Electromagnetic Theory Bo Thide 2007 CH1

ElectromagneticField Theory

BO THID

UPSILON BOOKS

ELECTROMAGNETIC FIELD THEORY

ElectromagneticField Theory

BO THIDSwedish Institute of Space Physics

Uppsala, Sweden

and

Department of Astronomy and Space PhysicsUppsala University, Sweden

and

LOIS Space CentreSchool of Mathematics and Systems Engineering

Vxj University, Sweden

UPSILON BOOKS UPPSALA SWEDEN

Also available

ELECTROMAGNETIC FIELD THEORYEXERCISES

byTobia Carozzi, Anders Eriksson, Bengt Lundborg,

Bo Thid and Mattias Waldenvik

Freely downloadable fromwww.plasma.uu.se/CED

This book was typeset in LATEX2 (based on TEX 3.141592 and Web2C 7.4.4) on an HP Visu-alize 90003600 workstation running HP-UX 11.11.

Copyright19972006 byBo ThidUppsala, SwedenAll rights reserved.

Electromagnetic Field TheoryISBN X-XXX-XXXXX-X

To the memory of professorLEV MIKHAILOVICH ERUKHIMOV (19361997)

dear friend, great physicist, poetand a truly remarkable man.

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Contents

Contents ix

List of Figures xiii

Preface xv

1 Classical Electrodynamics 11.1 Electrostatics 2

1.1.1 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 for E 262.1.2 The wave equation for B 272.1.3 The time-independent wave equation for E 27

ix

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 342.5 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 896.2.3 Magnetic dipole radiation 916.2.4 Electric quadrupole radiation 92

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 966.3.3 Bremsstrahlung 1046.3.4 Cyclotron and synchrotron radiation 1086.3.5 Radiation from charges moving in matter 115

6.4 Bibliography 1226.5 Examples 124

7 Relativistic Electrodynamics 1317.1 The special theory of relativity 131

7.1.1 The Lorentz transformation 1327.1.2 Lorentz space 1347.1.3 Minkowski space 139

7.2 Covariant classical mechanics 1427.3 Covariant classical electrodynamics 143

7.3.1 The four-potential 1437.3.2 The Linard-Wiechert potentials 1447.3.3 The electromagnetic field tensor 147

7.4 Bibliography 150

8 Electromagnetic Fields and Particles 1538.1 Charged particles in an electromagnetic field 153

8.1.1 Covariant equations of motion 1538.2 Covariant field theory 159

8.2.1 Lagrange-Hamilton formalism for fields and interactions 1608.3 Bibliography 1678.4 Example 169

F Formul 171F.1 The electromagnetic field 171

F.1.1 Maxwells equations 171F.1.2 Fields and potentials 171F.1.3 Force and energy 172

F.2 Electromagnetic radiation 172

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Contents

F.2.1 Relationship between the field vectors in a plane wave 172F.2.2 The far fields from an extended source distribution 172F.2.3 The far fields from an electric dipole 172F.2.4 The far fields from a magnetic dipole 173F.2.5 The far fields from an electric quadrupole 173F.2.6 The fields from a point charge in arbitrary motion 173

F.3 Special relativity 174F.3.1 Metric tensor 174F.3.2 Covariant and contravariant four-vectors 174F.3.3 Lorentz transformation of a four-vector 174F.3.4 Invariant line element 174F.3.5 Four-velocity 174F.3.6 Four-momentum 175F.3.7 Four-current density 175F.3.8 Four-potential 175F.3.9 Field tensor 175

F.4 Vector relations 175F.4.1 Spherical polar coordinates 176F.4.2 Vector formulae 176

F.5 Bibliography 178

M Mathematical Methods 179M.1 Scalars, vectors and tensors 179

M.1.1 Vectors 179M.1.2 Fields 181M.1.3 Vector algebra 184M.1.4 Vector analysis 186

M.2 Analytical mechanics 188M.2.1 Lagranges equations 188M.2.2 Hamiltons equations 189

M.3 Examples 190M.4 Bibliography 198

Index 199

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List of Figures

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 varying B field 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 1066.10 Radiation from a charge in circular motion 1096.11 Synchrotron radiation lobe width 1116.12 The perpendicular field of a moving charge 1136.13 Electron-electron scattering 1156.14 Vavilov-Cerenkov cone 120

7.1 Relative motion of two inertial systems 1337.2 Rotation in a 2D Euclidean space 1397.3 Minkowski diagram 140

8.1 Linear one-dimensional mass chain 160

M.1 Tetrahedron-like volume element of matter 190

xiii


Preface

This book is the result of a more than thirty year long love affair. In 1972, I tookmy first advanced course in electrodynamics at the Department of TheoreticalPhysics, Uppsala University. A year later, I joined the research group there andtook on the task of helping my supervisor, professor PER-OLOF FRMAN, withthe preparation of a new version of his lecture notes on the Theory of Electricity.These two things opened up my eyes for the beauty and intricacy of electrody-namics, already at the classical level, and I fell in love with it. Ever since thattime, I have on and off had reason to return to electrodynamics, both in my stud-ies, research and the teaching of a course in advanced electrodynamics at UppsalaUniversity some twenty odd years after I experienced the first encounter with thissubject.

The current version of the book is an outgrowth of the lecture notes that Iprepared for the four-credit course Electrodynamics that was introduced in theUppsala University curriculum in 1992, to become the five-credit course ClassicalElectrodynamics in 1997. To some extent, parts of these notes were based onlecture notes prepared, in Swedish, by BENGT LUNDBORGwho created, developedand taught the earlier, two-credit course Electromagnetic Radiation 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 usefulin higher university education anywhere in the world, it was produced within a

xv

Preface

World-WideWeb (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 makeit difficult to procure a printed copy of the book.

Thanks are due not only to Bengt Lundborg for providing the inspiration towrite this book, but also to professor CHRISTER WAHLBERG and professor GRANFLDT, 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, TOBIA CAROZZI and ROGER KARLSSON as well as ANDERS ERIKS-SON, 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 Academicianprofessor VITALIY LAZAREVICH GINZBURG, 2003 Nobel 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, STAFFAN RSBY, and to my fellow members of theCAPELLA PEDAGOGICA UPSALIENSIS.

Uppsala, Sweden BO THIDDecember, 2006 www.physics.irfu.se/bt

1At the time of publication of this edition, more than 500 000 downloads have been recorded.

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1Classical

Electrodynamics

Classical electrodynamics deals with electric and magnetic fields and interactionscaused by macroscopic distributions of electric charges and currents. This meansthat the concepts of localised electric charges and currents assume the validity ofcertain 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 macroscopic scales.

It took the genius of JAMES CLERK MAXWELL to unify electricity and mag-netism into a super-theory, electromagnetism or classical electrodynamics (CED),and to realise that optics is a subfield of this super-theory. Early in the 20th cen-tury, HENDRIK ANTOON LORENTZ took the electrodynamics theory further to themicroscopic scale and also laid the foundation for the special theory of relativity,formulated by ALBERT EINSTEIN in 1905. In the 1930s PAUL A. M. DIRAC ex-panded electrodynamics to a more symmetric form, including magnetic as wellas electric charges. With his relativistic quantum mechanics, he also paved theway for the development of quantum electrodynamics (QED) for which RICHARDP. FEYNMAN, JULIAN SCHWINGER, and SIN-ITIRO TOMONAGA in 1965 receivedtheir Nobel prizes in physics. Around the same time, physicists such as SHELDONGLASHOW, ABDUS SALAM, and STEVEN WEINBERG were able to unify electro-dynamics the weak interaction theory to yet another super-theory, electroweaktheory, an achievement which rendered them the Nobel prize in physics 1979.The modern theory of strong interactions, quantum chromodynamics (QCD), isinfluenced by QED.

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

1

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 Maxwellsequations 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 whichhas 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 withcharge q located at x, due to the presence of a charge q located at x. Accordingto Coulombs law this force is, in vacuum, given by the expression

F(x) =qq

4pi0

x x|x x|3 =

qq

4pi0(

1|x x|

)=

qq

4pi0(

1|x x|

)(1.1)

1The physicist and philosopher PIERRE DUHEM (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

FIGURE 1.1: Coulombs law describes how a static electric charge q, located ata point x relative to the origin O, experiences an electrostatic force from a static

electric charge q located at x.

where in the last step formula (F.71) on page 177 was used. In SI units, which weshall use throughout, the force F is measured in Newton (N), the electric charges qand q in Coulomb (C) [= Ampre-seconds (As)], and the length |x x| in metres(m). The constant 0 = 107/(4pic2) 8.8542 1012 Farad per metre (F/m) isthe vacuum permittivity and c 2.9979 108 m/s is the speed of light in vacuum.In CGS units 0 = 1/(4pi) 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 theconcept of a field and to describe the electrostatic interaction in terms of a staticvectorial electric field Estat defined by the limiting process

Estatdef lim

q0Fq

(1.2)

where F is the electrostatic force, as defined in equation (1.1) on page 2, from anet electric charge q on the test particle with a small electric net electric chargeq. Since the purpose of the limiting process is to assure that the test charge q doesnot distort the field set up by q, the expression for Estat does not depend explicitlyon q but only on the charge q and the relative radius vector x x. 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 177,we find that the electrostatic field Estat at the field point x (also known as theobservation point), due to a field-producing electric charge q at the source pointx, is given by

Estat(x) =q

4pi0

x x|x x|3 =

q

4pi0(

1|x x|

)=

q

4pi0(

1|x x|

)(1.3)

In the presence of several field producing discrete electric charges qi , locatedat the points xi , 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) =1

4pi0iqi

x xix xi3 (1.4)If the discrete electric charges are small and numerous enough, we introduce

the 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) =1

4pi0

V d3x (x)

x x|x x|3 =

14pi0

V d3x (x)

(1

|x x|)

= 14pi0

V d3x

(x)|x x|

(1.5)

where we used formula (F.70) on page 177 and the fact that (x) does not dependon the unprimed (field point) coordinates on which operates.

2In the preface to the first edition of the first volume of his book A Treatise on Electricity and Mag-netism, first published in 1873, 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 nonlinearity due to vacuum polarisation effects man-ifesting themselves in the momentary creation and annihilation of electron-positron pairs, but classicallythis nonlinearity is negligible.


Electrostatics

V

qi

q

O

xi

x xix

FIGURE 1.2: Coulombs law for a distribution of individual charges qi localisedwithin a volume V of limited extent.

We emphasise that under the assumption of linear superposition, equa-tion (1.5) on page 4 is valid for an arbitrary distribution of electric charges, in-cluding discrete charges, in which case is expressed in terms of Dirac deltadistributions:

(x) =iqi (x

xi) (1.6)

as illustrated in figure 1.2. Inserting this expression into expression (1.5) on page 4we recover expression (1.4) on page 4.

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 page 177, we find that

Estat(x) = 14pi0

V d3x (x)

x x|x x|3

= 14pi0

V d3x (x)

(1

|x x|)

= 14pi0

V d3x (x)2

(1

|x x|)

=10

V d3x (x) (x x) = (x)

0

(1.7)

which is the differential form of Gausss law of electrostatics.Since, according to formula (F.62) on page 177, [(x)] 0 for any 3D



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

Estat(x) = 14pi0

(V d3x

(x)|x x|

)= 0 (1.8)

i.e., that Estat is an irrotational field.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, magnetostatics deals withstationary 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 figure 1.3 on page 7, let F denote such a force acting on asmall loop C, with tangential line element dl, located at x and carrying a currentI in the direction of dl, due to the presence of a small loop C, with tangentialline element dl, located at x and carrying a current I in the direction of dl.According to Ampres law this force is, in vacuum, given by the expression

F(x) =0II

4pi

Cdl

Cdl x x

|x x|3

= 0II

4pi

Cdl

Cdl

(1

|x x|) (1.10)

In SI units, 0 = 4pi 107 1.2566 106 H/m is the vacuum permeability.From the definition of 0 and 0 (in SI units) we observe that

00 =107

4pic2(F/m) 4pi 107 (H/m) = 1

c2(s2/m2) (1.11)


Magnetostatics

C

C

I dlI dl

O

x

x x

x

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

through the tangential line element dl located at x. 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 page 6may 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 page 176, we can rewrite (1.10) as

F(x) = 0II

4pi

CdlCdl

(1

|x x|)

0II

4pi

C

C

x x|x x|3 dl 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

4pi

C

C

x x|x x|3 dl dl

(1.13)

which clearly exhibits the expected symmetry in terms of loops C and C.

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



can be defined as

dBstat(x)def 0I

4pidl 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 dl of stationary current I at the sourcepoint x. The SI unit for the magnetic field, sometimes called the magnetic fluxdensity or magnetic induction, is Tesla (T).

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

Bstat(x) =04pi

V d3x j(x) x x

|x x|3 = 04pi

V d3x j(x)

(1

|x x|)

=04pi

V d3x

j(x)|x x|

(1.15)

where we used formula (F.70) on page 177, formula (F.57) on page 177, and thefact that j(x) does not depend on the unprimed coordinates on which operates.Comparing equation (1.5) on page 4 with 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 of Bstat, equation (1.10) onpage 6 may we written

F(x) = ICdl Bstat(x) (1.16)

In order to assess the properties of Bstat, we determine its divergence and curl.Taking the divergence of both sides of equation (1.15) and utilising formula (F.63)on page 177, we obtain

Bstat(x) = 04pi (

V d3x

j(x)|x x|

)= 0 (1.17)

since, according to formula (F.63) on page 177, (a) vanishes for any vectorfield a(x).

Applying the operator bac-cab rule, formula (F.64) on page 177, the curl ofequation (1.15) can be written

Bstat(x) = 04pi

(

V d3x

j(x)|x x|

)=

= 04pi

V d3x j(x)2

(1

|x x|)+04pi

V d3x [j(x) ]

(1

|x x|)

(1.18)


Electrodynamics

In the first of the two integrals on the right-hand side, we use the representationof the Dirac delta function given in formula (F.73) on page 177, and integrate thesecond one by parts, by utilising formula (F.56) on page 177 as follows:

V d3x [j(x) ]

(1

|x x|)

= xkV d3x

{j(x)

[

xk

(1

|x x|)]}

V d3x

[ j(x)]( 1|x x|)

= xkS d2x n j(x)

xk

(1

|x x|)V d3x

[ j(x)]( 1|x x|)

(1.19)

Then we note that the first integral in the result, obtained by applying Gaussstheorem, vanishes when integrated over a large sphere far away from the localisedsource j(x), and that the second integral vanishes because j = 0 for stationarycurrents (no charge accumulation in space). The net result is simply

Bstat(x) = 0V d3x j(x)(x x) = 0j(x) (1.20)

1.3 ElectrodynamicsAs we saw in the previous sections, the laws of electrostatics and magnetostaticscan be summarised in two pairs of time-independent, uncoupled vector partialdifferential equations, namely the equations of classical electrostatics

Estat(x) = (x)0

(1.21a)

Estat(x) = 0 (1.21b)and the equations of classical magnetostatics

Bstat(x) = 0 (1.22a) Bstat(x) = 0j(x) (1.22b)

Since there is nothing a priori which connects Estat directly with Bstat, we mustconsider classical electrostatics and classical magnetostatics as two independenttheories.



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 chargeLet j(t, x) denote the time-dependent electric current density. In the simplest caseit can be defined as j = v where v is 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 v f(t, x, v) where f(t, x, v) is the (normalised) distribution function for

particle species with electric charge q.The electric charge conservation law can be formulated in the equation of

continuity

(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 density j(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 page 9, we would obtain the formal result

B(t, x) = 0V d3x j(t, x)(x x) + 0

4pi

t

V d3x (t, x)

(1

|x x|)

= 0j(t, x) + 0

t0E(t, x)

(1.24)


Electrodynamics

where, in the last step, we have assumed that a generalisation of equation (1.5) onpage 4 to time-varying fields allows us to make the identification4

14pi0

t

V d3x (t, x)

(1

|x x|)

=

t

[ 14pi0

V d3x (t, x)

(1

|x x|)]

=

t

[ 14pi0

V d3x

(t, x)|x x|

]=

tE(t, x)

(1.25)

The result is Maxwells source equation for the B field

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)/t is 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 j and E. Thisapproximation is called Ohms law:

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

where is the electric 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.



such as frequency mixing. Examples of media which are highly non-linear aresemiconductors and plasma. We draw the attention to the fact that even in caseswhen the linear relation between E and j is a good approximation, we still haveto use Ohms law with care. The conductivity is, in general, time-dependent(temporal dispersive media) but then it is often the case that equation (1.27) onpage 11 is valid for each individual Fourier component of the field.

If the current is caused by an applied electric field E(t, x), this electric fieldwill exert work on the charges in the medium and, unless the medium is super-conducting, there will be some energy loss. The rate at which this energy is ex-pended is j E per unit volume. If E is irrotational (conservative), j will decayaway with time. Stationary currents therefore require that an electric field whichcorresponds to an electromotive force (EMF) is present. In the presence of such afield EEMF, Ohms law, equation (1.27) on page 11, takes the form

j = (Estat + EEMF) (1.28)

The electromotive force is defined as

E =Cdl (Estat + EEMF) (1.29)

where dl is a tangential line element of the closed loop C.

1.3.4 Faradays law of inductionIn subsection 1.1.2 we derived the differential equations for the electrostatic field.In particular, on page 6we derived equation (1.8) which states that Estat(x) = 0and thus that Estat is a conservative field (it can be expressed as a gradient of ascalar field). This implies that the closed line integral of Estat in equation (1.29)above vanishes and that this equation becomes

E =Cdl EEMF (1.30)

It has been established experimentally that a nonconservative EMF field isproduced in a closed circuit C if the magnetic flux through this circuit varies withtime. This is formulated in Faradays law which, in Maxwells generalised form,reads

E(t, x) =Cdl E(t, x) = d

dtm(t, x)

= ddt

Sd2x n B(t, x) =

Sd2x n

tB(t, x)

(1.31)


Electrodynamics

d2x n

B(x) B(x)

v

dlC

FIGURE 1.4: A loopC which moves with velocity v in a spatially varying mag-netic field B(x) will sense a varying magnetic flux during the motion.

where m is the magnetic flux and S is the surface encircled by C which 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 waythat it links a magnetic field which varies during the movement. The convectivederivative is 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 page 12, we obtain

E(t, x) = ddt

Sd2x n B =

Sd2x n B

tSd2x n (v )B (1.34)



During spatial differentiation v is to be considered as constant, and equa-tion (1.17) on page 8 holds also for time-varying fields:

B(t, x) = 0 (1.35)(it is one ofMaxwells equations) so that, according to formula (F.59) on page 177,

(B v) = (v )B (1.36)allowing us to rewrite equation (1.34) on page 13 in the following way:

E(t, x) =Cdl EEMF = d

dt

Sd2x n B

= Sd2x n B

tSd2x n (B v)

(1.37)

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

E(t, x) =Cdl EEMF =

Sd2x n B

tCdl (B v) (1.38)

or, rearranging the terms,Cdl (EEMF v B) =

Sd2x n B

t(1.39)

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

(EEMF v B) = Bt

(1.40)

In the fixed system, an observer measures the electric field

E = EEMF v B (1.41)Hence, a moving observer measures the following Lorentz force on 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 a stationary observer, the Maxwell equation

E = Bt

(1.44)

is indeed valid even if the loop is moving.


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 fields E(t, x) and B(t, x). The equationsare

E = 0

(1.45a)

E = Bt

(1.45b)

B = 0 (1.45c) B = 00 E

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, and j(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 theclassical 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 and j to 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 determine E and B.

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 macroscopicscales. 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 displacement D and the magnetising field H. In themost general case, these derived fields are complicated nonlocal, nonlinear func-tionals of the primary fields E and B:

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



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 D and Hare

D = (t, x) (1.49a) E = B

t(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 chapter 4.

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 hoc assumption that there exist magnetic monopoles represented bya magnetic charge density, which we denote by m = m(t, x), and a magneticcurrent density, which we denote by jm = 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 = 00 E

t+ 0je (1.50d)


Electromagnetic duality

We shall call these equations Diracs symmetrised Maxwell equations or the elec-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 177, the diver-gence of a curl always vanishes. Using (1.50c) to rewrite this relation, we obtainthe magnetic 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 page 10.

We notice that the new equations (1.50) on page 16 exhibit the following sym-metry (recall that 00 = 1/c2):

E cB (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 ( = pi/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)c?e = ce cos + m sin (1.54c)?m = ce sin + m cos (1.54d)c?je = 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 as electromagnetodynamics), invariant. Since E and jeare (true or polar) vectors, B a pseudovector (axial vector), e a (true) scalar, thenm and , which behaves as a mixing angle in a two-dimensional charge space,must be pseudoscalars and jm a pseudovector.



The invariance of Diracs symmetrised Maxwell equations under the similaritytransformation means that the amount of magnetic monopole density m is irrele-vant for the physics as long as the ratio m/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 is the same for all particles.Such particles are referred to as dyons [14]. By varying the mixing angle we canchange 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 AND D. 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. LIFSHITZ, The Classical Theory of Fields, fourth revisedEnglish ed., vol. 2 of Course 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).

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.


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.

[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 AND R. C. MCPHEDRAN, Electromagnetic Processes in Dispersive Me-dia, Cambridge University Press, Cambridge . . . , 1991, ISBN 0-521-41025-8.

[12] W. K. H. PANOFSKY AND M. 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.

[15] J. SCHWINGER, L. L. DERAAD, JR., K. A. MILTON, AND W. TSAI, Classical Electrody-namics, Perseus Books, Reading, MA, 1998, ISBN 0-7382-0056-5.

[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.



1.6 Examples

BFARADAYS LAW AS A CONSEQUENCE OF CONSERVATION OF MAGNETIC CHARGEEXAMPLE 1.1

Postulate 1.1 (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-late that 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) =04pi

Vd3x m(x)

x x|x x|3 =

04pi

Vd3x m(x)

(1

|x x|)

= 04piVd3x

m(x)|x x|

(1.55)

[cf. equation (1.5) on page 4 for Estat] and, if magnetic currents exist, a Biot-Savart-like law forelectric fields [cf. equation (1.15) on page 8 for Bstat]:

Estat(x) = 04pi

Vd3x jm(x) x x

|x x|3 =04pi

Vd3x jm(x)

(1

|x x|)

= 04pi

Vd3x

jm(x)|x x|

(1.56)

Taking the curl of the latter and using the operator bac-cab rule, formula (F.59) on page 177,we find that

Estat(x) = 04pi

(

Vd3x

jm(x)|x x|

)=

=04pi

Vd3x jm(x)2

(1

|x x|) 04pi

Vd3x [jm(x) ]

(1

|x x|) (1.57)

Comparing with equation (1.18) on page 8 for Estat and the evaluation of the integrals there, weobtain

Estat(x) = 0Vd3x jm(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 page 177,the equation of continuity for magnetic charge, equation (1.52) on page 17, and the assumptionof the generalisation of equation (1.55) to time-dependent magnetic charge distributions, weobtain, formally,


Examples

E(t, x) = 0Vd3x jm(t, x)(x x) 0

4pi

t

Vd3x m(t, x)

(1

|x x|)

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

(1.59)

[cf. equation (1.24) on page 10] which we recognise as equation (1.50b) on page 16. A trans-formation of this electromagnetodynamic result by rotating into the electric realm of chargespace, thereby letting jm tend to zero, yields the electrodynamic equation (1.50b) on page 16,i.e., the Faraday law in the ordinary Maxwell equations. This process also provides an alter-native interpretation of the term B/t as a magnetic displacement current, dual to the electricdisplacement current [cf. equation (1.26) on page 11].

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.

C END OF EXAMPLE 1.1

BDUALITY OF THE ELECTROMAGNETODYNAMIC EQUATIONS EXAMPLE 1.2

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 +

1cm sin

)=

?e

0

(1.60)

?E + ?Bt

= (E cos + cB sin ) + t

(1cE sin + B cos

)= 0jm cos B

tcos + c0je sin +

1cEt

sin

1cEt

sin +Bt

cos = 0jm cos + c0je sin = 0(cje sin + jm cos ) = 0?jm

(1.61)

?B = (1cE sin + B cos ) =

e

c0sin + 0m cos

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



?B 1c2?Et

= (1cE sin + B cos ) 1

c2

t(E cos + cB sin )

=1c0jm sin +

1cBt

cos + 0je cos +1c2Et

cos

1c2Et

cos 1cBt

sin

= 0

(1cjm sin + je cos

)= 0

?je

(1.63)

QED


BDIRACS SYMMETRISED MAXWELL EQUATIONS FOR A FIXED MIXING ANGLEEXAMPLE 1.3Show that for a fixed mixing angle such 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 page 17 yields

?e = e cos +1cm sin = e cos +

1cce tan sin

=1

cos (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 ) =1

cos 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 page 17 yields

E = ?E cos c?B 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)


Examples

and

?B = 0?m = 0 (1.69)

so that

E = 1cos

e

0cos 0 =

e

0(1.70)

and so on for the other equations. QED


BCOMPLEX FIELD SIX-VECTOR FORMALISM EXAMPLE 1.4It is sometimes convenient to introduce the complex field six-vector, also known as the

Riemann-Silberstein vector

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

where E,B R3 and hence G C3. One fundamental property of C3 is that inner (scalar)products in this space are invariant just as they are inR3. However, as discussed in example M.3on page 193, the inner (scalar) product in C3 can be defined in two different ways. Consideringthe special case of the scalar product of G with itself, we have the following two possibilitiesof defining (the square of) the length of G:

1. The inner (scalar) product defined as G scalar 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 as G scalar 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 of G with 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 of G with the complex conjugate of itself



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.


BDUALITY EXPRESSED IN THE COMPLEX FIELD SIX-VECTOREXAMPLE 1.5Expressed in the Riemann-Silberstein complex field vector, introduced in example 1.4 on

page 23, the duality transformation equations (1.54) on page 17 become

?G = ?E + ic?B = E cos + cB sin iE sin + icB cos = E(cos i sin ) + icB(cos i sin ) = ei(E + icB) = eiG (1.77)

from which it is easy to see that

?G ?G = ?G2 = eiG eiG = |G|2 (1.78)while

?G ?G = e2iG G (1.79)

Furthermore, assuming that = (t, x), we see that the spatial and temporal differentiationof ?G leads to

t?G

?Gt

= 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 t?G transforms as ?G itself only if is time-independent, and that ?Gand ?G transform as ?G itself only if is space-independent.



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