetic fields are being determined. This can be done as follows.
First, assuming that the charge distribution moves in the negative x direction and assuming that dV' is in the xy plane, we see from Fig. 3.3 that the present position vector r0 of dV' can be expressed in terms of the retarded position vector r as

r0 = r - rv/c. Next, we write Eq. (3-1.18) as

(3-1.19)

SECTION 3-1 MOVING CHARGE DISTRIBUTIONS

55

[r-(r • v)/c] =[r-x 1v/c]

Adding and subtracting x' 2 and ,2v2/c2 to the right side of Eq. (3-
1. 20), we then have

[r-(r • v)/c]

(3-1.21)

Let us now collect the terms on the right of Eq. (3-1.21) into three groups:
(3-1.22)

(3-1.23)
and (3-1.24)

By Eq. (3-1.9), the first group represents x0'2, where x0' is the distance between the yz plane and the volume element dV' of the moving charge at its present position. The second group is simply y'2, where y' is the (constant) y coordinate of the volume element
dV'. And the third group is -y'2v2/c2. We can write therefore

[r-(r. v)/c] =(x62 +y'2 -y'2v21c2)112

(3-1.25)

But, as can be seen from Fig. 3.3, x0' 2 + y'2 = ra2, and y'2/(xa2 + y'2) = sin2 8, where 8 is the angle between r0 and the velocity
vector v. Therefore
[r-(r • v)/c] =r[l -(r • v)/rc] =r0{1 -(v 2/c 2)sin28} 112 , (3-1.26)
where all the quantities in the last expression are present time quantities. In obtaining Eqs. (3-1.25) and (3-1.26) we assumed that the volume element dV' of the moving charge was located in the xy plane. Clearly, however, the two equations are valid even

56

CHAPTER 3 FIELDS OF MOVING CHARGES

if dV' is not in that plane, provided that we replace in these
equations y'2 by y'2 + z'2.
Expressions involving the retarded position vector r and its magnitude r have a very peculiar and important property which should be kept in mind when dealing with moving charges and currents. As already mentioned, a moving charge is assumed to move through different but fixed points of space. Therefore neither the retarded position vector r nor its magnitude r explicitly appearing in retarded integrals is a function of time. On the other hand, in the case of moving charges and currents, the distance r
appearing in the retarded time t' = t - rlc is variable and
therefore is a function of time. The same applies to Eqs. (3-1.7) (3-1.17) presented above and to all similar expressions.
T
Example 3-1.1 Derive Eq. (3-1.16). Let us arrange a rectangular system of coordinates so that the
acceleration vector of the moving charge is in the xy plane and the velocity vector is in the negative x direction. Let the point of observation be at the origin. The position vector of the charge is
then r = - x'i - y'j. Using vector identity (V-7), we have

V['r--(-r 1-• v-)/=c] =

V'[r - (r • v)/c] [r-(r • v)/c]2

(3-1.27)

In differentiating the numerator in Eq. (3-1.27), we should remember that the numerator is retarded. However, as explained in Section 3-1, neither the position vector r nor its magnitude r appearing in retarded integrals is a function of time and therefore neither is affected by retardation (the charge moves through different but fixed points of space). The only quantity in the numerator affected by retardation is the velocity v which is a function of the retarded time t - rlc and does change as the charge moves. Hence we can write, making use of vector identity (V-5),

SECTION 3-1 MOVING CHARGE DISTRIBUTIONS

57

V'

1

=- V'r-V'[(r•v)/c]

[r-(r•v)/c]

[r-(r•v)/c]2

-ru -(1/c)V'[r • v]
=--------
[r-(r • v)/c]2

(3-1.28)

To evaluate V'[r • v], we first use vector identity (V-30), obtaining

V'[r • v] = [V'(r • v)] + ~ r [_ a(r_• _ v)J . C at

(3-1.29)

The first expression on the right can be evaluated with the help of vector identity (V-6). Note that in this expression V' operates upon unretarded quantities. Therefore we have

V'(r • v) =(r • V')v +r x(V' xv) +(v • V')r +vx(V' xr). (3-1.30)

Since all the quantities in this equation are unretarded, and since the
unretarded v does not depend on spatial coordinates, the first two
terms on the right of this equation vanish. Since V' x r = 0, the
last term vanishes also. By vector identity (V-4), the remaining
term is simply - v. We thus obtain

V' (r • v) = - v.

(3-1.31)

Taking into account that r in the last term of Eq. (3-1.29) is not a function of time, we have

-ru-
C

[-a=( r-. -v-)J
at

=

-ru[r.
C

- aatv]=

-rur[ v . "].
C

(3-1.32)

Combining Eqs. (3-1.28), (3-1.29), (3-1.31), and (3-1.32), factoring out r in the denominator, and multiplying the numerator and the denominator by r, we finally obtain

V'-=--_.,..-.1......,....,,. = r - rv/c + (r • v)r/c 2

[r-(r • v)/c]

r 3[1 -(r • v)/rc]2

(3-1.33)

58

CHAPTER 3 FIELDS OF MOVING CHARGES

Although all quantities in this equation refer to the retarded position

of the charge, to avoid an exceedingly cumbersome notation we do

not place them between the retardation brackets.

...

3-2. Correlation Between the Electric and the Magnetic Field of a Moving Charge Distribution

There are two special cases of moving charge distributions for which there exist simple correlations between the electric and the magnetic field produced by the distributions. The first case is that of an arbitrary charge distribution moving with constant velocity. The second case is that of a point charge moving with acceleration.
Consider first a charge distribution of arbitrary size and shape moving with constant velocity v. Let us form the vector product of e0v and Eq. (2-2.1). Since vis a constant vector, we can place it under the integral sign, so that

aJ] V X [V' p + - 1 -

eoV X E = - _!_ J

c 2 at dV'.

471"

r

(3-2.1)

If a charge distribution moves with constant velocity v, then by Eq. (3-1.4) the derivative oJ/ot is parallel to v. Therefore the product v x [oJ/ot] vanishes, and since v is not affected by retardation, Eq. (3-2.1) simplifies to

(3-2.2)

Using now the vector identity V'x(vp) = (V' X v)p - V X V'p

(3-2.3)

SECTION 3-2 CORRELATION BETWEEN FIELDS

59

and taking into account that V' x v = 0 and that vp = J, we obtain from Eq. (3-2.2)

J e0v X E = _l_ 41r

[V' X J] dV1, r

(3-2.4)

which, by Eq. (2-2.2), is the same as

Since µJI written as

(3-2.5)
B, and e0µ0 = 1/c2, this equation can also be

B = (v X E)/c 2 .

(3-2.6)

Observe that E in Eqs. (3-2.5) and (3-2.6) is the electric field produced by a moving charge distribution.
It is interesting to note that since, in the present case, the term lJJ/lJt in Eq. (3-2.1) makes no contribution to v x E, we can write Eq. (3-2.6), using Eq. (2-2.1), as

(3-2.7)

and, assuming that the velocity is along the x axis, so that v x
i = 0, as
(3-2.8)

where only the components of V' perpendicular to v occur. Furthermore, using Eq. (2-2.4) and taking into account that lJJIIJt
makes no contribution to v x E and that v x i = 0, we can
write Eq. (3-2.6) as

60

CHAPTER 3 FIELDS OF MOVING CHARGES

B =vx 1 J{[p] +_ 1 [aPll.(yj+zk)dV'
47re0c 2 r 3 r2c at lf
J{ = v x ~ [p] + _l [ap ll(yj + zk)dV'. 471" r 3 r 2c at lf

(3-2.9)

As it follows from Eqs. (3-1.7) and (3-1.8), for slowly moving charge distributions the retardation can be neglected, in which case Eq. (3-2.6) reduces to

B = (v x E)/c 2 ,

(3-2.10)

where E is the ordinary electrostatic field of the charge distribution under consideration. Likewise, Eqs. (3-2.7) - (3-2.9) reduce to the corresponding equations involving unretarded charge densities.
Consider now a point charge moving with acceleration. Let us assume that the retarded position of the point charge is given by the vector r, and let us form the cross product of r/(rµ,oe) and Eq. (2-2.12). Assuming that r for a moving point charge can be considered the same throughout the entire volume occupied by the charge, we can place r/r under the integral signs.2 Noting that r
x r = 0, we then obtain
rxE = 1 J{[J] +_l_[aJllxrdV', (3-2.11)
µ,0cr 41re0µ,0c 2 r 3 r 2c at lf

and, taking into account that BQ/LoC2 = 1 and using Eq. (2-2.5), we immediately obtain

(3-2.12)

or

B = rxE, er

(3-2.13)

REFERENCES AND REMARKS FOR CHAPTER 3

61

where r is the retarded position vector connecting the moving point charge with the point of observation. Equations (3-2.12) and (3-2.13) show that the magnetic field of a moving point charge is perpendicular to the electric field produced by the charge and to the radius vector joining the retarded position of the charge with the point of observation.3
It is interesting to note that for a point charge moving with constant velocity, Eqs. (3-2.5) and (3-2.6) as well as Eqs. (32.12) and (3-2.13) hold, because Eqs. (3-2.12) and (3-2.13) are true for any acceleration, including zero acceleration. However, it is important to remember that Eqs. (3-2.12) and (3-2.13) involve the retarded position vector r. If the acceleration is zero, Eq. (3-2.13) reduces to Eq. (3-2.6), as is shown in Example 41.1.

References and Remarks for Chapter 3

1. The retarded length should not be confused with the relativistic 11 Lorentz-contracted length. 11 See Section 9-1. 2. This procedure is generally applicable to stationary point charges only. For moving point charges its applicability depends on certain parameters of the system under consideration. See Section 4-7 (in particular Eqs. 4-7 .1 and 4-7.2) for details. 3. It is important to stress that Eqs. (3-2.12) and (3-2.13), although usually presented in the literature as perfectly true, are actually only approximately correct. See Section 4-7 for details.

4
ELECTRIC AND MAGNETIC FIELDS AND POTENTIALS OF MOVING POINT AND LINE CHARGES
The finite propagation speed of electric and magnetic fields has a profound effect on the electric and magnetic fields and potentials associated with moving charge distributions. In this chapter we shall use retarded integrals for determining electric and magnetic fields and potentials of the two simplest types of moving charge distributions: a moving point charge and a moving line charge.
4-1. The Electric Field of a Uniformly Moving Point Charge1
Any stationary charge distribution viewed from a sufficiently large distance constitutes a "point charge. "2 Consider a charge distribution of total charge q and density p confined to a small rectangular prism (Fig. 4.1) whose center is located at the point x', y' in the .xy plane of a rectangular system of coordinates, and whose sides l, a, and b are parallel to the x, y, and z axis, respectively. Let the point of observation be at the origin of the coordinates, and let the distance between the center of the prism and the origin be r0 ► a, b, l. Viewed from the origin, this
62

SECTION 4-1 UNIFORMLY MOVING POINT CHARGE 63
y
r0 >>a,b,l
X
Fig. 4.1 A charge of uniform density p is confined to a small rectangular prism. The total charge of the prism is q. The charge constitutes a point charge when viewed from a distance large compared to the linear dimensions of the prism.
charge distribution constitutes a point charge. 3 Let the charge
move with uniform velocity v = - vi. We want to find the
electric and magnetic fields of this charge at the point of observation.
To find the electric field produced by this charge, we shall use Eq. (2-2.1). First we eliminate from Eq. (2-2.1) the term with the current density J. We can do so with the help of Eq. (3-1.4). Since the velocity of our charge is v = v) = - vi, and since the
charge moves without acceleration so that v = 0, Eq. (3-1.4)
gives (4-1.1)
Substituting Eq. (4-1.1) into Eq. (2-2.1), we then have for the electric field of the charge

64 CHAPTER 4 MOVING POINT AND LINE CHARGES
- - - - I!---►
Fig. 4.2 When the charge shown in Fig. 4.1 is moving and is at a retarded position, its apparent length, shape, and thickness of its front and back surface layers are no longer the same as for the stationary charge. (All r's meet at the origin).
(4-1.2)
Observe that in this equation V' and atax' operate on the
unretarded p, so that in computing V' p and 8p/8x' we must use the ordinary, unretarded, shape and size of the prism. Since p is
constant within the prism, V' p = 0 within it, and the only
contribution to V'p comes from the surface layer of the prism, where p changes from p (inside the prism) to O(outside the prism). Let the thickness of the surface layer be w. Taking into account that V' p represents the rate of change of p in the direction of the
greatest rate of change, we then have V'p = (plw)n;n, where n;n is
a unit vector normal to the surface layer and pointing into the prism. Hence V' p for the right, left, top, bottom, front, and back surfaces of the charge (prism) are -(p/w)i, (p/w)i, -(p/w)j, (p/w)j, -(p/w)k, and (p/w)k, respectively. Likewise, 8p/8x' is zero in the interior of the charge and is different from zero only in the left and in the right surface layers of the charge, where

SECTION 4-1 UNIFORMLY MOVING POINT CHARGE 65
V
Fig. 4.3 The relations between r3, r4, and a* for the moving charge at a retarded position. (The two r's meet at the origin.)
op!ox' = p!w in the left surface layer and op!ox' = - p!w in the right surface layer.
The volume integral of Eq. (4-1.2) can be split therefore into six integrals, one over each of the six surface layers corresponding to the six surfaces of the charge (prism). However,
since the center of the charge is in the xy plane (z' = 0), the
integrals over the two surface layers parallel to the xy plane cancel each other, because V'p for one of the layers is opposite to that for the other layer, while r is the same for both layers. Thus only the four integrals over the layers parallel to the xz and yz planes remain. Let us designate the retarded distances from these layers to the point of observation as ri, ri, r3, and r4 (see Figs. 4.2 and 4.3). Since the linear dimensions of the charge are much smaller than r1, r2, r3, and r4, we can replace each integral over a surface layer by the product of the integrand and the volume of the corresponding layer. However, the integration in Eq. (4-1.2) is over the effective (retarded) volume of the charge, and therefore we must use not the true volume of the surface layers, but their effective volume. The effective volume of the surface layers is not the same as their actual volume, because, in accordance with Eq. (3-1.7), the length l of the two layers parallel to the xz plane must

66

CHAPTER 4 MOVING POINT AND LINE CHARGES

be replaced by

/*:::

/

1-(r • v)/rc'

(4-1.3)

and because, also in accordance with Eq. (3-1.7), the thickness w of the two layers parallel to the yz plane must be replaced by

w • ::: --:--,--w--,--,--
1 -(r • v)/rc
Equation (4-1.2) becomes therefore

(4-1.4)

E -- _-1- [-pl- wabW 1*(- I") +p_ /wabw2•·1 + p/_ wbl3• W (-"J)

41reo '1

'2

'3

(4-1.5)

+p_ /wbl4• W• J(+v2- )(-plwabW•·1Ip+/w-abW2*(-")I)~

r4

c2 '1

'2

or, substituting I* and w* from Eqs. (4-1.3) and (4-1.4),

E = - - -1 -[- -p-lw- , - abw( -I") + _ _ plw_.,...abW•I

41re0 r1 -r1 •v/c

r2 -r2 •v/c

+ Plw blw( -j) + Plw blwj

r3 -r3 • v /c

r4 -r4 • v /c

+

(-v
c

2 2

)

(- -pl- w -
r1 -r1 • v/c

ab W •

Ir+ 2 --rp2/-w•-v,/-c

abW

(

-

1"))~

,

(4-1.6)

which simplifies to

E = _ pb f1(1 _ v2)( 1 _ 1 )ai

41re0~ c 2 r2-r•2 v/c r1-r•1 v/c

)t] + (r

1
-r • v /c

-

1 r -r • v /c

J •

4 4

3 3

(4-1.7)

As can be seen from Figs. 4.2 and 4.3, the differences of the fractions in these equations are simply the increments of the function 1/(r - r • v/c) associated with the displacement of the tail

SECTION 4-1 UNIFORMLY MOVING POINT CHARGE 67
of r over the distances represented by the vector I* [in the i component of Eq. (4-1.7)] and by the vector a* [in the j component of Eq. (4-1.7)]. Therefore we can write Eq. (4-1.7) as4

E = -

t pb f1(1
41re0

-

ev22)~l(v'

1 ) r-r•vle

•

I* tf i

~(v' + l

r-r

1 •vl

e

)

•

a

·]1j}.

(4-1.8)

Substituting the gradient from Eq. (3-1.17) (remembering that v = 0) and substituting I* and a* from Eqs. (3-1.14) and (3-1.15),
we have

E = pb f1(1 _ v2)( r - rvle • i) la i 411"t) e 2 r 3(l -r • v lre)2 1 -r • v Ire

+

(- r 3(-1r-- -r r•- vvlel-re• )2

·
I

)

-
r(

y'vle
----l
l - r • v Ire)

l1J•

(4-1.9)

+ (- -r-- rvl-e - • J·) - -r--x'-vl-e l liJ·] . r 3(1 -r • v lre)2 r(l - r • v Ire)

Simplifying and taking into account that r • i = - x', r • j = - y', v • i = - v, v • j = 0, and r • v = x'v, we obtain

E =

pabl

r(1- vi)(-x'+rvle)i

l 47rt:0r 3[1-r•vlre]3 e 2

+ ( -x , +rv1e) -y '-viJe. + ( -y ') -r --x-'vJ le ·]

r

r

=

pabl

~(1- v2)(-x'i-rvle) +(1- v2)(-y)j],

47rt:0r 3[1 -r • vlre]3l e 2

e2

and finally, noting that r = - x'i - y'j, and that pabl = q,

(4-1.11)

68

CHAPTER 4 MOVING POINT AND LINE CHARGES

Equation (4-1.11) expresses E in terms of the retarded position of the charge specified by the retarded position vector r (see Fig. 3.4). Usually it is desirable to express E in terms of the present position of the charge specified by the present position vector r0 (see Fig. 3.4). We can convert Eq. (4-1.11) from r to rOby using Eqs. (3-1.19) and (3-1. 26). According to Eq. (3-1.19),

r-rv/c=r0 ,

(4-1.12)

so that the last factor in Eq. (4-1.11) is simply the present position vector r0. Substituting Eq. (4-1.12) and Eq. (3-1.26) into Eq. (41. 11), we obtain the desired equation for the electric field of a uniformly moving point charge expressed in terms of the present position of the charge

E _- _ _ _q_(l -_v2_/ c 2_) _ _r0 . 41re0r5{1-(v 2/c 2)sin28}312

(4-1.13)

This equation (in a different notation) was first derived by Oliver Heaviside in 1888 on the basis of Maxwell's equations by using the "operational calculus" that he invented.5

Fig. 4. 4 As was first noticed by

Heaviside, the electric field of a

. V

moving point charge concentrates itself in the direction

perpendicular to the direction of

motion of the charge and

decreases along the line of the

motion.

There are two interesting properties of Eq. (4-1.13). First, as was noted by Heaviside, with increasing velocity of the charge the electric field of the charge concentrates itself more and more

SECTION 4-1 UNIFORMLY MOVING POINT CHARGE 69

about the equatorial plane, 8 = 1r/2, and decreases along the line
of motion, 8 = 0. This effect is shown in Fig. 4.4. Second, the
electric field appears to originate at the charge in its present position. This, of course, is merely an illusion, because by supposition the distance between the charge and the point of observation is much greater than the linear dimensions of the charge, so that neither Eq. (4-1.11) nor Eq. (4-1.13) gives us any information concerning the structure of the field close to the charge. Note also that because of the finite speed of the propagation of the field signals and light signals one can never observe the charge at its present position. In fact, the charge could have stopped after sending the field signal from its retarded position, and even then Eq. (4-1.13) would remain valid, although in this case Eq. (4-1.13) would apply to the "projected," or
."anticipated," present position of the charge.
Example 4-1.1 Show that for a point charge moving without acceleration Eq. (3-2.13) reduces to (3-2.6).
According to Eq. (4-1.12), the retarded position vector of the charge can be expressed in terms of the present position as

r = r0 + rv/e. Substituting Eq. (4-1.14) into Eq. (3-2.13), we have

(4-1.14)

B = rxE = (r0 +rv/e)xE = r0 XE + (rv/e)xE. (4-1.lS)

er

er

er

er

Since, by Eq. (4-1.13), Eis directed along r0, r0 x E = 0, and we

are left with

B = (v x E)/e2 ,

(4-1.16)

which was to be proved.

Example 4-1.2 Equation (4-1.13) represents a "snapshot" of the electric field of a moving point charge, since it does not express the

70

CHAPTER 4 MOVING POINT AND LINE CHARGES

field as a function of time. Modify Eq. (4-1.13) so that it shows how the field changes as the charge moves.
Let us assume that the "snapshot" is for t = 0. If the charge
moves in the - x direction, the functional dependence of E on the
x coordinate will be preserved for t ~ 0 if we express Eq. (4-1. 13)
in terms of x0' and replace x0' by Xo' - vt. From Eqs. (3-1.26) and (3-1.25), we have

, 0{1-(v2/c2)sin20}112 = 0c,~2+y'2-y,2v2/c2)112 = [x~2 +(1 -v2/c2)y'2]112.

(4-1.17)

Replacing in Eq. (4-1. 17) x0' by x0' - vt, we obtain

where x0' is now the x coordinate of the point charge at t = 0.
Expressing r0 in terms of its components and replacing x0' by x0' -
vt, we similarly have r0 = -(?c,0' - vt)i - y'j. Therefore Eq. (4-
1.13) can be written as

q(l-v 2/c 2){(x~ -vt)i +y'j} E = - ---,-------
4no{0c-o/ - vt)2 +(1-v2/c2)y' 2p12 '

(4-1.19)

where the dependence of E on t is shown explicitly. This equation holds for the charge moving parallel to the x axis in the xy plane. If it moves parallel to the x axis anywhere in space, y'2 in this
equation should be replaced by (y'2 + z'2).

4-2. The Magnetic Field of a Uniformly Moving Point Charge
Although by using Eq. (2-2.2) or Eq. (2-2.5), we can find the magnetic field of a uniformly moving point charge in the same manner as we found the electric field in Section 4-1 (see Example 4-2.1), it is much easier to find it from the known electric field by using Eq. (3-2.5) or Eq. (3-2.6).

SECTION 4-2 MAGNETIC FIELD OF POINT CHARGE 71
Applying Eq. (3-2.5) to Eq. (4-1.11), we obtain for the magnetic field in terms of the retarded position of the charge

H = -...,q,....[1_-_vz_lc_z_J__,. [v x r] .
47rr3[1 -r • v /rc]3

(4-2.1)

Applying Eq. (3-2.5) to Eq. (4-1.13), we obtain for the magnetic field in terms of the present position of the charge

(4-2.2)

~
Example 4-2.1 Find the magnetic field of a uniformly moving point charge shown in Fig. 4.1 by using Eq. (2-2.2),
(4-2.3)
To use Eq. (4-2.3), we need to know V' x J associated with
the charge under consideration. The moving charge constitutes a
current density J = pv. Since vis not a function of x', y', z', we
have V' x J = V'p x v. But pis constant within the charge, and therefore the only contribution to V' x J comes from the surface
layer of the charge, where p changes from p (inside the charge) to 0 (outside the charge). Using the values for V' p obtained in Section
4-1, we then have for V' x J of the top, bottom, front, and back
surface layers of the charge (prism) -pv/wk, pv/wk, pv/wj, and -p/wj, respectively; the left and right surface layers make no
contribution to V' x J, because v and V'p are parallel (or antiparallel) there. Furthermore, since V' x J in the front surface layer is opposite to V' x J in the back surface layer, while both
surface layers are at the same distance r from the point of observation, the contributions of these two layers to the integral in

72 CHAPTER 4 MOVING POINT AND LINE CHARGES

Eq. (4-2.3) cancel each other, so that only the top and the bottom surface layers contribute to the magnetic field of the charge.
Since the linear dimensions of the charge are much smaller than r3 and r4, (see Fig. 4.3), we can replace the integrals over the two surface layers by the product of the integrand and the volumes of the corresponding layers. Using Eq. (4-2.3) and taking into account the effective volume of the boundary layers (see Sections 3-1 and 4-1), we have, as in Eqs. (4-1.5)-(4-1.7),

_l_[ H = -

pv/w wblk + pv!w wblk]

41r r3 -r3 •v/e

r4 -r4 •v/e

= _ pvbl[ 1

_

1 ]k

41r r3 -r3 •v/e r4 -r4 •vie •

(4-2.4)

The difference of the two fractions in the last expression is simply the increment of the function 1/(r - r • v/e) associated with the displacement of the tail of rover the distance represented by the vector a* (see Fig 4.3). Therefore, using Eqs. (3-1.17) and (31.15), we can write Eq. (4-2.4) as

H = -

pbvl[(- -r- -rv-/e- -

•

·) y'vle
I --,..,----,---,-ll

411" r 3(1 -r • v /re)2 r(l - r • v /re)

+( r-rv/e ·j) r-x'vle a]k

r 3(1 -r • v /re)2 r(l - r • v /re) •

(4-2.5)

Simplifying and taking into account that r • i = - x', r • j = y', v • i = - v, v • j = 0, and r • v = x'v, we obtain

H = -

qv

[(-x' +rvfe)y'vlre +(-y')(l -x'v/re)]k

41rr 3[1 -r • v /re]3

= qv[l -v 2/e 2]y' k 41rr3[1 -r • v /re]3 '

(4-2.6)

. which, noting that ry'k = v x r, is the same as Eq. (4-2.1). 6

SECTION 4-3 UNIFORMLY MOVING LINE CHARGE 73
4-3. The Electric and Magnetic Fields of a Line Charge Uniformly Moving Along its Length
Consider a line charge of finite length L, cross-sectional area
S, charge density p, and linear charge density A = pS moving
with constant velocity v parallel to the x axis of a rectangular system of coordinates in the negative direction of the axis and at a distance R above the axis (Fig. 4.5). Let the point of observation O be at the origin. What is the electric field at O at
the time t when the leading end of the charge is at a distance Li
from the y axis? We can find the electric field of the moving charge by using
Eq. (2-2.1) or Eq. (2-2.4) if we know the retarded position of the

y
-L ►I L ~ I .. 2 W/2"ff////////////A •::
--X2 ~-'I

vr/c

.. I v

·r x'1 ~" ~I

R

X
a = - - - - - - - - - - - - - - - - - - - ' -...

Fig. 4. 5 A line charge of linear density A is moving with constant velocity v. The retarded positions of the trailing and leading ends of the charge are x1' and xz', respectively. The present positions of
the two ends are L1 and Li, respectively. The distance between the
trajectory of the charge and the x axis is R. The point of observation O is at the origin. The "retarded, 11 or "effective, 11
length of the charge is longer than its true length.

74

CHAPTER 4 MOVING POINT AND LINE CHARGES

charge corresponding to the time for which the field is computed.

We can determine this position as follows.

First, let us determine the retarded position xi' of the leading

end of the charge corresponding to the time t, that is, the position

from which the leading end sends out its field signal which arrives

at O at the time t. If the retarded distance between O and the

leading end is r2, then the time it takes for the signal to travel

from the leading end to O is rife. During this time the charge

travels a distance v(rife). Therefore at the moment when the

leading end sends out its field signal, the position of the leading

end is

(4-3.1)

Next, let us find the retarded position xi' of the trailing end of the charge corresponding to the time t. If the retarded distance between O and the trailing end is r1, then the time it takes for the signal to travel from the trailing end to O is rife. During this time the charge travels a distance v(rife). Hence, at the moment when the trailing end sends out its signal, the position of the trailing end is
(4-3.2)

The x component of the electric field. We are now ready to find the electric field of the charge by using Eq. (2-2.1) or Eq. (22.4). The easiest way to find the x component of the electric field of the charge under consideration is to use Eq. (2-2.1). According to this equation, the x component of the field is due to the x components of [V'p] and [aJtat] of the moving charge. For the line charge under consideration, these components exist only at the leading and trailing ends of the charge and are the same as for the moving charged prism discussed in the preceding sections of this chapter: [V'p]x = (p/w)i for the leading end, and [V'Plx = - (plw)i for the trailing end, [aJtat]x = - (v2plw)i for the
leading end, and [aJ/at]x = (v2p/w)i for the trailing end, where w

SECTION 4-3 UNIFORMLY MOVING LINE CHARGE 75
is the thickness of the surface layer of the charge (this is the actual thickness, not the retarded one). Since the surface layer of the charge may be assumed as thin as one wishes, the retarded volume integral in Eq. (2-2.1), as far as the x component of the field is concerned, reduces to the product of the integrand and the volume of the surface layers of the leading and trailing ends of the charge at their retarded positions. By Eq. (4-1.4), for the leading end, this volume is, using the asterisk to indicate values evaluated at retarded positions,
(4-3.3)

and for the trailing end it is

wi' S

=

wS --,----,---.,--,---

1 -(r1 • v)/r1c

The x component of the electric field is therefore

(4-3.4)

or

E

A(l -v2/c2) ( 1

1 ) (4-3.6)

x = -

41reo

r2-xfv!c - r 1 -x(v!c •

Equation (4-3.6) gives the electric field in terms of the retarded position of the charge. We shall now convert it to the present position of the charge (that is, the actual position of the charge at the time t). The calculations are similar to those used for deriving Eqs. (3-1.20)-(3-1.26). First, we note that, by Eq. (43.1),
(4-3.7)

Next, we write the denominator of the first fraction inside the parentheses of Eq. (4-3.6) as

76 CHAPTER 4 MOVING POINT AND LINE CHARGES

Adding and subtracting x'2 and r/vlc- to the right side of Eq. (4-
3.8), we then have

r2 -x{vlc

(4-3.9)

.x,; =(r{ -2r~{ vie +x{ 2v2/c 2+ 2-x{ 2 +r{v 2/ c 2-r{v2/c 2) 112 .

Let us now collect the terms on the right of Eq. (4-3.9) into three groups:
(4-3.10)

(4-3.11)
and (4-3.12)
By Eq. (4-3.7), the first group represents L,/. The second group is simply R2(see Fig. 4.5). And the third group is - R2v2/c2.
Similar relations hold for the denominator of the second fraction inside the parentheses of Eq. (4-3.6). Therefore Eq. (43. 6) transforms to

E = A(l-v2/c 2)[

1

_

1

]

x 41recft (Lf IR2 + 1 -v21c2)112 (L;/R2 +1 -v2/c2)112 ,

(4-3.13)

where only the present time quantities appear.

The y component of the electric field. The easiest way to find the y component of the electric field of the charge under consideration is to use Eq. (2-2.4). Only the first integral of Eq. (2-2.4) makes a contribution to the y component of the field, because {JJ/<Jt has no y component. Separating this integral into two integrals, we then have

SECTION 4-3 UNIFORMLY MOVING LINE CHARGE 77

E = - _l_J [p]RdV' __l_J_l[0P]RdV1 • (4-3.14)

y

411"eo r 3

411"eo r 2c at

The first integral in Eq. (4-3.14) is the same as for a stationary
charge, except that the integration must be extended over the retarded (effective) length of the charge. Designating the
contribution of the first integral as E1y and noting that r = (x'2 +
R2)112, we obtain

E = - _l_J!!...RdV' = - ~ I x { R dx 1, (4-3.15)

ly

411"eo r3

411"eo x; (x'2 +R2)312

or

Yi -'i • E

}.. [ x(

Xi ] }.. (Xi x()

Iy = - 411"eoll (x(2 + R2)112 - (Xi+ R2)112 = 411"eoll

(4-3.16)

In order to evaluate the contribution of E2y of the second integral of Eq. (4-3.14) to the total field, we must determine the value of the derivative [op/at]. According to the notation convention for retarded quantities explained in Chapter 1, this derivative is the ordinary derivative op/at used at the retarded position of the moving charge. By Eq. (3-1.3), taking into account that for our charge v = - vi, [op/at] is then simply vap!ax'. Since p is constant within the line charge, only the leading and the trailing ends of the charge contribute to this expression, and the contributions are vp!w and - vp/w, respectively. The electric field E2y is therefore

E2 = - _R_J vplw dVi + _R_J vp/w dV(, (4-3.17)

:Y

411"e0c r;

4n0c rf

where the integration is over the surface layers of the leading and trailing ends of the charge at the retarded positions of the charge. Since the thickness of the surface layers is much smaller than r1 and r2, we can replace the integrals, as before for Ex, by the

78

CHAPTER 4 MOVING POINT AND LINE CHARGES

products of the integrands and the volumes of integration (the

volumes of the respective surface layers). Using the relations dVz'

= w2*S, dVi' = w1*S, and using Eqs. (4-3.3) and (4-3.4) , we

then have

s] E2y = - -R-[ - - - vp-lw- WS + - - - vp-lw- W

411'80e r{-r(r •v)le
2 2

rf-r(r ·v)le

I I

(4-3.18)

"'A.vR [

1

1 ]

= 411'80e r I(rI -x{ vie) - rz<r2-xf vie) •

Adding Eqs. (4-3.16) and (4-3.18), we obtain for the y

component of the field

I

I

E =_"'A._[-~+ R 2vle + X2 _

R 2vle ]

Y 411'80-T-l' r I r1(r1-X1' VIe) r2 r2(r2-X2' VIe) (4-3.19)

=_"'A._[xf (r2 -xf vie) -R 2vle _ x{ (r1 -x{ vie) -R 2vle]

411'8o-T-l·

rz<r2-x2I vie)

ri(rI -xII vie)

'

or

E=

-

},

,_-

I
[X2r

2

-X2/

2

vI e

-

R2vIe

_

x

I
1r

1

-

X

1/2vIe

-

R2

vIe

].

(4

-3

.2

0)

Y 47r8rft

rz<r2 - x { v l e )

r I(rI -x{vle)

But x I'2vle + R2vle = r/vle and x2'2vle + R.1-vle = r/vle.

Therefore

(x E =

A

I

I

-2---r2,vl-e - -x I---rI,vl-e)

Y 411'80-T-l' r2 -x21vle r I -x'Ivle •

(4-3.21)

Now, by Eq. (4-3.1), xz' - r2vle = Li, and by Eq. (4-3.2), x I' - r Ivle = L I. Substituting Li and L I into Eq. (4-3.21) and
transforming the denominators to the present position quantities by
means of Eqs. (4-3.7)-(4-3.12), just as we did in Eq. (4-3.6), we

finally obtain

E= A [

Lz

_

LI

]· (4-3.22)

Y 411'8rft.2 (Li IR2 + 1 -v21e2)112 (LflR2 + 1 -v21e2)112

SECTION 4-4 ACCELERATING POINT CHARGE

79

The magnetic field. Although we could find the magnetic field of the moving line charge from Eq. (2-2.2) or from Eq. (2-2.5), it is much simpler to find it from the electric field of the charge. According to Eq. (3-2.5), the magnetic field H of any uniformly moving charge distribution is always

(4-3.23)

where E is the electric field of the moving charge distribution.
Since v = - vi, the only non-vanishing component of the cross
product in Eq. (4-3.23) is the z component involving Ey only. Substituting v and Eq. (4-3.22) into Eq. (4-3.23) and denoting Av
as the current /, we obtain

4-4. The Electric Field of a Point Charge in Arbitrary Motion
As before, we consider a constant charge distribution of total charge q and density p confined to a small rectangular prism (Fig. 4.6) whose center is located at the point x', y' in the xy plane of a rectangular system of coordinates, and whose sides l, a, and b
are parallel to the x, y, and z axis, respectively. The point of
observation is at the origin. The distance of the center of the prism from the point of observation (the origin) is r0 ► a, b, l, so that the prism constitutes a point charge.2 We shall assume that at the retarded time t' the center of the prism moves with velocity
v in the negative x direction and has an acceleration v.
For a given present time t, the retarded times associated with different points of the prism are different, corresponding to different retarded distances of these points from the point of observation. Therefore the retarded velocities of the different

80

CHAPTER 4 MOVING POINT AND LINE CHARGES

y

r0 >>a,b,l
X
Fig. 4. 6 A charge of unifonn density p is confined to a small rectangular prism. The charge constitutes a point charge when viewed from a distance large compared to its linear dimensions.
points of the prism are also different. If the prism is sufficiently far from the point of observation, which we assume to be the case, the difference between the retarded times corresponding to different points of the prism is very small, and therefore the retarded acceleration of the prism may be assumed to have the
same value v for all points of the prism, even if in reality the
acceleration is variable. Therefore the velocities of the different points of the prism can be calculated from velocity formulas for motion with constant acceleration.
As we shall presently see, in addition to the velocity of the center of the prism, we only need the velocities of the right, left, top, and bottom surfaces of the prism. Let the distances of these surfaces from the point of observation be r1, r2, r3, and r4, as shown in Fig. 4.7. The time interval between the retarded time for the center of the prism and for its left or right surface is then approximately (r1-ri)/2c (see Section 3.1), and the time interval between the retarded time for the center of the prism and for its top or bottom surface is approximately (r3 -r4)/2c. Therefore the

SECTION 4-4 ACCELERATING POINT CHARGE

81

Fig. 4. 7 When the charge shown in Fig. 4.6 is in a state of accelerated motion and is at a retarded position, its apparent length, shape, and thickness of its surface layers are no longer the same as for the stationary charge. The distances from the center of the charge and from the four surface layers to the point of observation are represented by the vectors r, r 1, r2, r3, and r4. All five r's meet at the point ofobservation (origin of coordinates). The acceleration vector is in the xy plane.

(approximate) retarded velocities of the right, left, top, and
bottom surfaces of the prism are, respectively, v1 = v - v(r1 r2)/2c, v2 = v + v(r1 - r;J/2c, v3 = v - v(r3 - r4)/2c, and v4
= v + v(r3 - r4)/2c.
As was explained in Section 3-1, the apparent size and shape
of the prism in its retarded position is not the same as that of the
prism when it is at rest. In particular, if the prism moves in the - x direction, the prism appears to be longer, it appears to be
slanted, and the effective volume of the prism and of its surface
layers changes (Fig. 4.7). As a result, the following geometrical relations hold for the moving prism at its retarded position:
The apparent length of the prism is, by Eq. (3-1. 7),

l*

l = -1--r •-v/- rc

The apparent volume of the prism is, by Eq. (3-1.8),

(4-4.1)

82

CHAPTER 4 MOVING POINT AND LINE CHARGES

(abl) * = --,--a-b-l-,--1-r • v/rc

(4-4.2)

By the same equations, the apparent volume of the right surface layer (distance r1 from the origin) is

(abw); = abw
1 -r1 • v1/r1c

(4-4.3)

the apparent volume of the left surface layer (distance r2 from the

origin) is

(abw)i• = - -a-bw- 1 -r2 • v/r2c

(4-4.4)

the apparent volume of the top surface layer (distance r3 from the origin) is

(lbw)3• =

lbw

1 -r3 • v/r3c

(4-4.5)

and the apparent volume of the bottom surface layer (distance r4 from the origin) is

(lbw)4* =

lbw

1 -r4 • vir4c

(4-4.6)

We shall find the electric field of our accelerating point charge by using Eq. (2-2.1)

E

=

-

_l_f [V'p

+

_!_ aJ]
c2 at dV'.

41rt:0

r

(2-2.1)

Consider first the contribution of the gradient of the charge density, V 'p, to the field. Since p is constant within the charge,
V'p = 0 within it, so that the only contribution to V'p comes
from the surface layer of the charge, where p changes from 0 (outside the charge) to p (inside the charge). Let the actual thickness of the surface layer of the charge be w. Taking into account that V 'p represents the rate of change of p in the direction

SECTION 4-4 ACCELERATING POINT CHARGE

83

of the greatest rate of change, we then have V'p = (p/w)nin•
where nin is a unit vector normal to the surface layer and pointing into the charge.7 Since the center of the charge is in the xy plane
(z' = 0), the integrals over the two surface layers parallel to the
xy plane cancel each other, because V'p for one of the layers is opposite to that for the other layer, while r is the same for both layers. Thus, as far as V' p is concerned, only the four integrals over the layers parallel to the xz and yz planes remain. Referring to Figs. 4.6 and 4.7, they are the right, left, top, and bottom surface layers, and V' p associated with these surface layers is, respectively -(p/w)i, (p/w)i, -(p/w)j, and (p/w)j (these are the same relations that we used for finding the electric field of a uniformly moving point charge in Section 4.1).
Assuming that r 1, r2, r3, and r4 are much larger than l*, we can replace the integrals over the four layers by the products of the integrands and the retarded volumes of the layers, which gives

Let us designate the part of Eq. (4-4.7) which explicitly depends on p as EP. Using Eqs. (4-4.3)-(4-4.6) and cancelling w, we can write then

The differences of the fractions in this equation are simply the increments of the function 1/(r - r • v/c) associated with the displacement of the tail of r over a small distance represented by

84 CHAPTER 4 MOVING POINT AND LINE CHARGES

the vector I* [in the i component of Eq. (4-4.8)] and by the vector a* [in the j component of Eq. (4-4.8)]. Therefore, just as we did in the case of Eq. (4-1.7), we can write Eq. (4-4.8) as

+[(v' E = P

-

J!.!!.....{r(v'
411'Bo [

l r-r•v/c

)

·I*

lf_i

l ) ·a* ]tj}. (4-4. 9) r-r•v/c

Using Eqs. (3-1.16), (3-1.14), and (3-1.15), we now have

E = pb [(r - rv/c + (r • v)r/c 2 • i) la i

P 411'e0 r 3(1-r•v/rc)2

1-r•v/rc

+(-r ~- r-v/-c -+-(r-•,v,)r-/c-2

•

I ·

)

-

-

-

-

y 1vlc
,-,------,

-

-

-,-

alJ'

r 3(1-r•v/rc)2

r(l-r•v/rc)

(4-4.10)

+(r-rv/c+(r•v)r/c 2 .j) r-x'vlc atj].

r 3(1-r • v/rc)2

r(l - r • v/rc)

Simplifying and taking into account that r • i = - x', r • j =
- y', v • i = - v, v • j = 0, and r • v = x'v, we obtain

E =

pabl

{ r - x ' + r v / c - ( r • v ) x ' / c 2] i

P 411'e0r 3[l - r • v Ire]3

+ [-x' + rvlc -(r • v)x' /c 2] y'vlc j r
,I } +[-y' -(r•v)y'/c2]r-: vcj

(4-4.11)

=

pabl

(-x'i-rv/c-(r•v)x'/c2i

4'll'e0r 3[1-r • v/rc]3

+(v2y, /cz)j -y, j -(r • v)y' /cZj].

Since we are not interested in the acceleration-independent
field Ev (this field was found in Section 4-1), we shall drop in Eq.
(4-4.11) the terms that do not contain the acceleration v, and shall
designate the rest of the equations as E,4p, with the subscript "A"
standing for "acceleration." Noting that r = - x'i - y'j, and that
pabl = q, we then obtain

SECTION 4-4 ACCELERATING POINT CHARGE

85

(4-4.12)

Consider now the contribution of aJ1at to the field. By Eq. (31.4), we have

-a=J -a-(p=v)-(v•V,p)v+pav_=-(v•V,p)v+pv· . (4-4.13)

at at

at

However, because the retarded velocity is different in different

regions (points) of the charge, we must evaluate Eq. (4-4.13)

separately for each region under consideration. There are five

such regions: the interior of the charge, the right surface, the left

surface, the top surface, and the bottom surface.

In the interior of the charge, V'p = 0. Therefore for the

interior we have

aJ
at

= pv.

(4-4.14)

At the right surface, V'p=(ap/ax')i= -(p/w)i, and the velocity is v 1• By Eq. (4-4.13), for the right surface we therefore have

or

TaJ1t = (plw)(v1xv1 + wv1),

(4-4.16)

and since we can make was small as we please,

aJJ

- at

= (plw)v1 v1. X

(4-4.17)

At the left surface, V'p = ap1ax'i = p/wi, and the velocity is v2.Therefore, by the same reasoning as in the case of Eq. (44.16),

(4-4.18)

86 CHAPTER 4 MOVING POINT AND LINE CHARGES
At the top surface, V'p = op/oy'j = - plwj, and the velocity is v3. Therefore,
(4-4.19)
At the bottom surface, V'p = op/oy'j=plwj, and the velocity is
v4. Therefore (4-4.20)

Let us now designate the integral in Eq. (4-4.7) as EJ. Since, by supposition, all r's for the charge (prism) are much larger than the linear dimensions of the charge, we can replace the integration by the product of the respective integrands and the volumes of the five regions that contribute to oJ/ot. Using Eqs. (4-4.14), (44.17)-(4-4.20) and (4-4.2)-(4-4.6), we then have

41rec2E =P"( abl ) 0 J r 1-r • v/rc

(4-4.21)

or
(4-4.22)
Since the linear dimensions of the charge are very small compared to the r's, the difference of the fractions in the last two terms of Eq. (4-4.22) can be regarded as the total differential
(increment) df = (oflox')dx' + (of/oy')dy' of the functions

SECTION 4-4 ACCELERATING POINT CHARGE

87

vxv r-r•v/c

(4-4.23)

and

r-r • vie

(4-4.24)

corresponding to the displacements of the tail of r by I* and by
a*, respectively (see Fig. 4.7).
Using Eq. (3-1.16), noting that r = - x'i - y'j, noting that
vY = 0 (because vis parallel to the x axis), and remembering that v and v are functions of the retarded time t' = t - rlc, so that ov/ox' = (ov/ot')ot'/ox' = (ov/ot')x'lrc = vx'lrc with similar
expressions for ov/oy', ovlox', and ov/oy', we have for the needed
partial derivatives of the two functions

a(

vv

)

-x'-rv/c-(r•v)x'/c2

ox' r[l -(/. v)/rc] = vxv r 3[1 x-(r • v)/rc]2

(4-4.25)

(1\v + vxv)x' r 2c[l -(r • v)/rc]'

and
In evaluating Eq. (4-4.22) with the help of Eqs. (4-4.25)-(4-4.27),
we shall omit from Eq. (4-4.25) the terms not containing v, since
they only contribute to the acceleration-independent field Ev, which we already found in Section 4-1. Combining Eqs. (4-4.22), (4-4.25)-(4-4.27), (3-1.14), and (3-1.15), we then have, denoting the acceleration-dependent field as E1A ,

88 CHAPTER 4 MOVING POINT AND LINE CHARGES

41re0e2EJA

=

--=---q-,v-....,.. r(l -r • v/re)

+pab[ -vxv(r•v)x' _ (vxv+vx v)x'] ·----l-,-r 3e 2(1 -r • v /re)2 r 2e(l -r • v /re) (1 -r • v /re)

-pbl [

v vxI Y

ay 'vie
•

r 2e(l -r • v /re) r(l -r • v /re)

(4-4.28)

vyvy'

a(r-r • v/e)]

+ r 2e(l -r • v/re) • r(l -r • v/re) '

or

41re0e2EJA

qv
= ----,-,----,-..,...
r(l -r • v/re)

+ _ _ _q_ _ _ [

-vv(r·v)x' x

-v. n ,

r 2e(l-r•v/re)2 re(l-r•v/re) x

-vv. x'-

v v Y

x'y'v

-v. vy

1

+

v vy'(r Y

•

v)]

.

x

re

Y

re

(4-4.29)

Since r • v = x'v = - x'vx and since - vx x' - vY y' = v • r (see Figs. 4.6 and 4.7), Eq. (4-4.29) reduces to

41re e 2E =

qv

0 JA r(l -r · v/re)

(4-4 •30)

+--=----q-----=- [-v,(-r,-•-v-)-(-r-•-v,)- +(r • v)v +(r • v)v]' r 2e(l -r • v /re)2 re(l -r • v /re)

which after elementary simplifications becomes

E =

qv

+

q~·~v

JA 4,re0e 2r(l -r · v/re)2 41re0e 3r 2 (1 -r • v/re)3

(4-4.31)

Finally, in accordance with Eq. (4-4.7), subtracting Eq. (4-4.31) from Eq. (4-4.12), we obtain for EA