emft book on electro magnetism

8/12/2019 EMFT Book On Electro magnetism

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ElectromagneticField Theory

BOTHID

UPSILON BOOKS


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ELECTROMAGNETIC F IELDTHEORY


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ElectromagneticField Theory

BOTHIDSwedish Institute of Space Physics

Uppsala, Sweden

and

Department of Astronomy and Space PhysicsUppsala University, Sweden

and

LOIS Space Centre

School of Mathematics and Systems EngineeringVxj University, Sweden

UPSILON BOOKS

UPPSALA

SWEDEN


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Also available

ELECTROMAGNETICF IELDTHEORYEXERCISES

by

Tobia Carozzi, Anders Eriksson, Bengt Lundborg,Bo Thid and Mattias Waldenvik

Freely downloadable fromwww.plasma.uu.se/CED

This book was typeset in LATEX 2(based on TEX3.141592and Web2C7.4.4) on an HP Visu-alize90003600workstation running HP-UX11.11.

Copyright19972008byBo ThidUppsala, SwedenAll rights reserved.

Electromagnetic Field TheoryISBN X-XXX-XXXXX-X


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To the memory of professor

LEV M IKHAILOVICHERUKHIMOV(19361997)dear friend, great physicist, poet

and a truly remarkable man.


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CONTENTS

Contents ix

List of Figures xiii

Preface xv

1 Classical Electrodynamics 11.1 Electrostatics 21.1.1 Coulombs law 21.1.2 The electrostatic field 3

1.2 Magnetostatics 61.2.1 Ampres law 61.2.2 The magnetostatic field 7

1.3 Electrodynamics 91.3.1 Equation of continuity for electric charge 101.3.2 Maxwells displacement current 101.3.3 Electromotive force 111.3.4 Faradays law of induction 121.3.5 Maxwells microscopic equations 151.3.6 Maxwells macroscopic equations 15

1.4 Electromagnetic duality 161.5 Bibliography 181.6 Examples 20

2 Electromagnetic Waves 252.1 The wave equations 26

2.1.1 The wave equation forE 262.1.2 The wave equation forB 272.1.3 The time-independent wave equation forE 27

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Contents

2.2 Plane waves 302.2.1 Telegraphers equation 312.2.2 Waves in conductive media 32

2.3 Observables and averages 332.4 Bibliography 352.5 Example 36

3 Electromagnetic Potentials 393.1 The electrostatic scalar potential 393.2 The magnetostatic vector potential 403.3 The electrodynamic potentials 403.4 Gauge transformations 413.5 Gauge conditions 42

3.5.1 Lorenz-Lorentz gauge 433.5.2 Coulomb gauge 473.5.3 Velocity gauge 49

3.6 Bibliography 493.7 Examples 51

4 Electromagnetic Fields and Matter 534.1 Electric polarisation and displacement 53

4.1.1 Electric multipole moments 534.2 Magnetisation and the magnetising field 564.3 Energy and momentum 58

4.3.1 The energy theorem in Maxwells theory 584.3.2 The momentum theorem in Maxwells theory 59

4.4 Bibliography 624.5 Example 63

5 Electromagnetic Fields from Arbitrary Source Distributions 655.1 The magnetic field 675.2 The electric field 695.3 The radiation fields 715.4 Radiated energy 74

5.4.1 Monochromatic signals 745.4.2 Finite bandwidth signals 75

5.5 Bibliography 76

6 Electromagnetic Radiation and Radiating Systems 776.1 Radiation from an extended source volume at rest 77

6.1.1 Radiation from a one-dimensional current distribution 78

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6.1.2 Radiation from a two-dimensional current distribution 816.2 Radiation from a localised source volume at rest 85

6.2.1 The Hertz potential 856.2.2 Electric dipole radiation 906.2.3 Magnetic dipole radiation 916.2.4 Electric quadrupole radiation 93

6.3 Radiation from a localised charge in arbitrary motion 936.3.1 The Linard-Wiechert potentials 946.3.2 Radiation from an accelerated point charge 976.3.3 Bremsstrahlung 1056.3.4 Cyclotron and synchrotron radiation 1086.3.5 Radiation from charges moving in matter 116

6.4 Bibliography 1236.5 Examples 124

7 Relativistic Electrodynamics 1337.1 The special theory of relativity 133

7.1.1 The Lorentz transformation 134

7.1.2 Lorentz space 1367.1.3 Minkowski space 141

7.2 Covariant classical mechanics 1437.3 Covariant classical electrodynamics 145

7.3.1 The four-potential 1457.3.2 The Linard-Wiechert potentials 1467.3.3 The electromagnetic field tensor 148

7.4 Bibliography 152

8 Electromagnetic Fields and Particles 1558.1 Charged particles in an electromagnetic field 155

8.1.1 Covariant equations of motion 1558.2 Covariant field theory 161

8.2.1 Lagrange-Hamilton formalism for fields and interactions 1628.3 Bibliography 1698.4 Example 171

F Formul 173F.1 The electromagnetic field 173

F.1.1 Maxwells equations 173F.1.2 Fields and potentials 174F.1.3 Force and energy 174

F.2 Electromagnetic radiation 174

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Contents

F.2.1 Relationship between the field vectors in a plane wave 174F.2.2 The far fields from an extended source distribution 174F.2.3 The far fields from an electric dipole 175F.2.4 The far fields from a magnetic dipole 175F.2.5 The far fields from an electric quadrupole 175F.2.6 The fields from a point charge in arbitrary motion 175

F.3 Special relativity 176

F.3.1 Metric tensor 176F.3.2 Covariant and contravariant four-vectors 176F.3.3 Lorentz transformation of a four-vector 176F.3.4 Invariant line element 176F.3.5 Four-velocity 176F.3.6 Four-momentum 177F.3.7 Four-current density 177F.3.8 Four-potential 177F.3.9 Field tensor 177

F.4 Vector relations 177F.4.1 Spherical polar coordinates 178

F.4.2 Vector formulae 178F.5 Bibliography 180

M Mathematical Methods 181M.1 Scalars, vectors and tensors 181

M.1.1 Vectors 182M.1.2 Fields 183M.1.3 Vector algebra 187M.1.4 Vector analysis 189

M.2 Analytical mechanics 191M.2.1 Lagranges equations 191M.2.2 Hamiltons equations 192

M.3 Examples 194M.4 Bibliography 202

Index 203

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LIST OFFIGURES

1.1 Coulomb interaction between two electric charges 31.2 Coulomb interaction for a distribution of electric charges 51.3 Ampre interaction 71.4 Moving loop in a varyingBfield 13

5.1 Radiation in the far zone 73

6.1 Linear antenna 796.2 Electric dipole antenna geometry 806.3 Loop antenna 826.4 Multipole radiation geometry 876.5 Electric dipole geometry 896.6 Radiation from a moving charge in vacuum 946.7 An accelerated charge in vacuum 966.8 Angular distribution of radiation during bremsstrahlung 1056.9 Location of radiation during bremsstrahlung 1076.10 Radiation from a charge in circular motion 1096.11 Synchrotron radiation lobe width 1116.12 The perpendicular electric field of a moving charge 114

6.13 Electron-electron scattering 1166.14 Vavilov-Cerenkov cone 120

7.1 Relative motion of two inertial systems 1357.2 Rotation in a 2D Euclidean space 1417.3 Minkowski diagram 142

8.1 Linear one-dimensional mass chain 162

M.1 Tetrahedron-like volume element of matter 194

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PREFACE

This book is the result of a more than thirty-five year long love a ffair. In theautumn of1972, I took my first advanced course in electrodynamics at the De-partment of Theoretical Physics, Uppsala University. A year later, I joined theresearch group there and took on the task of helping the late professorPER OLO FFRMAN, who one year later become my Ph.D. thesis advisor, with the prepa-ration of a new version of his lecture notes on the Theory of Electricity. Thesetwo things opened up my eyes for the beauty and intricacy of electrodynamics,already at the classical level, and I fell in love with it. Ever since that time, I haveon and offhad reason to return to electrodynamics, both in my studies, researchand the teaching of a course in advanced electrodynamics at Uppsala University

some twenty odd years after I experienced the first encounter with this subject.The current version of the book is an outgrowth of the lecture notes that I pre-pared for the four-credit course Electrodynamics that was introduced in the Up-psala University curriculum in 1992, to become the five-credit course ClassicalElectrodynamics in1997. To some extent, parts of these notes were based on lec-ture notes prepared, in Swedish, by my friend and colleague BENGT LUNDBORG,who created, developed and taught the earlier, two-credit course ElectromagneticRadiation at our faculty.

Intended primarily as a textbook for physics students at the advanced under-graduate or beginning graduate level, it is hoped that the present book may beuseful for research workers too. It provides a thorough treatment of the theoryof electrodynamics, mainly from a classical field theoretical point of view, andincludes such things as formal electrostatics and magnetostatics and their uni-fication into electrodynamics, the electromagnetic potentials, gauge transforma-tions, covariant formulation of classical electrodynamics, force, momentum andenergy of the electromagnetic field, radiation and scattering phenomena, electro-magnetic waves and their propagation in vacuum and in media, and covariantLagrangian/Hamiltonian field theoretical methods for electromagnetic fields, par-ticles and interactions. The aim has been to write a book that can serve both asan advanced text in Classical Electrodynamics and as a preparation for studies inQuantum Electrodynamics and related subjects.

In an attempt to encourage participation by other scientists and students inthe authoring of this book, and to ensure its quality and scope to make it useful

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Preface

in higher university education anywhere in the world, it was produced within aWorld-Wide Web (WWW) project. This turned out to be a rather successful move.By making an electronic version of the book freely down-loadable on the net,comments have been received from fellow Internet physicists around the worldand from WWW hit statistics it seems that the book serves as a frequently usedInternet resource.1 This way it is hoped that it will be particularly useful forstudents and researchers working under financial or other circumstances that make

it difficult to procure a printed copy of the book.Thanks are due not only to Bengt Lundborg for providing the inspiration to

write this book, but also to professorCHRISTERWAHLBERGand professorGRANFLDT, Uppsala University, and professor YAKOV ISTOMIN, Lebedev Institute,Moscow, for interesting discussions on electrodynamics and relativity in generaland on this book in particular. Comments from former graduate studentsMATTIASWALDENVIK,TOBIACAROZZIandROGERKARLSSONas well asANDERSERIKS-SO N, all at the Swedish Institute of Space Physics in Uppsala and who all haveparticipated in the teaching on the material covered in the course and in this bookare gratefully acknowledged. Thanks are also due to my long-term space physicscolleague HELMUT KOPKA of the Max-Planck-Institut fr Aeronomie, Lindau,

Germany, who not only taught me about the practical aspects of high-power radiowave transmitters and transmission lines, but also about the more delicate aspectsof typesetting a book in TEX and LATEX. I am particularly indebted to AcademicianprofessorVITALIYLAZAREVICHGINZBURG,2003Nobel Laureate in Physics, forhis many fascinating and very elucidating lectures, comments and historical noteson electromagnetic radiation and cosmic electrodynamics while cruising on theVolga river at our joint Russian-Swedish summer schools during the 1990s, andfor numerous private discussions over the years.

Finally, I would like to thank all students and Internet users who have down-loaded and commented on the book during its life on the World-Wide Web.

I dedicate this book to my son MATTIAS, my daughter KAROLINA, myhigh-school physics teacher, S TAFFANRSBY, and to my fellow members of theCAPELLAPEDAGOGICAUPSALIENSIS .

Uppsala, Sweden BOTHIDJune,2008 www.physics.irfu.se/bt

1At the time of publication of this edition, more than 600 000 downloads have been recorded.

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1

CLASSICALELECTRODYNAMICS

Classical electrodynamics deals with electric and magnetic fields and interactionscaused bymacroscopicdistributions of electric charges and currents. This meansthat the concepts of localised electric charges and currents assume the validity of

certain mathematical limiting processes in which it is considered possible for thecharge and current distributions to be localised in infinitesimally small volumes ofspace. Clearly, this is in contradiction to electromagnetism on a truly microscopicscale, where charges and currents have to be treated as spatially extended objectsand quantum corrections must be included. However, the limiting processes usedwill yield results which are correct on small as well as large macroscopicscales.

It took the genius ofJAMESCLERKMAXWELLto consistently unify electricityand magnetism into a super-theory, electromagnetismor classical electrodynam-ics(CED), and to realise that optics is a subfield of this super-theory. Early inthe 20th century, HENDRIK A NTOON L ORENTZtook the electrodynamics theoryfurther to the microscopic scale and also laid the foundation for the special the-ory of relativity, formulated by ALBERT EINSTEINin 1905. In the 1930s PAULA. M. DIRAC expanded electrodynamics to a more symmetric form, includingmagnetic as well as electric charges. With his relativistic quantum mechanics,he also paved the way for the development ofquantum electrodynamics (QED)for whichR ICHARD P. FEYNMAN,JULIANS CHWINGER, andS IN-I TIROTOMON-AGA in1965received their Nobel prizes in physics. Around the same time, physi-cists such asSHELDONGLASHOW,ABDUSSALAM, andSTEVENWEINBERGwereable to unify electrodynamics the weak interaction theory to yet another super-theory,electroweak theory, an achievement which rendered them the Nobel prizein physics 1979. The modern theory of strong interactions,quantum chromody-namics(QCD), is influenced by QED.

In this chapter we start with the force interactions in classical electrostatics

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1. Classical Electrodynamics

and classical magnetostatics and introduce the static electric and magnetic fieldsto find two uncoupled systems of equations for them. Then we see how the con-servation of electric charge and its relation to electric current leads to the dynamicconnection between electricity and magnetism and how the two can be unifiedinto one super-theory, classical electrodynamics, described by one system ofeight coupled dynamic field equationsthe Maxwell equations.

At the end of this chapter we study Diracs symmetrised form of Maxwells

equations by introducing (hypothetical) magnetic charges and magnetic currentsinto the theory. While not identified unambiguously in experiments yet, mag-netic charges and currents make the theory much more appealing, for instance byallowing for duality transformations in a most natural way.

1.1 ElectrostaticsThe theory which describes physical phenomena related to the interaction be-tween stationary electric charges or charge distributions in a finite space which

has stationary boundaries is called electrostatics. For a long time, electrostatics,under the name electricity, was considered an independent physical theory of itsown, alongside other physical theories such as magnetism, mechanics, optics andthermodynamics.1

1.1.1 Coulombs lawIt has been found experimentally that in classical electrostatics the interactionbetween stationary, electrically charged bodies can be described in terms of amechanical force. Let us consider the simple case described by figure 1.1 onpage 3. Let F denote the force acting on an electrically charged particle withchargeq located at x, due to the presence of a charge qlocated at x. AccordingtoCoulombs lawthis force is, in vacuum, given by the expression

F(x) = qq

40

x x|x x|3 =

qq

40

1

|x x|

=

qq

40

1|x x|

(1.1)

1The physicist and philosopher P IERREDUHEM(18611916) once wrote:

The whole theory of electrostatics constitutes a group of abstract ideas and general propo-sitions, formulated in the clear and concise language of geometry and algebra, and con-nected with one another by the rules of strict logic. This whole fully satisfies the reason ofa French physicist and his taste for clarity, simplicity and order.. . .

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Electrostatics

q

q

O

x

x x

x

FIGURE1.1: Coulombs law describes how a static electric chargeq, located ata pointx relative to the origin O, experiences an electrostatic force from a static

electric chargeqlocated atx.

where in the last step formula (F.71) on page179was used. In SI units, which weshall use throughout, the forceFis measured in Newton (N), the electric chargesq

andqin Coulomb (C) [= Ampre-seconds (As)], and the length |x x| in metres(m). The constant 0 = 107/(4c2) 8.85421012 Farad per metre (F/m) isthevacuum permittivityandc2.9979 108 m/s is the speed of light in vacuum.In CGS units 0 = 1/(4) and the force is measured in dyne, electric charge instatcoulomb, and length in centimetres (cm).

1.1.2 The electrostatic fieldInstead of describing the electrostatic interaction in terms of a force action at adistance, it turns out that it is for most purposes more useful to introduce the

concept of a field and to describe the electrostatic interaction in terms of a staticvectorialelectric fieldEstat defined by the limiting process

Estat def lim

q0F

q(1.2)

where F is the electrostatic force, as defined in equation (1.1) on page 2, from anet electric charge qon the test particle with a small electric net electric chargeq. Since the purpose of the limiting process is to assure that the test charge qdoesnot distort the field set up by q, the expression forEstat does not depend explicitlyon q but only on the charge q and the relative radius vector xx. This meansthat we can say that any net electric charge produces an electric field in the space

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that surrounds it, regardless of the existence of a second charge anywhere in thisspace.2

Using(1.1) and equation (1.2)on page 3, and formula (F.70)on page 179,we find that the electrostatic field Estat at the field point x (also known as theobservation point), due to a field-producing electric charge qat thesource pointx, is given by

Estat(x) = q40

x x|x x|3 =

q40

1|x x|

= q

40

1|x x|(1.3)

In the presence of several field producing discrete electric charges qi , locatedat the pointsxi ,i =1, 2, 3, . . . , respectively, in an otherwise empty space, the as-sumption of linearity of vacuum3 allows us to superimpose their individual elec-trostatic fields into a total electrostatic field

Estat(x) = 140

i

qix xi

x xi

3 (1.4)

If the discrete electric charges are small and numerous enough, we introducethe electric charge density , measured in C/m3 in SI units, located at x withina volume V of limited extent and replace summation with integration over thisvolume. This allows us to describe the total field as

Estat(x) = 140

V

d3x(x) x x|x x|3 =

140

V

d3x(x)

1|x x|

= 1

40

V

d3x (x)|x x|

(1.5)

where we used formula(F.70) on page179and the fact that(x) does not depend

on the unprimed (field point) coordinates on which

operates.2In the preface to the first edition of the first volume of his bookA Treatise on Electricity and Mag-

netism, first published in1873, James Clerk Maxwell describes this in the following almost poetic manner[9]:

For instance, Faraday, in his minds eye, saw lines of force traversing all space where themathematicians saw centres of force attracting at a distance: Faraday saw a medium wherethey saw nothing but distance: Faraday sought the seat of the phenomena in real actionsgoing on in the medium, they were satisfied that they had found it in a power of action ata distance impressed on the electric fluids.

3In fact, vacuum exhibits a quantum mechanical nonlinearitydue tovacuum polarisation effectsman-ifesting themselves in the momentary creation and annihilation of electron-positron pairs, but classicallythis nonlinearity is negligible.



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Electrostatics

V

qi

q

O

xi

x xix

FIGURE 1.2: Coulombs law for a distribution of individual chargesqilocalisedwithin a volumeVof limited extent.

We emphasise that under the assumption of linear superposition, equa-tion (1.5) on page 4is valid for an arbitrary distribution of electric charges, in-

cluding discrete charges, in which case is expressed in terms of Dirac deltadistributions:

(x) = i

qi(x xi ) (1.6)

as illustrated in figure 1.2.Inserting this expression into expression (1.5)onpage4we recover expression (1.4)on page4.

Taking the divergence of the general Estat expression for an arbitrary electriccharge distribution, equation(1.5) on page 4, and using the representation of theDirac delta distribution, formula (F.73) on page179,we find that

Estat(x) = 140

V

d3x(x) x x|x x|3

= 140

V

d3x(x)

1|x x|

= 140

V

d3x(x) 2

1|x x|

=

10

V

d3x(x) (x x) = (x)0

(1.7)

which is the differential form ofGausss law of electrostatics.Since, according to formula (F.62) on page 179, [(x)] 0 for any 3D



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R3 scalar field(x), we immediately find that in electrostatics

Estat(x) = 140

V

d3x (x)|x x|

= 0 (1.8)

i.e., thatEstat is an irrotationalfield.To summarise, electrostatics can be described in terms of two vector partial

differential equations

Estat(x) = (x)0

(1.9a)

Estat(x) = 0 (1.9b)

representing four scalar partial differential equations.

1.2 MagnetostaticsWhile electrostatics deals with static electric charges, magnetostaticsdeals with

stationary electric currents,i.e., electric charges moving with constant speeds, andthe interaction between these currents. Here we shall discuss this theory in somedetail.

1.2.1 Ampres lawExperiments on the interaction between two small loops of electric current haveshown that they interact via a mechanical force, much the same way that electriccharges interact. In figure1.3on page7, let Fdenote such a force acting on asmall loopC, with tangential line element dl, located at x and carrying a currentIin the direction of dl, due to the presence of a small loop C, with tangential

line element dl, located at x and carrying a current I in the direction of dl.According toAmpres lawthis force is, in vacuum, given by the expression

F(x) = 0II

4

C

dl

Cdl

x x|x x|3

= 0II

4

C

dl

Cdl

1

|x x|

(1.10)In SI units, 0 = 4107 1.2566106 H/m is the vacuum permeability.From the definition of0and 0(in SI units) we observe that

00 = 107

4c2 (F/m) 4 107 (H/m) = 1

c2 (s2/m2) (1.11)



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Magnetostatics

C

C

IdlIdl

O

x

x x

x

FIGURE 1.3: Ampres law describes how a small loopC, carrying a staticelectric current Ithrough its tangential line element dllocated at x, experiencesa magnetostatic force from a small loop C, carrying a static electric current I

through the tangential line element dllocated atx. The loops can have arbitraryshapes as long as they are simple and closed.

which is a most useful relation.At first glance, equation (1.10) on page6may appear unsymmetric in terms of

the loops and therefore to be a force law which is in contradiction with Newtonsthird law. However, by applying the vector triple product bac-cab formula (F.51)on page178,we can rewrite(1.10)as

F(x) = 0II

4

C

dl

Cdl

1

|x x|

0II

4

C

C

x x|x x|3dl dl

(1.12)

Since the integrand in the first integral is an exact differential, this integral van-

ishes and we can rewrite the force expression, equation (1.10) on page 6,in thefollowing symmetric way

F(x) = 0II

4

C

C

x x|x x|3dl dl

(1.13)

which clearly exhibits the expected symmetry in terms of loopsCand C.

1.2.2 The magnetostatic fieldIn analogy with the electrostatic case, we may attribute the magnetostatic interac-tion to a static vectorial magnetic fieldBstat. It turns out that the elemental Bstat



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can be defined as

dBstat(x)def 0I

4dl

x x|x x|3 (1.14)

which expresses the small element dBstat(x) of the static magnetic field set up atthe field point x by a small line element dlof stationary current Iat the sourcepointx. The SI unit for the magnetic field, sometimes called the magnetic fluxdensityormagnetic induction, is Tesla (T).

If we generalise expression (1.14) to an integrated steady state electric currentdensityj(x), measured in A/m2 in SI units, we obtain Biot-Savarts law:

Bstat(x) = 0

4

V

d3xj(x) x x|x x|3 =

0

4

V

d3xj(x)

1|x x|

=

0

4

V

d3x j(x)|x x|

(1.15)

where we used formula(F.70)on page 179,formula (F.57) on page179, and thefact thatj(x) does not depend on the unprimed coordinates on which operates.

Comparing equation (1.5) on page4with equation (1.15), we see that there existsa close analogy between the expressions for Estat and Bstat but that they differin their vectorial characteristics. With this definition ofBstat, equation(1.10)onpage6may we written

F(x) = I

Cdl Bstat(x) (1.16)

In order to assess the properties ofBstat, we determine its divergence and curl.Taking the divergence of both sides of equation(1.15) and utilising formula(F.63)on page179, we obtain

Bstat(x) =

0

4

Vd3x

j(x)

|x x| = 0 (1.17)since, according to formula (F.63) on page179, ( a) vanishes for any vectorfielda(x).

Applying the operator bac-cab rule, formula (F.64)on page 179,the curl ofequation(1.15)can be written

Bstat(x) = 0

4

V

d3x j(x)|x x|

=

= 04

V

d3xj(x) 2

1|x x|

+

0

4

V

d3x[j(x) ]

1|x x|

(1.18)



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However, when we include time-dependence, these theories are unified intoone theory,classical electrodynamics. This unification of the theories of electric-ity and magnetism is motivated by two empirically established facts:

1. Electric charge is a conserved quantity and electric current is a transport ofelectric charge. This fact manifests itself in the equation of continuity and,as a consequence, in Maxwells displacement current.

2. A change in the magnetic flux through a loop will induce an EMF electricfield in the loop. This is the celebrated Faradays law of induction.

1.3.1 Equation of continuity for electric chargeLetj(t, x) denote the time-dependent electric current density. In the simplest caseit can be defined as j = vwhere vis the velocity of the electric charge den-sity . In general, j has to be defined in statistical mechanical terms as j(t, x) =q

d3v vf(t, x, v) where f(t, x, v) is the (normalised) distribution function for

particle specieswith electric chargeq.The electric charge conservation law can be formulated in the equation ofcontinuity

(t, x)t

+ j(t, x) = 0 (1.23)

which states that the time rate of change of electric charge (t, x) is balanced by adivergence in the electric current densityj(t, x).

1.3.2 Maxwells displacement currentWe recall from the derivation of equation (1.20) on page 9 that there we used thefact that in magnetostatics j(x) = 0. In the case of non-stationary sourcesand fields, we must, in accordance with the continuity equation (1.23), set j(t, x) = (t, x)/t. Doing so, and formally repeating the steps in the derivationof equation (1.20) on page9,we would obtain the formal result

B(t, x) = 0

V

d3xj(t, x)(x x) + 04

t

V

d3x(t, x)

1|x x|

= 0j(t, x) +0

t0E(t, x)

(1.24)



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Electrodynamics

where, in the last step, we have assumed that a generalisation of equation (1.5)onpage4to time-varying fields allows us to make the identification4

140

t

V

d3x(t, x)

1|x x|

=

t 140 V

d3x(t, x) 1

|x

x

|=

t

1

40

V

d3x (t, x)|x x|

=

tE(t, x)

(1.25)

The result is Maxwells source equation for theBfield

B(t, x) = 0

j(t, x) +

t0E(t, x)

= 0j(t, x) +

1c2

tE(t, x) (1.26)

where the last term 0E(t, x)/tis the famous displacement current. This termwas introduced, in a stroke of genius, by Maxwell [8] in order to make the righthand side of this equation divergence free when j(t, x) is assumed to represent thedensity of the total electric current, which can be split up in ordinary conduc-

tion currents, polarisation currents and magnetisation currents. The displacementcurrent is an extra term which behaves like a current density flowing in vacuum.As we shall see later, its existence has far-reaching physical consequences as itpredicts the existence of electromagnetic radiation that can carry energy and mo-mentum over very long distances, even in vacuum.

1.3.3 Electromotive forceIf an electric field E(t, x) is applied to a conducting medium, a current densityj(t, x) will be produced in this medium. There exist also hydrodynamical andchemical processes which can create currents. Under certain physical conditions,

and for certain materials, one can sometimes assume, that, as a first approxima-tion, a linear relationship exists between the electric current density jandE. Thisapproximation is calledOhms law:

j(t, x) = E(t, x) (1.27)

whereis theelectric conductivity(S/m). In the most general cases, for instancein an anisotropic conductor,is a tensor.

We can view Ohms law, equation (1.27) above, as the first term in a Taylorexpansion of the law j[E(t, x)]. This general law incorporates non-linear effects

4Later, we will need to consider this generalisation and formal identification further.



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Electrodynamics

d2x n

B(x) B(x)

v

dl

C

FIGURE 1.4: A loop Cwhich moves with velocityvin a spatially varying mag-netic fieldB(x) will sense a varying magnetic flux during the motion.

where m is the magnetic flux and S is the surface encircled by Cwhich canbe interpreted as a generic stationary loop and not necessarily as a conductingcircuit. Application of Stokes theorem on this integral equation, transforms itinto the differential equation

E(t, x) = tB(t, x) (1.32)

which is valid for arbitrary variations in the fields and constitutes the Maxwellequation which explicitly connects electricity with magnetism.

Any change of the magnetic flux m will induce an EMF. Let us thereforeconsider the case, illustrated if figure 1.4,that the loop is moved in such a way

that it links a magnetic field which varies during the movement. Theconvectivederivativeis evaluated according to the well-known operator formula

ddt

=

t + v (1.33)

which follows immediately from the rules of differentiation of an arbitrary differ-entiable function f(t, x(t)). Applying this rule to Faradays law, equation(1.31)on page12, we obtain

E(t) = ddt

S

d2x n B =

Sd2x n B

t

Sd2x n (v )B (1.34)



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During spatial differentiation v is to be considered as constant, and equa-tion (1.17) on page8holds also for time-varying fields:

B(t, x) = 0 (1.35)(it is one of Maxwells equations) so that, according to formula (F.59)onpage 179,

(B v) = (v

)B (1.36)

allowing us to rewrite equation (1.34) on page13in the following way:

E(t) =

Cdl EEMF = d

dt

S

d2x n B

=

Sd2x n B

t

Sd2x n (B v)

(1.37)

With Stokes theorem applied to the last integral, we finally get

E(t) =

Cdl EEMF =

S

d2x n Bt

Cdl (B v) (1.38)

or, rearranging the terms,C

dl (EEMF v B) =

Sd2x n B

t(1.39)

whereEEMF is the field which is induced in the loop, i.e., in themovingsystem.The use of Stokes theorem backwards on equation (1.39) above yields

(EEMF v B) = Bt

(1.40)

In thefixedsystem, an observer measures the electric field

E = EEMF v B (1.41)

Hence, a moving observer measures the following Lorentz forceon a charge q

qEEMF = qE + q(v B) (1.42)

corresponding to an effective electric field in the loop (moving observer)

EEMF = E + v B (1.43)

Hence, we can conclude that for astationaryobserver, the Maxwell equation

E = Bt

(1.44)

is indeed valid even if the loop is moving.



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Electrodynamics

1.3.5 Maxwells microscopic equationsWe are now able to collect the results from the above considerations and formulatethe equations of classical electrodynamics valid for arbitrary variations in time andspace of the coupled electric and magnetic fieldsE(t, x) andB(t, x). The equationsare

E =

0(1.45a)

E = Bt

(1.45b)

B = 0 (1.45c)

B = 00E

t+0j(t, x) (1.45d)

In these equations(t, x) represents the total, possibly both time and space depen-dent, electric charge,i.e., free as well as induced (polarisation) charges, andj(t, x)represents the total, possibly both time and space dependent, electric current, i.e.,conduction currents (motion of free charges) as well as all atomistic (polarisation,magnetisation) currents. As they stand, the equations therefore incorporate the

classical interaction between all electric charges and currents in the system andare called Maxwells microscopic equations. Another name often used for themis the Maxwell-Lorentz equations. Together with the appropriate constitutive re-lations, which relate andjto the fields, and the initial and boundary conditionspertinent to the physical situation at hand, they form a system of well-posed partialdifferential equations which completely determineEandB.

1.3.6 Maxwells macroscopic equationsThe microscopic field equations (1.45) provide a correct classical picture for arbi-trary field and source distributions, including both microscopic and macroscopic

scales. However, for macroscopic substances it is sometimes convenient to intro-duce new derived fields which represent the electric and magnetic fields in which,in an average sense, the material properties of the substances are already included.These fields are the electric displacementDand the magnetising fieldH. In themost general case, these derived fields are complicated nonlocal, nonlinear func-tionals of the primary fields EandB:

D = D[t, x;E,B] (1.46a)

H = H[t, x;E, B] (1.46b)

Under certain conditions, for instance for very low field strengths, we may assumethat the response of a substance to the fields may be approximated as a linear one



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so that

D = E (1.47)

H = 1B (1.48)

i.e., that the derived fields are linearly proportional to the primary fields and thatthe electric displacement (magnetising field) is only dependent on the electric

(magnetic) field.The field equations expressed in terms of the derived field quantities DandH

are

D = (t, x) (1.49a)

E = Bt

(1.49b)

B = 0 (1.49c)

H = D

t +j(t, x) (1.49d)

and are called Maxwells macroscopic equations. We will study them in moredetail in chapter4.

1.4 Electromagnetic dualityIf we look more closely at the microscopic Maxwell equations (1.45), we see thatthey exhibit a certain, albeit not complete, symmetry. Let us follow Dirac andmake the ad hocassumption that there exist magnetic monopolesrepresented bya magnetic charge density, which we denote by m = m(t, x), and a magneticcurrent density, which we denote byjm = jm(t, x). With these new quantities in-

cluded in the theory, and with the electric charge density denoted e and the elec-tric current density denoted je, the Maxwell equations will be symmetrised intothe following two scalar and two vector, coupled, partial differential equations:

E = e

0(1.50a)

E = Bt

0jm (1.50b) B = 0m (1.50c)

B = 00E

t+0j

e (1.50d)



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Electromagnetic duality

We shall call these equationsDiracs symmetrised Maxwell equationsor theelec-tromagnetodynamic equations.

Taking the divergence of (1.50b), we find that

( E) = t

( B) 0 jm 0 (1.51)

where we used the fact that, according to formula (F.63)on page 179, the diver-

gence of a curl always vanishes. Using (1.50c) to rewrite this relation, we obtainthemagnetic monopole equation of continuity

m

t + jm = 0 (1.52)

which has the same form as that for the electric monopoles (electric charges) andcurrents, equation (1.23) on page10.

We notice that the new equations (1.50) on page16exhibit the following sym-metry (recall that00 = 1/c2):

EcB (1.53a)

cB E (1.53b)ce m (1.53c)m ce (1.53d)cje jm (1.53e)jm cje (1.53f)

which is a particular case ( = /2) of the general duality transformation, alsoknown as the Heaviside-Larmor-Rainich transformation (indicted by the Hodgestar operator )

E = E cos + cB sin (1.54a)

c

B = E sin + cB cos (1.54b)ce = ce cos +m sin (1.54c)m = ce sin +m cos (1.54d)cje = cje cos +jm sin (1.54e)jm = cje sin +jm cos (1.54f)

which leaves the symmetrised Maxwell equations, and hence the physics theydescribe (often referred to aselectromagnetodynamics), invariant. SinceEandje

are (true or polar) vectors, Ba pseudovector (axial vector),e a (true) scalar, thenm and, which behaves as a mixing anglein a two-dimensional charge space,must be pseudoscalars andjm a pseudovector.



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The invariance of Diracs symmetrised Maxwell equations under the similaritytransformation means that the amount of magnetic monopole densitym is irrele-vant for the physics as long as the ratiom/e = tan is kept constant. So whetherwe assume that the particles are only electrically charged or have also a magneticcharge with a given, fixed ratio between the two types of charges is a matter ofconvention, as long as we assume that this fraction isthe same for all particles.Such particles are referred to asdyons[14]. By varying the mixing anglewe can

change the fraction of magnetic monopoles at will without changing the laws ofelectrodynamics. For= 0 we recover the usual Maxwell electrodynamics as weknow it.5

1.5 Bibliography[1] T. W. BARRETT ANDD. M. GRIMES,Advanced Electromagnetism. Foundations, Theory

and Applications, World Scientific Publishing Co., Singapore, 1995, ISBN 981-02-2095-2.

[2] R. BECKER, Electromagnetic Fields and Interactions, Dover Publications, Inc.,New York, NY, 1982, ISBN 0-486-64290-9.

[3] W. GREINER, Classical Electrodynamics, Springer-Verlag, New York, Berlin, Heidel-berg, 1996, ISBN 0-387-94799-X.

[4] E. HALLN,Electromagnetic Theory, Chapman & Hall, Ltd., London, 1962.

[5] J . D . JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc.,New York, NY . . . , 1999, ISBN 0-471-30932-X.

[6] L. D. LANDAU AND E. M. L IFSHITZ, The Classical Theory of Fields, fourth revisedEnglish ed., vol. 2 ofCourse of Theoretical Physics, Pergamon Press, Ltd., Oxford ...,

1975, ISBN 0-08-025072-6.[7] F. E. LOW,Classical Field Theory, John Wiley & Sons, Inc., New York, NY ..., 1997,

ISBN 0-471-59551-9.

[8] J. C. MAXWELL, A dynamical theory of the electromagnetic field, Royal Society Trans-actions, 155(1864). 11

5As Julian Schwinger (19181994) put it[15]:

...there are strong theoretical reasons to believe that magnetic charge exists in nature,and may have played an important role in the development of the universe. Searches formagnetic charge continue at the present time, emphasising that electromagnetism is veryfar from being a closed object.

http://-/?-http://-/?-


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Bibliography

[9] J. C. MAXWELL,A Treatise on Electricity and Magnetism, third ed., vol. 1, Dover Publi-cations, Inc., New York, NY, 1954, ISBN 0-486-60636-8. 4

[10] J. C. MAXWELL,A Treatise on Electricity and Magnetism, third ed., vol. 2, Dover Publi-cations, Inc., New York, NY, 1954, ISBN 0-486-60637-8.

[11] D. B. MELROSE ANDR. C. MCPHEDRAN,Electromagnetic Processes in Dispersive Me-dia, Cambridge University Press, Cambridge ..., 1991, ISBN 0-521-41025-8.

[12] W. K. H. PANOFSKY ANDM. PHILLIPS, Classical Electricity and Magnetism, second ed.,Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702-6.

[13] F. ROHRLICH,Classical Charged Particles, Perseus Books Publishing, L.L.C., Reading,MA ..., 1990, ISBN 0-201-48300-9.

[14] J. SCHWINGER, A magnetic model of matter,Science, 165(1969), pp. 757761. 18

[15] J. SCHWINGER, L. L. DERAA D, J R., K. A. M ILTON,AN DW. TSA I,Classical Electrody-namics, Perseus Books, Reading, MA, 1998, ISBN 0-7382-0056-5. 18,121

[16] J . A. STRATTON, Electromagnetic Theory, McGraw-Hill Book Company, Inc., NewYork, NY and London, 1953, ISBN 07-062150-0.

[17] J. VANDERLINDE, Classical Electromagnetic Theory, John Wiley & Sons, Inc., NewYork, Chichester, Brisbane, Toronto, and Singapore, 1993, ISBN 0-471-57269-1.

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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1.6 Examples

FARADAYS LAW AS A CONSEQUENCE OF CONSERVATION OF MAGNETIC CHARGEEXAMPLE1.1

Postulate1.1(Indestructibility of magnetic charge). Magnetic charge exists and is indestruc-

tible in the same way that electric charge exists and is indestructible. In other words we postu-latethat there exists an equation of continuity for magnetic charges:

m(t, x)t

+ jm(t, x) = 0

Use this postulate and Diracs symmetrised form of Maxwells equations to derive Fara-days law.

The assumption of the existence of magnetic charges suggests a Coulomb-like law for mag-netic fields:

Bstat(x) = 0

4

V

d3xm(x) x x|x x |3 =

0

4

V

d3xm(x)

1

|x x|

=

0

4 Vd3x

m(x)

|x x |

(1.55)

[cf.equation (1.5)on page4forEstat] and, if magnetic currents exist, a Biot-Savart-like law forelectric fields [cf.equation (1.15)on page8forBstat]:

Estat(x) = 04

V

d3xjm(x) x x|x x|3 =

0

4

V

d3xjm(x)

1

|x x |

= 04

V

d3x jm(x)|x x|

(1.56)

Taking the curl of the latter and using the operator bac-cab rule, formula (F.59) on page179,we find that

Estat(x) = 04

V

d3x jm(x)|x x|

=

= 0

4

V

d3xjm(x)2

1|x x|

0

4

V

d3x[jm(x) ]

1|x x|

(1.57)Comparing with equation (1.18) on page8forEstat and the evaluation of the integrals there, weobtain

Estat(x) = 0

Vd3xjm(x) (x x) = 0jm(x) (1.58)

We assume that formula (1.56) above is valid also for time-varying magnetic currents.Then, with the use of the representation of the Dirac delta function, equation (F.73)on page179,the equation of continuity for magnetic charge, equation(1.52)on page17,and the assumptionof the generalisation of equation (1.55) to time-dependent magnetic charge distributions, weobtain, formally,



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Examples

E(t, x) = 0

Vd3xjm(t, x)(x x) 0

4

t

V

d3xm(t, x)

1|x x|

= 0jm(t, x) tB(t, x)

(1.59)

[cf. equation(1.24)on page10] which we recognise as equation(1.50b) on page16. A trans-formation of this electromagnetodynamic result by rotating into the electric realm of charge

space, thereby letting jm tend to zero, yields the electrodynamic equation (1.50b) on page16,i.e., the Faraday law in the ordinary Maxwell equations. This process also provides an alter-native interpretation of the termB/tas a magnetic displacement current, dual to theelectricdisplacement current[cf. equation(1.26)on page11].

By postulating the indestructibility of a hypothetical magnetic charge, we have thereby beenable to replace Faradays experimental results on electromotive forces and induction in loops asa foundation for the Maxwell equations by a more appealing one.

END OF EXAMPLE1.1

DUALITY OF THE ELECTROMAGNETODYNAMIC EQUATIONS EXAMPLE1.2

Show that the symmetric, electromagnetodynamic form of Maxwells equations (Diracssymmetrised Maxwell equations), equations(1.50) on page 16,are invariant under the dualitytransformation(1.54).

Explicit application of the transformation yields

E = (E cos + cB sin ) = e

0cos + c0m sin

= 10

e cos +1

cm sin

=

e

0(1.60)

E +

B

t = (E cos + cB sin ) +

t

1

cE sin + B cos

= 0jm cos Bt

cos + c0je sin +1c

E

tsin

1c

E

tsin +

B

tcos = 0jm cos + c0je sin

= 0(cje sin +jm cos ) = 0jm

(1.61)

B = (1cE sin + B cos ) =

e

c0sin +0

m cos

= 0(ce sin +m cos ) = 0m(1.62)



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B 1

c2E

t = (1

cE sin + B cos ) 1

c2

t(E cos + cB sin )

=1c

0jm sin +

1c

B

t cos +0j

e cos + 1c2

E

t cos

1c2

E

t cos 1

c

B

t sin

= 01cjm sin +je cos

= 0je

(1.63)

QED

END OF EXAMPLE1.2

DIRACS SYMMETRISEDMAXWELL EQUATIONS FOR A FIXED MIXING ANGLEEXAMPLE1.3

Show that for a fixed mixing anglesuch that

m = ce tan (1.64a)

jm = cje tan (1.64b)

the symmetrised Maxwell equations reduce to the usual Maxwell equations.Explicit application of the fixed mixing angle conditions on the duality transformation

(1.54)on page17yields

e = e cos +1c

m sin = e cos +1c

ce tan sin

= 1cos

(e cos2 +e sin2 ) = 1cos

e(1.65a)

m = ce sin + ce tan cos = ce sin + ce sin = 0 (1.65b)je = je cos +je tan sin =

1cos

(je cos2 +je sin2 ) = 1cos

je (1.65c)

jm = cje sin + cje tan cos = cje sin + cje sin = 0 (1.65d)Hence, a fixed mixing angle, or, equivalently, a fixed ratio between the electric and magneticcharges/currents, hides the magnetic monopole influence (m and jm) on the dynamic equa-tions.

We notice that the inverse of the transformation given by equation(1.54)on page17yields

E = E cos cB sin (1.66)This means that

E = E cos c B sin (1.67)Furthermore, from the expressions for the transformed charges and currents above, we find that

E =e

0=

1cos

e

0(1.68)



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Examples

and

B = 0m = 0 (1.69)

so that

E = 1cos

e

0cos 0 =

e

0(1.70)

and so on for the other equations. QED

END OF EXAMPLE1.3

COMPLEX FIELD SIX-VECTOR FORMALISM EXAMPLE1.4

It is sometimes convenient to introduce the complex field six-vector, also known as theRiemann-Silberstein vector

G(t, x) = E(t, x) + icB(t, x) (1.71)

where E ,B

R3 and hence G

C3. One fundamental property ofC3 is that inner (scalar)

products in this space are invariant just as they are in R3. However, as discussed in exam-pleM.3on page197, the inner (scalar) product in C3 can be defined in two different ways.Considering the special case of the scalar product ofG with itself, we have the following twopossibilities of defining (the square of) the length ofG:

1. The inner (scalar) product defined asGscalar multiplied with itself

G G = (E + icB) (E + icB) =E2 c2B2 + 2icE B (1.72)Since this is an invariant scalar quantity, we find that

E2 c2B2 = Const (1.73a)E B = Const (1.73b)

2. The inner (scalar) product defined asGscalar multiplied with the complex conjugate ofitself

G G = (E + icB) (E icB) = E2 + c2B2 (1.74)which is also an invariant scalar quantity. As we shall see later, this quantity is propor-tional to the electromagnetic field energy, which indeed is a conserved quantity.

3. As with any vector, the cross product ofGwith itself vanishes:

G G = (E + icB) (E + icB)

= E E c2B B + ic(E B) + ic(B E)= 0 + 0 + ic(E B) ic(E B) = 0

(1.75)

4. The cross product ofGwith the complex conjugate of itself



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G G = (E + icB) (E icB)= E E + c2B B ic(E B) + ic(B E)= 0 + 0 ic(E B) ic(E B) = 2ic(E B)

(1.76)

is proportional to the electromagnetic power flux, to be introduced later.

END OF EXAMPLE1.4

DUALITY EXPRESSED IN THE COMPLEX FIELD SIX-VECTOREXAMPLE1.5

Expressed in the Riemann-Silberstein complex field vector, introduced in exam-ple1.4on page23,the duality transformation equations (1.54) on page17become

G = E + icB = E cos + cB sin iE sin + icB cos = E(cos isin ) + icB(cos isin ) = ei(E + icB) = eiG (1.77)

from which it is easy to see that

G G =

G

2

= eiG eiG = |G|2 (1.78)

whileG G = e2iG G (1.79)

Furthermore, assuming that = (t, x), we see that the spatial and temporal differentiationofGleads to

tG

G

t= i(t)eiG + eitG (1.80a)

G G = iei G + ei G (1.80b) G G = iei G + ei G (1.80c)

which means that tGtransforms as Gitself only ifis time-independent, and that Gand Gtransform as Gitself only ifis space-independent.

END OF EXAMPLE1.5



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2

ELECTROMAGNETICWAVES

In this chapter we investigate the dynamical properties of the electromagnetic fieldby deriving a set of equations which are alternatives to the Maxwell equations. It

turns out that these alternative equations are wave equations, indicating that elec-tromagnetic waves are natural and common manifestations of electrodynamics.

Maxwells microscopic equations [cf.equations (1.45) on page15] are

E = (t, x)0

(Gausss law) (2.1a)

E = Bt

(Faradays law) (2.1b)

B = 0 (No free magnetic charges) (2.1c)

B = 0j(t, x) + 00E

t (Maxwells law) (2.1d)

and can be viewed as an axiomatic basis for classical electrodynamics. They de-scribe, in scalar and vector differential equation form, the electric and magneticfieldsEandBproduced by given, prescribed charge distributions (t, x) and cur-rent distributionsj(t, x) with arbitrary time and space dependences.

However, as is well known from the theory of differential equations, these fourfirst order, coupled partial differential vector equations can be rewritten as two un-coupled, second order partial equations, one for Eand one forB. We shall derivethese second order equations which, as we shall see are wave equations, and thendiscuss the implications of them. We show that for certain media, theBwave fieldcan be easily obtained from the solution of the Ewave equation.

25


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2. Electromagnetic Waves

2.1 The wave equationsWe restrict ourselves to derive the wave equations for the electric field vector Eand the magnetic field vector Bin an electrically neutral region, i.e., a volumewhere there is no net charge, = 0, and no electromotive force EEMF = 0.

2.1.1 The wave equation forEIn order to derive the wave equation forE we take the curl of (2.1b) and use(2.1d),to obtain

( E) = t

( B) = 0

t

j + 0

tE

(2.2)

According to the operator triple product bac-cab rule equation (F.64)onpage 179

( E) = ( E) 2E (2.3)

Furthermore, since = 0, equation (2.1a) on page25yields

E = 0 (2.4)

and sinceEEMF = 0, Ohms law, equation(1.28)on page 12,allows us to use theapproximation

j = E (2.5)

we find that equation(2.2) above can be rewritten

2E 0

t

E + 0

tE

= 0 (2.6)

or, also using equation (1.11) on page6and rearranging,

2E 0E

t 1

c22E

t2 = 0 (2.7)

which is the homogeneous wave equation for E in a uncharged, conducting mediumwithout EMF. For waves propagating in vacuum (no charges, no currents), thewave equation forEis

2E 1c2

2E

t2 = 2E = 0 (2.8)

where 2 is the dAlembert operator, defined according to formula (M.97) onpage199.



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The wave equations

2.1.2 The wave equation forBThe wave equation for B is derived in much the same way as the wave equationforE. Take the curl of (2.1d) and use Ohms lawj = Eto obtain

( B) = 0 j + 00

t( E) = 0 E + 00

t( E)

(2.9)

which, with the use of equation (F.64) on page179and equation(2.1c) on page25can be rewritten

( B) 2B = 0B

t 00

2

t2B (2.10)

Using the fact that, according to (2.1c), B = 0 for any medium and rearranging,we can rewrite this equation as

2B 0B

t 1

c22B

t2 = 0 (2.11)

This is the wave equation for the magnetic field. For waves propagating in vacuum

(no charges, no currents), the wave equation forBis

2B 1c2

2B

t2 = 2B = 0 (2.12)

We notice that for the simple propagation media considered here, the waveequations for the magnetic field Bhas exactly the same mathematical form as thewave equation for the electric field E, equation (2.7) on page26.Therefore, it suf-fices to consider only the Efield, since the results for the Bfield follow trivially.For EM waves propagating in more complicated media, containing, eg., inhomo-geneities, the wave equation forE and for B do not have the same mathematicalform.

2.1.3 The time-independent wave equation forEIf we assume that the temporal dependence ofE(andB) is well-behaved enoughthat it can be represented by a sum of a finite number of temporal spectral (Fourier)components, i.e., in the form of a temporal Fourier series, then it is sufficient torepresent the electric field by one of these Fourier components

E(t, x) = E0(x)cos(t) = E0(x)Re

eit

(2.13)

since the general solution is obtained by a linear superposition (summation) of theresult for one such spectral (Fourier) component, often called a time-harmonic



44/223


wave. When we insert this, in complex notation, into equation (2.7) on page 26we find that

2E0(x)eit 0

tE0(x)e

it 1c2

2

t2E0(x)e

it

= 2E0(x)eit 0(i)E0(x)eit 1c2

(i)2E0(x)eit(2.14)

or, dividing out the common factor eit and rewriting,

2E0 +2

c2

1 + i

0

E0 = 0 (2.15)

Multiplying byeit and introducing therelaxation time = 0/of the mediumin question, we see that the differential equation for the time-harmonic wave canbe written

2E(t, x) + 2

c2

1 +

i

E(t, x) = 0 (2.16)

In the limit of very many frequency components the Fourier sum goes overinto aFourier integral. To illustrate this general case, let us introduce the FouriertransformofE(t, x)

F[E(t, x)] def Ew(x) = 12

dtE(t, x) eit (2.17)

and the corresponding inverse Fourier transform

F1[E(x)]def E(t, x) =

dE(x) eit (2.18)

Then we find that the Fourier transform ofE(t, x)/tbecomes

FE(t, x)

t

def 12

dtE(t, x)

t

eit

=12

E(t, x) eit

=0

i 12

dtE(t, x) eit

= iE(x)

(2.19)

and that, consequently,

F

2E(t, x)t2

def 1

2

dt

2E(t, x)

t2

eit = 2E(x) (2.20)



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The wave equations

Fourier transforming equation (2.7)on page 26and using (2.19) and (2.20), weobtain

2E +2

c2

1 +

i

E = 0 (2.21)

A subsequent inverse Fourier transformation of the solution E of this equation

leads to the same result as is obtained from the solution of equation (2.16) onpage 28. I.e., by considering just one Fourier component we obtain the resultswhich are identical to those that we would have obtained by employing the heavymachinery of Fourier transforms and Fourier integrals. Hence, under the assump-tion of linearity (superposition principle) there is no need for the heavy, time-consuming forward and inverse Fourier transform machinery.

In the limit of long, (2.16) tends to

2E + 2

c2E = 0 (2.22)

which is atime-independent wave equationfor E, representing undamped propa-

gating waves. In the short limit we have instead

2E + i0E = 0 (2.23)

which is atime-independent diffusion equationforE.For most metals1014 s, which means that the diffusion picture is good for

all frequencies lower than optical frequencies. Hence, in metallic conductors, thepropagation term 2E/c2t2 is negligible even for VHF, UHF, and SHF signals.Alternatively, we may say that the displacement current 0E/tis negligible rel-ative to the conduction current j = E.

If we introduce the vacuum wave number

k=

c (2.24)

we can write, using the fact that c = 1/

00 according to equation(1.11)onpage6,

1

=

0 =

0

1ck

=

k

0

0=

kR0 (2.25)

where in the last step we introduced the characteristic impedancefor vacuum

R0 =

0

0376.7 (2.26)



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2.2 Plane wavesConsider now the case where all fields depend only on the distanceto a givenplane with unit normal n. Then thedeloperator becomes

= n

= n (2.27)

and Maxwells equations attain the form

n E

= 0 (2.28a)

n E

= B

t(2.28b)

n B

= 0 (2.28c)

n B

= 0j(t, x) + 00

E

t= 0E + 00

E

t(2.28d)

Scalar multiplying (2.28d) by n, we find that

0 = n n B = n 0 + 00 tE (2.29)which simplifies to the first-order ordinary differential equation for the normalcomponentEnof the electric field

dEndt

+

0En = 0 (2.30)

with the solution

En = En0 et/0 = En0 e

t/ (2.31)

This, together with (2.28a), shows that the longitudinal componentofE, i.e., thecomponent which is perpendicular to the plane surface is independent ofand has

a time dependence which exhibits an exponential decay, with a decrement givenby the relaxation time in the medium.Scalar multiplying (2.28b) by n, we similarly find that

0 = n n

E

= n B

t (2.32)

or

n Bt

= 0 (2.33)

From this, and (2.28c), we conclude that the only longitudinal component ofBmust be constant in both time and space. In other words, the only non-staticsolution must consist oftransverse components.



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Plane waves

2.2.1 Telegraphers equationIn analogy with equation (2.7) on page26,we can easily derive the equation

2E

20

E

t 1

c22E

t2 = 0 (2.34)

This equation, which describes the propagation of plane waves in a conductingmedium, is called thetelegraphers equation. If the medium is an insulator so that = 0, then the equation takes the form of the one-dimensional wave equation

2E

2 1

c22E

t2 = 0 (2.35)

As is well known, each component of this equation has a solution which can bewritten

Ei = f( ct) + g( + ct), i = 1, 2, 3 (2.36)where fandgare arbitrary (non-pathological) functions of their respective argu-ments. This general solution represents perturbations which propagate along,where the fperturbation propagates in the positivedirection and thegperturba-tion propagates in the negativedirection.

If we assume that our electromagnetic fields E and B are time-harmonic,i.e., that they can each be represented by a Fourier component proportional toexp{it}, the solution of equation (2.35) above becomes

E = E0ei(tk) = E0ei(kt) (2.37)

By introducing the wave vector

k = kn =

cn =

ck (2.38)

this solution can be written as

E = E0ei(kxt) (2.39)

Let us consider the lower sign in front ofkin the exponent in (2.37). Thiscorresponds to a wave which propagates in the direction of increasing . Insertingthis solution into equation (2.28b) on page30,gives

n E

= iB = ikn E (2.40)

or, solving forB,

B = k

n E =

1k E =

1ck E =

00 n E (2.41)

Hence, to each transverse component ofE, there exists an associated magneticfield given by equation (2.41) above. IfEand/orBhas a direction in space whichis constant in time, we have aplane wave.



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2.2.2 Waves in conductive mediaAssuming that our medium has a finite conductivity , and making the time-harmonic wave Ansatz in equation (2.34) on page 31, we find that the time-independent telegraphers equationcan be written

2E

2

+ 002E + i0E =

2E

2

+ K2E = 0 (2.42)

where

K2 = 002

1 + i

0

=

2

c2

1 + i

0

= k2

1 + i

0

(2.43)

where, in the last step, equation (2.24) on page29was used to introduce the wavenumberk. Taking the square root of this expression, we obtain

K= k

1 + i

0 = + i (2.44)

Squaring, one finds that

k2

1 + i

0

= (2 2) + 2i (2.45)

or

2 = 2 k2 (2.46)

= k2

20 (2.47)

Squaring the latter and combining with the former, one obtains the second orderalgebraic equation (in2)

2(2 k2) = k4

2

4202 (2.48)

which can be easily solved and one finds that

= k

1 +

0

2+ 1

2 (2.49a)

= k

1 +

0

2 1

2 (2.49b)



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Observables and averages

As a consequence, the solution of the time-independent telegraphers equation,equation (2.42) on page32,can be written

E = E0eei(t) (2.50)

With the aid of equation (2.41) on page 31 we can calculate the associated mag-netic field, and find that it is given by

B = 1

Kk E = 1

(k E)( + i) = 1

(k E) |A| ei (2.51)

where we have, in the last step, rewritten +i in the amplitude-phase form|A| exp{i}. From the above, we immediately see that E, and consequently also B,is damped, and thatEandBin the wave are out of phase.

In the limit0, we can approximate Kas follows:

K= k

1 + i

0

12

= k

i

0

1 i 0

12k(1 + i)

20

=

00(1 + i) 20

= (1 + i)02

(2.52)

In this limit we find that when the wave impinges perpendicularly upon the medium,the fields are given, insidethe medium, by

E = E0exp

0

2

exp

i

0

2 t

(2.53a)

B = (1 + i)

0

2 (n E) (2.53b)

Hence, both fields fall offby a factor 1/eat a distance

= 20

(2.54)

This distanceis called theskin depth.

2.3 Observables and averagesIn the above we have used complex notationquite extensively. This is for mathe-matical convenience only. For instance, in this notation differentiations are almosttrivial to perform. However, every physical measurablequantity is always real



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valued. I.e., Eobservable = Re {Emathematical }.1 It is particularly important to re-member this when one works with products of observable physical quantities. Forinstance, if we have two physical vectors FandGwhich both are time-harmonic,i.e., can be represented by Fourier components proportional to exp{it}, then wemust make the following interpretation

F(t, x)G(t, x) = Re

{F}

Re{G

}= ReF0(x) eit ReG0(x) eit(2.55)

Furthermore, letting denote complex conjugate, we can express the real part ofthe complex vector Fas

Re {F} = ReF0(x) e

it = 12

[F0(x) eit + F0(x) e

it] (2.56)

and similarly for G. Hence, the physically acceptable interpretation of the scalarproduct of two complex vectors, representing physical observables, is

F(t, x)G(t, x) = ReF0(x) eit ReG0(x) eit

=12

[F0(x) eit + F0(x) e

it] 12

[G0(x) eit + G0(x) e

it]

=14

F0G0 + F0G0 + F0G0e2it + F0G0e2it

=

12

ReF0G0 + F0G0e2it

=

12

ReF0e

it G0eit + F0G0e2it

=12

ReF(t, x) G(t, x) + F0G0e2it

(2.57)

Often in physics, we measure temporal averages ( ) of our physical observ-ables. If so, we see that the average of the product of the two physical quantitiesrepresented byFandGcan be expressed as

F G Re {F} Re {G} = 12

Re {F G} (2.58)

since the temporal average of the oscillating function exp{2it}vanishes.

1Note that this is different from quantum physics where observable= |mathematical|



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Bibliography

2.4 Bibliography[1] J . D . JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc.,

New York, NY . . . , 1999, ISBN 0-471-30932-X.

[2] W. K. H. PANOFSKY ANDM. PHILLIPS,Classical Electricity and Magnetism, second ed.,Addison-Wesley Publishing Company, Inc., Reading, MA ..., 1962, ISBN 0-201-05702-6.



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2.5 Example

WAVE EQUATIONS IN ELECTROMAGNETODYNAMICSEXAMPLE2.1

Derive the wave equation for theEfield described by the electromagnetodynamic equations(Diracs symmetrised Maxwell equations) [cf.equations (1.50)on page16]

E = e0

(2.59a)

E = Bt

0jm (2.59b) B = 0m (2.59c)

B = 00E

t+0j

e (2.59d)

under the assumption of vanishing net electric and magnetic charge densities and in the absenceof electromotive and magnetomotive forces. Interpret this equation physically.

Taking the curl of(2.59b) and using(2.59d), and assuming, for symmetry reasons, thatthere exists a linear relation between the magneticcurrent density jm and the magnetic field B(the magnetic dual of Ohms law for electriccurrents,je = eE)

jm = mB (2.60)

one finds, noting that 00 = 1/c2, that

( E) = 0 jm t

( B) = 0m B t

0j

e + 1c2

E

t

= 0m

0eE +

1c2

E

t

0e E

t 1

c22E

t2

(2.61)

Using the vector operator identity ( E) = ( E) 2E, and the fact that E = 0for a vanishing net electric charge, we can rewrite the wave equation as

2E 0

e +m

c2

E

t 1

c22E

t220meE = 0 (2.62)

This is the homogeneous electromagnetodynamic wave equation forEwe were after.

Compared to the ordinary electrodynamic wave equation forE, equation (2.7)on page26,we see that we pick up extra terms. In order to understand what these extra terms mean phys-ically, we analyse the time-independent wave equation for a single Fourier component. Thenour wave equation becomes

2E + i0

e +m

c2

E +

2

c2E 20meE

= 2E + 2

c2

1 1

20

0me

+ i

e + m/c2

0

E = 0

(2.63)

Realising that, according to formula(2.26) on page 29, 0/0 is the square of the vacuumradiation resistanceR0, and rearranging a bit, we obtain the time-independent wave equation in



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Example

Diracs symmetrised electrodynamics

2E + 2

c2

1 R

20

2me

1 + i e + m/c20

1 R202

meE = 0 (2.64)

From this equation we conclude that the existence of magnetic charges (magnetic monopoles),and non-vanishing electric and magnetic conductivities would lead to a shift in the e ffectivewave number of the wave. Furthermore, even if the electric conductivity e vanishes, theimaginary term does not necessarily vanish and the wave might therefore experience damping(or growth) according as m is positive (or negative). This would happen in a hypotheticalmedium which is a perfect insulator for electric currents but which can carry magnetic currents.

Finally, we note that in the particular case that = R0

me, the wave equation becomesa (time-independent) diffusion equation

2E + i0

e +m

c2

E = 0 (2.65)

and, hence, no waves exist at all!

END OF EXAMPLE2.1



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55/223


3

ELECTROMAGNETICPOTENTIALS

As an alternative to expressing the laws of electrodynamics in terms of electricand magnetic fields, it turns out that it is often more convenient to express thetheory in terms of potentials. This is particularly true for problems related to ra-diation and relativity. In this chapter we will introduce and study the properties ofsuch potentials and shall find that they exhibit some remarkable properties whichelucidate the fundamental aspects of electromagnetism and lead naturally to thespecial theory of relativity.

3.1 The electrostatic scalar potentialAs we saw in equation (1.8)on page6,the electrostatic fieldEstat(x) is irrotational.Hence, it may be expressed in terms of the gradient of a scalar field. If we denotethis scalar field by stat(x), we get

Estat(x) = stat(x) (3.1)Taking the divergence of this and using equation (1.7) on page 5,we obtain Pois-sons equation

2stat(x) = Estat(x) = (x)0

(3.2)

A comparison with the definition ofEstat, namely equation (1.5) on page4,showsthat this equation has the solution

stat(x) = 140

V

d3x (x)|x x| + (3.3)

39


56/223


57/223

Gauge transformations

From equation (1.45c)on page 15we note that also in electrodynamics thehomogeneous equation B(t, x) = 0 remains valid. Because of this divergence-free nature of the time- and space-dependent magnetic field, we can express it asthe curl of anelectromagnetic vector potential:

B(t, x) = A(t, x) (3.6)

Inserting this expression into the other homogeneous Maxwell equation(1.32)on

page13, we obtain

E(t, x) = t

[ A(t, x)] = tA(t, x) (3.7)

or, rearranging the terms,

E(t, x) +

tA(t, x)

= 0 (3.8)

As before we utilise the vanishing curl of a vector expression to write thisvector expression as the gradient of a scalar function. If, in analogy with the elec-trostatic case, we introduce theelectromagnetic scalar potentialfunction (t, x),equation (3.8)becomes equivalent to

E(t, x) +

tA(t, x) = (t, x) (3.9)

This means that in electrodynamics, E(t, x) is calculated from the potentials ac-cording to the formula

E(t, x) = (t, x) tA(t, x) (3.10)

and B(t, x) from formula (3.6) above. Hence, it is a matter of taste whether wewant to express the laws of electrodynamics in terms of the potentials (t, x) andA(t, x), or in terms of the fields E(t, x) andB(t, x). However, there exists an im-portant difference between the two approaches: in classical electrodynamics the

only directly observable quantities are the fields themselves (and quantities de-rived from them) and not the potentials. On the other hand, the treatment becomessignificantly simpler if we use the potentials in our calculations and then, at thefinal stage, use equation (3.6) and equation(3.10)above to calculate the fields orphysical quantities expressed in the fields.

3.4 Gauge transformationsWe saw in section3.1on page39and in section3.2on page40that in electrostat-ics and magnetostatics we have a certain mathematicaldegree of freedom, up to



58/223

3. Electromagnetic Potentials

terms of vanishing gradients and curls, to pick suitable forms for the potentials andstill get the samephysicalresult. In fact, the way the electromagnetic scalar poten-tial(t, x) and the vector potential A(t, x) are related to the physically observablesgives leeway for similar manipulation of them also in electrodynamics.

If we transform (t, x) and A(t, x) simultaneously into new ones (t, x) andA(t, x) according to the mapping scheme

(t, x)(t, x) = (t, x) + (t, x)t

(3.11a)

A(t, x) A(t, x) = A(t, x) (t, x) (3.11b)

where (t, x) is an arbitrary, differentiable scalar function called the gauge func-tion, and insert the transformed potentials into equation (3.10) on page41for theelectric field and into equation(3.6)on page 41 for the magnetic field, we obtainthe transformed fields

E = A

t = ()

t A

t +

()t

= At

(3.12a)

B =

A =

A (

) =

A (3.12b)

where, once again equation (F.62) on page 179 was used. We see that the fieldsare unaffected by the gauge transformation (3.11). A transformation of the poten-tials and Awhich leaves the fields, and hence Maxwells equations, invariantis called a gauge transformation. A physical law which does not change under agauge transformation is said to be gauge invariant. It is only those quantities (ex-pressions) that are gauge invariant that have experimental significance. Of course,the EM fields themselves are gauge invariant.

3.5 Gauge conditionsInserting (3.10) and (3.6) on page41into Maxwells equations(1.45)on page15we obtain, after some simple algebra and the use of equation (1.11) on page6

2 = (t, x)0

t

( A) (3.13a)

2A 1c2

2A

t2 ( A) = 0j(t, x) +

1c2

t(3.13b)



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Gauge conditions

which can be rewritten in the following, more symmetric, form ofgeneral inho-mogeneous wave equations

1c2

2

t2 2 = (t, x)

0+

t

A + 1

c2

t

(3.14a)

1c2

2A

t2 2A = 0j(t, x) A +

1c2

t (3.14b)These two second order, coupled, partial differential equations, representing in allfour scalar equations (one for and one each for the three components Ai, i =1, 2, 3 ofA) are completely equivalent to the formulation of electrodynamics interms of Maxwells equations, which represent eight scalar first-order, coupled,partial differential equations.

As they stand, equations (3.13) on page42and equations (3.14) look compli-cated and may seem to be of limited use. However, if we write equation (3.6) onpage41 in the form A(t, x) = B(t, x) we can consider this as a specificationof A. But we know from Helmholtz theoremthat in order to determine the(spatial) behaviour ofAcompletely, we must also specify A. Since this diver-gence does not enter the derivation above,we are free to choose

Ain whatever

way we like and still obtain the same physical results!

3.5.1 Lorenz-Lorentz gaugeIf we choose Ato fulfil the so called Lorenz-Lorentz gauge condition1

A + 1c2

t= 0 (3.15)

the coupled inhomogeneous wave equation (3.14) on page 43simplify into thefollowing set ofuncoupled inhomogeneous wave equations:

2def

1c2

2

t2 2

=

1c2

2

t2 2 = (t, x)

0(3.16a)

2Adef

1c2

2

t2 2

A =

1c2

2A

t2 2A = 0j(t, x) (3.16b)

where 2 is the dAlembert operatordiscussed in example M.5on page199.Each of these four scalar equations is an inhomogeneous wave equation of the

1In fact, the Dutch physicist Hendrik Antoon Lorentz, who in 1903 demonstrated the covariance ofMaxwells equations, was not the original discoverer of this condition. It had been discovered by the Danishphysicist Ludvig V. Lorenz already in 1867[6]. In the literature, this fact has sometimes been overlookedand the condition was earlier referred to as the Lorentz gauge condition.



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following generic form:

2(t, x) = f(t, x) (3.17)

where is a shorthand for either or one of the components Aiof the vector po-tentialA, and fis the pertinent generic source component, (t, x)/0or 0ji(t, x),respectively.

We assume that our sources are well-behaved enough in time tso that theFourier transformpair for the generic source function f

F1[f(x)]def f(t, x) =

d f(x) eit (3.18a)

F[f(t, x)] def f(x) = 12

dt f(t, x) eit (3.18b)

exists, and that the same is true for the generic potential component :

(t, x) =

d (x) e

it (3.19a)

(x) = 12

dt(t, x) eit (3.19b)

Inserting the Fourier representations (3.18a)and (3.19a) into equation (3.17) above,and using the vacuum dispersion relation for electromagnetic waves

= ck (3.20)

the generic 3D inhomogeneous wave equation, equation (3.17), turns into

2(x) + k2(x) = f(x) (3.21)which is a 3D inhomogeneous time-independent wave equation, often called the3Dinhomogeneous Helmholtz equation.

As postulated byHuygens principle, each point on a wave front acts as a point

source for spherical wavelets of varying amplitude (weight). A new wave front isformed by a linear superposition of the individual wavelets from each of the pointsources on the old wave front. The solution of (3.21) can therefore be expressedas a weighted superposition of solutions of an equation where the source term hasbeen replaced by a single point source

2G(x, x) + k2G(x, x) = (x x) (3.22)and the solution of equation (3.21) above which corresponds to the frequency is given by the superposition

(x) =

Vd3x f(x)G(x, x) (3.23)



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Gauge conditions

where f(x) is the wavelet amplitude at the source point x. The functionG(x, x)is called theGreen functionor thepropagator.

Due to translational invariance in space,G(x, x) = G(x x). Furthermore, inequation(3.22) on page44, the Dirac generalised function (x x), which repre-sents the point source, depends only on x xand there is no angular dependencein the equation. Hence, the solution can only be dependent on r= |x x| and noton the direction ofx

x. If we interpretras the radial coordinate in a spherically

polar coordinate system, and recall the expression for the Laplace operator in sucha coordinate system, equation (3.22) on page44becomes

d2

dr2(rG) + k2(rG) = r(r) (3.24)

Away fromr= |x x| = 0,i.e., away from the source pointx, this equation takesthe form

d2

dr2(rG) + k2(rG) = 0 (3.25)

with the well-known general solution

G = C+ eikr

r+ C eikr

r C+eik|xx ||x x| + C

eik|xx ||x x| (3.26)

whereCare constants.In order to evaluate the constants C, we insert the general solution, equa-

tion (3.26), into equation (3.22) on page 44 and integrate over a small volumearoundr= |x x| = 0. Since

G(x x)C+ 1|x x| + C 1|x x| , x x0 (3.27)

The volume integrated equation (3.22) on page 44 can under this assumption beapproximated by

C+ + C

V

d3x 2

1|x x|

+ k2

C+ + C

V

d3x 1|x x|=

V

d3x(x x) (3.28)

In virtue of the fact that the volume element d3x in spherical polar coordinatesis proportional to|x x|2, the second integral vanishes when|x x| 0. Fur-thermore, from equation(F.73) on page179,we find that the integrand in the firstintegral can be written as 4(|x x|) and, hence, that

C+ + C = 14

(3.29)



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Insertion of the general solution equation (3.26)onpage 45 into equation (3.23)on page44gives

(x) = C+

V

d3x f(x)eik|xx

|

|x x| + C

Vd3x f(x)

eik|xx|

|x x| (3.30)

The inverse Fourier transform of this back to the tdomain is obtained by insertingthe above expression for (x) into equation (3.19a) on page44:

(t, x) = C+

Vd3x

d f(x

)expit k|xx|

|x x|

+ C

Vd3x

d f(x

)expi

t+ k|xx

|

|x x|

(3.31)

If we introduce the retarded time tretand the advanced time tadvin

emft book on electro magnetism

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