Electromagnetic Field Theory [eBook]

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Bo Thidé

ELECTROMAGNETIC FIELD THEORY

Alsoavailable

ELECTROMAGNETIC FIELD THEORYEXERCISES

by

TobiaCarozzi,AndersEriksson,BengtLundborg,Bo ThidéandMattiasWaldenvik

ELECTROMAGNETIC

FIELD THEORY

Bo Thidé

Departmentof SpaceandPlasmaPhysicsUppsalaUniversity

and

SwedishInstituteof SpacePhysicsUppsalaDivision

Sweden

ΣIpsum

This bookwastypesetin LATEX 2on anHP9000/700seriesworkstationandprintedonanHP LaserJet5000GNprinter.

Copyright c

1997,1998byBo ThidéUppsala,SwedenAll rightsreserved.

ElectromagneticField TheoryISBN X-XXX-XXXXX-X

CONTENTS

Preface xi

1 ClassicalElectrodynamics 11.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Coulomb’s law . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Theelectrostaticfield . . . . . . . . . . . . . . . . . . . . 3

1.2 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Ampère’s law . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Themagnetostaticfield . . . . . . . . . . . . . . . . . . . 6

1.3 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 Equationof continuity . . . . . . . . . . . . . . . . . . . 91.3.2 Maxwell’sdisplacementcurrent . . . . . . . . . . . . . . 91.3.3 Electromotive force . . . . . . . . . . . . . . . . . . . . . 101.3.4 Faraday’s law of induction . . . . . . . . . . . . . . . . . 111.3.5 Maxwell’smicroscopicequations . . . . . . . . . . . . . 131.3.6 Maxwell’smacroscopicequations . . . . . . . . . . . . . 14

1.4 Electromagnetodynamics. . . . . . . . . . . . . . . . . . . . . . 15Example1.1 Invarianceof theelectromagnetodynamicequations 16Example1.2 Maxwell fromDirac-Maxwellequationsfor afixed

mixing angle . . . . . . . . . . . . . . . . . . . 17Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 ElectromagneticWaves 212.1 Thewave equation . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Planewaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Telegrapher’sequation . . . . . . . . . . . . . . . . . . . 252.2.2 Wavesin conductivemedia. . . . . . . . . . . . . . . . . 26

2.3 Observablesandaverages. . . . . . . . . . . . . . . . . . . . . . 27

i

ii

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 ElectromagneticPotentials 313.1 Theelectrostaticscalarpotential . . . . . . . . . . . . . . . . . . 313.2 Themagnetostaticvectorpotential . . . . . . . . . . . . . . . . . 323.3 Theelectromagneticscalarandvectorpotentials. . . . . . . . . . 32

3.3.1 Electromagneticgauges . . . . . . . . . . . . . . . . . . 34Lorentzequationsfor theelectromagneticpotentials . . . 34Gaugetransformations. . . . . . . . . . . . . . . . . . . 35

3.3.2 Solutionof the Lorentzequationsfor theelectromagneticpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . 36Theretardedpotentials . . . . . . . . . . . . . . . . . . . 39

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 TheElectromagneticFields 434.1 Themagneticfield . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Theelectricfield . . . . . . . . . . . . . . . . . . . . . . . . . . 47Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Relativistic Electrodynamics 535.1 Thespecialtheoryof relativity . . . . . . . . . . . . . . . . . . . 53

5.1.1 TheLorentztransformation . . . . . . . . . . . . . . . . 545.1.2 Lorentzspace. . . . . . . . . . . . . . . . . . . . . . . . 55

Metric tensor . . . . . . . . . . . . . . . . . . . . . . . . 56Radiusfour-vectorin contravariantandcovariantform . . 56Scalarproductandnorm . . . . . . . . . . . . . . . . . . 57Invariantline elementandpropertime . . . . . . . . . . . 57Four-vectorfields . . . . . . . . . . . . . . . . . . . . . . 58TheLorentztransformationmatrix . . . . . . . . . . . . . 59TheLorentzgroup . . . . . . . . . . . . . . . . . . . . . 59

5.1.3 Minkowski space. . . . . . . . . . . . . . . . . . . . . . 605.2 Covariantclassicalmechanics . . . . . . . . . . . . . . . . . . . 625.3 Covariantclassicalelectrodynamics . . . . . . . . . . . . . . . . 63

5.3.1 Thefour-potential . . . . . . . . . . . . . . . . . . . . . 645.3.2 TheLiénard-Wiechertpotentials. . . . . . . . . . . . . . 655.3.3 Theelectromagneticfield tensor . . . . . . . . . . . . . . 67

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

iii

6 Interactionsof FieldsandParticles 736.1 ChargedParticlesin anElectromagneticField . . . . . . . . . . . 73

6.1.1 Covariantequationsof motion . . . . . . . . . . . . . . . 73Lagrangeformalism . . . . . . . . . . . . . . . . . . . . 73Hamiltonianformalism . . . . . . . . . . . . . . . . . . . 76

6.2 CovariantField Theory . . . . . . . . . . . . . . . . . . . . . . . 806.2.1 Lagrange-Hamiltonformalismfor fieldsandinteractions. 80

Theelectromagneticfield . . . . . . . . . . . . . . . . . . 84Example6.1 Field energy differenceexpressedin thefield tensor 84Otherfields . . . . . . . . . . . . . . . . . . . . . . . . . 88

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7 Interactionsof FieldsandMatter 917.1 Electricpolarisationandtheelectricdisplacementvector . . . . . 91

7.1.1 Electricmultipolemoments . . . . . . . . . . . . . . . . 917.2 Magnetisationandthemagnetisingfield . . . . . . . . . . . . . . 947.3 Energy andmomentum. . . . . . . . . . . . . . . . . . . . . . . 95

7.3.1 Theenergy theoremin Maxwell’s theory . . . . . . . . . 967.3.2 Themomentumtheoremin Maxwell’s theory . . . . . . . 97

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8 ElectromagneticRadiation 1038.1 Theradiationfields . . . . . . . . . . . . . . . . . . . . . . . . . 1038.2 Radiatedenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8.2.1 Monochromaticsignals. . . . . . . . . . . . . . . . . . . 1068.2.2 Finite bandwidthsignals . . . . . . . . . . . . . . . . . . 106

8.3 Radiatingsystems. . . . . . . . . . . . . . . . . . . . . . . . . . 1088.3.1 Simplegeometries . . . . . . . . . . . . . . . . . . . . . 108

Linearantenna . . . . . . . . . . . . . . . . . . . . . . . 1088.3.2 Multipole expansion . . . . . . . . . . . . . . . . . . . . 110

TheHertzpotential . . . . . . . . . . . . . . . . . . . . . 110Electricdipoleradiation . . . . . . . . . . . . . . . . . . 113Magneticdipoleradiation . . . . . . . . . . . . . . . . . 114Electricquadrupoleradiation. . . . . . . . . . . . . . . . 116

8.3.3 Radiationfrom chargesmoving in vacuum . . . . . . . . 117Uniformly moving charges . . . . . . . . . . . . . . . . . 119Acceleratedcharges . . . . . . . . . . . . . . . . . . . . 122Radiationfor smallvelocities . . . . . . . . . . . . . . . 126Bremsstrahlung. . . . . . . . . . . . . . . . . . . . . . . 127

iv

Example8.1 Bremsstrahlungat low speedsandshortaccelera-tion times . . . . . . . . . . . . . . . . . . . . . 130

Cyclotronandsynchrotronradiation . . . . . . . . . . . . 132Radiationin thegeneralcase. . . . . . . . . . . . . . . . 137Theconvectionpotentialandtheconvectionforce . . . . . 138Virtual photons . . . . . . . . . . . . . . . . . . . . . . . 141

8.3.4 Radiationfrom chargesmoving in matter . . . . . . . . . 143Vavilov-Cerenkov radiation . . . . . . . . . . . . . . . . 145

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

F Formulae 153F.1 TheElectromagneticField . . . . . . . . . . . . . . . . . . . . . 153

F.1.1 Maxwell’sequations . . . . . . . . . . . . . . . . . . . . 153Constitutive relations . . . . . . . . . . . . . . . . . . . . 153

F.1.2 Fieldsandpotentials . . . . . . . . . . . . . . . . . . . . 154Vectorandscalarpotentials . . . . . . . . . . . . . . . . 154Lorentz’ gaugeconditionin vacuum . . . . . . . . . . . . 154

F.1.3 Forceandenergy . . . . . . . . . . . . . . . . . . . . . . 154Poynting’svector . . . . . . . . . . . . . . . . . . . . . . 154Maxwell’sstresstensor. . . . . . . . . . . . . . . . . . . 154

F.2 ElectromagneticRadiation . . . . . . . . . . . . . . . . . . . . . 154F.2.1 Relationshipbetweenthefield vectorsin aplanewave . . 154F.2.2 Thefar fieldsfrom anextendedsourcedistribution . . . . 154F.2.3 Thefar fieldsfrom anelectricdipole . . . . . . . . . . . . 155F.2.4 Thefar fieldsfrom amagneticdipole . . . . . . . . . . . 155F.2.5 Thefar fieldsfrom anelectricquadrupole. . . . . . . . . 155F.2.6 Thefieldsfrom apoint chargein arbitrarymotion . . . . . 155F.2.7 Thefieldsfrom apoint chargein uniformmotion . . . . . 156

F.3 SpecialRelativity . . . . . . . . . . . . . . . . . . . . . . . . . . 156F.3.1 Metric tensor . . . . . . . . . . . . . . . . . . . . . . . . 156F.3.2 Covariantandcontravariantfour-vectors. . . . . . . . . . 156F.3.3 Lorentztransformationof a four-vector . . . . . . . . . . 156F.3.4 Invariantline element. . . . . . . . . . . . . . . . . . . . 157F.3.5 Four-velocity . . . . . . . . . . . . . . . . . . . . . . . . 157F.3.6 Four-momentum . . . . . . . . . . . . . . . . . . . . . . 157F.3.7 Four-currentdensity . . . . . . . . . . . . . . . . . . . . 157F.3.8 Four-potential. . . . . . . . . . . . . . . . . . . . . . . . 157F.3.9 Field tensor . . . . . . . . . . . . . . . . . . . . . . . . . 157

F.4 VectorRelations. . . . . . . . . . . . . . . . . . . . . . . . . . . 158F.4.1 Sphericalpolarcoordinates. . . . . . . . . . . . . . . . . 158

v

Basevectors . . . . . . . . . . . . . . . . . . . . . . . . 158Directedline element. . . . . . . . . . . . . . . . . . . . 158Solidangleelement. . . . . . . . . . . . . . . . . . . . . 158Directedareaelement . . . . . . . . . . . . . . . . . . . 158Volumeelement . . . . . . . . . . . . . . . . . . . . . . 158

F.4.2 Vectorformulae. . . . . . . . . . . . . . . . . . . . . . . 159Generalrelations . . . . . . . . . . . . . . . . . . . . . . 159Specialrelations . . . . . . . . . . . . . . . . . . . . . . 160Integral relations . . . . . . . . . . . . . . . . . . . . . . 160

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

M MathematicalMethods 165M.1 Scalars,VectorsandTensors . . . . . . . . . . . . . . . . . . . . 165

M.1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Radiusvector . . . . . . . . . . . . . . . . . . . . . . . . 165

M.1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Scalarfields . . . . . . . . . . . . . . . . . . . . . . . . . 167Vectorfields . . . . . . . . . . . . . . . . . . . . . . . . 167Tensorfields . . . . . . . . . . . . . . . . . . . . . . . . 168ExampleM.1 Tensorsin 3D space . . . . . . . . . . . . . . 170

M.1.3 Vectoralgebra . . . . . . . . . . . . . . . . . . . . . . . 173Scalarproduct . . . . . . . . . . . . . . . . . . . . . . . 173ExampleM.2 Scalarproduct,normandmetricin Lorentzspace 173ExampleM.3 Metric in generalrelativity . . . . . . . . . . . 173Dyadicproduct . . . . . . . . . . . . . . . . . . . . . . . 174Vectorproduct . . . . . . . . . . . . . . . . . . . . . . . 174

M.1.4 Vectoranalysis . . . . . . . . . . . . . . . . . . . . . . . 175Thedel operator . . . . . . . . . . . . . . . . . . . . . . 175ExampleM.4 Thefour-del operatorin Lorentzspace . . . . . 176Thegradient . . . . . . . . . . . . . . . . . . . . . . . . 176ExampleM.5 Gradientsof scalarfunctionsof relativedistances

in 3D . . . . . . . . . . . . . . . . . . . . . . . 176Thedivergence . . . . . . . . . . . . . . . . . . . . . . . 177ExampleM.6 Divergencein 3D . . . . . . . . . . . . . . . 177TheLaplacian. . . . . . . . . . . . . . . . . . . . . . . . 178ExampleM.7 TheLaplacianandtheDiracdelta . . . . . . . 178Thecurl . . . . . . . . . . . . . . . . . . . . . . . . . . . 178ExampleM.8 Thecurl of agradient . . . . . . . . . . . . . 178ExampleM.9 Thedivergenceof acurl . . . . . . . . . . . . 179

M.2 AnalyticalMechanics. . . . . . . . . . . . . . . . . . . . . . . . 180

M.2.1 Lagrange’sequations. . . . . . . . . . . . . . . . . . . . 180M.2.2 Hamilton’sequations. . . . . . . . . . . . . . . . . . . . 181

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

vi

L IST OF FIGURES

1.1 Coulombinteraction . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Ampèreinteraction . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Moving loop in avaryingB field . . . . . . . . . . . . . . . . . . 12

5.1 Relativemotionof two inertial systems . . . . . . . . . . . . . . 545.2 Rotationin a2D Euclideanspace. . . . . . . . . . . . . . . . . . 605.3 Minkowski diagram. . . . . . . . . . . . . . . . . . . . . . . . . 61

6.1 Linearone-dimensionalmasschain. . . . . . . . . . . . . . . . . 80

8.1 Radiationin thefar zone . . . . . . . . . . . . . . . . . . . . . . 1058.2 Radiationfrom amoving chargein vacuum . . . . . . . . . . . . 1178.3 A uniformly moving chargein vacuum. . . . . . . . . . . . . . . 1208.4 An acceleratedchargein vacuum. . . . . . . . . . . . . . . . . . 1258.5 Angulardistributionof radiationduringbremsstrahlung. . . . . . 1288.6 Locationof radiationduringbremsstrahlung. . . . . . . . . . . . 1298.7 Radiationfrom achargein circularmotion . . . . . . . . . . . . . 1338.8 Synchrotronradiationlobewidth . . . . . . . . . . . . . . . . . . 1368.9 Theperpendicularfield of amoving charge. . . . . . . . . . . . . 1418.10 Vavilov-Cerenkov cone . . . . . . . . . . . . . . . . . . . . . . . 147

M.1 Tetrahedron-likevolumeelementof matter. . . . . . . . . . . . . 171

vii

To thememoryofLEV M IKHAILOVICH ERUKHIMOV

dearfriend,remarkablephysicistanda truly greathuman

PREFACE

This book is the resultof a twenty-five yearlong love affair. In 1972,I took myfirst advancedcoursein electrodynamicsat the TheoreticalPhysics department,UppsalaUniversity. Shortly thereafter, I joinedtheresearchgroupthereandtookon the task of helping my supervisor, professorPER-OLOF FRÖMAN, with thepreparationof a new versionof his lecturenoteson Electricity Theory. Thesetwothingsopenedupmy eyesfor thebeautyandintricacy of electrodynamics,alreadyat theclassicallevel, andI fell in lovewith it.

Ever sincethat time, I have off andon hadreasonto returnto electrodynamics,both in my studies,researchand teaching,and the currentbook is the result ofmy own teachingof a coursein advancedelectrodynamicsat UppsalaUniversitysometwenty odd yearsafter I experiencedthe first encounterwith this subject.The book is the outgrowth of the lecturenotesthat I preparedfor the four-creditcourseElectrodynamicsthatwasintroducedin theUppsalaUniversitycurriculumin 1992,to becomethe five-creditcourseClassicalElectrodynamicsin 1997. Tosomeextent,partsof thesenoteswerebasedonlecturenotesprepared,in Swedish,by BENGT LUNDBORG who created,developedandtaughttheearlier, two-creditcourseElectromagneticRadiationat our faculty.

Intendedprimarily as a textbook for physics studentsat the advancedunder-graduateor beginninggraduatelevel, I hopethebookmaybeusefulfor researchworkers too. It providesa thoroughtreatmentof the theoryof electrodynamics,mainly from a classicalfield theoreticalpoint of view, and includessuchthingsaselectrostaticsandmagnetostaticsandtheir unificationinto electrodynamics,theelectromagneticpotentials,gaugetransformations,covariant formulationof clas-sical electrodynamics,force,momentumandenergy of theelectromagneticfield,radiationandscatteringphenomena,electromagneticwavesandtheir propagationin vacuumandin media,andcovariantLagrangian/Hamiltonianfield theoreticalmethodsfor electromagneticfields,particlesandinteractions.Theaimhasbeentowrite a bookthatcanserve bothasanadvancedtext in ClassicalElectrodynamicsandasapreparationfor studiesin QuantumElectrodynamicsandrelatedsubjects.

xi

xii PREFACE

In an attemptto encourageparticipationby otherscientistsandstudentsin theauthoringof this book, and to ensureits quality and scopeto make it useful inhigheruniversityeducationanywherein theworld, it wasproducedwithin aWorld-Wide Web (WWW) project. This turnedout to be a rathersuccessfulmove. Bymakinganelectronicversionof thebook freely down-loadableon thenet, I havenotonly receivedcommentson it from fellow Internetphysicistsaroundtheworld,but know, from WWW ‘hit’ statisticsthat at the time of writing this, the bookservesasa frequentlyusedInternetresource.This way it is my hopethat it willbeparticularlyusefulfor studentsandresearchersworkingunderfinancialor othercircumstancesthatmake it difficult to procurea printedcopy of thebook.

I amgratefulnotonly to Per-Olof FrömanandBengtLundborg for providing theinspirationfor my writing thisbook,but alsoto CHRISTER WAHLBERG atUppsalaUniversity for interestingdiscussionson electrodynamicsin generalandon thisbook in particular, and to my former graduatestudentsMATTIAS WALDENVIK

andTOBIA CAROZZI aswell asANDERS ERIKSSON, all at theSwedishInstituteof SpacePhysics,UppsalaDivision,andwhohaveparticipatedin theteachingandcommentedonthematerialcoveredin thecourseandin thisbook.Thanksarealsodueto my long-termspacephysicscolleagueHELMUT KOPKA of theMax-Planck-Institut für Aeronomie,Lindau,Germany, who not only taughtmeabouttheprac-tical aspectsof theof high-power radiowave transmittersandtransmissionlines,but alsoaboutthe moredelicateaspectsof typesettinga book in TEX andLATEX.I am particularlyindebtedto AcademicianprofessorV ITALIY L. GINZBURG forhis many fascinatingandvery elucidatinglectures,commentsandhistoricalfoot-noteson electromagneticradiationwhile cruising on the Volga river during ourjoint Russian-Swedishsummerschools.

Finally, I would like to thankall studentsand Internetuserswho have down-loadedandcommentedon thebookduringits life on theWorld-WideWeb.

I dedicatethis book to my son MATTIAS, my daughterKAROLINA, myhigh-schoolphysicsteacher, STAFFAN RÖSBY, andto my fellow membersof theCAPELLA PEDAGOGICA UPSALIENSIS.

Uppsala,Sweden BO THIDÉ

December, 1998

CHAPTER 1

ClassicalElectrodynamics

Classicalelectrodynamicsdealswith electricandmagneticfieldsandinteractionscausedby macroscopicdistributionsof electricchargesandcurrents.This meansthat the conceptsof localisedchargesandcurrentsassumethe validity of certainmathematicallimiting processesin which it is consideredpossiblefor the chargeandcurrentdistributionsto belocalisedin infinitesimallysmallvolumesof space.This is in obviouscontradictionto electromagnetismonamicroscopicscale,wherechargesand currentsare known to be spatially extendedobjects. However, thelimiting processesyield resultswhicharecorrectonamacroscopicscale.

In thisChapterwe startwith theforceinteractionsin classicalelectrostaticsandclassicalmagnetostaticsandintroducethe staticelectricandmagneticfields andfind two uncoupledsystemsof equationsfor them. Thenwe seehow theconser-vation of electriccharge and its relation to electriccurrentleadsto the dynamicconnectionbetweenelectricityandmagnetismandhow the two canbeunified inonetheory, classicalelectrodynamics,describedby onesystemof coupleddynamicfield equations.

1.1 Electrostatics

The theorythat describesphysical phenomenarelatedto the interactionbetweenstationaryelectricchargesor chargedistributionsin spaceis calledelectrostatics.

1

2 CHAPTER 1. CLASSICAL ELECTRODYNAMICS

x

x x

x

Figure1.1. Coulomb’s law describeshow a staticelectriccharge , locatedat apoint x relative to the origin

, experiencesan electrostaticforce from a static

electriccharge locatedat x .1.1.1 Coulomb’s law

It has beenfound experimentallythat in classicalelectrostaticsthe interactionbetweentwo stationaryelectricallychargedbodiescanbe describedin termsofa mechanicalforce. Let usconsiderthesimplecasedepictedin Figure1.1 whereF denotestheforceactingon a chargedparticlewith charge locatedat x, duetothepresenceof acharge locatedatx . Accordingto Coulomb’s law this forceis,in vacuum,givenby theexpression

F x 4 0

x x x x 3

4 0

1x x (1.1)

wherewe have usedresultsfrom ExampleM.5 on page177. In SI units, whichwe shall use throughout,the force F is measuredin Newton (N), the charges and in Coulomb(C), and the length

x x in metres(m). The constant 0 107 4 2 8 8542 10 12 Faradpermetre(F/m) is thevacuumpermit-

tivity and 2 9979 108 m/s is the speedof light in vacuum. In CGSunits 0 1 4 andthe force is measuredin dyne,the charge in statcoulomb,andlengthin centimetres(cm).

1.1. ELECTROSTATICS 3

1.1.2 Theelectrostaticfield

Insteadof describingthe electrostaticinteractionin termsof a “force actionat adistance,” it turnsout that it is oftenmoreconvenientto introducetheconceptofa field and to describethe electrostaticinteractionin termsof a static vectorialelectricfield Estat definedby thelimiting process

Estatdef

lim! 0

F (1.2)

whereF is theelectrostaticforce,asdefinedin Equation(1.1)on thefacingpage,from anetcharge onthetestparticlewith asmallelectricnetcharge . Sincethepurposeof thelimiting processis to assurethatthetestcharge doesnot influencethe field, the expressionfor Estat doesnot dependexplicitly on but only on thecharge andthe relative radiusvectorx x . This meansthat we cansaythatany net electric charge producesan electric field in the spacethat surroundsit,regardlessof theexistenceof asecondchargeanywherein this space.1

Using formulae(1.1) and(1.2), we find that the electrostaticfield Estat at thefieldpointx (alsoknown astheobservationpoint), dueto afield-producingcharge at thesourcepoint x , is givenby

Estat x 4 0

x x x x 3"

4 0

1x x (1.3)

In thepresenceof severalfield producingdiscretecharges# , at x # , $% 1 & 2 & 3 &''' ,respectively, theassumptionof linearityof vacuum2 allowsusto superimposetheirindividualE fieldsinto a total E field

1In theprefaceto thefirst editionof thefirst volumeof his bookA Treatiseon Electricity andMag-netism, first publishedin 1873,JamesClerk Maxwell describesthis in thefollowing, almostpoetic,manner[5]:

“For instance,Faraday, in hismind’seye,saw linesof forcetraversingall spacewherethemathematicianssaw centresof forceattractingat a distance:Faradaysaw a me-diumwherethey saw nothingbut distance:Faradaysoughttheseatof thephenomenain realactionsgoingon in themedium,they weresatisfiedthatthey hadfoundit in apowerof actionata distanceimpressedon theelectricfluids.”

2In fact, vacuumexhibits a quantummechanical nonlinearity due to vacuumpolarisation effectsmanifestingthemselves in the momentarycreationandannihilationof electron-positronpairs,butclassicallythisnonlinearityis negligible.


Estat x )( # #4 0

x x #x x # 3 (1.4)

If the discretechargesaresmall andnumerousenough,we introducethe chargedensity* locatedat x andwrite thetotal field as

Estat x 14 0 + , * x x x

x x 3 - 3. 1

4 0 + , * x 1x x - 3. (1.5)

where,in the last step,we usedformula Equation(M.60) on page177. We em-phasisethatEquation(1.5) above is valid for anarbitrarydistribution of charges,including discretecharges,in which case* canbe expressedin termsof oneormoreDirac deltafunctions.

Since,accordingto formulaEquation(M.70) on page179, / 10 x 32 0

for any 3D 4 3 scalarfield0 x , we immediatelyfind thatin electrostatics Estat x 1

4 0 + , * x 65 1x x 7 - 3. 0 (1.6)

I.e., Estat is an irr otationalfield.Takingthedivergenceof thegeneralEstat expressionfor anarbitrarychargedis-

tribution, Equation(1.5),andusingtherepresentationof theDirac deltafunction,Equation(M.65) onpage178,we find that98

Estat x 98 14 0 + , * x x x

x x 3 - 3. " 1

4 0 + , * x 98: 1x x - 3.

" 14 0 + , * x ; 2 1

x x - 3. 1 0 + , * x < x x - 3. * x 0

(1.7)

which is Gauss’s law in differentialform.

1.2. MAGNETOSTATICS 5

1.2 MagnetostaticsWhile electrostaticsdealswith staticcharges,magnetostaticsdealswith stationarycurrents,i.e., chargesmoving with constantspeeds,and the interactionbetweenthesecurrents.

1.2.1 Ampère’s law

Experimentson the interactionbetweentwo small currentloopshave shown thatthey interactvia a mechanicalforce,muchthesameway thatchargesinteract.LetF denotesucha forceactingon a small loop = carryinga current > locatedat x,dueto thepresenceof asmallloop =? carryingacurrent> locatedatx . Accordingto Ampère’s law this forceis, in vacuum,givenby theexpression

F x @ 0 >%> 4 A B A BDC - l - l x x

x x 3" @ 0 >%> 4 A B A BDC - l 65 - l 1

x x 7(1.8)

Here- l and - l aretangentiallineelementsof theloops = and = , respectively, and,in SI units, @ 0 4E 10 7 1 2566 10 6 H/m is thevacuumpermeability.Fromthedefinitionof 0 and @ 0 (in SI units)weobserve that 0 @ 0 107

4 2 (F/m) 4F 10 7 (H/m) 1 2 (s2/m2) (1.9)

which is ausefulrelation.At first glance,Equation(1.8)above appearsto beunsymmetricin termsof the

loopsandthereforeto be a force law that is in contradictionwith Newton’s thirdlaw. However, by applyingthevectortriple product“bac-cab”Formula(F.56) onpage159,wecanrewrite (1.8) in thefollowing way

F x " @ 0 >%>4 A B A BDC 5 - l 8 1

x x 7 - l @ 0 >G> 4 A B A BDC x x

x x 3 - l 8 - l (1.10)

Recognisingthefacttheintegrandin thefirst integral is anexactdifferentialsothatthis integral vanishes,we canrewrite the force expression,Equation(1.8) above,in thefollowing symmetricway

F x H @ 0 >G>4 A B A BDC x x

x x 3 I l 8 - l (1.11)

This clearlyexhibits theexpectedsymmetryin termsof loops = and = .


JlK L

L K

x x x

Jl

x

Figure1.2. Ampère’slaw describeshow asmallloopK

, carryingastaticelectriccurrent L throughits tangentialline element

Jl locatedat x, experiencesa mag-

netostaticforcefrom asmallloopK , carryingastaticelectriccurrentL through

thetangentialline elementJl locatedatx . Theloopscanhavearbitraryshapes

aslongasthey aresimpleandclosed.

1.2.2 Themagnetostaticfield

In analogywith theelectrostaticcase,we mayattributethemagnetostaticinterac-tion to a vectorialmagneticfield field Bstat. I turnsout that Bstat canbe definedthrough

- Bstat x def @ 0 >4 - l x x

x x 3 (1.12)

which expressesthe small element- Bstat x of the staticmagneticfield setup atthefield point x by a small line element- l of stationarycurrent > at thesourcepoint x . The SI unit for the magneticfield, sometimescalledthe magneticfluxdensityor magneticinduction, is Tesla(T).

Generalisingexpression(1.12)to an integratedsteadystatecurrentdistributionj x , weobtainBiot-Savart’s law:

Bstat x @ 0

4 + , j x x x x x 3 - 3.

@ 0

4 + , j x 1x x - 3. (1.13)

ComparingEquation(1.5)onpage4 with Equation(1.13),weseethatthereexistsa closeanalogybetweenthe expressionsfor Estat andBstat but that they differ in

1.2. MAGNETOSTATICS 7

theirvectorialcharacteristics.With thisdefinitionof Bstat, Equation(1.8)onpage5maywewritten

F x "> A B - l Bstat x (1.14)

In orderto studythepropertiesof Bstat, we investigateits divergenceandcurl.Taking the divergenceof both sidesof Equation(1.13) on the facing page,weobtain98

Bstat x @ 0

4 98 + , j x 1x x - 3.

@ 0

4 + , j x 8 5 1x x 7 - 3. 0 (1.15)

whereEquation(M.70) on page179for / 10 x M2 wasused.

Applying theoperator“bac-cab”rule, Formula(F.69) on page159, thecurl ofEquation(1.13)on thefacingpagecanbewritten Bstat x " @ 0

4 + , j x 1x x - 3.

" @ 0

4 + , j x ; 2 1x x - 3. N @ 0

4 + , / j x 8 2 1x x - 3. (1.16)

If, in thefirst of thetwo integralson theright handside,we usetherepresentationof theDirac deltafunctionEquation(M.65) on page178,andintegratethesecondoneby parts,by utilising Formula(F.61)onpage159asfollows:

+ , / j x 8: 2 1x x - 3.

x O + , 8QP j x 5FRR . O 1x x 7TS - 3.

+ , U 8 j x V 1x x - 3.

x O + W j x RR . O 1x x 8 - S + , U 8 j x V 1

x x - 3. 0 (1.17)

Herethefirst integral,obtainedby applyingGauss’stheorem,vanisheswheninteg-ratedover a largespherefar away from the localisedsourcej x , andthesecond


onevanishesbecause98

j 0 for stationarycurrents(nochargeaccumulationinspace).Thenetresultis simply Bstat x @ 0 + , j x < x x - 3. @ 0j x (1.18)

1.3 Electrodynamics

As we saw in theprevious sections,the laws of electrostaticsandmagnetostaticscanbesummarisedin two pairsof time-independent,uncoupledvectordifferentialequations,namelytheequationsof classicalelectrostatics98

Estat x * x 0(1.19a) Estat x 0 (1.19b)

andtheequationsof classicalmagnetostatics98Bstat x 0 (1.20a) Bstat x @ 0j x (1.20b)

Sincethereis nothinga priori thatconnectsEstat directly with Bstat, we mustcon-siderclassicalelectrostaticsandclassicalmagnetostaticsastwo independentthe-ories.

However, whenwe includetime-dependence,thesetheoriesareunifiedinto onetheory: classicalelectrodynamics. This unificationof the theoriesof electricityandmagnetismis motivatedby two empiricallyestablishedfacts:

1. Electriccharge is a conservedquantityandcurrentis a transportof electriccharge. This fact manifestsitself in the equationof continuity and, as aconsequence,in Maxwell’sdisplacementcurrent.

2. A changein the magneticflux througha loop will inducean EMF electricfield in theloop. This is thecelebratedFaraday’s law of induction.

1.3. ELECTRODYNAMICS 9

1.3.1 Equationof continuity

Let j denotethe electric currentdensity(A/m2). In the simplestcaseit can bedefinedas j v * wherev is thevelocity of thechargedensity. In general,j hasto bedefinedin statisticalmechanicaltermsas j XY& x Z [\ [ ] v ^ [ X_& x & v - 3

where ^ [ X_& x & v is the (normalised)distribution function for particle species0

with electricalcharge [ .Theelectriccharge conservationlaw canbeformulatedin theequationof con-

tinuity R * XY& x R X N 98j X_& x 0 (1.21)

which statesthatthetime rateof changeof electriccharge * XY& x is balancedby adivergencein theelectriccurrentdensityj X_& x .1.3.2 Maxwell’sdisplacementcurrent

We recall from thederivationof Equation(1.18)on the facingpagethatwe usedthefact that in magnetostatics

a8j x 0. In thecaseof non-stationarysources

andfields,we must,in accordancewith thecontinuityEquation(1.21)above, setb8j X_& x c R * X_& x R X . Doing so, and formally repeatingthe stepsin the

derivationof Equation(1.18)onthefacingpage,wewouldobtaintheformalresult

B X_& x @ 0 + , j X_& x < x x - 3. N @ 0

4 RR X + , * XY& x 1x x - 3.

@ 0j X_& x N @ 0 RR X 0E XY& x (1.22)

where,in thelaststep,we have assumedthata generalisationof Equation(1.5)onpage4 to time-varyingfieldsallowsusto make theidentification

14 0

RR X + , * XY& x 1x x - 3.

14 0

RR X + , * XY& x 1x x - 3.

dRR X E XY& x (1.23)

Later, we will needto considerthis formal resultfurther. Theresultis Maxwell’ssourceequationfor theB field


B X_& x @ 0j X_& x N RR X 0E XY& x (1.24)

wherethe last term R 0E X_& x R X is the famousdisplacementcurrent. This termwasintroduced,in a stroke of genius,by Maxwell in orderto make theright handsideof thisequationdivergencefreewhenj X_& x is assumedto representthedens-ity of the total electric current,which can be split up in “ordinary” conductioncurrents,polarisationcurrentsandmagnetisationcurrents.Thedisplacementcur-rent is anextra termwhich behaveslike a currentdensitywhich flows in vacuumand,asweshallseelater, its existencehasveryfar-reachingphysicalconsequencesasit predictstheexistenceof electromagneticradiationthatcancarryenergy andmomentumover very longdistances,evenin vacuum.

1.3.3 Electromotive force

If an electricfield E XY& x , is appliedto a conductingmedium,a currentdensityj XY& x will be producedin this medium. Thereexist also hydrodynamicalandchemicalprocesseswhich cancreatecurrents.Undercertainphysicalconditions,andfor certainmaterials,onecansometimesassumea linearrelationshipbetweenthecurrentdensityj andE, calledOhm’s law:

j XY& x fe E XY& x (1.25)

where e is theelectricconductivity(S/m). In themostgeneralcases,for instancein ananisotropicconductor, e is a tensor.

We canview Ohm’s law, Equation(1.25),asthe first term in a Taylor expan-sion of the law j /E XY& x 32 . This generallaw incorporatesnon-lineareffectssuchas frequency mixing. Examplesof mediawhich arehighly non-lineararesemi-conductorsandplasma.We draw theattentionto thefact thateven in caseswhenthe linear relationbetweenE and j is a goodapproximation,we still have to useOhm’s law with care.Theconductivity e is, in general,time-dependent(temporaldispersivemedia) but thenit is often thecasethat Equation(1.25)above is validfor eachindividual Fourier componentof thefield. We shall not, however, dwelluponsuchcomplicatedcaseshere.

If thecurrentis causedby anappliedelectricfield E XY& x , thiselectricfield willexert work on thechargesin themediumand,unlessthemediumis superconduct-ing, therewill besomeenergy loss. Therateat which this energy is expendedisj8E per unit volume. If E is irrotational(conservative), j will decayaway with

time. Stationarycurrentsthereforerequirethatanelectricfield which corresponds


to anelectromotiveforce(EMF) is present.In thepresenceof sucha field EEMF,Ohm’s law, Equation(1.25)on thefacingpage,takestheform

j fe EstatN

EEMF (1.26)

Theelectromotive forceis definedasg A B EstatN

EEMF 8 - l (1.27)

where - l is a tangentialline elementof theclosedloop = .

1.3.4 Faraday’s law of induction

In Subsection1.1.2we derivedthedifferentialequationsfor theelectrostaticfield.In particular, wederivedEquation(1.6)onpage4whichstatesthat

Estat x 0andthusthatEstat is aconservativefield (it canbeexpressedasagradientof ascalarfield). This impliesthattheclosedline integralof Estat in Equation(1.27)vanishesandthatthis equationbecomesg A B EEMF

8 - l (1.28)

It wasfoundexperimentallythata nonconservative EMF field is producedin aclosedcircuit = if themagneticflux throughthis circuit varieswith time. This isformulatedin Faraday’s law which, in Maxwell’sgeneralisedform, readsg X_& x A B E X_& x 8 - l -- X Φm X_& x -- X + W B X_& x 8 - S + W - S 8 RR X B X_& x (1.29)

whereΦm is the magnetic flux and h is the surfaceencircledby = which canbe interpretedasa genericstationary“loop” andnot necessarilyasa conductingcircuit. Applicationof Stokes’ theoremonthis integralequation,transformsit intothedifferentialequation E X_& x fRR X B X_& x (1.30)

which is valid for arbitrary variationsin the fields and constitutesthe Maxwellequationwhichexplicitly connectselectricitywith magnetism.


B i x jkSv

v

B i x jkl

l

Figure1.3. A loop m which moveswith velocity v in a spatiallyvaryingmag-neticfield B n x o will senseavaryingmagneticflux duringthemotion.

An EMF is inducedby all changesof the magneticflux Φm. Let us thereforeconsiderthecase,illustratedif Figure1.3, that the“loop” is movedin sucha waythat it links a magneticfield which variesduring the movement. The convectivederivativeis evaluatedaccordingto thewell-known operatorformulapprqtscuu qwv v xy (1.31)

which follows immediatelyfrom therulesof differentiationof anarbitrarydiffer-entiablefunction z q_| x q ~ . Applying this rule to Faraday’s law, Equation(1.29)on thepreviouspage,weobtain qY| x s pprq B x p S

s p S x uu q B v xy B x p S (1.32)

In spatialdifferentiationv is to beconsideredasconstant,andEquation(1.15)on page7 holdsalsofor time-varyingfields:yx B q_| x s 0 (1.33)

(it is oneof Maxwell’sequations)sothat


B v v 8: B (1.34)

allowing usto rewrite Equation(1.32)on theprecedingpagein thefollowing way:g X_& x A B EEMF8 - l -- X + W B8 - S + W RR X B 8 - S + W B v 8 - S (1.35)

With Stokes’ theoremappliedto thelastintegral,we finally getg X_& x A B EEMF8 - l + W RR X B 8 - S A B B v 8 - l (1.36)

or, rearrangingtheterms,

+ B EEMF v B 8 - l " + W RR X B 8 - S (1.37)

whereEEMF is thefield which is inducedin the“loop,” i.e., in themoving system.Theuseof Stokes’ theorem“backwards”onEquation(1.37)yields EEMF v B " RR X B (1.38)

In thefixedsystem,anobservermeasurestheelectricfield

E EEMF v B (1.39)

Hence,a moving observermeasuresthefollowing Lorentzforceona charge EEMF f E N v B (1.40)

correspondingto an“effective” electricfield in the“loop” (moving observer)

EEMF EN v B (1.41)

Hence,we canconcludethatfor astationaryobserver, theMaxwell equation E RR X B (1.42)

is indeedvalid evenif the“loop” is moving.

1.3.5 Maxwell’smicroscopicequations

Wearenow ableto collecttheresultsfrom theaboveconsiderationsandformulatetheequationsof classicalelectrodynamicsvalid for arbitraryvariationsin timeandspaceof thecoupledelectricandmagneticfieldsE XY& x andB X_& x . Theequations


are 98E * X_& x 0

(1.43a) EN R BR X 0 (1.43b)98

B 0 (1.43c) B 0 @ 0 R ER X @ 0j X_& x (1.43d)

In theseequations* X_& x representsthetotal,possiblybothtimeandspacedepend-ent,electriccharge, i.e., freeaswell asinduced(polarisation)charges,andj X_& x representsthetotal, possiblyboth time andspacedependent,electriccurrent,i.e.,conductioncurrents(motionof freecharges)aswell asall atomistic(polarisation,magnetisation)currents. As they stand,the equationsthereforeincorporatetheclassicalinteractionbetweenall electric chargesand currentsin the systemandarecalledMaxwell’s microscopicequations. Anothernameoften usedfor themis the Maxwell-Lorentzequations. Togetherwith the appropriateconstitutivere-lations, which relate * and j to thefields,andthe initial andboundaryconditionspertinentto thephysicalsituationathand,they form asystemof well-posedpartialdifferentialequationswhichcompletelydetermineE andB.

1.3.6 Maxwell’smacroscopicequations

The microscopicfield equations(1.43) provide a correctclassicalpicturefor ar-bitraryfield andsourcedistributions,includingbothmicroscopicandmacroscopicscales.However, for macroscopicsubstancesit is sometimesconvenientto intro-ducenew derivedfieldswhich representtheelectricandmagneticfieldsin which,in anaveragesence,thematerialpropertiesof thesubstancesarealreadyincluded.Thesefields arethe electric displacementD andthe magnetisingfield H. In themostgeneralcase,thesederivedfields arecomplicatednonlocal,nonlinearfunc-tionalsof theprimaryfieldsE andB:

D D / X_& x; E & B 2 (1.44a)

H H / X_& x; E & B 2 (1.44b)

Undercertainconditions,for instancefor very low field strenghts,wemayassumethattheresponseof asubstanceis linearsothat

D X_& x E (1.45)

H @ 1 X_& x B (1.46)

i.e., that the derived fields arelinearly proportionalto the primary fields andthattheelecricdisplacement(magnetisingfield) is only dependentontheelectric(mag-

1.4. ELECTROMAGNETODYNAMICS 15

netic)field.The field equationsexpressedin termsof the derived field quantitiesD andH

are 98D * X_& x (1.47a) E

N R BR X 0 (1.47b)98B 0 (1.47c) H R DR X j XY& x (1.47d)

andarecalledMaxwell’smacroscopicequations.

1.4 Electromagnetodynamics

If we look morecloselyat themicroscopicMaxwell equations(1.48),we seethatthey exhibit some,albeit not a complete,symmetry. Let us for explicitnessde-notetheelectricchargedensity ** XY& x by * e andtheelectriccurrentdensityj j X_& x by je. We further make the ad hoc assumptionthat thereexist mag-neticmonopolesrepresentedby amagneticchargedensity, denoted* m * m XY& x ,anda magneticcurrentdensity, denotedjm jm X_& x . With thesenew quantitiesincludedin thetheory, theMaxwell equationscanbewritten98

E * e 0(1.48a) E

N R BR X " @ 0jm (1.48b)98B @ 0 * m (1.48c) B 1 2 R ER X @ 0je (1.48d)

Weshallcall theseequationstheDirac-Maxwellequationsor theelectromagneto-dynamicequations

Takingthedivergenceof (1.48b),we find that98 E RR X 98B @ 0

8jm

0 (1.49)

wherewe usedthefact that,accordingto Formula(M.74) on page180,thediver-genceof a curl alwaysvanishes.Using (1.48c)to rewrite this relation,we obtaintheequationof continuityfor magneticmonopoles


R * mR X N 8jm 0 (1.50)

whichhasthesameformasthatfor theelectricchargesandcurrents,Equation(1.21)on page9.

Wenoticethatthenew Equations(1.48)onthepreviouspageexhibit thefollow-ing symmetry(recallthat 0 @ 0 1 2):

E B (1.51a)

B 1 E (1.51b)* e 1 * m (1.51c)* m '* e (1.51d)

je 1 jm (1.51e)

jm je (1.51f)

which is apartiularcase( f 2) of thegeneralduality transformationE E cos N B sin (1.52a)B " 1 E sin N B cos (1.52b) * e * e cos N 1 * m sin (1.52c) * m "w* e sin N * m cos (1.52d)je je cos N 1 jm sin (1.52e)jm "w je sin N jm cos (1.52f)

which leaves the Dirac-Maxwell equations,andhencethe physics they describe(oftenreferredto aselectromagnetodynamics), invariant. SinceE andje are(trueor polar) vectors,B a pseudovector(axial vector), * e a (true) scalar, then * m and , which behavesasa mixinganglein a two-dimensional“chargespace,” mustbepseudoscalarsandjm apseudovector.

INVARIANCE OF THE ELECTROMAGNETODYNAMIC EQUATIONSEXAMPLE 1.1

Show thatthesymmetric,electromagnetodynamicform of Maxwell’sequations(theDirac-Maxwell equations),Equations(1.48)ontheprecedingpageareinvariantunderthedualitytransformation(1.52).

Explicit applicationof thetransformationyields

1.4. ELECTROMAGNETODYNAMICS 17

H'E 6: E cos B sin e

0cos 0 m sin 1

0 e cos 1 m sin ¢¡F e0

(1.53)

¤£ E ¦¥ B¥¨§ ¤£©

E cosª B sin Q«¥¥¬§ 1 E sin B cos ¡ ¥ B¥¨§ cos 0jm cos 1 ¥ E¥¨§ sin 0je sin 1 ¥ E¥¨§ sin ¥ B¥¨§ cosw 0jm cosª 0je sin 0 je sin jm cos 0

jm (1.54)

andanalogouslyfor theothertwo Dirac-Maxwellequations. QED END OF EXAMPLE 1.1 ®

Theinvarianceof theDirac-Maxwellequationsunderthesimilarity transforma-tion meansthat theamountof magneticmonopoledensity * m is irrelevant for thephysicsaslongastheratio * m

* e tan is keptconstant.Sowhetherwe assumethattheparticlesareonly electricallychargedor have alsoa magneticchargewitha given,fixedratio betweenthetwo typesof chargesis a matterof convention,aslong aswe assumethat this fraction is thesamefor all particles. By varying themixing angle we canchangethefractionof magneticmonopolesat will withoutchangingthe laws of electrodynamics.For 0 we recover theusualMaxwellelectrodynamicsaswe know it.

MAXWELL FROM DIRAC-MAXWELL EQUATIONS FOR A FIXED MIXING ANGLE EXAMPLE 1.2

Show thatfor afixedmixing angle suchthat

m F e tan (1.55a)

jm F je tan (1.55b)

theDirac-Maxwellequationsreduceto theusualMaxwell equations.

Explicit applicationof the fixed mixing angleconditionson the duality transformation

(1.52)on page16yields e e cos 1 m sin ¯ e cos 1 e tan sin 1cos e cos2 e sin2 1

cos e (1.56a) m e sin ª e tan cos¯ e sin ª e sin w 0 (1.56b)je je cos je tan sin w 1

cos je cos2 je sin2 1cos je (1.56c)

jm je sin ª je tan cosw je sin je sin w 0 (1.56d)

Hence,afixedmixing angle,or, equvalently, afixedratiobetweentheelectricandmagneticcharges/currents,“hides” the magneticmonopoleinfluence( m and jm) on the dynamicequations.Furthermore,wenoticethatH'

E H E cos 6 B sin 6 E cos 0 m sin H E cos 2 0 e tan sin w 6 E cos e0

tan sin H E cos e

0

sin2 cos 1

cos e0

(1.57)

or HE 1

cos2 e0

1 sin2 e

0(1.58)

andsoon for theotherequations. QED END OF EXAMPLE 1.2 ®

18

BIBLIOGRAPHY 1

[1] RichardBecker. ElectromagneticFields and Interactions. Dover Publications,Inc.,New York, NY, 1982. ISBN 0-486-64290-9.

[2] Erik Hallén. ElectromagneticTheory. Chapman& Hall, Ltd., London,1962.

[3] JohnD. Jackson.ClassicalElectrodynamics. Wiley & Sons,Inc., New York, NY . . . ,secondedition,1975. ISBN 0-471-43132-X.

[4] Lev Davidovich LandauandEvgeniyMikhailovich Lifshitz. TheClassicalTheoryofFields, volume2 of Courseof TheoreticalPhysics. PergamonPress,Ltd., Oxford . . . ,fourth revisedEnglishedition,1975. ISBN 0-08-025072-6.

[5] JamesClerk Maxwell. A Treatiseon Electricity and Magnetism, volume1. DoverPublications,Inc., New York, NY, third edition,1954. ISBN 0-486-60636-8.

[6] David Blair MelroseandR. C. McPhedran.ElectromagneticProcessesin DispersiveMedia. CambridgeUniversityPress,Cambridge. . . , 1991. ISBN 0-521-41025-8.

[7] WolfgangK. H. Panofsky andMelba Phillips. ClassicalElectricity and Magnetism.Addison-Wesley PublishingCompany, Inc., Reading,MA . . . , third edition, 1962.ISBN 0-201-05702-6.

[8] JuliusAdamsStratton.ElectromagneticTheory. McGraw-Hill Book Company, Inc.,New York, NY andLondon,1953. ISBN 07-062150-0.

[9] JackVanderlinde.ClassicalElectromagneticTheory. JohnWiley & Sons,Inc., NewYork, Chichester, Brisbane,Toronto,andSingapore,1993. ISBN 0-471-57269-1.

19

CHAPTER 2

ElectromagneticWaves

Maxwell’s microscopicequations(1.43)on page14, which areusuallywritten inthefollowing form98

E * X_& x 0(2.1a) E " RR X B (2.1b)98

B 0 (2.1c) B @ 0j X_& x N 0 @ 0 RR X E (2.1d)

canbe viewed asan axiomaticbasisfor classicalelectrodynamics.In particular,theseequationsarewell suitedfor calculatingthe electricandmagneticfields EandB from given,prescribedchargedistributions * XY& x andcurrentdistributionsj X_& x of arbitrarytime-andspace-dependentform.

However, asis well known from thetheoryof differentialequations,thesefourfirst order, coupledpartialdifferentialvectorequationscanberewrittenastwo un-coupled,secondorderpartialequations,onefor E andonefor B. We shallderivethe secondorderequationfor E, which, aswe shall seeis a homogeneouswaveequation, andthendiscussthe implicationsof this equation.We shall alsoshowhow theB field canbeeasilycalculatedfrom thesolutionof theE equation.

2.1 Thewave equation

Let us considera volumewith no net charge, *F 0, andno electromotive forceEEMF 0. Takingthecurl of (2.1b)andusing(2.1d),weobtain

21

22 CHAPTER 2. ELECTROMAGNETIC WAVES

E RR X B @ 0 RR X j N 0 RR X E (2.2)

Accordingto theoperatortripleproduct“bac-cab”ruleEquation(F.69)onpage159 E 98E ; 2E (2.3)

Furthermore,since*° 0, Equation(2.1a)on thepreviouspageyields98E 0 (2.4)

andsinceEEMF 0, Ohm’s law, Equation(1.26)on page11,yields

j fe E (2.5)

we find thatEquation(2.2)canberewritten; 2E @ 0 RR X e EN 0 RR X E 0 (2.6)

or, alsousingEquation(1.9)onpage5,; 2E @ 0 eRR X E 1 2 R 2R X 2E 0 (2.7)

which is thehomogeneouswaveequationfor E.We look for asolutionin theform of a time-harmonicwave, andmake therefore

thefollowing Fourier componentAnsatz

E E0 x ± Q² ³¬´ (2.8)

Insertingthis into Equation(2.7),weobtain

; 2E @ 0 eRR X E0 x ± Q² ³¨´ 1 2 R 2R X 2E0 x ± Q² ³¬´ ; 2E @ 0 e $¶µ E0 x ± Q² ³¬´ 1 2 ·¸µ 2E0 x ± Q² ³¨´ ; 2E @ 0 e ·¸µ E 1 2 w·¹µ 2E 0 (2.9)

whichwe canrewrite as; 2EN µ 2 2 1 N · e 0 µ E ; 2E

N µ 2 2 1 N ·º µ E 0 (2.10)

Thequantity º ) 0 e is calledtherelaxationtimeof themediumin question.In

thelimit of long º , Equation(2.10)tendsto

2.2. PLANE WAVES 23

; 2EN µ 2 2 E 0 (2.11)

which is a time-independentwaveequationfor E, representingweakly dampedpropagatingwaves.In theshort º limit wehave instead

; 2EN ·¸µ @ 0 e E 0 (2.12)

which is a time-independentdiffusionequationfor E.For mostmetalsº©» 10 14 s,whichmeansthatthediffusionpictureis goodfor

all frequencieslower thanoptical frequencies.Hence,in metallicconductors,thepropagation term R 2E 2 R X 2 is negligible even for VHF, UHF, andSHF signals.Alternatively, we maysaythatthedisplacementcurrent 0 R E R X is negligible rel-ative to theconductioncurrentj e E.

If we introducethevacuumwavenumber¼ µ (2.13)

we can write, using the fact that ½ 1¿¾ 0 @ 0 accordingto Equation(1.9) onpage5,

1ÀrÁÂ ÃÄ0ÁÂÅÃÄ

0

1Æ¢Ç ÂÈÃ ÇÊÉ Ë 0Ä0ÂÃ ÇÍÌ 0 (2.14)

wherein thelaststepwe introducedthecharacteristicimpedancefor vacuum

Ì 0 ÂÎÉ Ë 0Ä0 Ï 376Ð 7Ω (2.15)

2.2 Planewaves

Considernow the casewhereall fields dependonly on the distanceÑ to a givenplanewith unit normal ˆÒ . Thenthedel operatorbecomesÓ Â ˆÒHÔÔ Ñ (2.16)

andMaxwell’sequationsattaintheform


ˆÒÖÕ Ô EÔ Ñ Â 0 (2.17a)

ˆÒ)× Ô EÔ Ñ Â"Ø Ô BÔÚÙ (2.17b)

ˆÒÖÕÚÔ BÔ Ñ Â 0 (2.17c)

ˆÒ)×ÛÔ BÔ Ñ Â Ë 0j Ü ÙYÝ x Þ ß Ä 0 Ë 0Ô EÔàÙ Â Ë 0 Ã E ß Ä 0 Ë 0

Ô EÔÚÙ (2.17d)

Scalarmultiplying (2.17d)by ˆÒ , wefind that

0 Â ˆÒÖÕ á ˆÒ)×¦Ô BÔ Ñ â Â ˆÒÖÕ á Ë 0 Ã ß Ä 0 Ë 0ÔÔàÙ â E (2.18)

which simplifies to the first-orderordinary differential equationfor the normalcomponentãä of theelectricfield- ã ä- Ù ß ÃÄ

0ãä Â 0 (2.19)

with thesolutionãä Â ãwä 0 å¨æQç¢èêéë 0 Â ãä 0 å¨æìèêéMí (2.20)

This, togetherwith (2.17a),shows that the longitudinal componentof E, i.e., thecomponentwhich is perpendicularto theplanesurfaceis independentof Ñ andhasatimedependencewhichexhibitsanexponentialdecay, with adecrementgivenbytherelaxationtime À in themedium.

Scalarmultiplying (2.17b)by ˆÒ , wesimilarly find that

0 Â ˆÒÖÕ á ˆÒ)×¦Ô EÔ Ñ â ÂØ ˆÒÖÕQÔ BÔàÙ (2.21)

or

ˆÒÖÕ Ô BÔàÙ Â 0 (2.22)

From this, and (2.17c),we concludethat the only longitudinal componentof Bmustbeconstantin bothtime andspace.In otherwords,theonly non-staticsolu-tion mustconsistof transversecomponents.

2.2. PLANE WAVES 25

2.2.1 Telegrapher’sequation

In analogywith Equation(2.7)onpage22,we caneasilyderive theequationÔ 2EÔ Ñ 2 Ø Ë 0 Ã Ô EÔàÙ Ø 1Æ 2 Ô 2EÔÚÙ 2 Â 0 (2.23)

This equation,which describesthe propagation of planewaves in a conductingmedium,is calledthe telegrapher’sequation. If themediumis aninsulatorsothatÃ Â 0, thentheequationtakestheform of theone-dimensionalwaveequationÔ 2EÔ Ñ 2 Ø 1Æ 2 Ô 2EÔÚÙ 2 Â 0 (2.24)

As is well known, eachcomponentof this equationhasa solutionwhich canbewrittenã î Â)ï Ü Ñ Ø Æ Ù Þ ßð Ü ÑÊß Æ Ù Þ Ý ñ Â 1 Ý 2 Ý 3 (2.25)

where ï and ð arearbitrary(non-pathological)functionsof their respective argu-ments. This generalsolution representsperturbationswhich propagatealong Ñ ,wherethe ï perturbationpropagatesin thepositive Ñ directionandthe ð perturba-tion propagatesin thenegative Ñ direction.

If weassumethatE is time-harmonic,i.e., canberepresentedby aFouriercom-ponentproportionalto exp Øwò Á Ù , thesolutionof Equation(2.24)becomes

E Â E0 å¨æQóõô÷ö¨èêøùYúüû (2.26)

By introducingthewavevector

k Â Ç ˆÒ Â Á Æ ˆÒ Â Á Æ ˆý (2.27)

this solutioncanbewrittenas

E Â E0 å¢óõô k þ x æìö¬èÿû (2.28)

Let usconsidertheminussign in theexponentin Equation(2.26)above. Thiscorrespondsto a wave which propagatesin thedirectionof increasingÑ . Insertingthissolutioninto Equation(2.17b)on thefacingpage,gives

ˆÒ)×ÛÔ EÔ Ñ Âfò Á B Âò Ç ˆÒ × E (2.29)

or, solvingfor B,

B Â ÇÁ ˆÒ)× E Â 1Á k × E Â 1Æ ˆý × E Â Ä0 Ë 0 ˆÒ)× E (2.30)

Hence,to eachtransversecomponentof E, thereexistsanassociatedmagneticfieldgiven by Equation(2.30)above. If E and/orB hasa directionin spacewhich isconstantin time,wehave aplanepolarisedwave(or linearly polarisedwave).


2.2.2 Wavesin conductivemedia

Assumingthat our medium has a finite conductivity Ã , and making the time-harmonicwave Ansatzin Equation(2.23)on theprecedingpage,we find that thetime-independenttelegrapher’sequationcanbewrittenÔ 2EÔ Ñ 2 ß Ä 0 Ë 0

Á 2E ß ò Ë 0 Ã Á E Â Ô 2EÔ Ñ 2 ß 2E Â 0 (2.31)

where 2 Â Ä 0 Ë 0Á 2 á 1 ß ò ÃÄ

0Á â Â Á 2Æ 2 á 1 ß ò ÃÄ

0Á â Â Ç 2 á 1 ß ò ÃÄ

0Á â (2.32)

where,in thelaststep,Equation(2.13)on page23 wasusedto introducethewavenumberÇ . Takingthesquarerootof this expression,we obtain Â Ç É 1 ß ò ÃÄ

0ÁÂ ß ò (2.33)

where

FÂ Ç É 1 ß çë 0 ö 2 ß 1

2(2.34a)

1Â Ç É 1 ß çë 0 ö 2 Ø 1

2(2.34b)

Hence,thesolutionof thetime-independenttelegrapher’sequation,Equation(2.31),canbewritten

E Â E0 å¨æ ú'å¢óõô¬ú_æìö¬èêû (2.35)

With theaid of Equation(2.30)on thepreviouspagewe cancalculatetheassoci-atedmagneticfield, andfind thatit is givenby

B Â 1Á ˆý × E Â 1Á Ü ˆý × E ÞÜ ß ò Þ Â 1Á Ü ˆý × E Þ å¢ó (2.36)

wherewe have, in the last step, rewritten ß ò in the amplitude-phaseform exp ò . Fromtheabove,we immediatelyseethatE is dampedandthatE andB in thewave areoutof phase.

In thecasethat Ä 0Á Ã , we canapproximate asfollows: Â Ç á 1 ß ò ÃÄ

0Á â 1

2 Â Ç ò ÃÄ0Á 1 Ø1ò Ä 0

ÁÃ 1

2 Ï Ç Ü 1 ß ò Þ É Ã2Ä 0Á

Â Ä0 Ë 0Á Ü 1 ß ò Þ É Ã2Ä 0

ÁÂ Ü 1 ß ò Þ É Ë 0 Ã Á2(2.37)

From this analysiswe concludethat whenthe wave impingesperpendicularlyuponthemedium,thefieldsaregiven,insidethismedium,by

E Â E0 exp ! Ø É Ë 0 Ã Á2Ñ#" exp ! ò á É Ë 0 Ã Á2

Ñ Ø Á Ù â " (2.38a)

B Â Ü 1 ß ò Þ É Ë 0 Ã2Á Ü ˆÒ × E Þ (2.38b)

Hence,bothfieldsfall off by a factor1$ å atadistance% Â 2Ë 0 Ã Á (2.39)

This distance%

is calledtheskindepth.

2.3 ObservablesandaveragesIn theabovewehaveusedcomplex notationquiteextensively. This is for mathem-atical convenienceonly. For instance,in this notationdifferentiationsarealmosttrivial to perform.However, everyphysicalmeasurablequantityis alwaysrealval-ued. I.e., “Ephysical Â Re Emathematical .” It is particularlyimportantto rememberthiswhenoneworkswith productsof physicalquantities.

Generallyspeaking,we tendto measuretemporalaveragesÜ'&)(!Þ of our physicalobservables. If we have two physical vectorsF andG which both aretime har-monic,i.e., canberepresentedby Fouriercomponentsproportionalto exp Øò Á Ù ,it is easyto show that the averageof the productof the two physical quantitiesrepresentedby F andG canbeexpressedas& Re F Õ Re G ( Â 1

2Re F Õ G * Â 1

2Re F * Õ G (2.40)

where * denotescomplex conjugate.

27

BIBLIOGRAPHY 2


29

CHAPTER 3

ElectromagneticPotentials

Insteadof expressingthelawsof electrodynamicsin termsof electricandmagneticfields, it turnsout that it is often moreconvenientto expressthe theoryin termsof potentials. In this Chapterwe will introduceandstudythe propertiesof suchpotentials.

3.1 TheelectrostaticscalarpotentialAs wesaw in Equation(1.6)onpage4, theelectrostaticfield EstatÜ x Þ is irrotational.Hence,it maybeexpressedin termsof thegradientof a scalarfield. If we denotethisscalarfield by Ø,+ statÜ x Þ , we get

EstatÜ x Þ ÂØ Ó + statÜ x Þ (3.1)

Takingthedivergenceof this andusingEquation(1.7)on page4, we obtainPois-sons’equation-

2 + statÜ x Þ ÂØ Ó Õ EstatÜ x Þ ÂØ/. Ü x ÞÄ0

(3.2)

A comparisonwith thedefinitionof Estat, namelyEquation(1.5) on page4, aftertheÓ

hasbeenmovedoutof theintegral,showsthatthisequationhasthesolution+ statÜ x Þ Â 140 Ä 0 1 2 . Ü x Þ x Ø x - 3. ¨ß (3.3)

wheretheintegrationis takenover all sourcepointsx at which thechargedensity. Ü x Þ is non-zeroand is an arbitraryquantitywhich hasa vanishinggradient.An exampleof sucha quantityis a scalarconstant.Thescalarfunction + statÜ x Þ inEquation(3.3)above is calledtheelectrostaticscalarpotential.

31

32 CHAPTER 3. ELECTROMAGNETIC POTENTIALS

3.2 ThemagnetostaticvectorpotentialConsidertheequationsof magnetostatics(1.20)onpage8. FromEquation(M.74)on page180we know thatany 3D vectora hasthepropertythat

Ó Õ Ü Ó × aÞ 3 0andin thederivationof Equation(1.15)onpage7 in magnetostaticswe foundthatÓ Õ BstatÜ x Þ Â 0. We thereforerealisethatwe canalwayswrite

BstatÜ x Þ Â Ó × AstatÜ x Þ (3.4)

whereAstatÜ x Þ is calledthemagnetostaticvectorpotential.We saw above that the electrostaticpotential (as any scalarpotential) is not

unique: we may, without changingthe physics,add to it a quantitywhosespa-tial gradientvanishes. A similar arbitrarinessis true also for the magnetostaticvectorpotential.

In themagnetostaticcase,we maystartfrom Biot-Savart’s law asexpressedbyEquation(1.13)onpage6 and“move the

Óoutof theintegral:”

BstatÜ x Þ Â Ë 0

40 1 2 j Ü x Þ × x Ø x x Ø x 3 - 3. ÂØ Ë 0

40 1 2 j Ü x Þ × Ó á 1 x Ø x â - 3. Â Ó × Ë 0

40 1 2 j Ü x Þ x Ø x - 3. (3.5)

An identificationof termsallowsusto definethestaticvectorpotentialas

AstatÜ x Þ Â Ë 0

40 1 2 j Ü x Þ x Ø x - 3. ¨ß a Ü x Þ (3.6)

wherea Ü x Þ is anarbitraryvectorfield whosecurl vanishes.FromEquation(M.70)on page179we know thatsucha vectorcanalwaysbewritten asthegradientof ascalarfield.

3.3 Theelectromagneticscalarandvectorpotentials

Let usnow generalisethestaticanalysisabove to theelectrodynamiccase,i.e., thecasewith temporalandspatialdependentsources. Ü ÙYÝ x Þ and j Ü ÙYÝ x Þ , andcorres-pondingfieldsE Ü Ù_Ý x Þ andB Ü ÙYÝ x Þ , asdescribedby Maxwell’s equations(1.43)onpage14. In otherwords, let us study the electromagneticpotentials + Ü Ù_Ý x Þ andA Ü ÙYÝ x Þ .

3.3. THE ELECTROMAGNETIC SCALAR AND VECTOR POTENTIALS 33

From Equation(1.43c) on page14 we note that also in electrodynamicsthehomogeneousequation

Ó Õ B Ü Ù_Ý x Þ Â 0 remainsvalid. Becauseof thisdivergence-freenatureof the time- andspace-dependentmagneticfield, we canexpressit asthecurl of anelectromagneticvectorpotential:

B Ü Ù_Ý x Þ Â Ó × A Ü ÙYÝ x Þ (3.7)

Insertingthis expressioninto the other homogeneousMaxwell equation,Equa-tion (1.30)on page11,weobtainÓ × E Ü Ù_Ý x Þ ÂØ ÔÔàÙ / Ó × A Ü ÙYÝ x Þ54 ÂØ Ó ×«ÔÔàÙ A Ü Ù_Ý x Þ (3.8)

or, rearrangingtheterms,Ó × á E Ü Ù_Ý x Þ ß ÔÔàÙ A Ü ÙYÝ x Þ â Â 0 (3.9)

As beforeweutilise thevanishingcurl of avectorexpressionto write thisvectorexpressionasthegradientof a scalarfunction. If, in analogywith theelectrostaticcase,we introducethe electromagneticscalar potential function Ø6+ Ü ÙYÝ x Þ , Equa-tion (3.9)becomesequivalentto

E Ü Ù_Ý x Þ ß ÔÔàÙ A Ü ÙYÝ x Þ ÂHØ Ó + Ü ÙYÝ x Þ (3.10)

This meansthatin electrodynamics,E Ü Ù_Ý x Þ canbecalculatedfrom theformula

E Ü Ù_Ý x Þ ÂØ Ó + Ü Ù_Ý x Þ Ø ÔÔàÙ A Ü ÙYÝ x Þ (3.11)

andB Ü Ù_Ý x Þ from Equation(3.7) above. Hence,it is a matterof tastewhetherwewant to expressthe laws of electrodynamicsin termsof thepotentials+ Ü Ù_Ý x Þ andA Ü Ù_Ý x Þ , or in termsof thefieldsE Ü Ù_Ý x Þ andB Ü Ù_Ý x Þ . However, thereexistsan im-portantdifferencebetweenthe two approaches:in classicalelectrodynamicstheonly directly observable quantitiesare the fields themselves (and quantitiesde-rivedfrom them)andnot thepotentials.On theotherhand,thetreatmentbecomessignificantlysimpler if we usethe potentialsin our calculationsandthen,at thefinal stage,useEquation(3.7)andEquation(3.11)above to calculatethefieldsorphysicalquantitiesexpressedin thefields.

Inserting(3.11)and(3.7)into Maxwell’sequations(1.43)onpage14weobtain,


aftersomesimplealgebraandtheuseof Equation(1.9)on page5,-2 + ß ÔÔÚÙ Ü Ó Õ A Þ ÂØ/. Ü ÙYÝ x ÞÄ

0(3.12a)-

2A Ø 1Æ 2 Ô 2ÔàÙ 2A Ø Ó á Ó Õ A ß 1Æ 2 ÔÔàÙ + â ÂØ Ë 0j Ü Ù_Ý x Þ (3.12b)

Thesetwo secondorder, coupled,partialdifferentialequations,representingin allfour scalarequations(onefor + andoneeachfor the . 1 Ý . 2, and . 3 componentsof A) arecompletelyequivalentto theformulationof electrodynamicsin termsofMaxwell’sequations,which representsix scalarfirst-order, coupled,partialdiffer-entialequations(oneequationfor eachof thecomponentsof E andB).

3.3.1 Electromagneticgauges

Lorentzequationsfor theelectromagneticpotentials

As they stand,Equations(3.12) look complicatedand may seemto be of lim-ited use. However, if we write Equation(3.7) on the precedingpagein the formÓ × A Ü Ù_Ý x Þ Â B Ü ÙYÝ x Þ we canconsiderthis asa specificationof

Ó × A. But weknow from Helmholtz’theorem that in orderto determinethe (spatialbehaviour)of A completely, wemustalsospecify

Ó Õ A. Sincethisdivergencedoesnotenterthederivationabove,weare freeto choose

Ó Õ A in whateverwaywelikeandstillobtain the samephysicalresults! This is somewhat analogousto the freedomofaddinganarbitraryscalarconstant(whosegradvanishes)to thepotentialenergy inclassicalmechanicsandstill getthesameforce.

With ajudiciouschoiceofÓ Õ A, thecalculationscanbesimplifiedconsiderably.

LorentzintroducedÓ Õ A ß 1Æ 2 ÔÔÚÙ + Â 0 (3.13)

which is calledtheLorentzgaugecondition, becausethischoicesimplifiesthesys-temof coupledequations(3.12)above into thefollowing setof uncoupledpartialdifferentialequations:7 2 + def3 1Æ 2 Ô 2ÔàÙ 2 + Ø - 2 + Â8. Ü ÙYÝ x ÞÄ

0(3.14a)7 2A

def3 1Æ 2 Ô 2ÔàÙ 2A Ø - 2A Â Ë 0j Ü ÙYÝ x Þ (3.14b)

where7 2 is thed’Alembertoperator discussedin ExampleM.4 on page176. We

shallcall (3.14)theLorentzequationsfor theelectromagneticpotentials.


Gauge transformations

We saw in Section3.1 on page31 andin Section3.2 on page32 that in electro-staticsandmagnetostaticswe have a certainmathematicaldegreeof freedom,upto termsof vanishinggradientsanddivergences,to pick suitableformsfor thepo-tentialsandstill get thesamephysicalresult. In fact,theway theelectromagneticscalarpotential+ Ü Ù_Ý x Þ andthevectorpotentialA Ü Ù_Ý x Þ arerelatedto thephysicallyobservablesgivesleewayfor similar “manipulation”of themalsoin electrodynam-ics. If we transform+ Ü Ù_Ý x Þ andA Ü ÙYÝ x Þ simultaneouslyinto new ones+ Ü Ù_Ý x Þ andA Ü Ù_Ý x Þ accordingto thescheme+ Ü Ù_Ý x Þ Â9+ Ü ÙYÝ x Þ ß ÔÔÚÙ Γ Ü Ù_Ý x Þ (3.15a)

A Ü Ù_Ý x Þ Â A Ü Ù_Ý x Þ Ø Ó Γ Ü Ù_Ý x Þ (3.15b)

whereΓ Ü Ù_Ý x Þ is anarbitrary, differentiablescalarfunctioncalledthegauge func-tion, andinsertthetransformedpotentialsinto Equation(3.11)on page33 for theelectricfield andinto Equation(3.7) on page33 for themagneticfield, we obtainthetransformedfields

E Â"Ø Ó + Ø ÔÔàÙ A ÂØ Ó + Ø ÔÔàÙ Ó Γ Ø ÔÔÚÙ A ß ÔÔàÙ Ó Γ

Â"Ø Ó + Ø ÔÔàÙ A (3.16a)

B Â Ó × A Â Ó × A Ø Ó × Ó Γ Â Ó × A (3.16b)

where,onceagain Equation(M.70) on page179 wasused.Comparingtheseex-pressionswith (3.11)and(3.7) we seethat thefieldsareunaffectedby thegaugetransformation(3.15). A transformationof the potentials+ andA which leavesthefields,andhenceMaxwell’s equations,invariantis calleda gauge transforma-tion. A physical law which doesnot changeundera gaugetransformationis saidto be gauge invariant. By definition, the fields themselvesare,of course,gaugeinvariant.

The potentials + Ü ÙYÝ x Þ and A Ü ÙYÝ x Þ calculatedfrom (3.12) on the facing page,with anarbitrarychoiceof

Ó Õ A, canbe furthergaugetransformedaccordingto(3.15)above. If, in particular, wechoose

Ó Õ A accordingto theLorentzcondition,Equation(3.13)on thefacingpage,andapply thegaugetransformation(3.15)ontheresultingLorentzequations(3.14)on theprecedingpage,theseequationswillbetransformedinto

1Æ 2 Ô 2ÔàÙ 2 +©Ø - 2 + Ø ÔÔàÙ á 1Æ 2 Ô 2ÔÚÙ 2 Γ Ø - 2Γ â Â . Ü Ù_Ý x ÞÄ0

(3.17a)

1Æ 2 Ô 2ÔÚÙ 2A Ø - 2A Ø Ó á 1Æ 2 Ô 2ÔàÙ 2 Γ Ø - 2Γ â Â Ë 0j Ü Ù_Ý x Þ (3.17b)


We noticethat if we requirethat the gaugefunction Γ Ü Ù_Ý x Þ itself be restrictedtofulfil thewave equation

1Æ 2 Ô 2ÔàÙ 2Γ Ø - 2Γ Â 0 (3.18)

thesetransformedLorentzequationswill keeptheir original form. Thesetof po-tentialswhich have beengaugetransformedaccordingto Equation(3.15)on theprecedingpagewith a gaugefunction Γ Ü ÙYÝ x Þ which is restrictedto fulfil Equa-tion (3.18)above, i.e., thosegaugetransformedpotentialsfor which the Lorentzequations(3.14)areinvariant,comprisestheLorentzgauge.

Otherusefulgaugesare: The radiation gauge, also known as the transverse gauge, defined byÓ Õ A Â 0.: TheCoulombgauge, definedby + Â 0,Ó Õ A Â 0.: The temporal gauge, alsoknown astheHamiltongauge, definedby + Â 0.: Theaxial gauge, definedby 3 Â 0.

Theprocessof choosingaparticulargaugeconditionis referredto asgaugefixing.

3.3.2 Solutionof the Lorentzequationsfor the electromag-neticpotentials

As wesee,theLorentzequations(3.14)onpage34for + Ü Ù_Ý x Þ andA Ü ÙYÝ x Þ representa setof uncoupledequationsinvolving four scalarequations(oneequationfor +andoneequationfor eachof thethreecomponentsof A). Eachof thesefour scalarequationsis an inhomogeneouswaveequationof thefollowing genericform:7 2Ψ Ü ÙYÝ x Þ Â)ï Ü ÙYÝ x Þ (3.19)

whereΨ is ashorthandfor either + or oneof thevectorcomponentsof A, and ï isthepertinentgenericsourcecomponent.

Weassumethatoursourcesarewell-behavedenoughin time Ù sothattheFouriertransformpair for thegenericsourcefunction; æ 1 / ï ö Ü x Þ54 def3 ï Ü ÙYÝ x Þ Â 1 <æ < ï ö Ü x Þ å¨æQó ö¬è - Á (3.20a); / ï Ü Ù_Ý x Þ54 def3 ï ö Ü x Þ Â 1

20 1 <æ < ï Ü ÙYÝ x Þ å¢ó ö¬è - Ù (3.20b)

exists,andthatthesameis truefor thegenericpotentialcomponent:


Ψ Ü ÙYÝ x Þ Â 1 <æ < Ψ ö Ü x Þ å¨æQó ö¨è - Á (3.21a)

Ψ ö Ü x Þ Â 120 1 <æ < Ψ Ü ÙYÝ x Þ å¢ó ö¬è - Ù (3.21b)

Insertingthe Fourier representations(3.20a)and(3.21a)into Equation(3.19) ontheprecedingpage,andusingthevacuumdispersionrelationfor electromagneticwavesÁ Â Æ:Ç (3.22)

thegeneric3D inhomogeneouswave equationEquation(3.19)on thefacingpageturnsinto-

2Ψ ö Ü x Þ ß Ç 2Ψ ö Ü x Þ ÂHØÊï ö Ü x Þ (3.23)

which is a 3D inhomogeneoustime-independentwaveequation, often called the3D inhomogeneousHelmholtzequation.

As postulatedby Huygen’s principle, eachpoint on a wave front actsasa pointsourcefor sphericalwaveswhich form a new wave from a superpositionof theindividualwavesfrom eachof thepointsourcesontheold wavefront. Thesolutionof (3.23)canthereforebeexpressedasa superpositionof solutionsof anequationwherethesourcetermhasbeenreplacedby apoint source:-

2 = Ü x Ý x Þ ß Ç 2 = Ü x Ý x Þ Â"Ø % Ü x Ø x Þ (3.24)

andthesolutionof Equation(3.23)abovewhichcorrespondsto thefrequency Á isgivenby thesuperposition

Ψ ö Â 1 ï ö Ü x Þ = Ü x Ý x Þ - 3. (3.25)

Thefunction = Ü x Ý x Þ is calledtheGreen’s functionor thepropagator.In Equation(3.24), the Dirac generalisedfunction

% Ü x Ø x Þ , which representsthe point source,dependsonly on x Ø x andthereis no angulardependenceinthe equation.Hence,the solutioncanonly be dependenton > Â x Ø x . If weinterpret > as the radial coordinatein a sphericallypolar coordinatesystem,the“sphericallysymmetric” = Ü > Þ is givenby thesolutionof- 2- > 2 Ü > = Þ ß Ç 2 Ü > = Þ ÂØ > % Ü > Þ (3.26)

Away from > Â x Ø x Â 0, i.e., away from the sourcepoint x , this equationtakestheform


- 2- > 2 Ü > = Þ ß Ç 2 Ü > = Þ Â 0 (3.27)

with thewell-known generalsolution= Â@?BA å ó ùDC> ß ? æ å æQó ùDC> 3 ?BA å ó÷ùFE x æ x G E x Ø x ß ? æ å æQó÷ùFE x æ x G E x Ø x (3.28)

where ? ø areconstants.In order to evaluatethe constants? ø , we insert the generalsolution Equa-

tion (3.28) above into Equation(3.24) on the precedingpageand integrateovera smallvolumearound> Â x Ø x Â 0. Since= Ü HH x Ø x IHH Þ J ?BA 1 x Ø x ß ? æ 1 x Ø x Ý HH x Ø x KHHML 0 (3.29)

Equation(3.24)on thepreviouspagecanunderthis assumptionbeapproximatedby N ?BA ß ? æO 1 -

2 á 1 x Ø x â - 3. ß Ç 2

N ?BA ß ? æO 1 1 x Ø x - 3. Â"Ø 1 % Ü HH x Ø x PHH Þ - 3. (3.30)

In virtue of thefact that thevolumeelement- 3. in sphericalpolarcoordinatesisproportionalto x Ø x 2, thesecondintegralvanisheswhen x Ø x L 0. Further-more, from Equation(M.65) on page178, we find that the integrandin the firstintegral canbewritten as Ø 40 % Ü' x Ø x Þ and,hence,that?QA ß ? æ Â 1

40 (3.31)

Insertionof the generalsolution Equation(3.28) into Equation(3.25) on thepreviouspagegives

Ψ ö Ü x Þ ÂR?QA 1 ï ö Ü x Þ å ó÷ùFE x æ x G E x Ø x - 3. ¨ß ? æ 1 ï ö Ü x Þ å æQó ùFE x æ x G E x Ø x - 3. (3.32)

The Fourier transformto ordinary Ù domainof this is obtainedby insertingtheabove expressionfor Ψ ö Ü x Þ into Equation(3.21a)on theprecedingpage:

Ψ Ü Ù_Ý x Þ Â@?BA 1/1 ï ö Ü x Þ exp S Øò Á Ù Ø ùFE x æ x G Eö UT x Ø x - Á - 3. ß ? æ 1/1 ï ö Ü x Þ exp S Øwò Á Ù ß ùFE x æ x G Eö UT x Ø x - Á - 3. (3.33)

If we introducethe retardedtime Ù ret andtheadvancedtime Ù adv in the followingway [using the fact that in vacuum Ç $ Á Â 1$ Æ , accordingto Equation(3.22)on

]:

Ù ret Â Ù ret Ü Ù_Ý HH x Ø x KHH Þ Â Ù Ø Ç x Ø x Á Â Ù Ø x Ø x Æ (3.34a)

Ù adv Â Ù adv Ü Ù_Ý HH x Ø x KHH Þ Â Ù ß Ç x Ø x Á Â Ù ß x Ø x Æ (3.34b)

anduseEquation(3.20a)on page36,we obtain

Ψ Ü ÙYÝ x Þ Â@?BA 1 ï Ü Ù ret Ý x Þ x Ø x - 3. ¨ß ? æ 1 ï Ü Ù advÝ x Þ x Ø x - 3. (3.35)

This is a solutionof the genericinhomogeneouswave equationfor the potentialcomponentsEquation(3.19) on page36. We note that the solution at time Ù atthe field point x is dependenton the behaviour at other times Ù of the sourceatx andthatbothretardedandadvancedÙ aremathematicallyacceptablesolutions.However, if weassumethatcausalityrequiresthatthepotentialat Ü ÙYÝ x Þ is setupbythesourceatanearliertime,i.e., at Ü Ù ret Ý x Þ , wemustin Equation(3.35)set ? æ Â 0andtherefore,accordingto Equation(3.31)on theprecedingpage,? A Â 1$rÜ 40 Þ .Theretardedpotentials

From the above discussionon the solutionof the inhomogeneouswave equationwe concludethatundertheassumptionof causalitytheelectromagneticpotentialsin vacuumcanbewritten+ Ü Ù_Ý x Þ Â 1

40 Ä 0 1 . Ü Ù ret Ý x Þ x Ø x - 3. (3.36a)

A Ü Ù_Ý x Þ Â Ë 0

40 1 j Ü Ù ret Ý x Þ x Ø x - 3. (3.36b)

Sincetheseretardedpotentialswereobtainedassolutionsto theLorentzequations(3.14)onpage34they arevalid in theLorentzgaugebut maybegaugetransformedaccordingto theschemedescribedin Subsection3.3.1on page35. As they stand,we shallusethemfrequentlyin thefollowing.

39

BIBLIOGRAPHY 3

[1] L. D. Fadeev and A. A. Slavnov. Gauge Fields: Introduction to QuantumThe-ory. Number50 in Frontiersin Physics: A LectureNote andReprintSeries.Ben-jamin/CummingsPublishingCompany, Inc., Reading,MA . . . , 1980. ISBN 0-8053-9016-2.

[2] Mike Guidry. Gauge Field Theories: An Introduction with Applications.Wiley & Sons,Inc., New York, NY . . . , 1991. ISBN 0-471-63117-5.



41

CHAPTER 4

TheElectromagneticFields

While, in principle, the electric and magneticfields can be calculatedfrom theMaxwell equationsin Chapter1, or evenfrom thewave equationsin Chapter2, itis oftenphysicallymorelucid to calculatethemfrom theelectromagneticpotentialsderivedin Chapter3. In thisChapterwewill derivetheelectricandmagneticfieldsfrom thepotentials.

We recall that in order to find the solution (3.35) for the genericinhomogen-eouswave equation(3.19)on page36 we presupposedtheexistenceof a Fouriertransformpair (3.20a)onpage36 for thegenericsourceterm

ï Ü Ù_Ý x Þ Â 1 <æ < ï ö Ü x Þ å¨æQó ö¬è - Á (4.1a)

ï ö Ü x Þ Â 120 1 <æ < ï Ü Ù_Ý x Þ å¢ó ö¨è - Ù (4.1b)

andfor thegenericpotentialcomponentV Ü ÙYÝ x Þ Â 1 <æ < V ö Ü x Þ å¨æQó ö¨è - Á (4.2a)V ö Ü x Þ Â 120 1 <æ < V Ü Ù_Ý x Þ å¢ó ö¬è - Ù (4.2b)

Thatsuchtransformpairsexists is true for mostphysical variableswhich arenotstrictly monotonicallyincreasinganddecreasingwith time. For chargesandcur-rentsvarying in time we cantherefore,without lossof generality, work with in-dividual Fourier components.Strictly speaking,the existenceof a singleFouriercomponentassumesa monochromaticsource,which in turn requiresthattheelec-tric and magneticfields exist for infinitely long times. However, by taking theproperlimits, we canstill usethis approacheven for sourcesandfields of finiteduration.

43

44 CHAPTER 4. THE ELECTROMAGNETIC FIELDS

This is themethodwe shallutilise in this Chapterin orderto derive theelectricandmagneticfields in vacuumfrom arbitrarygiven charge densities. Ü Ù_Ý x Þ andcurrentdensitiesj Ü Ù_Ý x Þ , definedby theFouriertransformpairs. Ü ÙYÝ x Þ Â 1 <æ < . ö Ü x Þ å¨æQó ö¨è - Á (4.3a). ö Ü x Þ Â 1

20 1 <æ < . Ü ÙYÝ x Þ å¢ó ö¬è - Ù (4.3b)

and

j Ü ÙYÝ x Þ Â 1 <æ < j ö Ü x Þ å¨æQó ö¬è - Á (4.4a)

j ö Ü x Þ Â 120 1 <æ < j Ü Ù_Ý x Þ å¢ó ö¨è - Ù (4.4b)

undertheassumptionthatonly retardedpotentialsproducephysically acceptablesolutions.1

TheFouriertransformpair for theretardedvectorpotentialcanthenbewritten+ Ü Ù_Ý x Þ Â 1 <æ < + ö Ü x Þ å¨æQó ö¬è - Á (4.5a)+ ö Ü x Þ Â 120 1 <æ < + Ü Ù_Ý x Þ å¢ó ö¬è - Ù Â 1

40 Ä 0 1 . ö Ü x Þ å ó÷ùFE x æ x G E x Ø x - 3. (4.5b)

wherein thelaststep,wemadeuseof theexplicit expressionfor theFouriertrans-form of thegenericpotentialcomponent

V ö Ü x Þ , Equation(3.32)on page38. Sim-ilarly, thefollowing Fouriertransformpair for thevectorpotentialmustexist:

A Ü ÙYÝ x Þ Â 1 <æ < A ö Ü x Þ å¨æQó ö¨è - Á (4.6a)

A ö Ü x Þ Â 120 1 <æ < A Ü ÙYÝ x Þ å¢ó ö¬è - Ù Â Ë 0

40 1 j ö Ü x Þ å ó÷ùFE x æ x G E x Ø x - 3. (4.6b)

Clearly, wemustrequirethat

A ö Â A * æìö Ý + ö ÂW+ * æìö (4.7)

in order that all physical quantitiesbe real. Similar transformpairsandrequire-mentsof realvaluednessexist for thefieldsthemselves.

In thelimit thatthesourcescanbeconsideredasbeingmonochromatic,wehave

1In fact, JohnA. WheelerandRichardP. Feynmanderived in 1945a fully self-consistentelectro-dynamicsusingboththeretardedandtheadvancedpotentials[6]; Seealso[1].

4.1. THE MAGNETIC FIELD 45

themuchsimplerexpressions. Ü Ù_Ý x Þ Â . 0 Ü x Þ å¨æQó ö¬è (4.8a)

j Ü ÙYÝ x Þ Â j0 Ü x Þ å¨æQó ö¬è (4.8b)+ Ü Ù_Ý x Þ ÂW+ 0 Ü x Þ å¨æQó ö¨è (4.8c)

A Ü Ù_Ý x Þ Â A0 Ü x Þ å¨æQó ö¬è (4.8d)

whereagain the real-valuednessof all thesequantitiesis implied. As discussedabove, we cansafelyassumethat all formulasderived for a generalFourier rep-resentationof the source(generaldistribution of frequenciesin the source)arevalid for thesesimple limiting cases.We notethat in this context, we canmakethe formal identification . ö Â . 0

% Ü Á Ø Á Þ , j ö Â j0% Ü Á Ø Á Þ etc., andthat we

thereforeoftenidentify . ö with . 0, j ö with j0 andsoon.

4.1 Themagneticfield

Let usnow computethemagneticfield from thevectorpotential,definedby Equa-tion (4.6a)andEquation(4.6b)ontheprecedingpage,andFormula(3.7)onpage33:

B Ü Ù_Ý x Þ Â Ó × A Ü ÙYÝ x Þ (4.9)

The calculationsare much simplified if we work in Á spaceand, at the finalstage,Fourier transformbackto ordinary Ù space.We areworking in theLorentzgaugeandnotethatin Á spacetheLorentzcondition,Equation(3.13)on page34,takestheformÓ Õ A ö ØFò Ç Æ + ö Â 0 (4.10)

whichprovidesarelationbetween(theFouriertransformsof) thevectorandscalarpotentials.

UsingtheFouriertransformedversionof Equation(4.9)andEquation(4.6b)ontheprecedingpage,we obtain

B ö Ü x Þ Â Ó × A ö Ü x Þ Â Ë 0

40 Ó × 1 2 j ö Ü x Þ å ó÷ùFE x æ x G E x Ø x X 3. (4.11)

UsingFormula(F.62)on page159,wecanrewrite thisas


B ö Ü x Þ ÂØ Ë 0

40 1 2 j ö Ü x Þ ×ZY Ó\[ å ó÷ùFE x æ x G E x Ø x ]B^_X 3. ÂØ Ë 0

40 Y 1 2 j ö Ü x Þ × [ Ø x Ø x x Ø x 3 ] å¢ó ùFE x æ x G E X 3. ß 1 2 j ö Ü x Þ × á ò Ç x Ø x x Ø x å¢ó ùFE x æ x G E â 1 x Ø x X 3. ^

Â Ë 0

40 Y 1 2 j ö Ü x Þ å ó÷ùFE x æ x G E × Ü x Ø x Þ x Ø x 3 X 3. ß 1 2 Ü Øò Ç Þ j ö Ü x Þ å ó÷ùFE x æ x G E × Ü x Ø x Þ x Ø x 2 X 3. ^ (4.12)

From this expressionfor the magneticfield in the frequency ( Á ) domain,weobtainthemagneticfield in thetemporal( Ù ) domainby takingthe inverseFouriertransform(usingtheidentity Øwò Ç ÂØwò Á $ Æ ):

B Ü ÙYÝ x Þ Â 1 <æ < B ö Ü x Þ å¨æQó ö¨è X ÁÂ Ë 0

40a` 1 2 S b j ö Ü x Þ å óõôõùFE x æ x G E æìö¬èÿû X Á T × Ü x Ø x Þ x Ø x 3 X 3. ß 1Æ 1 2 S b Ü Øwò Á Þ j ö Ü x Þ å ó¹ôõùFE x æ x G E æìö¬èÿû X Á T × Ü x Ø x Þ x Ø x 2 X 3. Kc

Â Ë 0

40 1 2 j Ü Ù ret Ý x Þ × Ü x Ø x Þ x Ø x 3 X 3. d e f gInductionfieldß Ë 0

40 Æ 1 2 j Ü Ù ret Ý x Þ × Ü x Ø x Þ x Ø x 2 X 3. d e f gRadiationfield

(4.13)

where

j Ü Ù ret Ý x Þ def3 á Ô jÔàÙ â èihrè Gret

(4.14)

The first term, the inductionfield, dominatesnearthe currentsourcebut falls offrapidly with distancefrom it, while thesecondterm,the radiationfield or the farfield, dominatesat largedistancesandrepresentsenergy that is transportedout toinfinity. Notehow thespatialderivatives(

Ó) gave riseto a timederivative (˙)!

4.2. THE ELECTRIC FIELD 47

4.2 TheelectricfieldTo calculatethe electric field, we use the Fourier transformedversion of For-mula (3.11)on page33, insertingEquations(4.5b)and(4.6b)asthe explicit ex-pressionsfor theFouriertransformsof + andA:

E ö Ü x Þ ÂØ Ó + ö Ü x Þ ß ò Á A ö Ü x ÞÂØ 140 Ä 0

Ó 1 2 . ö Ü x Þ å ó÷ùFE x æ x G E x Ø x X 3. ¨ß ò Ë 0Á

40 1 2 j ö Ü x Þ å ó÷ùFE x æ x G E x Ø x X 3. Â 1

40 Ä 0

Y 1 2 . ö Ü x Þ å ó÷ùFE x æ x G E Ü x Ø x Þ x Ø x 3 X 3. Øò Ç 1 2 á . ö Ü x Þ'Ü x Ø x Þ x Ø x Ø j ö Ü x ÞÆ â å ó ùjE x æ x G E x Ø x X 3. ^ (4.15)

UsingtheFouriertransformof thecontinuityequation(1.21)onpage9Ó Õ j ö Ü x Þ Ø1ò Á . ö Ü x Þ Â 0 (4.16)

we seethatwe canexpress. ö in termsof j ö asfollows. ö Ü x Þ ÂHØ òÁ Ó Õ j ö Ü x Þ (4.17)

Doing so in the last term of Equation(4.15) above, andalsousing the fact thatÇ Â Á $ Æ , we canrewrite thisEquationas

E ö Ü x Þ Â 140 Ä 0

Y 1 2 . ö Ü x Þ å ó÷ùFE x æ x G E Ü x Ø x Þ x Ø x 3 X 3. Ø 1Æ 1 2 á Ó Õ j ö Ü x ÞÜ x Ø x Þ x Ø x ØFò Ç j ö Ü x Þ â å ó ùjE x æ x G E x Ø x X 3. d e f g

I ö^

(4.18)

Thelastintegral canbefurtherrewritten in thefollowing way:

I ö Â 1 2 á Ó Õ j ö Ü x ÞÜ x Ø x Þ x Ø x Øò Ç j ö Ü x Þ â å ó ùjE x æ x G E x Ø x X 3. Â 1 2 á Ôk ömlÔ . l . n Ø . n x Ø x ØFò Ç k ö n Ü x Þ â ˆo nàå ó ùjE x æ x G E x Ø x X 3. (4.19)

But, since


ÔÔ . l [ k öpl . n Ø . n x Ø x 2 å¢ó ùFE x æ x G E ] Â á Ôk öplÔ . l â . n Ø . n x Ø x 2 å¢ó÷ùFE x æ x G Eß k öpl ÔÔ . l [ . n Ø . n x Ø x 2 å¢ó÷ùFE x æ x G E ] (4.20)

we canrewrite I ö as

I ö Â"Ø 1 2 Y k öpl ÔÔ . l [ . n Ø . n x Ø x 2 ˆo n å¢ó ùFE x æ x G E ] ß ò Ç j ö å ó÷ùFE x æ x G E x Ø x ^ X 3. ß 1 2 ÔÔ . l [ k öpl . n Ø . n x Ø x 2 ˆo n å¢ó ùFE x æ x G E ] X 3. (4.21)

where,accordingto Gauss’s theorem,thelasttermvanishesif j ö is assumedto belimited andtendsto zeroat largedistances.Furtherevaluationof thederivative inthefirst termmakesit possibleto write

I ö Â"Ø 1 2 [ Ø j ö å ó ùFE x æ x G E x Ø x 2 ß 2 x Ø x 4 q j ö Õ Ü x Ø x Þ r Ü x Ø x Þ å¢ó÷ùFE x æ x G E ] X 3. Øò Ç 1 2 [ Ø / j ö Õ Ü x Ø x Þs4¿Ü x Ø x Þ x Ø x 3 å¢ó÷ùFE x æ x G E ß j ö å ó ùFE x æ x G E x Ø x ] X 3. (4.22)

Using the triple product“bac-cab”Formula (F.56) on page159 backwards,andinsertingtheresultingexpressionfor I ö into Equation(4.18)on thepreviouspage,we arrive at the following final expressionfor the Fourier transformof the totalE-field:

E ö Ü x Þ ÂØ 140 Ä 0

Ó 1 2 . ö Ü x Þ å ó ùjE x æ x G E x Ø x X 3. ¨ß ò Ë 0Á

40 1 2 j ö Ü x Þ å ó ùjE x æ x G E x Ø x X 3. Â 1

40 Ä 0

Y 1 2 . ö Ü x Þ å ó ùjE x æ x G E Ü x Ø x Þ x Ø x 3 X 3. ß 1Æ 1 2 / j ö Ü x Þ å ó÷ùFE x æ x G E Õ Ü x Ø x Þs4MÜ x Ø x Þ x Ø x 4 X 3. ß 1Æ 1 2 / j ö Ü x Þ å ó÷ùFE x æ x G E × Ü x Ø x Þs4 × Ü x Ø x Þ x Ø x 4 X 3. Ø ò ÇÆ 1 2 / j ö Ü x Þ å ó÷ùFE x æ x G E × Ü x Ø x Þs4 × Ü x Ø x Þ x Ø x 3 X 3. ^ (4.23)

Taking the inverseFourier transformof Equation(4.23),onceagain usingthevacuumrelation t Â Ç Æ , we find, at last, the expressionin time domainfor the

total electricfield:

E Ü Ù_Ý x Þ Â 1 <æ < E ö Ü x Þ å¨æQó ö¬è X ÁÂ 140 Ä 0 1 2 . Ü Ù ret Ý x Þ'Ü x Ø x Þ x Ø x 3 X 3. d e f g

RetardedCoulombfieldß 140 Ä 0

Æ 1 2 / j Ü Ù ret Ý x Þ Õ Ü x Ø x Þs4MÜ x Ø x Þ x Ø x 4 X 3. d e f gIntermediatefieldß 1

40 Ä 0Æ 1 2 / j Ü Ù ret Ý x Þ × Ü x Ø x Þ54 × Ü x Ø x Þ x Ø x 4 X 3. d e f g

Intermediatefieldß 140 Ä 0

Æ 2 1 2 / j Ü Ù ret Ý x Þ × Ü x Ø x Þs4 × Ü x Ø x Þ x Ø x 3 X 3. d e f gRadiationfield

(4.24)

Here,thefirst termrepresentstheretardedCoulombfield andthelasttermrepres-entsthe radiationfield which carriesenergy over very largedistances.Theothertwo termsrepresentan intermediatefield which contributesonly in thenearzoneandmustbetakeninto accountthere.

With thiswehaveachievedourgoalof findingclosed-formanalyticexpressionsfor theelectricandmagneticfields whenthe sourcesof thefields arecompletelyarbitrary, prescribeddistributionsof chargesandcurrents. The only assumptionmadeis that the advancedpotentialshave beendiscarded;recall the discussionfollowing Equation(3.35)onpage39 in Chapter3.

49

BIBLIOGRAPHY 4

[1] Sir FredHoyle andJayantV. Narlikar. Lectureson Cosmology andActionat a Dis-tanceElectrodynamics. World Scientific PublishingCo. Pte.Ltd, Singapore,NewJersey, LondonandHongKong,1996. ISBN 9810-02-2573-3(pbk).




[5] JuliusAdamsStratton.ElectromagneticTheory. McGraw-Hill Book Company, Inc.,New York, NY andLondon,1953. ISBN 07-062150-0.

[6] JohnArchibaldWheelerandRichardPhillipsFeynman.Interactionwith theabsorberasamechanismfor radiation.Reviewsof ModernPhysics, 17,1945.

51

CHAPTER 5

RelativisticElectrodynamics

We saw in Chapter3 how thederivationof theelectrodynamicpotentialsled, in amostnaturalway, to theintroductionof acharacteristic,finite speedof propagationthatequalsthespeedof light Æ Â 1$ Ä 0 Ë 0 andwhich canbeconsideredasa con-stantof nature.To takethisfinite speedof propagationof informationinto account,andto ensurethat our laws of physicsbe independentof any specificcoordinateframe,requiresa treatmentof electrodynamicsin a relativistically covariant(co-ordinateindependent)form. This is theobjectof thecurrentchapter.

5.1 Thespecialtheoryof relativityAn inertial system, or inertial referenceframe, is a systemof reference,or rigidcoordinatesystem,in which the law of inertia (Galileo’s law, Newton’s first law)holds. In otherwords,an inertial systemis a systemin which free bodiesmoveuniformly anddo not experienceany acceleration.Thespecialtheoryof relativitydescribeshow physicalprocessesareinterrelatedwhenobservedin differentiner-tial systemsin uniform, rectilinearmotion relative to eachotherandis basedontwo postulates:

Postulate1 (Relativity principle; Poincaré,1905) All lawsof physics(exceptthelaws of gravitation) are independentof the uniform translationalmotion of thesystemon which they operate.

Postulate2 (Einstein, 1905) Thevelocityof light is independentof themotionofthesource.

A consequenceof the first postulateis that all geometricalobjects(vectors,tensors)in anequationdescribingaphysicalprocessmusttransformin acovariantmanner, i.e., in thesameway.

53

54 CHAPTER 5. RELATIVISTIC ELECTRODYNAMICS

v

uDv w x vzyijy|~yi w x vyiy|'yiP

Σ Σ | |

Figure 5.1. Two inertial systemsΣ and Σ in relative motion with velocity valongthe axis.At time 0 theorigin of Σ coincidedwith theorigin of Σ. At time , theinertial systemΣ hasbeentranslateda distancealongthe axis in Σ. An event representedby Ps K in Σ is represented

by K K in Σ ..

5.1.1 TheLorentztransformation

Let usconsidertwo three-dimensionalinertial systemsΣ andΣ in vacuumwhicharein rectilinearmotion relative to eachother in sucha way that Σ moveswithconstantvelocity v alongthe . axisof theΣ system.Thetimesandthespatialco-ordinatesasmeasuredin thetwo systemsare Ù and Ü . ÝP ÝD Þ , and Ù and Ü . ÝP ÝD Þ ,respectively. At time Ù Â Ù Â 0 theorigins and andthe . and . axesof thetwo inertial systemscoincideandata latertime Ù they have therelative locationasdepictedin Figure5.1.

For convenience,let usintroducethetwo quantities1Â Æ (5.1) Â 11 Ø 2

(5.2)

where Â v . In the following, we shall make frequentuseof theseshorthandnotations.

As shown by Einstein,the two postulatesof specialrelativity requirethat thespatialcoordinatesandtimesasmeasuredby anobserver in Σ andΣ , respectively,areconnectedby thefollowing transformation:

5.1. THE SPECIAL THEORY OF RELATIVITY 55

Æ Ù Â Ü Æ Ù Ø . Þ (5.3a). Â Ü . Ø Ù Þ (5.3b) Â (5.3c) Â (5.3d)

Takingthedifferencebetweenthesquareof (5.3a)andthesquareof (5.3b)wefindthat Æ 2 Ù 2 Ø . 2 Â 2

N Æ 2 Ù 2 Ø 2. Æ Ù ß . 2 2 Ø . 2 ß 2. Ù Ø 2 Ù 2 OÂ 1

1 Ø 2Æ 2 Æ 2 Ù 2 á 1 Ø 2Æ 2 â Ø . 2 á 1 Ø 2Æ 2 â Â Æ 2 Ù 2 Ø . 2 (5.4)

From Equation(5.3) on the facingpagewe seethat the and coordinatesareunaffectedby thetranslationalmotionof theinertial systemΣ alongthe . axisofsystemΣ. Usingthisfact,wefind thatwecangeneralisetheresultin Equation(5.4)above toÆ 2 Ù 2 Ø . 2 Ø 2 Ø 2 Â Æ 2 Ù 2 Ø . 2 Ø 2 Ø 2 (5.5)

which meansthat if a light wave is transmittedfrom thecoincidingorigins and at time Ù Â Ù Â 0 it will arrive at anobserver at Ü . ÝP ÝD Þ at time Ù in Σ andanobserver at Ü . ÝP ÝD Þ at time Ù in Σ in sucha way thatbothobserversconcludethatthespeed(spatialdistancedividedby time)of light in vacuumis Æ . Hence,thespeedof light in Σ andΣ is thesame.A linear coordinatetransformationwhichhasthispropertyis calleda (homogeneous)Lorentztransformation.

5.1.2 Lorentzspace

Let us introduceanorderedquadrupleof realnumbers,enumeratedwith thehelpof upperindices Ë Â 0 Ý 1 Ý 2 Ý 3, wherethe zerothcomponentis Æ Ù ( Æ is the speedof light and Ù is time), andthe remainingcomponentsarethe componentsof theordinary ¡ 3 radiusvectorx definedin Equation(M.1) onpage166:. ¢ Â Ü . 0 Ý . 1 Ý . 2 Ý . 3 Þ Â Ü Æ Ù_Ý . ÝP ÝD Þ 3 Ü Æ Ù_Ý x Þ (5.6)

Wetheninterpretthisquadruple. ¢ as(the Ë th componentof) a radiusfour-vectorin a real,linear, four-dimensionalvectorspace.


Metric tensor

We wantour spaceto bea Riemannianspace, i.e., a spacewherea distanceandascalarproductaredefined.Wethereforeneedto definein thisspaceametrictensor,alsoknown asthe fundamentaltensor, which we shall denoteð ¢ £ andchooseas(in matrixnotation):

Ü ð ¢ £ Þ Â ¤¥¥¦ 1 0 0 00 Ø 1 0 00 0 Ø 1 00 0 0 Ø 1

§ ¨¨© (5.7)

i.e., with a maindiagonalwith signsequence,or signature, ß Ý Ø Ý Ø Ý Ø .1Radiusfour-vectorin contravariantandcovariant form

Theradiusfour-vector . ¢ Â Ü . 0 Ý . 1 Ý . 2 Ý . 3 Þ Â Ü Æ Ù_Ý x Þ , asdefinedin Equation(5.6)on thepreviouspage,is, by definition,theprototypeof a contravariantvector(or,moreaccurately, a vector in contravariant componentform). The correspondingcovariant vector . ¢ is obtainedas(theupperindex Ë in . ¢ is summedover andisthereforeadummyindex andmaybereplacedby anotherdummyindex ª ):. ¢ Â ð ¢ £ .«£ (5.8)

This processis an exampleof contraction and is often called the “lowering” ofindex.

Index loweringof thecontravariantradiusfour-vector. ¢ amountsto multiplyingthecolumnvectorrepresentationof . ¢ from theleft by thematrix representation,Equation(5.7),of ð ¢ £ to obtainthecolumnvectorrepresentationof . ¢ . Thesimplediagonalform of ð ¢ £ , Equation(5.7)above, meansthat theindex loweringopera-tion in ourchosenflat 4D spaceis nearlytrivial:¤¥¥¦ . 0.

1.2.3

§ ¨¨© Â ¤¥¥¦ 1 0 0 00 Ø 1 0 00 0 Ø 1 00 0 0 Ø 1

§ ¨¨© ¤¥¥¦ . 0. 1. 2. 3

§ ¨¨© Â ¤¥¥¦ . 0Ø . 1Ø . 2Ø . 3

§ ¨¨© (5.9)

whichwe candescribeas. ¢ Â ð ¢ £ .«£ Â Ü Æ ÙYÝ Ø x Þ (5.10)

i.e., the covariantradiusfour-vector . ¢ is obtainedfrom the contravariantradiusfour-vector . ¢ simply by changingthe sign of the last threecomponents.These1Without changingthe physics,onecanalternatively choosea signature¬~¯® ° ® ° ® ° ± . The latterhastheadvantagethat the transitionfrom 3D to 4D becomessmooth,while it will introducesomeannoying minussignsin thetheory. In currentphysicsliterature,thesignature¬ ° ®I¯®I²®I³± seemsto bethemostcommonlyusedone.


componentsarereferredto asthespacecomponents; thezerothcomponentis re-ferredto asthe timecomponent.

Scalarproductandnorm

Takingthescalarproductof . ¢ with itself, weget,by definition,ð ¢ £ .U£:. ¢ Â . ¢ . ¢ Â Ü . 0 Ý . 1 Ý . 2 Ý . 3 Þ Õ Ü . 0 Ý . 1 Ý . 2 Ý . 3 Þ Â Ü Æ ÙYÝ Ø x Þ Õ Ü Æ ÙYÝ x ÞÂ Ü Æ ÙYÝ . ÝPàÝD Þ Õ Ü Æ ÙYÝ Ø . Ý Ø Ý Ø Þ Â Æ 2 Ù 2 Ø . 2 Ø 2 Ø 2 (5.11)

which actsasa norm or distancein our 4D space.We see,by comparingEqua-tion (5.11)andEquation(5.5) on page55, that this normis conserved(invariant)duringa Lorentztransformation.We noticefurther from Equation(5.11)thatourspacehasanindefinitenormwhichmeansthatwedealwith anon-Euclideanspace.Thefour-dimensionalspace(or space-time) with thesepropertiesis calledLorentzspaceandis denoted 4. Thecorrespondingreal, linear4D spacewith a positivedefinitenorm which is conserved during ordinaryrotationsis a Euclideanvectorspacewhichwe denote¡ 4.

The ´ 4 metric tensorEquation(5.7) on the precedingpagehasa numberofinterestingproperties:Firstly, we seethat this tensorhasa traceTr Ü ð ¢ £ Þ Â Ø 2whereasin ¡ 4, asin any vectorspacewith definitenorm,thetraceequalsthespacedimensionality. Secondly, wefind, aftertrivial algebra,thatthefollowing relationsbetweenthecontravariant,covariantandmixedformsof themetrictensorhold:ð ¢ £ Â ð £ ¢ (5.12a)ð ¢ £ Â ð ¢ £ (5.12b)ð £ µ ð µ¶¢ Â ð ¢£ Â % ¢£ (5.12c)ð £ µ ð µ¶¢ Â ð £¢ Â % £¢ (5.12d)

Herewe have introducedthe4D versionof theKronecker delta% ¢£ , a mixedfour-

tensorof rank2 which fulfils% ¢£ Â % £¢ Â ! 0 if Ë¸·Â ª1 if Ë Â ª (5.13)

Invariant line elementandpropertime

ThedifferentialdistanceX ¹ betweenthetwo points . ¢ and . ¢ ß X . ¢ in ´ 4 canbecalculatedfrom theRiemannianmetric, givenby thequadratic differential formX ¹ 2 Â ð ¢ £ X .«£ X . ¢ Â X . ¢ X . ¢ Â Ü X . 0 Þ 2 Ø Ü X . 1 Þ 2 Ø Ü X . 2 Þ 2 Ø Ü X . 3 Þ 2Â Æ 2 X Ù 2 Ø X . 2 Ø X 2 Ø X 2 (5.14)

wherethemetrictensoris asin Equation(5.7)on thefacingpage.Thesquarerootof thisexpressionis the invariant line element


X ¹ Â Æ X Ù 1 Ø 1Æ 2 Y á X .X Ù â 2 ß á X X Ù â 2 ß á X X Ù â 2 ^Â Æ É 1 Ø 2Æ 2 X Ù Â Æ 1 Ø 2 X Ù Â Æ X Ù Â Æ X À (5.15)

wherewe introducedX À Â X Ù $ (5.16)

Since X À measuresthe time whenno spatialchangesarepresent,it is calledthepropertime.

ExpressingEquation(5.5)onpage55 in termsof thedifferentialinterval X ¹ andcomparingwith Equation(5.14)on theprecedingpage,we find thatX ¹ 2 Â Æ 2 X Ù 2 Ø X . 2 Ø X 2 Ø X 2 (5.17)

is invariantduringaLorentztransformation.Conversely, wemaysaythateveryco-ordinatetransformationwhichpreserve this differentialinterval is aLorentztrans-formation.

If in someinertial systemX . 2 ß X 2 ß X 2 º Æ 2 X Ù 2 (5.18)X ¹ is a time-like interval, but ifX . 2 ß X 2 ß X 2 » Æ 2 X Ù 2 (5.19)X ¹ is aspace-like interval, whereasX . 2 ß X 2 ß X 2 Â Æ 2 X Ù 2 (5.20)

is a light-like interval; we mayalsosaythat in this casewe areon the light cone.A vector which hasa light-like interval is called a null vector. The time-like,space-likeor light-likeaspectsof aninterval X ¹ is invariantundera Lorentztrans-formation.

Four-vectorfields

Any quantitythatrelativeto any coordinatesystemhasaquadrupleof realnumbersand which transformin the sameway as the radius four-vector . ¢ , is called afour-vector. I analogywith the notationfor the radiusfour-vectorwe introducethe notation ¼ ¢ Â Ü ¼ 0 Ý aÞ for a generalcontravariant four-vectorfield in ´ 4 and


find that the “loweringof index” rule, Equation(M.20) on page169, for suchanarbitraryfour-vectoryieldsthedualcovariant four-vectorfield¼ ¢ Ü . µ Þ Â ð ¢ £ ¼ £ Ü . µ Þ Â Ü ¼ 0 Ü . µ Þ Ý Ø a Ü . µ Þ!Þ (5.21)

Thescalarproductbetweenthis four-vectorfield andanotherone ½ ¢ Ü . µ Þ isð ¢ £ ¼ £ Ü . µ Þ ½ ¢ Ü . µ Þ Â Ü ¼ 0 Ý Ø aÞ Õ Ü ½ 0 Ý b Þ Â ¼ 0 ½ 0 Ø a Õ b (5.22)

which is a scalar field, i.e., an invariantscalarquantity Ü . µ Þ which dependsontime andspace,asdescribedby . µ Â Ü Æ Ù_Ý . ÝP ÝD Þ .TheLorentztransformationmatrix

Introducingthetransformationmatrix

Ü Λ ¢ £ Þ Â ¤¥¥¦ Ø¾³ 0 0Ø¾³ 0 00 0 1 00 0 0 1

§ ¨¨© (5.23)

thelinearLorentztransformation(5.3)onpage55, i.e., thecoordinatetransforma-tion . ¢ L . ¢ Â . ¢ Ü . 0 Ý . 1 Ý . 2 Ý . 3 Þ , from oneinertial systemΣ to anotherinertialsystemΣ , canbewritten. ¢ Â Λ ¢ £ .«£ (5.24)

Theinversetransformthentakestheform.U£ Â Ü Λ ¢ £ Þ æ 1 . ¢ (5.25)

TheLorentzgroup

It is easyto show, by meansof direct algebra,that two successive Lorentztrans-formationsof the type in Equation(5.25) above, anddefinedby the speedpara-meters 1 and 2, respectively, correspondto a singletransformationwith speedparameterFÂ 1 ß 2

1 ß 1 2(5.26)

This meansthat the nonemptysetof Lorentztransformationsconstitutea closedalgebraic structure with a binary operationwhich is associative. Furthermore,onecanshow that this setpossessesat leastone identityelementandat leastoneinverseelement. In otherwords, this setof Lorentz transformationsconstituteamathematicalgroup. However tempting,we shall not make any further useofgrouptheory.


¿ 1

À 0

¿Á 1À Á 0

ÂÂFigure5.2. Minkowski spacecanbe consideredan ordinaryEuclideanspacewherea Lorentz transformationfrom ÃiÄ 1 Å Æ 0 ÇÉÈÊPËKÌ to ÃiÄMÍ 1 Å Æ Í 0 ÇÉÈÊIËKÌ cor-respondsto an ordinary rotation throughan angle Î . This rotation leaves the

EuclideandistanceÏiÄ 1 Ð 2 Ñ Ï Æ 0 Ð 2 Ç Ä 2 Ò Ê 2 Ë 2 invariant.

5.1.3 Minkowski space

Specifyinga point ÓÕÔ×Ö Ø Ó 0 Ù Ó 1 Ù Ó 2 Ù Ó 3 Ú in 4D space-timeis a way of sayingthat “somethingtakes placeat a certain time ÛÜÖÝÓ 0 Þ ß and at a certainplaceØ Ó ÙPà«ÙDá Ú Ö Ø Ó 1 Ù Ó 2 Ù Ó 3 Ú .” Sucha point is thereforecalledan event. The traject-ory for aneventasafunctionof timeandspaceis calledaworld line. For instance,theworld line for a light raywhichpropagatesin vacuumis thetrajectoryÓ 0 ÖÓ 1.

If we introduceâ 0 ÖäãÓ 0 Öã ß Û (5.27a)åæÜç ã å è (5.27b)

whereé ç ê ë1, we canrewrite Equation(5.14)onpage57asåæ 2 ç ì å â 0 í 2 î ì åðï 1 í 2 î ì åðï 2 í 2 î ì åðï 3 í 2 (5.28)

i.e., asa 4D differentialform which is positivedefinitejust asis ordinary3D Euc-lideanspaceñ 3. We shallcall the4D Euclideanspaceconstructedin this way theMinkowskispaceò 4.

As before,it sufficesto considerthe simplified casewherethe relative motionbetweenΣ andΣ ó is alongthe

ïaxes.Thenåæ 2 ç ì å â 0 í 2 î ì åðï 1 í 2 (5.29)

andwe considerâ 0 and

ï 1 asorthogonalaxesin an Euclideanspace.As in allEuclideanspaces,every interval is invariantundera rotationof the

â 0 ï 1 plane


ô 0 õ9öø÷öø÷

ô ù 0ô ù 1ú õ ú ù ô 1 õ ôû

ûü

ýý ù

ô 0 õ ô 1

Figure 5.3. Minkowski diagramdepicting geometricallythe transformation(5.31) from the unprimedsystemto the primed system. Here þ denotestheworld line for aneventandtheline ÿ 0 ÿ 1 ÿ theworld line for a lightray in vacuum.Note that theevent is simultaneouswith all pointson the ÿ 1

axis( 0), includingtheorigin while theevent , which is alsosimultan-eouswith all pointsonthe ÿ axis,including , to anobserveratrestin theprimedsystem,is not simultaneouswith in the unprimedsystembut occurs

thereat time .throughanangle into

â ó 0 ï ó 1:â ó 0 ç ë ï 1 sin î â 0 cos (5.30a)ï ó 1 çäï 1 cos î â 0 sin (5.30b)

SeeFigure5.2.If we introducetheangle ç ë é , oftencalledtherapidityor theLorentzboost

parameter, andtransformbackto the original spaceandtime variablesby usingEquation(5.27b)on theprecedingpagebackwards,we obtain ó ç ë ï

sinh î cosh (5.31a)ï ó çäïcosh ë sinh (5.31b)

whichareidenticalto thetransformationequations(5.3)onpage54 if we let

sinh ç (5.32a)

cosh ç (5.32b)

tanh ç (5.32c)


It is thereforepossibleto envisagethe Lorentz transformationas an “ordinary”rotationin the4D Euclideanspace 4. Equation(5.26)on page59 for successiveLorentztransformationthencorrespondsto thetanhadditionformula

tanh 1 2 tanh 1 tanh 2

1 tanh 1 tanh 2(5.33)

The useof "!# and 4, which leadsto the interpretationof the Lorentztrans-formationasan “ordinary” rotation,may, at best,be illustrative, but is not veryphysical. Besides,if we leave the flat $ 4 spaceand enter the curved spaceofgeneralrelativity, the “ "!# ” trick will turn out to be an impasse.Let us thereforeimmediatelyreturnto $ 4 whereall componentsarerealvalued.

5.2 Covariantclassicalmechanics

The measure% & of the differential “distance” in $ 4 allows us to definethe four-velocity'( % )% & ( +* , 1 - u!/. 01 12

1 354 26 2 - u! 2 1 374 26 2 89 (5.34)

which,whenmultipliedwith thescalarinvariant : 0 ! 2 yieldsthefour-momentum;<( : 0 ! 2 % )% & ( +* : 0 ! !=- u 01 : 0 ! 221 354 26 2 - : 0 ! u2

1 374 26 2 89 (5.35)

Fromthisweseethatwe canwrite

p : u (5.36)

where: >* : 0 : 021 374 26 2 (5.37)

i.e., that Lorentz covarianceimplies that the mass-like term in the ordinary 3Dlinearmomentumis not invariant.

5.3. COVARIANT CLASSICAL ELECTRODYNAMICS 63

Thezeroth(time) componentof thefour-momentum; ( is givenby; 0 >* : 0 ! 2 : 0 ! 221 374 26 2 :?! 2 (5.38)

We interpretthisastheenergy @ , i.e., wecanwrite;( @A-B! p (5.39)

Scalarmultiplying this four-momentumwith itself, we obtain; ( ;( ð ( C ; C ;<( ; 0 2 3 ; 1 2 3 ; 2 2 3 ; 3 2 @A-D3E! p F @A-B! p @ 2 3G! 2 H p H 2 : 0 ! 2 2 (5.40)

Sincethis is an invariant,this equationholdsin any inertial frame,particularlyintheframewherep 0 wherewe thushave@ : 0 ! 2 (5.41)

which is probablythemostwell-known physicsformulaever.

5.3 CovariantclassicalelectrodynamicsIn therestinertialsystemthechargedensityis I 0. Thefour-vector(in contravariantcomponentform)J ( I 0

% ) (% & , I-BI v!K. (5.42)

wherewe introducedI >* I 0 (5.43)

is calledthefour-current.As is shown in ExampleM.4 onpage176,thed’Alembertoperator is thescalar

productof thefour-delwith itself:L 2 NM ( M ( NM ( M ( 1! 2 M 2M # 2 3 O 2 (5.44)

Sinceit hasthecharacteristicsof afour-scalar, thed’Alembertoperatoris invariantand,hence,thehomogeneouswave equationis Lorentzcovariant.


5.3.1 Thefour-potential

If we introducethefour-potentialP ( QR-B! A (5.45)

where Q is the scalarpotentialandA the vectorpotential,definedin Section3.3on page32, we canwrite theinhomogeneouswave equations(Lorentzequations)Equation(3.14)onpage34 in thefollowing compact(andcovariant)way:L 2 P ( J (S

0(5.46)

With thehelpof theabove, we canformulateour electrodynamicequationsco-variantly. For instance,the covariant form of the equationof continuity, Equa-tion (1.21)on page9 isM ( J (UT M J (M ) ( 0 (5.47)

andtheLorentzgaugecondition, Equation(3.13)onpage34,canbewrittenM ( P ( T M P (M ) ( 0 (5.48)

Thegaugetransformations(3.15)onpage35 in covariantform aresimplyPWV ( P ( 3 M ( / ! Γ ) C X P ( 3 MM ) ( ! Γ ) C (5.49)

If only one dimensionLorentz contracts(for instance,due to relative motionalongthe ) direction),a3D spatialvolumetransformsaccordingto%Y % 3) 1* %Y 0 %Y 0 Z 1 3\[ 2 %Y 0 ] 1 3_^ 2! 2 (5.50)

thenfrom Equation(5.43)on theprecedingpageweseethatI`%Y I 0 %Y 0 (5.51)

i.e., thechargein a givenvolumeis conserved. We cantherforeconcludethat theelectronchargeis auniversalconstant.


5.3.2 TheLiénard-Wiechertpotentials

Let usnow solve theLorentzequation(theinhomogeneouswave equation)(3.14)on page34 in vacuumfor thecaseof a well-localisedcharge a V at a sourcepointdefinedby theradiusfour-vector ) V ( ) V 0 !Db V - ) V 1 - ) V 2 - ) V 3 . Thefield (obser-vation)point is denotedby theradiusfour-vector ) ( ) 0 !Dbc- ) 1 - ) 2 - ) 3 .

In therestsystemwe know thatthesolutionis simply P ( 0 d a V4e S 0

1H x 3 xV H

0- 0f (5.52)

where H x 3 xV H

0 is theusualdistancefrom thesourcepoint to thefield point, eval-uatedin therestsystem(signifiedby theindex “0”).

Let usintroducetherelativeradiusfour-vectorbetweenthesourcepointandthefield point:g ( ) ( 3 ) V ( ! bR3Gb V - x 3 x

V (5.53)

Scalarmultiplying this relative four-vectorwith itself, weobtaing ( g ( ! bh3ib V - x 3 xV F ! bD3jb V -D3 x 3 x

V B ! 2 bh3ib V 2 3lkk x 3 xV kk 2(5.54)

We know that in vacuumthesignal(field) from thecharge a V at ) V ( propagatesto ) ( with thespeedof light ! sothatkk x 3 x

V kk ! bm3Gb V (5.55)

Insertingthis into Equation(5.54),weseethatg ( g ( 0 (5.56)

or thatEquation(5.53)abovecanbewritteng ( kk x 3 xV kk - x 3 x

V (5.57)

Now we want to find the correspondenceto the rest systemsolution, Equa-tion (5.52), in an arbitrary inertial system. We note from Equation(5.34) onpage62 thatin therestsystem '( 0 01 12

1 3N4 26 2 - u! 2 1 374 26 2 89 nu o 0 p 4 o 0q 1 - 0 (5.58)

and g ( 0 kk x 3 xV kk - x 3 x

V 0 kk x 3 xV kk 0 - x 3 x

V 0 (5.59)

Like all scalarproducts,' ( g ( is invariant,sowe canevaluateit in any inertial


system.If weevaluateit in therestsystemtheresultis:'( g ( '( g ( 0 '( 0 g ( 0 1 - 0 F kk x 3 xV kk 0 - x 3 x

V 0 kk x 3 xV kk 0 (5.60)

We thereforeseethattheexpressionP ( a V4e S 0

' (' C g C (5.61)

subjectto thecondition

g ( g ( 0hasthepropertransformationproperties(propertensorform) andreduces,in therestsystem,to thesolutionEquation(5.52)on theprecedingpage.It is thereforethecorrectsolution,valid in any inertial system.

Accordingto Equation(5.34)on page62 andEquation(5.57)on theprecedingpage ' C g C r* , 1 - u!/. F kk x 3 x

V kk -D3 x 3 xV sr* d kk x 3 x

V kk 3 x 3 xV F u! f (5.62)

Introducing& kk x 3 xV kk 3 x 3 x

V F u! (5.63)

we canwrite' C g C >* & (5.64)

and ' (' C g C d 1& - u! & f (5.65)

from whichwe seethatthesolution(5.61)canbewrittenP ( ) t a V4e S 0

d 1& - u! & f QR-B! A (5.66)

wherein thelaststepthedefinitionof thefour-potential,Equation(5.45)onpage64,wasused.Writing thesolutionin theordinary3D-way, weconcludethatfor averylocalisedchargevolume,moving relative anobserver with a velocity u, thescalar


andvectorpotentialsaregivenby theexpressionsQ b- x 14e S 0

a V& 14e S 0

a VH x 3 xV H 3 n

x u x vwqyx u6 (5.67a)

A b- x 14e S 0 ! 2 a V u& 1

4e S 0 ! 2 a V uH x 3 xV H 3 n

x u x v qzx u6 (5.67b)

ThesepotentialsarecalledtheLiénard-Wiechertpotentials.

5.3.3 Theelectromagneticfield tensor

Consideravectorial(cross)productc betweentwo ordinaryvectorsa andb:

c a b 7| ~ch ~ ˆ 2 3 3 3 2 ˆ 1 3 1 3 1 3 ˆ 2 1 2 3 2 1 ˆ 3 (5.68)

Wenoticethatthe th componentof thevectorc canberepresentedas! + ~ 3 ~ ! ~ 3E! ~ - - J (5.69)

In otherwords,thepseudovectorc a b canbeconsideredasanantisymmetrictensorof ranktwo!

Thesameis truefor thecurl operator7 . For instance,theMaxwell equation E 3 M BM b (5.70)

canin this tensornotationbewrittenM @ ~M ) 3 M @ M ) ~ 3 M ~M b (5.71)

Weknow from Chapter3 thatthefieldscanbederivedfrom theelectromagneticpotentialsin thefollowing way:

B A (5.72a)

E 3jQ3 MM b P (5.72b)

In componentform, thiscanbewritten ~ M P ~M ) 3 M P M ) ~ NM P ~ 3 M ~ P (5.73a)@ 3 M QM ) 3 M P M b 3 M Q3 M P (5.73b)

Fromthis, we noticethecleardifferencebetweentheaxial vector(pseudovector)B andthepolar vector(“ordinaryvector”)E.

Our goal is to expressthe electricandmagneticfields in a tensorform wherethe componentsare functionsof the covariant form of the four-potential,Equa-tion (5.45)on page64:P ( QR-D3E! A (5.74)

Inspectionof (5.74)andEquation(5.73)on thepreviouspagemakesit naturaltodefinethecovariantfour-tensor ( C M P CM ) ( 3 M P (M ) C NM ( P C 3 M C P ( (5.75)

Thisanti-symmetric(skew-symmetric), covariantfour-tensorof rank2 is calledtheelectromagneticfield tensor. In matrix representation,the covariant field tensorcanbewritten ( C NM ( P C 3 M C P ( 01 0 @ @ @W3E@ 0 3E! ! 3E@ ! 0 3! 3W@E 3! ! 0

8 9(5.76)

Thematrix representationfor thecontravariantfield tensoris ( C NM ( P C 3 M C P ( 01 0 3W@ 3W@ 3W@ @ 0 3! ! @ ! 0 3E! @E 3E! ! 0

8 9(5.77)

It is perhapsinterestingto notethat the field tensoris a sort of four-dimensionalcurl of thefour-potentialvector

P ( .Onecanshow thatthetwo Maxwell sourceequations F E IS

0(5.78) B 0 d j S 0

M EM b f 0 d I u S 0M EM b f (5.79)

correspondtoM C (M ) C J (S0

(5.80)

andthatthetwo Maxwell “field” equations E 3 M BM b (5.81) F B 0 (5.82)

correspondto

68

5.3. COVARIANT CLASSICAL ELECTRODYNAMICS 69M ( CM ) t M C tM ) ( M t (M ) C 0 (5.83)

Hence,Equation(5.80) on the facingpageandEquation(5.83) above constituteMaxwell’sequationsin four-dimensionalformalism.

BIBLIOGRAPHY 5

[1] J. Aharoni. TheSpecialTheoryof Relativity. Dover Publications,Inc., New York,second,revisededition,1985. ISBN 0-486-64870-2.

[2] Asim O. Barut. DynamicsandClassicalTheoryof FieldsandParticles. Dover Pub-lications,Inc., New York, NY, 1980. ISBN 0-486-64038-8.

[3] WalterT. Grandy. Introductionto ElectrodynamicsandRadiation. AcademicPress,New York andLondon,1970. ISBN 0-12-295250-2.


[5] C. Møller. TheTheoryof Relativity. Oxford University Press,Glasgow . . . , secondedition,1972.


[7] J. J. Sakurai. AdvancedQuantumMechanics. Addison-Wesley PublishingCom-pany, Inc.,Reading,MA . . . , 1967. ISBN 0-201-06710-2.

[8] Barry Spain. TensorCalculus. Oliver andBoyd, Ltd., Edinburgh andLondon,thirdedition,1965. ISBN 05-001331-9.

71

CHAPTER 6

Interactionsof FieldsandParticles

In this Chapterwe studythe interactionbetweenelectricandmagneticfieldsandelectricallychargedparticles.The analysisis basedLagrangianandHamiltonianmethods,is fully covariant,andyieldsresultswhicharerelativistically correct.

6.1 ChargedParticlesin anElectromagneticFieldWe first establisha relativistic correct theory describingthe motion of chargedparticlesin prescribedelectricandmagneticfields. Fromtheseequationswe maythencalculatethechargedparticledynamicsin themostgeneralcase.

6.1.1 Covariantequationsof motion

We canobtainanequationof motionif we find a Lagrangefunctionin 4D for ourproblemandthenapply a variationalprincipleor if we find a Hamiltonianin 4Dandsolve thecorrespondingHamilton’sequations.We shalldoboth.

Lagrange formalism

Call the4D Lagrangefunction n 4q andassumethat it fulfils thevariationalprin-ciple

1 0

n 4q ) ( - '( % & 0 (6.1)

where% & is theinvariantline elementgivenby Equation(5.15)onpage58,andtheendpointsarefixed.

Wemustrequirethat n 4q fulfils thefollowing conditions:

73

74 CHAPTER 6. INTERACTIONS OF FIELDS AND PARTICLES

1. The Lagrangefunctionmustbe invariant. This implies that n 4q mustbe ascalar.

2. TheLagrangefunctionmustbe linear. This implies that n 4q mustnot con-tain higherthanthesecondpowerof thefour-velocity ' ( .

Accordingto Formula(M.76) on page180 the ordinary3D Lagrangianis thedifferencebetweenthe kinetic and potentialenergies. A free particle hasonlykinetic energy. If theparticlemassis : 0 thenin 3D thekinetic energy is : 0 ^ 2 2.This suggeststhatin 4D theLagrangianfor a freeparticleshouldbe freen

4q :?! 22

'(' ( (6.2)

For an interactionwith theelectromagneticfield we canintroducethe interactionwith the help of the four-potentialgiven by Equation(5.74) on page68 in thefollowing way n 4q :?! 2

2'(' ( ¡ ' ( P ( ) t (6.3)

We call this thefour-Lagrangian.Thevariationprinciple(6.1)with the4D Lagrangian(6.3) inserted,leadsto

1 0

n 4q ) ( - '( % & 1

0

d : 0 ! 22

'(`' ( ¡ '( P ( f % & 1

0 ¢ : 0 ! 22

MM ' ( '(' ( '( ¡ d P ( '( '( M P (M ) C ) C f £ % & 1

0 ¢ :?! 2 ' ( '( ¤ d P ( '( '( M P (M ) C ) C f £ % & 0 (6.4)

Accordingto Equation(5.34)onpage62, thefour-velocity is'( % )% & ( (6.5)

whichmeansthatwecanwrite thevariationof ' ( asa totalderivativewith respectto & : '( d % )% & ( f %% & ) ( (6.6)

Insertingthis into thefirst two termsin thelastintegral in Equation(6.4),weobtain

6.1. CHARGED PARTICLES IN AN ELECTROMAGNETIC FIELD 75 1

0

n 4q ) ( - '( % & 1

0

d :?! 2 ' ( %% & ) ( ¤ P ( %% & ) ( ¡ '( M P (M ) C ) C f % & (6.7)

Partial integrationin thetwo first termsin theright handmemberin (6.7)gives 1

0

n 4q ) ( - '( % & 1

0

d 3E: 0 ! 2 % ' (% & ) ( 3 % P (% & ) ( ¡ '( M P (M ) C ) C f % & (6.8)

wherethe integratedpartsdo not contribute sincethe variationsat the endpointsvanishes.Changeof irrelevantsummationindex from to ¥ in thefirst two termsin theright handmemberin (6.8)yields,afterbreakingoutacommonfactor

1 0

n 4q ) ( - '( % & 1

0

d 3E: 0 ! 2 % ' C% & 3 % P C% & ¡ '( M P (M ) C f ) C % & (6.9)

Applying well-known rulesof differentiationandthe expression(5.34) for thefour-velocity, we canreformtheexpressionfor % P C % & asfollows:% P C% & M P CM ) ( % ) (% & M P CM ) ( '( (6.10)

By insertingthis expression(6.10) into the right-handmemberof Equation(6.9)above, andmoving out a commonfactor ' ( , we obtainthefinal variationalprin-cipleexpression

1 0

n 4q ) ( - '( % & 1

0 ¢ 3E: 0 ! 2 % ' C% & '( d M P (M ) C 3 M P CM ) ( f £ ) C % & (6.11)

Since,accordingto thevariationalprinciple, this expressionshall vanishand

) Cis arbitraryalong the world-line from ) (0 tp ) (1 , the expressioninside ¦ X in the


integrandin theright handmemberof (6.11)mustvanish.In otherwords,wehavefoundanequationof motion:: 0 ! 2 % ' C% & + '( d M P (M ) C 3 M P CM ) ( f (6.12)

With thehelpof Equation(5.77)onpage68 we canexpressthisequationin termsof theelectromagneticfield tensorin thefollowing way:: 0 ! 2 % ' C% & + '( C ( (6.13)

This is thefinal equationof motionfor a particlein anelectromagneticfield. It isoftenreferredto astheMinkowskiequation.

Hamiltonianformalism

The usualHamilton equationsfor a 3D spacearegiven by Equation(M.81) onpage181in ChapterM. Thesesix first-orderpartialdifferentialequationsareM§M ; % %¨b (6.14a)M§M 3 % ; %¨b (6.14b)

where § ; - -Bb ; ˙ 3¡ - ˙ -Bb is theordinary3D Hamiltonian, is a gen-eralisedcoordinateand; is its canonicallyconjugatemomentum.

We seeka similar setof equationsin 4D space.To this endwe utilise thefour-velocity ' ( , asgivenby Equation(5.34)on page62, andthefour-momentum; ( ,asgivenby Equation(5.35)on page62, to introducethe four-Hamiltonian§ n 4q ;<(' ( 3 n 4q (6.15)

where; ( is consideredasthecanonicallyconjugatefour-momentum;<( M n 4qM ' ( (6.16)

where n 4q is asin Equation(6.3) on page74. With thehelpof these,the radiusfour-vector ) ( , consideredasthegeneralisefour-coordinate, andtheinvariantlineelement % & , definedin Equation(5.15) on page58, we introducethe following

6.1. CHARGED PARTICLES IN AN ELECTROMAGNETIC FIELD 77

eightpartialdifferentialequations:M§ n 4qM ; ( % ) (% & (6.17a)M§ n 4qM ) ( 3 % ; (% & (6.17b)

which form the four-dimensionalHamiltonequations.Wenotethatby solving(6.17)weobtain; ( which,accordingto Equation(5.39)

onpage63,hasazeroth(time)componentwhichweidentify with thetotalenergy.Hencewe mustrequirethat this component; 0 solvesthe ordinary3D Hamiltonequations(6.14).

Using to thedefinitionof § n 4q , Equation(6.15)on theprecedingpage,andtheexpressionfor n 4q , Equation(6.3)on page74,weobtain§ n 4q ;<(`' ( 3 n 4q ;(' ( 3 : 0 ! 2

2'(' ( 3 '( P ( ) C (6.18)

Furthermore,from the definition (6.16)of the conjugatefour-momentum; ( , weseethat;( M n 4qM ' ( MM ' ( d : 0 ! 2

2'(' ( ¡ '( P ( ) C f : 0 ! 2 '( ¡ P ( (6.19)

Insertingthis into (6.18),weobtain§ n 4q : 0 ! 2 '(' ( P (' ( 3 : 0 ! 22

'(' ( 3 '( P ( ) C 12: 0 ! 2 '(' ( (6.20)

Sincethe four-velocity scalar-multiplied by itself is ' ( ' ( 1, we clearly seefrom Equation(6.20)above that § n 4q is ascalarinvariant,whosevalueis simply§ n 4q 1

2: 0 ! 2 (6.21)

However, at thesametime(6.19)providesthealgebraicrelationship'( 1: 0 ! 2 ;<( 3 P ( (6.22)

andif this is usedin (6.20)to eliminate' ( , onegets


§ n 4q : 0 ! 22 d 1: 0 ! 2 ;<( 3 P ( 1: 0 ! 2 ; ( 3 P ( f 1

2: 0 ! 2 ;<( 3 P ( ; ( 3 P ( 12: 0 ! 2 © ;<(; ( 3 2 P (h; ( ¡ 2 P ( P (`ª (6.23)

That this four-Hamiltonianyields thecorrectcovariantequationof motioncanbeseenby insertingit into thefour-dimensionalHamilton’sequations(6.17)andusingtherelation(6.22):M§ n 4qM ) ( 3 : 0 ! 2 ; C 3 P C M P CM ) ( 3 : 0 ! 2 : 0 ! 2 ' C M P CM ) ( 3 ' C M P CM ) ( 3 % ; (% & 3:?! 2 % ' (% & 3 M P (M ) C ' C (6.24)

wherein thelaststepEquation(6.19)ontheprecedingpagewasused.Rearrangingterms,andusingEquation(5.77)on page68,we obtain: 0 ! 2 % ' (% & + ' C d M P CM ) ( 3 M P (M ) C f + ' C ( C (6.25)

which is identicalto thecovariantequationof motionEquation(6.13)on page76.We canthensafelyconcludethattheHamiltonianin questionis correct.

Settingexpression(6.23) above for § n 4q equalto the scalarvalue : 0 ! 2 2, asderivedabove,andusingthefactthat;<( ; 0 -B! p P ( QR-B! A ;<(; ( ; 0 2 3! 2 p 2P (; ( Q ; 0 3G! 2 p F A P ( P ( Q 2 3G! 2 A 2we obtaintheequation: 0 ! 2

2 12: 0 ! 2 ; 0 2 3G! 2 p 2 3 2 Q ; 0 2 ! 2 p F A ¤ 2 Q 2 3 2 ! 2 A 2

(6.27)

which is asecondorderalgebraicequationin ; 0

6.1. CHARGED PARTICLES IN AN ELECTROMAGNETIC FIELD 79 ; 0 2 3 2 Q ; 0 3G! 2 p 2 2 ! 2 p F A 3 2 ! 2 A 2 ¤ 2 Q 2 3 2 : 0 ! 2 2 ; 0 2 3 2 Q ; 0 3! 2 « p 2 3 2 p F A ¡ 2 A 2 ¬ ¤ 2 Q 2 3G: 20 ! 4 ; 0 2 3 2 Q ; 0 3! 2 p 3 A 2 2 Q 2 3G: 2

0 ! 4 0 (6.28)

with thesolution; 0 + Q®¡! 2 p 3 A 2 : 20 ! 2 (6.29)

Sincethe fourth component(time component); 0 of the four-momentum; ( isthetotalenergy, thepositivesolutionin (6.29)mustbeidentifiedwith theordinaryHamiltonfunction § . Thismeansthat§ T; 0 + Q ! 2 p 3 A 2 : 2

0 ! 2 (6.30)

is theordinary3D Hamilton function for a chargedparticlemoving in scalarandvectorpotentialsassociatedwith prescribedelectricandmagneticfields.

TheordinaryLagrangeandHamiltonfunctions and § arerelatedto eachotherby the3D transformation[cf. the4D transformation(6.15)between n 4q and § n 4q ] p F u 3 § (6.31)

Furthermore,wemake theidentification

p 3 A : 0u21 37¯ 26 2 : u (6.32)

Togetherwith (6.30)and(6.31),givestheordinary3D Lagrangefunction 3 Q ¤ A F u 3G: 0 ! 2 ] 1 3 ' 2! 2 (6.33)

for a chargedparticlemoving in scalarandvectorpotentialsassociatedwith pre-scribedelectricandmagneticfields.


°±³²µ´

1±³² ±³² ¶

1· · ·· ¸ ¸ ¸ ¸ ¹¸ ° ° °Figure6.1. A one-dimensionalchainconsistingof º discrete,identicalmasspoints » , connectedto their neighbourswith identical,idealspringswith springconstants¼ . Theequilibriumdistancebetweentheneighbouringmasspointsis½ and ¾¿ À 1 ÁÃÂÅÄ , ¾h¿ ÁÃÂÅÄ , ¾h¿ Æ 1 ÁÃÂÅÄ arethe instantaneousdeviations,alongthe Ç axis,

of positionsof the ÁyÈ É 1Ä th, È th, and ÁyÈ¨Ê 1Ä th masspoint, respectively.

6.2 CovariantFieldTheory

Sofar, wehave consideredtwo classesof problems.Eitherwe have calculatedthefieldsfrom given,prescribeddistributionsof chargesandcurrents,or we have de-rivedtheequationsof motionfor chargedparticlesin given,prescribedfields. Letusnow put thefieldsandtheparticlesonanequalfootingandpresenta theoreticaldescriptionwhich treatsthefields,theparticles,andtheir interactionsin a unifiedway. This involvestransitionto a field picturewith an infinite numberof degreesof freedom.We shall first considera simplemechanicalproblemwhosesolutionis well known. Then,drawing inferencesfrom this modelproblem,we apply asimilar view on theelectromagneticproblem.

6.2.1 Lagrange-Hamiltonformalism for fields and interac-tions

ConsiderË identicalmasspoints,eachwith mass: andconnectedto its neigh-bouralonga one-dimensionalstraightline, which we chooseto be the ) axis,byidenticalidealspringswith springconstants . At equilibriumthemasspointsareat rest,distributedevenly with a distance to their two nearestneighbours.Afterperturbation,themotionof masspoint will beaone-dimensionaloscillatorymo-tion along ˆ . Let usdenotethemagnitudeof thedeviation for masspoint fromits equilibriumpositionby Ì b ˆ .

Thesolutionto this mechanicalproblemcanbeobtainedif we canfind a Lag-rangian(Lagrange function) whichsatisfiesthevariationalequation

6.2. COVARIANT FIELD THEORY 81 Ì - ˙Ì -Bb %¨b 0 (6.34)

Accordingto Equation(M.76) on page180,theLagrangianis >Í 3¡Y whereÍ denotesthekineticenergy and Y thepotentialenergy of a classicalmechanicalsystemwith conservativeforces. In ourcasetheLagrangianis 1

2 ÎÏ o 1

« : ˙Ì 2 3 Ì Ð 1 3GÌ 2 ¬ (6.35)

Let uswrite theLagrangian,asgivenby Equation(6.35),in thefollowing way: ÎÏ o 1

`Ñ (6.36)

Here, Ñ 12 Ò : ˙Ì 2 3¡ d Ì Ð 1 3GÌ f 2 Ó

(6.37)

is the so called linear Lagrange density. If we now let Ë Ô Õ and, at thesametime, let the springsbecomeinfinitesimally shortaccordingto the follow-ing scheme: ÔÖ% ) (6.38a): Ô (linear massdensity) (6.38b) ÔØ× (Young’smodulus) (6.38c)Ì Ð 1 3GÌ Ô Ù ÚÙ (6.38d)

we obtain Ñ % ) (6.39)

whereÑ d Ì- M ÌM b - M ÌM ) -Bb f 12 Ò d M ÌM b f 2 3× d M ÌM ) f 2 Ó

(6.40)

Notice how we madea transitionfrom a discretedescription,in which the masspointswereidentifiedby a discreteintegervariable 1 - 2 -DÛDÛDÛ-sË , to a continu-ousdescription,wherethe infinitesimalmasspointswere insteadidentifiedby acontinuousrealparameter) , namelytheirpositionalong ˆ .


A consequenceof this transitionis that the numberof degreesof freedomforthe systemwent from the finite number Ë to infinity! Anotherconsequenceisthat Ñ hasnow becomedependentalsoon thepartialderivative with respectto )of the “field coordinate” Ì . But, aswe shall see,the transitionis well worth thepricebecauseit allowsusto treatall fields,beit classicalscalaror vectorialfields,or wave functions,spinorsandotherfields thatappearin quantumphysics,on anequalfooting.

Underthe assumptionof time independenceandfixed endpoints,the variationprinciple(6.34)on thepreviouspageyields: Ü b Ñ d Ì- M ÌM b - M ÌM ) f % ) %¨b ÝÞ MÑM Ì Ì MÑM , Ù ÚÙ .

d M ÌM b f MÑM , Ù ÚÙ . d M ÌM ) f ßà % ) %¨b 0 (6.41)

Thelastintegral canbeintegratedby parts.This resultsin theexpression ÝÞ M<ÑM Ì 3 MM b 01 MÑM , Ù ÚÙ . 89 3 MM ) 01 MÑM , Ù ÚÙ . 89 ßà Ìá% ) %¨b 0 (6.42)

wherethe variation is arbitrary (and the endpointsfixed). This meansthat theintegranditself mustvanish.If we introducethe functionalderivative Ñ Ì MÑM Ì 3 MM ) 01 MÑM , Ù ÚÙ . 89 (6.43)

we canexpressthisas Ñ Ì 3 MM b 01 MÑM , Ù ÚÙ . 89 0 (6.44)

which is theone-dimensionalEuler-Lagrangeequation.Insertingthelinearmasspoint chainLagrangiandensity, Equation(6.40)on the

precedingpage,into Equation(6.44)above, we obtaintheequationof motion forourone-dimensionallinearmechanicalstructure.It is:

6.2. COVARIANT FIELD THEORY 83

M 2M b 2 Ìâ3× M 2M ) 2 Ì d × M 2M b 2 3 M 2M ) 2 f Ì 0 (6.45)

i.e., the one-dimensionalwave equationfor compressionwaveswhich propagatewith phasespeed^äã Z × alongthelinearstructure.

A generalisationof the above 1D resultsto a three-dimensionalcontinuumisstraightforward.For this 3D casewegetthevariationalprinciple Ü b Ñ % 3)WÜ b Ñ d Ì- M ÌM ) ( f % 4) å ÝÞ M<ÑM Ì 3 MM ) ( 01 MÑM , Ù ÚÙ . 89 ßà

Ìá% 4) 0 (6.46)

wherethevariation

Ì is arbitraryandtheendpointsarefixed.Thismeansthattheintegranditself mustvanish:MÑM Ì 3 MM ) ( 01 MÑM , Ù ÚÙ . 89 0 (6.47)

This constitutesthethree-dimensionalEuler-Lagrangeequations.Introducingthethree-dimensionalfunctionalderivative Ñ Ì MÑM Ì 3 MM ) æ MÑM Ù ÚÙ ç è (6.48)

we canexpressthis as Ñ Ì 3 MM b 01 M<ÑM , Ù ÚÙ . 89 0 (6.49)

I analogywith particlemechanics(finite numberof degreesof freedom),wemayintroducethecanonicallyconjugatemomentumdensitye ) ( e bc- x MÑM , Ù ÚÙ . (6.50)

anddefinetheHamiltondensityé d e-BÌ- M ÌM ) ; b f e M ÌM b 3 Ñ d Ì- M ÌM b - M ÌM ) f (6.51)

If, asusual,we differentiatethis expressionandidentify terms,we obtainthefol-


lowing HamiltondensityequationsM éM e M ÌM b (6.52a) é Ì 3 M eM b (6.52b)

TheHamiltondensityfunctionsarein many wayssimilar to theordinaryHamiltonfunctionsandleadto similar results.

Theelectromagneticfield

Above, when we describedthe mechanicalfield, we useda scalarfield Ì b- x .If we want to describethe electromagneticfield in termsof a LagrangedensityÑ andEuler-Lagrangeequations,it comesnaturalto expressÑ in termsof thefour-potential

P ( ) t .The entiresystemof particlesandfields consistsof a mechanicalpart, a field

part andan interactionpart. We thereforeassumethat the total LagrangedensityÑ tot for this systemcanbeexpressedasÑ tot +Ñ mech Ñ inter Ñ field (6.53)

wherethemechanicalpart hasto do with theparticlemotion (kinetic energy). Itis givenby n 4q Y where n 4q is givenby Equation(6.2)on page74 and Y is thevolume. Expressedin the restmassdensityI 0

m, themechanicalLagrange densitycanbewrittenÑ mech 1

2I 0

m ! 2 '(`' ( (6.54)

The Ñ inter part which describesthe interactionbetweenthe chargedparticlesandtheexternalelectromagneticfield. A convenientexpressionfor this interactionLagrange densityisÑ inter J ( P ( (6.55)

For thefield part Ñ field wechoosethedifferencebetweenmagneticandelectricenergy density(in analogywith thedifferencebetweenkineticandpotentialenergyin amechanicalfield). Usingthefield tensor, weexpressthisfieldLagrangedensityas Ñ field 1

4S

0 ( C ( C (6.56)

sothatthetotal LagrangiandensitycanbewrittenÑ tot 12I 0

m ! 2 '(' ( J ( P ( 14S

0 ( C ( C (6.57)

6.2. COVARIANT FIELD THEORY 85ê FIELD ENERGY DIFFERENCE EXPRESSED IN THE FIELD TENSOR EXAMPLE 6.1Show, by explicit calculation,thatëìíDë ìîíï 2 Ázð 2 ñ 2 É ò 2 Ä (6.58)

FromFormula(5.77)onpage68 werecallthatëìí ïó ì=ôõí É ó íhôöì ïø÷ùú 0 É òüû É òý É òöþòüû 0 É ð ñ þ ð ñ ýòý ð ñ þ 0 É ð ñ ûòöþ É ð ñ ý ð ñ û 0

ÿ (6.59)

andfrom Formula(5.76)on page68 thatë ìíEïóîì ô í É óäí ô ìï ÷ùú 0 òüû òý òöþÉ òüû 0 É ð ñ þ ð ñ ýÉ òý ð ñ þ 0 É ð ñ ûÉ òþ É ð ñ ý ð ñ û 0

ÿ (6.60)

where denotestherow numberand thecolumnnumber. Then,EinsteinsummationanddirectsubstitutionyieldsëìíDë ìîíEï ë 00ë

00 Ê ë 01ë01 Ê ë 02ë

02 Ê ë 03ë03Ê ë 10ë

10 Ê ë 11ë11 Ê ë 12ë

12 Ê ë 13ë13Ê ë 20ë

20 Ê ë 21ë21 Ê ë 22ë

22 Ê ë 23ë23Ê ë 30ë

30 Ê ë 31ë31 Ê ë 32ë

32 Ê ë 33ë33ï 0 É ò 2û É ò 2ý É ò 2þÉ ò 2û Ê 0 Ê ð 2 ñ 2þ Ê ð 2 ñ 2ýÉ ò 2ý Ê ð 2 ñ 2þ Ê 0 Ê ð 2 ñ 2ûÉ ò 2þ Ê ð 2 ñ 2ý Ê ð 2 ñ 2û Ê 0ï É 2ò 2û É 2ò 2ý É 2ò 2þ Ê ð 2 ñ 2û Ê ð 2 ñ 2ý Ê ð 2 ñ 2þï É 2ò 2 Ê 2ð 2 ñ 2 ï 2 Ázð 2 ñ 2 É ò 2 Ä (6.61)

QED END OF EXAMPLE 6.1

Using Ñ tot in the 3D Euler-Lagrangeequations,Equation(6.47) on page83(with Ì replacedby

P C ), we canderive the dynamicsfor the whole system.Forinstance,theelectromagneticpartof theLagrangiandensityÑ EM >Ñ inter Ñ field J C P C 1

4S

0 ( C ( C (6.62)

insertedinto the Euler-Lagrangeequations,expression(6.47) on page83, yieldstwo of Maxwell’s equations.To seethis,we notefrom Equation(6.62)above and


theresultsin Example6.1thatM<Ñ EMM P C J C (6.63)

Furthermore,M ( ¢ M<Ñ EMM M ( P C £ S0

4 M ( ¢ MM M ( P C , t t . £ S0

4 M ( MM M ( P C M t P 3 M P t M t P 3 M P t S0

4 M ( MM M ( P C ¢ M t P M t P 3 M t P M P t3 M P t M t P ¤M P t M P t £ S0

2 M ( ¢ MM M ( P C , M t P M t P 3 M t P M P t . £(6.64)

But MM M ( P C , M t P M t P . NM t P MM M ( P C M t P NM t P MM M ( P C M t P NM t P MM M ( P C M t P NM t P MM M ( P C ð¨t M ð P NM t P MM M ( P C M t P ð¨t ð M t P MM M ( P C M P NM t P MM M ( P C M t P NM P MM M ( P C M P 2M ( P C (6.65)

Similarly,

6.2. COVARIANT FIELD THEORY 87MM M ( P C , M t P M P t . 2M C P ( (6.66)

sothatM ( ¢ MÑ EMM M ( P C £ S 0 M ( M ( P C 3 M C P ( S 0M ( CM ) ( (6.67)

ThismeansthattheEuler-Lagrangeequations,expression(6.47)onpage83,fortheLagrangiandensityÑ EM andwith

P C asthefield quantitybecomeMÑ EMM P C 3 M ( ¢ MÑ EMM M ( P C £ J C 3 S 0M ( CM ) ( 0 (6.68)

or M ( CM ) ( J CS0

(6.69)

Explicitly, setting ¥ 0 in this covariantequationandusing the matrix rep-resentationFormula (5.77) on page68 for the covariantcomponentform of theelectromagneticfield tensor

( C , weobtainM 00M ) 0 M 10M ) 1 M 20M ) 2 M 30M ) 3 0 M @ M ) M @ M M @EM F E IS0

(6.70)

which is the Maxwell sourceequationfor the electricfield, Equation(1.43a)onpage14. For ¥ 1 wegetM 01M ) 0 M 11M ) 1 M 21M ) 2 M 31M ) 3 3 1! M @ M b 0 ! M M ! M M I ' S

0(6.71)

or, using S 0 0 1 ! 2 andidentifying I ' J ,MK M 3 M M 3 S 0 0M @ M b 0

J (6.72)

andsimilarly for ¥ 2 - 3. In summary, in three-vector form, we canwrite theresultas B 3 S 0 0

M EM b 0j b- x (6.73)

which is theMaxwell sourceequationfor themagneticfield, Equation(1.43d)onpage14.

Otherfields

In general,thedynamicequationsfor mostany fields,andnotonly electromagneticones,canbederivedfrom aLagrangiandensitytogetherwith avariationalprinciple(theEuler-Lagrangeequations).Both linearandnon-linearfieldsarestudiedwiththis technique.As a simpleexample,considera real,scalarfield Ì which hasthefollowing Lagrangedensity:Ñ 1

2 © M ( Ì M ( Ìâ3G: 2 Ì 2 ª 12 d M ÌM ) ( M ÌM ) ( 3G: 2 Ì 2 f (6.74)

Insertioninto the1D Euler-Lagrangeequation,Equation(6.44)on page82,yieldsthedynamicequation L 2 3G: 2 Ì 0 (6.75)

with thesolutionÌ n k x x u"!yq u"#%$ x $H x H (6.76)

whichdescribestheYukawamesonfield for ascalarmesonwith mass: . Withe 1! 2 M ÌM b (6.77)

we obtaintheHamiltondensityé 12 ! 2 e 2 Ì 2 : 2 Ì 2 (6.78)

which is positivedefinite.AnotherLagrangiandensitywhichhasattractedquitesomeinterestis theProca

LagrangianÑ EM >Ñ inter Ñ field J C P C 14S

0 ( C ( C : 2 P ( P ( (6.79)

which leadsto thedynamicequationM ( CM ) ( : 2 P C J CS0

(6.80)

This equationdescribesan electromagneticfield with a mass,or, in otherwords,massivephotons. If massive photonswould exist, large-scalemagneticfields, in-cludingthoseof theearthandgalacticspiralarms,wouldbesignificantlymodifiedto yield measurablediscrepancesfrom their usualform. Spaceexperimentsof thiskind onboardsatelliteshave led to stringentupperboundson thephotonmass.Ifthephotonreally hasa mass,it will have animpacton electrodynamicsaswell ason cosmologyandastrophysics.

88

BIBLIOGRAPHY 6

[1] Asim O. Barut. DynamicsandClassicalTheoryof FieldsandParticles. Dover Pub-lications,Inc., New York, NY, 1980. ISBN 0-486-64038-8.

[2] HerbertGoldstein.ClassicalMechanics. Addison-Wesley PublishingCompany, Inc.,Reading,MA . . . , secondedition,1981. ISBN 0-201-02918-9.

[3] WalterT. Grandy. Introductionto ElectrodynamicsandRadiation. AcademicPress,New York andLondon,1970. ISBN 0-12-295250-2.



[6] J. J. Sakurai. AdvancedQuantumMechanics. Addison-Wesley PublishingCom-pany, Inc.,Reading,MA . . . , 1967. ISBN 0-201-06710-2.

89

CHAPTER 7

Interactionsof FieldsandMatter

The microscopicMaxwell equations(1.43) derived in Chapter1 arevalid on allscaleswhereaclassicaldescriptionis good.However, whenmacroscopicmatterispresent,it is sometimesconvenientto usethecorrespondingmacroscopicMaxwellequations(in astatisticalsense)in whichauxiliary, derivedfieldsareintroducedinorderto incorporateeffectsof macroscopicmatterwhenthis is immersedfully orpartially in anelectromagneticfield.

7.1 Electricpolarisationandtheelectricdisplacementvector

7.1.1 Electricmultipolemoments

The electrostaticpropertiesof a spatialvolume containingelectric chargesandlocatedneara pointx0 canbecharacterizedin termsof thetotal charge or electricmonopolemoment&' ( ) * + x , - . 3/ , (7.1)

wherethe * is the charge density introducedin Equation(1.7) on page4), theelectricdipolemomentvector

p ' ( ) + x ,0 x0 - * + x , - . 3/ , (7.2)

with components1 2 , 3 ' 1 4 2 4 3, theelectricquadrupolemomenttensor

Q ' ( ) + x ,0 x0 - + x ,50 x0 - * + x , - . 3/ , (7.3)

with components6 2 7 4384:9 ' 1 4 2 4 3, andhigherorderelectricmoments.

91

92 CHAPTER 7. INTERACTIONS OF FIELDS AND MATTER

In particular, theelectrostaticpotentialEquation(3.3)onpage31 from a chargedistribution locatednearx0 canbeTaylorexpandedin thefollowing way:;

stat+ x - ' 1

4<>= 0 ? &@x 0 x0

@ A 1@x 0 x0

@ 2 1 2 + x 0 x0 - 2@x 0 x0

@ A1@

x 0 x0@ 3 6 2 7 B 3

2

+ x 0 x0 -C2@x 0 x0

@ + x 0 x0 - 7@x 0 x0

@ 0 12 D 2 7 E AGFFFIH

(7.4)

whereEinstein’s summationconventionover 3 and 9 is implied. As canbe seenfrom this expression,only thefirst few termsareimportantif thefield point (ob-servationpoint) is far away from x0.

For a normalmedium,the major contributionsto the electrostaticinteractionscomefrom thenetchargeandthelowestorderelectricmultipolemomentsinducedby the polarisationdue to an appliedelectricfield. Particularly importantis thedipole moment.Let P denotethe electricdipole momentdensity(electricdipolemomentperunit volume;unit: C/m2), alsoknown astheelectricpolarisation, insomemedium.In analogywith thesecondtermin theexpansionEquation(7.4)onpage92,theelectricpotentialfrom thisvolumedistributionP + x , - of electricdipolemomentsp at thesourcepoint x , canbewritten;

p+ x - ' 1

4<>= 0

( ) P + x , - J x 0 x ,@x 0 x , @ 3 . 3/ ,' 0 1

4<K= 0

( ) P + x , - JML B 1@x 0 x , @ E . 3/ ,' 1

4<>= 0

( ) P + x , - JMLN, B 1@x 0 x , @ E . 3/ , (7.5)

Using the expressionEquation(M.63) on page178 andapplyingthe divergencetheorem,we canrewrite thisexpressionfor thepotentialasfollows:;

p+ x - ' 1

4<>= 0 O ( ) LN,J B P + x , -@x 0 x , @ E . 3/ ,0 ( ) L , J P + x , -@

x 0 x , @ . 3/ ,QP' 14<>= 0 O R S P + x , - J n@

x 0 x , @ . 2/ 0 ( ) L , J P + x , -@x 0 x , @ . 3/ , P

(7.6)

wherethe first term, which describesthe effects of the induced,non-cancellingdipole momenton the surfaceof the volume,canbe neglected,unlessthereis a

7.1. ELECTRIC POLARISATION AND THE ELECTRIC DISPLACEMENT VECTOR 93

discontinuityin P J ˆT at thesurface.Doing so,we find that thecontribution fromtheelectricdipolemomentsto thepotentialis givenby;

p' 1

4<K= 0

( ) 0UL , J P + x , -@x 0 x , @ . 3/ , (7.7)

Comparingthisexpressionwith expressionEquation(3.3)onpage31 for theelec-trostaticpotentialfrom astaticchargedistribution * , weseethat 0VLWJ P + x - hasthecharacteristicsof achargedensityandthat,to thelowestorder, theeffectivechargedensitybecomes* + / - 0XLYJ P + / - , in which thesecondtermis apolarisationterm.

Theversionof equationEquation(1.7)onpage4 where“true” andpolarisationchargesareseparatedthusbecomesLZJ E ' * + x - 0[LZJ P + x -= 0

(7.8)

Rewriting thisequation,andat thesametimeintroducingtheelectricdisplacementvector(C/m2)

D ' = 0E A P (7.9)

we obtainLZJ + = 0E A P- ' LZJ D 'G*true

+ x - (7.10)

where *true is the “true” charge density in the medium. This is one of Max-

well’s equationsand is valid also for time varying fields. By introducing thenotation *

pol' 0VL\J P for the “polarised” charge densityin the medium,and*

total'G*

true A *pol for the“total” chargedensity, wecanwrite down thefollowing

alternativeversionof Maxwell’sequation(7.23a)onpage95LZJ E ' *total

+ x -= 0(7.11)

Often, for low enoughfield strengths@E@, the linear and isotopic relationship

betweenP andE

P ' = 0 ] E (7.12)

is a goodapproximation.Thequantity ] is theelectricsusceptibilitywhich is ma-terial dependent.For electromagneticallyanisotropicmediasuchasa magnetisedplasmaor a birefringentcrystal,thesusceptibilityis a tensor. In general,therela-tionshipis not of a simplelinear form asin Equation(7.12)but non-lineartermsareimportant. In sucha situationtheprincipleof superpositionis no longervalidandnon-lineareffectssuchasfrequency conversionandmixing canbeexpected.


Insertingtheapproximation(7.12)into Equation(7.9)on thepreviouspage,wecanwrite thelatter

D ' = E (7.13)

where,approximately,= ' = 0+ 1 A ] - (7.14)

7.2 MagnetisationandthemagnetisingfieldAn analysisof thepropertiesof stationarymagneticmediaandtheassociatedcur-rentsshows thatthreesuchtypesof currentsexist:

1. In analogywith “true” chargesfor the electric case,we may have “true”currentsj true, i.e., aphysicaltransportof truecharges.

2. In analogywith electricpolarisationP theremaybea form of chargetrans-port associatedwith thechangesof thepolarisationwith time. We call suchcurrentsinducedby anexternalfield polarisationcurrents. Weidentify themwith ^ P_ ^` .

3. Theremayalsobe intrinsic currentsof a microscopic,oftenatomic,naturethatareinaccessibleto directobservation,but whichmayproduceneteffectsat discontinuitiesandboundaries.We shallcall suchcurrentsmagnetisationcurrentsanddenotethemjm.

No magneticmonopoleshavebeenobservedyet. Sothereis nocorrespondencein themagneticcaseto theelectricmonopolemoment(7.1).Thelowestordermag-neticmoment,correspondingto theelectricdipolemoment(7.2), is themagneticdipolemoment

m ' 12( ) + x ,50 x0 - a j + x , - . 3/ , (7.15)

For a distribution of magneticdipolemomentsin a volume,we maydescribethisvolumein termsof themagnetisation, or magneticdipolemomentperunit volume,M . Via thedefinitionof thevectorpotentialonecanshow that themagnetisationcurrentandthemagnetisationis simply related:

jm' Lba M (7.16)

In a stationarymediumwe thereforehave a total currentwhich is (approxim-ately)thesumof thethreecurrentsenumeratedabove:

7.3. ENERGY AND MOMENTUM 95

j total' j true A ^ P^` A Lba M (7.17)

We thenobtaintheMaxwell equationLca B ' d0 B j true A ^ P^` A Lba M E (7.18)

Moving theterm Lea M to theleft handsideandintroducingthemagnetisingfield(magneticfield intensity, Ampère-turndensity) as

H ' Bd00 M (7.19)

andusingthedefinitionfor D, Equation(7.9)on page93, we canwrite this Max-well equationin thefollowing formLca H ' j true A ^ D^` (7.20)

We may, in analogywith the electriccase,introducea magneticsusceptibilityfor themedium.Denotingit ] m, we canwrite

H ' Bd (7.21)

where,approximately,d ' d0+ 1 A ] m - (7.22)

7.3 Energy andmomentumAs mentionedin Chapter1, Maxwell’sequationsexpressedin termsof thederivedfield quantitiesD andH canbewrittenLZJ D 'f* + `4 x - (7.23a)LZJ B ' 0 (7.23b)Lca E ' 0 ^ B^` (7.23c)Lca H ' j + `g4 x - A ^^` D (7.23d)

andarecalledMaxwell’s macroscopicequations. Theseequationsareconvenientto usein certainsimplecases.Togetherwith theboundaryconditionsandthecon-stitutive relations,they describeuniquely(but only approximately!)thepropertiesof the electricandmagneticfields in matter. We shall usethemin the followingconsiderationson the energy andmomentumof the electromagneticfield anditsinteractionwith matter.


7.3.1 Theenergy theoremin Maxwell’s theory

Scalarmultiplying (7.23c)by H, (7.23d)by E andsubtracting,weobtain

H J + Lba E - 0 E J + Lca H - ' LZJ + E a H -' 0 H J ^ B^` 0 E J j 0 E J ^ D^` ' 0 12

^^` + H J B A E J D - 0 j J E (7.24)

Integrationover the entirevolume h andusingGauss’s theorem(the divergencetheorem),weobtain0 ^^` ( ) 1

2+ H J B A E J D - . 3/ , ' ( ) j J E . 3/ , A ( S + E a H - Ji. S, (7.25)

But, accordingto Ohm’s law in the presenceof an electromotive force field,Equation(1.26)onpage11:

j 'Gj + E A EEMF - (7.26)

whichmeansthat( ) j J E . 3/ , ' ( ) 9 2j . 3/ ,0 ( ) j J EEMF . 3/ , (7.27)

Insertingthis into Equation(7.25)( ) j J EEMF . 3/ ,k l m nApplied electricpower

' ( ) 9 2j . 3/ ,k l m nJouleheatA ^^` ( ) 1

2+ E J D A H J B -k l m n

Field energy

. 3/ ,A ( S + E a H - Ji. S,k l m n

Radiatedpower

(7.28)

whichis theenergytheoremin Maxwell’stheoryalsoknown asPoynting’stheorem.It is convenientto introducethefollowing quantities:o

e' 1

2( ) E J D . 3/ , (7.29)o

m' 1

2( ) H J B . 3/ , (7.30)

S ' E a H (7.31)

whereo

e is theelectricfieldenergy,o

m is themagneticfieldenergy, bothmeasuredin J,andS is thePoyntingvector(powerflux), measuredin W/m2.

7.3. ENERGY AND MOMENTUM 97

7.3.2 Themomentumtheoremin Maxwell’s theory

Let us now investigate the momentumbalance(force actions)in the casethat afield interactswith matterin a non-relativistic way. For this purposewe considerthe force densitygiven by the Lorentzforce per unit volume * E A j a B. UsingMaxwell’sequations(7.23)andsymmetrising,we obtain* E A j a B ' + LZJ D - E A B Lba H 0 ^ D^` E a B' E + LZJ D - A + Lba H - a B 0 ^ D^` a B' E + LZJ D - 0 B a + Lca H -0 ^^` + D a B - A D a ^ B^`' E + LZJ D - 0 B a + Lca H -0 ^^` + D a B - 0 D a + Lca E - A H + LZJ Bk l m np 0

-' qE + LZJ D - 0 D a + Lca E -:r A qH + LZJ B - 0 B a + Lca H -Cr0 ^^` + D a B - (7.32)

Oneverifieseasily that the 3 th vectorcomponentsof the two termsin squarebracketsin theright handmemberof (7.32)canbeexpressedasqE + LZJ D - 0 D a + Lba E -Crs2' 1

2B E J ^ D^ / 2 0 D J ^ E^ / 2 E A ^^ / 7 B t 2 u 7 0 1

2E J D D 2 7 E (7.33)

and qH + LZJ B - 0 B a + Lca H -Cr 2' 12

B H J ^ B^ / 2 0 B J ^ H^ / 2 E A ^^ / 7 B v 2 wx7 0 12

B J H D 2 7 E (7.34)

respectively.

Usingthesetwo expressionsin the 3 th componentof Equation(7.32)above and


re-shuffling terms,weget+ * E A j a B - 2 0 12 ? B E J ^ D^ / 2 0 D J ^ E^ / 2 EA B H J ^ B^ / 2 0 B J ^ H^ / 2 E H A ^^` + D a B - 2' ^^ / 7 B t 2 u 7 0 1

2E J D D 2 7 A v 2 wx7 0 1

2H J B D 2 7 E (7.35)

IntroducingtheelectricvolumeforceFev via its 3 th component+ Fev - 2 ' + * E A j a B - 20 12 ? B E J ^ D^ / 2 0 D J ^ E^ / 2 E A B H J ^ B^ / 2 0 B J ^ H^ / 2 E H (7.36)

andtheMaxwellstresstensory 2 7 ' t 2 u 7 0 12

E J D D 2 7 A v 2 wx7 0 12

H J B D 2 7 (7.37)

we finally obtaintheforceequationO Fev A ^^` + D a B - P 2 ' ^ y 2 7^ / 7 (7.38)

If we introducetherelativeelectricpermittivity z andtherelativemagneticper-meability z m as

D ' z = 0E ' = E (7.39)

B ' z md

0H ' d H (7.40)

we canrewrite (7.38)as^ y 2 7^ / 7 ' B Fev A zz m 2

^ S^` E 2 (7.41)

whereS is thePoynting vectordefinedin Equation(7.31)on page96. Integrationover theentirevolume h yields( ) Fev . 3/ ,k l m n

Forceon thematter

A ..` ( ) zz m 2 S . 3/ ,k l m nField momentum

' ( S T ˆ| . 2/ ,k l m nMaxwell stress

(7.42)

which expressesthe balancebetweenthe force on the matter, the rateof changeof theelectromagneticfield momentumandtheMaxwell stress.This equationiscalledthemomentumtheoremin Maxwell’s theory.

In vacuum(7.42)becomes( ) * + E A v a B - . 3/ , A 1 2

..` ( ) S . 3/ , ' ( S T ˆ| . 2/ , (7.43)

or ..` pmech A ..` pfield ' ( S T ˆ| . 2/ , (7.44)

99

BIBLIOGRAPHY 7


101

CHAPTER 8

ElectromagneticRadiation

In this chapterwe will developthetheoryof electromagneticradiation,andthere-fore studyelectricandmagneticfields which arecapableof carryingenergy andmomentumover largedistances.In Chapter3 we wereableto derive generalex-pressionsfor thescalarandvectorpotentialsfrom whichwethen,in Chapter4 cal-culatedthetotal electricandmagneticfieldsfrom arbitrarydistributionsof chargeandcurrentsources.Theonly limitation in thecalculationof thefieldswasthattheadvancedpotentialswerediscarded.

Weshallnow studythesefieldsfurtherundertheassumptionthattheobserver islocatedin the far zone, i.e., very far away from thesourceregion(s).We thereforestudytheradiationfieldswhicharedominatingin thiszone.

8.1 TheradiationfieldsFromEquation(4.13)on page46 andEquation(4.24)on page49,which give thetotal electricandmagneticfields,we obtaintheradiationfields

Brad+ `4 x - ' ( ~ Brad + x - ~ .' d

0

4< ( ) j + ` ,ret 4 x , - a + x 0 x , -@x 0 x , @ 2 . 3/ , (8.1)

Erad+ `4 x - ' ( ~ Erad + x - ~ .' 1

4<K= 0 2

( ) q j + ` ,ret 4 x , - a + x 0 x , -Cr a + x 0 x , -@x 0 x , @ 3 . 3/ , (8.2)

where

103

104 CHAPTER 8. ELECTROMAGNETIC RADIATION

j + `C,ret 4 x , - def B ^ j^` E p ret

(8.3)

Insteadof studyingthefieldsin thetimedomain,wecanoftenmakeaspectrumanalysisinto thefrequency domainandstudyeachFouriercomponentseparately.A superpositionof all thesecomponentsand a transformationback to the timedomainwill thenyield thecompletesolution.

The Fourier representationof the radiationfields (8.1) (8.2) were includedinEquation(4.12)on page46 andEquation(4.23)on page48, respectively andareexplicitly givenby

Brad + x - ' 12< ( ~ Brad

+ `4 x - .`' 0 d 0

4< ( ) j + x , - a + x 0 x , -@x 0 x , @ 2 x ~ x . 3/ ,' 0 d 0

4< ( ) j + x , - a k@x 0 x , @ Q x ~ x . 3/ , (8.4)

Erad + x - ' 12< ( ~ Erad

+ `4 x - .`' 04<>= 0

( ) q j + x , - a + x 0 x , -:r a + x 0 x , -@x 0 x , @ 3 Q x ~ x . 3/ ,' 0 1

4<>= 0 ( ) q j + x , - a k r a + x 0 x , -@

x 0 x , @ 2 QM x ~ x . 3/ , (8.5)

If thesourceis locatedinsidea volume h nearx0 andhassucha limited spatialextentthatmax

@x , 0 x0

@ @x 0 x , @ , andtheintegrationsurface , centeredonx0,

hasa large enoughradius@x 0 x0

@ max

@x , 0 x0

@, we seefrom Figure8.1 that

we canapproximate x 0 x , k J + x 0 x , - @x 0 x0

@ 0 k J + x ,0 x0 - (8.6)

and

ˆ J ˆT@x 0 x0

@ 2 .X. Ω (8.7)

where. ' @x 0 x0

@ 2 . Ω, wasused.

8.1. THE RADIATION FIELDS 105

x x x0

S ˆ 2

ˆ¡x x0

x0

x x ¢

£x ¤Figure8.1. Relationbetweenthesurfacenormalandthek vectorfor radiationgeneratedat sourcepointsx ¥ nearthepoint x0 in thesourcevolume ¦ . At dis-tancesmuchlargerthantheextentof ¦ , theunit vector ˆ§ , normalto thesurface¨

which hasits centreat x0, andtheunit vector ˆ© of theradiationk vectorfromx ¥ arenearlycoincident.

Within approximation(8.6)theexpressions(8.4)and(8.5)for theradiationfieldscanbeapproximatedas

Brad + x - ª0« d 0

4< QM x ~ x0 ( ) j + x , - a k@

x 0 x , @ ~ k ¬® x ~ x0 ¯ . 3/ ,ª0« d 0

4< QM x ~ x0@

x 0 x0@ ( ) q j + x , - a k r ~ k ¬® x ~ x0 ¯ . 3/ , (8.8)

Erad + x - ª0« 14<K= 0

Q x ~ x0 ( ) q j + x , - a k r a + x 0 x , -@

x 0 x , @ 2 ~ k ¬® x ~ x0 ¯ . 3/ ,f 14<K= 0

x ~ x0@

x 0 x0@ + x 0 x0 -@

x 0 x0@ a ( ) q j + x , - a k r ~ k ¬® x ~ x0 ¯ . 3/ ,

(8.9)

I.e., if max@x , 0 x0

@ @x 0 x , @ , thenthefields canbe approximatedasspherical

wavesmultiplied by dimensionalandangularfactors,with integralsover pointsinthesourcevolumeonly.


8.2 Radiatedenergy

Let usconsidertheenergy thatcarriedin theradiationfieldsBrad, (8.1),andErad,Equation(8.2) on page103. We have to treat signalswith limited lifetime andhencefinite frequency bandwidthdifferentlyfrom monochromaticsignals.

8.2.1 Monochromaticsignals

If thesourceis strictly monochromatic,we canobtainthetemporalaverageof theradiatedpower ° directly, simplyby averagingover oneperiodsothat±

S² ' ±E a H ² ' 1

2d 0Re ³ E a B ´ µ ' 1

2d 0Re ¶ E ~ a + B ~ - ´ ·' 1

2d 0Re ¶ E a B ´ ~ · ' 1

2d 0Re ³ E a B ´ µ (8.10)

Using the far-field approximations(8.8) and(8.9) andthe fact that1_ ¸' ¹ = 0d

0

and º 0'¼» d

0 _ = 0 accordingto thedefinition(2.15)onpage23,we obtain±S² ' 1

32< 2 º 01@

x 0 x0@ 2 ( ) q½+ j a k -:r ~ k ¬Q x ~ x0 ¯ . 3/ , 2 x 0 x0@

x 0 x0@ (8.11)

or, makinguseof (8.7)onpage104,.°. Ω' 1

32< 2 º 0 ( ) q½+ j a k -:r ~ k ¬Q x ~ x0 ¯ . 3/ , 2 (8.12)

which is theradiatedpowerperunit solidangle.

8.2.2 Finite bandwidthsignals

A signal with finite pulsewidth in time ( ` ) domainhasa certainspreadin fre-quency ( ) domain. To calculatethe total radiatedenergy we needto integrateover the whole bandwith. The total energy transmittedthrougha unit areais the

8.2. RADIATED ENERGY 107

time integralof thePoyntingvector:( ~ S+ ` - .` ' ( ~ + E a H - .`' ( ~ . ( ~ .¾, ( ~ + E a H - ~ ¿ ¯ .` (8.13)

If wecarryout thetemporalintegrationfirst andusethefactthat( ~ ~ ¿ ¯ .` ' 2< D + A ¾, - (8.14)

Equation(8.13)canbewritten [cf. Parseval’s identity]( ~ S+ ` - .` ' 2< ( ~ + E a H ~ - .' 2< B ( 0

+ E a H ~ - . A ( 0~ + E a H ~ - . E' 2< B ( 0

+ E a H ~ - .À0 ( 0

+ E ~ a H - . E' 2<d0

( 0

+ E a B ~ A E ~ a B - .' 2<d0

( 0

+ E a B ´ A E ´ a B - . (8.15)

wherethe laststepfollows from thereal-valuednessof E andB . We inserttheFouriertransformsof thefield componentswhichdominateat largedistances,i.e.,theradiationfields(8.4)and(8.5).

Theresult,afterintegrationover thearea of a largespherewhichenclosesthesource,iso ' 1

4<ÂÁ d0= 0

( S ( 0

( ) j a k@x 0 x , @ Q x ~ x . 3/ , 2 ˆ J ˆT .Ã. (8.16)

Insertingtheapproximations(8.6)and(8.7) into Equation(8.16)aboveandalsointroducingo ' (

0

o . (8.17)

and recalling the definition (2.15) on page23 for the vacuumresistanceº 0 weobtain. o . Ω

.Ä 14< º 0 ( ) + j a k - ~ k ¬® x ~ x0 ¯ . 3/ , 2 . (8.18)

which, at large distances,is a goodapproximationto the energy that is radiatedper unit solid angle . Ω in a frequency band . . It is important to notice that


Formula(8.18) includesonly sourcecoordinates.This meansthat the amountofenergy that is beingradiatedis independenton thedistanceto thesource(aslongasit is large).

8.3 Radiatingsystems

As shown above,onecan,at leastin principle,calculatetheradiatedfields,Poynt-ing flux andenergy for anarbitrarycurrentdensityFouriercomponent.However, inpractice,it is oftendifficult to evaluatetheintegralsunlessthecurrenthasasimpledistribution in space.In thegeneralcase,onehasto resortto approximations.Weshallconsiderboththesesituations.

8.3.1 Simplegeometries

Certainradiationsystemshave a geometrywhich is one-dimensional,symmetricor in any otherway simpleenoughthat a direct calculationof the radiatedfieldsandenergy is possible.This is for instancethecasewhenthecurrentflows in onedirection in spaceonly and is limited in extent. An exampleof this is a linearantenna.

Linear antenna

Let usapplyEquation(8.12)on page106for calculatingthepower from a linear,transmittingantenna,fed acrossa small gap at its centrewith a monochromaticsource.Theantennais a straight,thin conductorof length Å which carriesa one-dimensionaltime-varyingcurrentsothatit produceselectromagneticradiation.

We assumethat theconductorresistanceandtheenergy lossdueto theelectro-magneticradiationarenegligible. Sincewe canassumethat the antennawire isinfinitely thin, thecurrentvanishesat theendpoint. Thecurrentthereforeformsastandingwave with wave number ' _ andcanbewritten

j0+ x , - 'fÆ

0 D + / ,1 - D + / ,2 - sinq + Å _ 2 0 @ / ,3 @ -:rsin+ Å _ 2- ˆÇ 3 (8.19)

wheretheamplitudeÆ 0 is constant.In orderto evaluateFormula(8.12)onpage106

8.3. RADIATING SYSTEMS 109

with theexplicit monochromaticcurrent(8.19)inserted,we needtheexpression ( ) + j0 a k - ~ k ¬Q x ~ x0 ¯ . 3/ , 2' (ÄÈ É 2È É 2 Æ0sinq + Å _ 2 0 @ / ,3 @ -:r

sin+ Å _ 2- sin Ê ~Q Ë 3 ÌÎÍÏÑÐ ~Q Ë 0 ÌÑÍÏÑÐ . / ,3 2'GÆ 20 2 sin2 Ê

sin2 + Å _ 2- Q Ë 0 cosÐ 2 2 ( È É 20

sinq + Å _ 2 0 / ,3 -:r cos+ / ,3 cosÊ - . / ,3 2' 4Æ 20 B cosq½+ Å _ 2- cosÊ r 0 cos+ Å _ 2-

sin Ê sin+ Å _ 2- E 2

(8.20)

insertingthis expressionand . Ω ' 2< sin Ê . Ê into Formula(8.12)on page106andintegratingover Ê , we find thatthetotal radiatedpower from theantennais° + Å - ' º 0

Æ 20

14< ( Ò

0B cosq½+ Å _ 2- cosÊ r 0 cos+ Å _ 2-

sin Ê sin+ Å _ 2- E 2

sin Ê . Ê (8.21)

Onecanshow that

lim È Ó 0° + Å - ' <

12B Å Ô E º 0 Õ 2

0 (8.22)

whereÔ

is thevacuumwavelength.

Thequantityº rad+ Å - ' ° + Å -Æ 2

eff

' ° + Å -12Æ 2

0

' º 0<6

B Å Ô E 2 197 B Å Ô E 2

Ω (8.23)

is calledtheradiationresistance. For thetechnologicallyimportantcaseof a half-wave antenna,i.e., for Å ' Ô _ 2 or Å ' < , Formula(8.21)reducesto° + Ô _ 2- ' º 0

Æ 20

14< ( Ò

0

cos2 Ö Ò2 cosÊ ×sin Ê . Ê (8.24)

The integral in (8.24) can be evaluatednumerically. It can also be evaluated


analyticallyasfollows:( Ò0

cos2 Ö Ò2 cosÊ ×sin Ê . Ê ' Ø cosÊ ÙÛÚ Ü ' ( 1~ 1

cos2 Ö Ò2 Ú×1 0XÚ 2 .Ú 'Ý

cos2 Þ <2Úß ' 1 A cos+ <>Ú -

2 à' 12( 1~ 1

1 A cos+ <KÚ -+ 1 A Ú - + 1 0XÚ - .Ú' 14( 1~ 1

1 A cos+ <>Ú -+ 1 A Ú - .Ú A 14( 1~ 1

1 A cos+ <KÚ -+ 1 0XÚ - .Ú' 12( 1~ 1

1 A cos+ <>Ú -+ 1 A Ú - .Ú 'âá 1 A ÚãÙ ä<xå' 12( 2Ò

0

1 0 cosää . ä ' 12q æ A ln 2<ã0 Ci + 2< -Cr 1 F 22 (8.25)

wherein thelaststeptheEuler-Mascheroni constantæç' 0 F 5772 FFF andthecosineintegral Ci + / - wereintroduced.Insertingthis into theexpressionEquation(8.24)on thepreviouspagewe obtainthevalue º rad

+ Ô _ 2- 73 Ω.

8.3.2 Multipole expansion

In thegeneralcase,andwhenweareinterestedin evaluatingtheradiationfar fromthe sourcevolume, we can introducean approximationwhich leadsto a multi-poleexpansionwhereindividual termscanbeevaluatedanalytically. We shalluseHertz’ methodto obtainthisexpansion.

TheHertzpotential

Let usconsiderthecontinuityequation,which, accordingto expression(1.21)onpage9, canbewritten^ * + `4 x -^` A LZJ j + `4 x - ' 0 (8.26)

If we introduceavectorfield p + `4 x - suchthatLZJ p ' 0 *true (8.27)^ p^` ' j true (8.28)

andcomparewith Equation(8.26)above,we seethatp + `g4 x - satisfiesthis continu-ity equation.Furthermore,if we comparewith theelectricpolarisation[cf. Equa-


tion (7.9)on page93], we seethatthequantityp is relatedto the“true” chargesinthesameway asP is relatedto polarisedcharge. Therefore,p is referredto asthepolarisationvector.

We introducea furtherpotentialZ with thefollowing propertyLZJ Z ' 0 ;(8.29a)

1 2

^ Z^` ' A (8.29b)

where;

andA arethe electromagneticscalarandvectorpotentials,respectively.As weseeZ actsasa“super-potential” in thesensethatit is apotentialfrom whichwe canobtainotherpotentials.It is calledtheHertz’ vectoror polarisationpoten-tial and,ascanbe seenfrom (8.28) and(8.29b), it satisfiesthe inhomogeneouswave equationè

2Z ' 1 2

^ 2^` 2Z 0 é 2Z ' p= 0(8.30)

This equationis of thesametypeasEquation(3.19)on page36,andhasthere-fore theretardedsolution

Z + `4 x - ' 14<>= 0

( p + ` ,ret 4 x , -@x 0 x , @ . 3/ , (8.31)

with Fouriercomponents

Z + x - ' 14<K= 0

( p + x , - QM x ~ x @x 0 x , @ . 3/ , (8.32)

If we introducethehelpvectorC suchthat

C ' Lca Z (8.33)

we seethatwe cancalculatethemagneticandelectricfields, respectively, asfol-lows

B ' 1 2

^ C^` (8.34a)

E ' Lca C (8.34b)

wherethelastequationis valid only outsidethesourcevolume,where LZJ E ' 0.Sincewe aremainly interestedin the fields in the far zone,a long distancefromthesourceregion, this is no essentiallimitation.

Assumethat the sourceregion is a limited volumearoundsomecentralpointx0 far away from the field (observation) point x. Under theseassumptions,we


canexpandexpression(8.31)on theprecedingpagetheHertz’ vector, dueto thepresenceof non-vanishingp + ` , 4 x , - in thevicinity of x0, in a formalseries.For thispurposewe recallfrom potentialtheorythat Q x ~ x @

x 0 x , @ ' êë p 0

+ 2ì A 1- ° ë + cosΘ - 9 ë + x ,0 x0 - í 1ë + @x 0 x0

@ - (8.35)

where

Θ is theanglebetweenx ,0 x0 andx 0 x0° ë + cosΘ - is theLegendrepolynomialof order ì9 ë + x ,0 x0 - is thesphericalBesselfunctionof thefirst kind of order ìí 1ë + @x 0 x0

@ - is thesphericalHankel functionof thefirst kind of order ìAccordingto theadditiontheoremfor Legendrepolynomials,wecanwrite° ë + cosΘ - ' ëêî p ~ ë + 0 1- î ° îë + cosÊ - ° ~ îë + cosÊ , - î ðï ~ ï ¯ (8.36)

where ° îë is an associatedLegendre polynomialand,in sphericalpolar coordin-ates,

x ,0 x0' + x ,0 x0 4 Ê ,ñ4 ; , - (8.37a)

x 0 x0' + @ x 0 x0

@ 4 Ê 4 ; - (8.37b)

InsertingEquation(8.35)above,togetherwith Equation(8.36),intoEquation(8.32)on thepreviouspage,we canin a formally exactway expandtheFouriercompon-entof theHertz’ vectoras

Z ' 4<K= 0

êë p 0

ëêî p ~ ë + 0 1- î í 1ë + @x 0 x0

@ - ° îë + cosÊ - î ïa ( ) p + x , - 9 ë + x ,0 x0 - ° ~ îë + cosÊ , - . 3/ , (8.38)

We noticethat thereis no dependenceon x 0 x0 insidethe integral; the integrandis only dependenton therelativesourcevectorx , 0 x0.

We areinterestedin the casewherethe field point is many wavelengthsawayfrom thesources,i.e., whenthefollowing inequalities x ,0 x0 1

@x 0 x0

@(8.39)

hold. Thenwe mayto a goodapproximationreplaceí 1ë with thefirst termin itsasymptoticexpansion:


í 1ë + 0 - ë x ~ x0@

x 0 x0@ (8.40)

andreplace9 ë with thefirst termin its power seriesexpansion:9 ë + x ,0 x0 - 2ë ì !+ 2ì A 1- ! Ö x ,0 x0 × ë (8.41)

Insertingtheseexpansionsinto Equation(8.38)on the facingpage,we obtainthemultipoleexpansionof theFouriercomponentof theHertz’ vector

Z êë p 0

Z ë ¯ (8.42a)

where

Z ë ¯ ' + 0« - ë 14<>= 0

Q x ~ x0@

x 0 x0@ 2

ë ì !+ 2ì - ! ( ) p + x , - + x ,0 x0 - ë ° ë + cosΘ - . 3/ ,(8.42b)

This expressionis approximatelycorrectonly if certaincareis exercised;if manytermsareneededfor anaccurateresult,theexpansionsof thesphericalHankel andBesselfunctionsusedabove maynot beconsistentandmustbereplacedby moreaccurateexpressions.Taking the inverseFourier transformof Z ò will yield theHertz’ vectorin timedomain,which insertedinto Equation(8.33)onpage111willyield C. Theresultingexpressionis thenin turn to beinsertedinto Equation(8.34)onpage111in orderto obtaintheradiationfields.

For a linearsourcedistributionalongthepolaraxis,Θ ' Ê in expression(8.42b)above,and ° ë + cosÊ - givestheangulardistributionof theradiation.In thegeneralcase,however, the angulardistribution must be computedwith the help of For-mula(8.36)on thefacingpage.Let usnow studythelowestordercontributionstotheexpansionof Hertz’ vector.

Electric dipoleradiation

Choosingì ' 0 in expression(8.42b)above,we obtain

Z 0 ' Q x ~ x0

4<K= 0@x 0 x0

@ ( ) p + x , - . 3/ , ' 14<K= 0

x ~ x0@

x 0 x0@ p 0 (8.43)

wherep 0 ' p "ó 1 is the Fourier componentof the electric dipole moment; cf.Equation(7.2)onpage91whichdescribesthestaticdipolemoment.If asphericalcoordinatesystemis chosenwith its polaraxisalongp ó 1, thecomponentsof Z 0


are ôöõ ' ô 0 cosÊ ' 14<>= 0

Q x ~ x0@

x 0 x0@ 1 ó 1 cosÊ (8.44a)ô Ð ' 0 ô 0 sin Ê ' 0 1

4<K= 0

Q x ~ x0@

x 0 x0@ 1 ó 1 sin Ê (8.44b)

ô ï ' 0 (8.44c)

EvaluatingFormula(8.33)on page111for thehelpvectorC, with thespheric-ally polarcomponents(8.44)of Z 0 inserted,weobtain

C 'ø÷ 0"ó ï ˆù ' 14<K= 0

B 1@x 0 x0

@ 0X E Q x ~ x0@

x 0 x0@ 1 ó 1 sin Ê ˆù (8.45)

Applying this to Equation(8.34)on page111,we obtaindirectly theFouriercom-ponentsof thefields:

B ' 0« d 0

4< B 1@x 0 x0

@ 0À E Q x ~ x0@

x 0 x0@ 1 ó 1 sin Ê ˆù (8.46)

E ' 14<K= 0 ? 2 ú 1@

x 0 x0@ 2 0 @

x 0 x0@ û cosÊ x 0 x0@

x 0 x0@ (8.47)A ú 1@

x 0 x0@ 2 0 @

x 0 x0@ 0 2 û sin Ê ˆü H Q x ~ x0

@x 0 x0

@ 1 "ó 1 (8.48)

Keepingonly the partswhich dominateat large distances(radiationfield) andrecallingthat the wave vectork ' + x 0 x0 -_ @

x 0 x0@where ' _ , we can

now write down theFouriercomponentsof theradiationpartsof themagneticandelectricfieldsfrom thedipole:

Brad ' 0 d 0

4< Q x ~ x0@

x 0 x0@ + p ó 1 a k - (8.49a)

Erad ' 0 14<>= 0

Q x ~ x0@

x 0 x0@ q½+ p ó 1 a k - a k r (8.49b)

Thesefieldsconstitutetheelectricdipoleradiation, alsoknown asE1radiation.

Magneticdipoleradiation

Thenext termin theexpression(8.42b)on thepreviouspagefor theexpansionoftheFouriertransformof theHertz’ vectoris for ì ' 1:


Z 1 ' 0« Q x ~ x0

4<>= 0@x 0 x0

@ ( ) x ,0 x0 p + x , - cosΘ . 3/ , (8.50)' 0« 14<K= 0

Q x ~ x0@

x 0 x0@ 2 ( ) q½+ x 0 x0 - J + x ,0 x0 -:r p + x , - . 3/ , (8.51)

Here,theterm q½+ x 0 x0 - J + x , 0 x0 -:r p + x , - canberewrittenq½+ x 0 x0 - J + x ,0 x0 -:r p + x , - ' + / 2 0 /0 ó 2 - + / ,50 /

0 ó 2 - p + x , - (8.52)

andintroducingý 2 ' / 2 0 /0 ó 2 (8.53a)ý ,2 ' / ,2 0 /0 ó 2 (8.53b)

the 9 th componentof theintegrandin Z 1 canbebrokenup into³ q½+ x 0 x0 - J + x ,0 x0 -:r p + x , -gµ 2 ' 12

ý 2 Ö 1 ó 7 ý ,2 A 1 ó 2 ý ,7 × (8.54a)A 12

ý 2 Ö 1 ó 7 ý ,2 0þ1 ó 2 ý ,7 × (8.54b)

i.e., asthesumtwo parts,thefirst beingsymmetricandthesecondantisymmetricin theindices384:9 . Wenotethattheantisymmetricpartcanbewritten as

12

ý 2 Ö 1 ó 7 ý ,2 0ç1 ó 2 ý ,7 × ' 121 ó 7 + ý 2 ý ,2 - 0 ý ,7 + ý 2 1 ó 7 -' 1

2q p + ÿ J ÿ - 0 ÿ + ÿ J p -:r 7' 1

2¶ + x 0 x0 - a q p a + x ,0 x0 -:r · 7 (8.55)

FromEquation(8.28)on page110andthefact thatwe areconsideringa singleFouriercomponent

p + `4 x - ' p ~ (8.56)

whichallowsusto expressp in j as

p ' j ò (8.57)

we canwrite theantisymmetricpartof theintegral in Formula(8.51)above as

12+ x 0 x0 - a ( ) p + x , - a + x ,50 x0 - h ,' 1

2 + x 0 x0 - a ( ) j + x , - a + x ,0 x0 - . 3/ ,' 0 1 + x 0 x0 - a m ò (8.58)

wherewe introducedtheFouriertransformof themagneticdipolemoment


m ò ' 12( ) + x ,0 x0 - a j + x , - . 3/ , (8.59)

The final result is that the antisymmetric,magneticdipole, part of Z 1 canbewritten

Z 1 ó antisym ' 0 4<K= 0 x ~ x0

@x 0 x0

@ 2 + x 0 x0 - a m (8.60)

In analogywith theelectricdipolecase,weinsertthisexpressionintoEquation(8.33)onpage111to evaluateC, with whichEquations(8.34)onpage111thengivestheB andE fields. Discarding,asbefore,all termsbelongingto the nearfields andtransitionfields andkeepingonly the termsthat dominateat large distances,weobtain

Brad + x - ' 0 d0

4< QM x ~ x0@

x 0 x0@ + m a k - a k (8.61a)

Erad + x - ' 4<K= 0

x ~ x0@

x 0 x0@ m a k (8.61b)

whicharethefieldsof themagneticdipoleradiation(M1 radiation).

Electric quadrupoleradiation

ThesymmetricpartZ 1 ó sym of the ì ' 1 contribution in theEquation(8.42b)onpage113 for theexpansionof the Hertz’ vectorcanbe expressedin termsof theelectricquadrupoletensor, which is definedin accordancewith Equation(7.3)onpage91:

Q + ` - ' ( ) + x ,0 x0 - + x ,50 x0 - * + `g4 x , - . 3/ , (8.62)

Again we use this expressionin Equation(8.33) on page111 to calculatethefields via Equations(8.34) on page111. Tedious,but fairly straightforward al-gebra(which we will not presenthere),yields the resultingfields. The radiationcomponentsof thefieldsin thefar field zone(wavezone)aregivenby

Brad + x - ' d 0 8< x ~ x0

@x 0 x0

@ + k J Q - a k (8.63a)

Erad + x - ' 8<K= 0

Q x ~ x0@

x 0 x0@ qs+ k J Q - a k r a k (8.63b)

This typeof radiationis calledelectricquadrupoleradiationor E2radiation.


x

x x u x S

Figure8.2. Signalswhichareobservedat thefield point x weregeneratedatsourcepoints x onasphere,centredonx andexpanding,astime increases,with thevelocity c outward from thecenter. Thesourcechargeelementmoveswith anarbitraryvelocityu andgivesriseto asource“leakage”outof thesource

volume 3 .8.3.3 Radiationfrom chargesmoving in vacuum

The derivation of the radiationfields for the caseof the sourcemoving relativeto the observer is considerablymorecomplicatedthanthe stationarycasesstud-ied above. In order to handlethis non-stationarysituation,we usethe retardedpotentials(3.36)on page39 in Chapter3 !#"

x $ % 14&(' 0

) * + !-,ret"x, $.

x / x, . 0 31 , (8.64a)

A !#"

x $ % 2 0

4& ) * j ! ,

ret"x, $.

x / x, . 0 31 , (8.64b)

andconsiderasourceregionwith suchlimited extentthatthechargesandcurrentsarewell localised.Specifically, we considera charge 3 , , for instanceanelectron,with unspecifiedbut limited spatialextent. Classically, anelectronis a localised,unstructuredchargedistributionwith asmall,finite radius.Thepartof this “chargedistribution” 0 3 , whichweareconsideringis locatedin 054 , % 0 31 , in thesphereinFigure8.2.Sinceweassumethattheelectron(or any otherothercharge)is movingwith a velocity u whosedirectionis arbitraryandwhosemagnitudecanbealmostcomparableto the speedof light, we cannotsay that the charge and current to


beusedin (8.64)is 6 ) * + ` ,ret 4 x , - . 3/ , and 6 ) u * + ` ,ret 4 x , - . 3/ , , respectively, becauseduring the finite time interval that the observed signal is generated,part of thechargedistributionwill “leak” outof thevolumeelement. 3/ , .

Thechargedistributionwhichcontributesto thefield at + `4 x - is locatedat + ` , 4 x , -on a spherewith radius 7 ' @

x 0 x , @ ' + `V0` , - . The radius interval of thisspherefrom which radiationis receivedat thefield pointx duringthetime interval+ `4I` A .` - is + 7 47 A .87 - andthenetamountof chargein this radialinterval is. & , 'G* + `C,ret 4 x , - .¸.87%0 * + ` ,ret 4 x , - + x 0 x , - J u@

x 0 x , @ .Ã.` (8.65)

wherethe last termrepresentstheamountof “sourceleakage”dueto thefact thatthechargedistribution moveswith velocity u. Since .` ' .87 _ and .¸.87 ' . 3/ ,we canrewrite this expressionfor thenetchargeas. & , 'f* + ` ,ret 4 x , - . 3/ ,0 * + ` ,ret 4 x , - + x 0 x , - J u @ x 0 x , @ . 3/ ,'f* + ` ,ret 4 x , - B 1 0 + x 0 x , - J u @ x 0 x , @ E . 3/ , (8.66)

or * + `C,ret 4 x , - . 3/ , ' . & ,1 0 x ~ x ¯ ¬ uÌ x ~ x (8.67)

which leadsto theexpression* + ` ,ret 4 x , -@x 0 x , @ . 3/ , ' . & ,@

x 0 x , @ 0 x ~ x ¯ ¬ uÌ (8.68)

This is theexpressionto beusedin theexpressions(8.64)for theretardedpoten-tials. Theresultis; + `g4 x - ' 1

4<>= 0

( . & ,@x 0 x , @ 0 x ~ x ¯ ¬ uÌ (8.69a)

A + `g4 x - ' d0

4< ( u . & ,@x 0 x , @ 0 x ~ x ¯ ¬ uÌ (8.69b)

For a sufficiently small andwell localisedcharge distribution we can,assumingthat the integrandsdo not changesign in the integration volume, usethe mean


valuetheoremandthefactthat 6 ) . & , 'G& , , evaluatetheseexpressionsto become; + `4 x - ' 14<K= 0

& ,@x 0 x , @ 0 x ~ x ¯ ¬ uÌ ' 1

4<K= 0

& ,9 (8.70a)

A + `4 x - ' 14<K= 0

2

& , u@x 0 x , @ 0 x ~ x ¯ ¬ uÌ ' 1

4<K= 0 2

& , u9 (8.70b)

where9 ' x 0 x , 0 + x 0 x , - J u (8.71)

is theretardedrelativedistance. ThesepotentialsarepreciselytheLiénard-Wiechertpotentialswhichwederivedin Equation(8.70)aboveusingacovariantformalism.It is importantto realisethat in thecomplicatedderivationpresentedheretheob-server is in a coordinatesystemwhichhasan“absolute”meaningandthevelocityu is that of the particle,whereasin the covariantderivation two framesof equalstandingweremoving relativeto eachotherwith u. Expressedin thefour-potential,Equation(5.45)onpage64, theLiénard-Wiechertpotentialsbecome:<; + / = - ' & ,

4<>= 0B 19 4 u 9 E ' + ; 4 A - (8.72)

The Liénard-Wiechertpotentialsare applicablein all problemswherea spa-tially localisedcharge emits electromagneticradiation,and we shall now studysuchemissionproblems.

Uniformlymovingcharges

Generallyspeaking,theLiénard-Wiechertpotentials(8.70),which give riseto ra-diation observed at the field point + `g4 x - , canonly be determinedasfunctionsoftheretardedquantitiesat + ` ,ret 4 x , - ; thereis usuallyno way thatanobserver at time` canobtainany knowledgeaboutwhathashappenedto thechargeafter ` ,ret. Thechargemayevenhaveceasedto exist! However, if thechargemovesin afield-free,isolatedspacewe know that it will not beaffectedby any externalforcesandwillthereforemoveuniformly in astraightline. Thisgivesusthepossibilityto estimate(but not to observe!) its positionat time ` . To computethefieldsfrom a uniformlymoving chargeis thereforerelatively easy.

Assumethatwe have a chargedparticlewhich movesin vacuumwith uniformvelocity u. At time ` , , when the particle passespoint x , , it emits a field signal


>? @BA

x C? @EDFA

xD C G

x H xD GI u

? @BAx0 C u

x H xD

Figure 8.3. Signalswhich were generatedat the sourcepoint JK ¥L x ¥NM are ob-servedat thefield point JK L x M . Theparticle,whichmoveswith constant,uniform

velocityu, hasthenreachedthe“simultaneous”coordinatex0.

which propagateswith speed andarrivesat the observation point at time ` . Inotherwords, x 0 x , ' + ` 0X` , - (8.73)

Under the sametime interval `«0` , , the particle hasmoved in a straight line adistance+ `ö0X` , - Ú ' @

x 0 x , @ Ú (8.74)

to the “simultaneous”position + `4 x0 - . The radiusvectorof this computedpointrelative to thefield point + `g4 x - is

x 0 x0' x 0 x ,0 @

x 0 x , @ u (8.75)

Accordingto Equation(8.71)on the previous page,the squareof the retardedrelativedistance9 is9 2 ' x 0 x , 2 0 2 x 0 x , + x 0 x , - J u A B + x 0 x , - J u E 2

(8.76)

Combiningthiswith theidentity x 0 x , 2 Ú 2 ' q½+ x 0 x , - J u r 2 A q½+ x 0 x , - a u r 2 (8.77)

and,from Formula(8.75),+ x 0 x , - a u ' + x 0 x0 - a u (8.78)

we canexpress9 in termsof the“simultaneous”coordinatex0:


9 ' @x 0 x0

@ 2 0 B + x 0 x0 - a u E 2 ' @x 0 x0

@ Á 1 0 Ú 2 2 sin2 O (8.79)

where O is the anglebetweenu and x 0 x0. Consequently, in the specialcaseof a uniform motion, the retardeddistance9 in the Liénard-Wiechertpotentials(8.70)canbeexpressedin termsof the “simultaneous”coordinate,viz., thepointat which theparticlehasarrivedat the time ` whenwe obtainthefirst knowledgeof its existenceat thesourcepointx , at theretardedtime ` , .

Theelectricfield is calculatedfrom thepotentialsin theusualway:

B + `4 x - ' Lba A + `4 x - (8.80a)

E + `4 x - ' 0VL ; + `4 x - 0 ^ A + `g4 x -^` (8.80b)

Becauseof theuniformity of thevelocityu we cantransformtemporaldifferencesinto spatialdifferences,andvice versa. A stationaryobserver at the observation(field) point x measuresat time ` A .` a field which, at time ` existedat a point adistance0 u .` away from x sothat

A + ` A .`4 x - 0 A + `4 x - ' A + `4 x 0 u .` - 0 A + `g4 x - (8.81)

Thismeansthatpartialtimederivativescanbereplacedbypartialspacederivatives:^^` A Ù 0 u JML A (8.82)

wherewenoticethedyadicproduct.The insertionof the Liénard-Wiechertpotentials(8.70)andthe relation(8.82)

into Formula(8.80b)abovegives

E ' & ,8<K= 0

9 3Þ 1 0 uu 2 JQßL 9 2 (8.83)

With 9 givenby Formula(8.79),this canberewritten as

E ' & ,4<>= 0

9 3 B 1 0 Ú 2 2 E + x 0 x0 -' & ,4<>= 0

1 0QP 2Ì 2@x 0 x0

@ 3 Þ 1 0 P 2Ì 2 sin2 O ß 32

+ x 0 x0 - (8.84)

We seethat E is directedalongthe vector from the “simultaneous”sourcepoint+ `4 x0 - to the field (observation)point + `4 x - . In a similar way, the magneticfieldcanbecalculatedandonefindsthat

B ' d0& ,

4<K= 09 3 B 1 0 Ú 2 2 E u a + x 0 x0 - ' 1 2u a E (8.85)

Fromtheseexplicit formulasfor theE andB fieldswe seethat


1. ÚãÙ 0 R E goesover into theCoulombfield ECoulomb

2. ÚãÙ 0 R B goesover into theBiot-Savart field

3. ÚãÙ R E becomesdependenton O4. ÚãÙ 4 sin O 0 R E Ù + 1 0XÚ 2 _ 2 - ECoulomb

5. ÚãÙ 4 sin O 1 R E Ù + 1 0XÚ 2 _ 2 - ~ 1É 2ECoulomb

Acceleratedcharges

Considera point charge & , andassumethat its trajectoryis known asa functionofretardedtime

x , ' x , + ` , - (8.86)

This meansthatwe know thesourcepointx , at whicha signalis emittedat time ` ,in orderto arrive at thefield point x at time ` . We obtaintheretardedvelocity andaccelerationfrom

u + `C, - ' . x ,.` , (8.87a)

a + ` , - ' u + ` , - ' . u.` , ' . 2x ,.` , 2 (8.87b)

If wechoosethefield pointasfixed,(8.87)yieldsfor therelativevectorx 0 x , :..` , + x 0 x , - ' 0 u + `C, - (8.88a). 2.` , 2 + x 0 x , - ' 0 u + `C, - (8.88b)

Theretardedtime ` , can,at leastin principle,beobtainedfrom theimplicit relation` 0X` , ' @x + ` - 0 x , + ` , - @ (8.89)

The fields aredetermined,asusual,from the potentials,Formulae(8.80) on thepreviouspage.In theseformulaetheunprimedL , i.e., thespatialderivativediffer-entiationoperatorL ' ^^ / 2 ˆÇ 2 (8.90)

meansthat we differentiatewith respectto the coordinatesof x whenwe keep `fixed, andthe unprimedtime derivative operator _ ^` that we differentiatewith


respectto ` while keepingx fixed. But the Liénard-Wiechertpotentials,Equa-tions(8.70)on page119,areexpressedin theu givenby Equation(8.87b)on theprecedingpageandthe retardedrelative distance9 given by Equation(8.71) onpage119.Thismeansthatourpotentials

;andA areexpressedin ` , ! Sincewecan

only know thefateof our sourcesat ` , andearlier, we thereforeneedto expressallouroperatorsin ` , .

We introducethe convention that an index x or ` on the operatorsmeansthatthey shouldbeappliedat constantx and ` , respectively. With this convention,wefind thatB ^^` , E x

@x 0 x , @ ' x 0 x , @ x 0 x , @ J B ^^` , E x

+ x 0 x , - ' 0 + x 0 x , - J u @ x 0 x , @ (8.91)

Furthermore,by applyingtheoperator+ ^ _ ^` - x to Equation(8.89)ontheprecedingpagewefind that

1 0 B ^` ,^` E x

' B ^^` E x

@x 0 x , @ ' O B ^^` , E x

@x 0 x , @ P B ^` ,^` E x' 0 + x 0 x , - J u @ x 0 x , @ B ^` ,^` E x

(8.92)

whichwecanrearrangeintoB ^` ,^` E x

' @x 0 x , @@

x 0 x , @ 0 + x 0 x , - J u _ ' @x 0 x , @9 (8.93)

Here 9 is theretardedrelativedistancegivenby Equation(8.71)onpage119.Mak-ing useof this,we obtainthefollowing operatorrelationB ^^` E x

' B ^` ,^` E xB ^^` , E x

' @x 0 x , @9 B ^^` , E x

(8.94)

By applying + L - to Equation(8.89)on theprecedingpagewe obtain0 + L - ` , ' + L - x 0 x , ' x 0 x ,@x 0 x , @ J + L - + x 0 x , -' x 0 x ,@

x 0 x , @ 0 + x 0 x , - J u@x 0 x , @ + L - `C, (8.95)

from whichweseethat+ L - ` , ' 0 x 0 x , 9 (8.96)

Thisgivesthefollowing operatorrelationwhen + L - is actingonanarbitraryfunc-tion of / and ` , :

124 CHAPTER 8. ELECTROMAGNETIC RADIATION+ L - ' + L - ATS + L - ` ,VU B ^^` , E x

' + L - 0 x 0 x , 9 B ^^` , E x(8.97)

With thehelpof therules(8.97)and(8.94)we areableto replace by ` , in theoperationswhichwe needto perform.Wefind, for instance,thatL ; + L - ; ' L B 1

4<>= 0

& ,9 E' 0 & ,4<>= 0

9 2 O x 0 x ,@x 0 x , @ 0 u 0 x 0 x , 9 B ^^` , E x

9 P (8.98a)^^` A B ^^` E xA ' ^^` B d

0

4< & , u9 E' & ,4<K= 0

2 9 3 O x 0 x , 9 u 0 x 0 x , u B ^^` , E x

9 P (8.98b)

Applying this to thecalculationof theE field with theuseof theLiénard-Wiechertpotentials,Equations(8.70)onpage119,we obtain

E + `g4 x - ' 0UL ; 0 ^ A^`' & ,4<>= 0

9 2 ? + x 0 x , - 0 @x 0 x , @ u _ @

x 0 x , @0 + x 0 x , - 0 @x 0 x , @ u _ 9 B ^^` , E x

9 0 @x 0 x , @ u 2

H (8.99)

In analogywith theuniform motion,we canintroducecertainusefulquantitiesasdepictedin Figure8.4. During thearbitrarymotion,we interpretx 0 x0 astheradiusvectorof thefield point relativeto thevirtual simultaneouspositionx0. Thisis thepositionthechargedparticlewouldhavehadif at ` , all externalforceswouldhave beenswitchedoff sothatthetrajectorywould have beena straightline in thedirectionof the tangentat x , . The particlewould, from that point onwards,havecontinuedwith theconstantvelocityu + ` , - from x , to x0.

In orderto simplify expression(8.99)above further, we usethefactthatB ^^` , E x

9 ' Ú 2 0 + x 0 x , - J u@x 0 x , @ 0 + x 0 x , - J u (8.100)

andfind thatwe canwrite theelectricfield from anarbitrarily moving particleat+ ` , 4 x , - is givenby theexpression


WXYx Z

WXE[FYx[ Z \

x ] x[ \^ u

WXYx0 Z u

WX [ Zx ] x

[Figure8.4. Signalswhichareobservedat thefield point JK L x M weregeneratedatthesourcepoint JKÑ¥ L x ¥ M . After time KÑ¥ theparticle,whichmoveswith nonuniformvelocity, hasfollowed a yet unknown trajectory. Extrapolatingthe trajectoryfrom ( K ¥L x ¥M , basedonthevelocityu JK ¥M , definesthe“virtual simultaneousradius

vector” x0.

E + `4 x - ' & ,4<K= 0

9 3 _ B + x 0 x , - 0 @x 0 x , @ u E B 1 0 Ú 2 2

Ek l m nCoulombfield when `ba 0A x 0 x , 2 a O B + x 0 x , - 0 @

x 0 x , @ u E a uPk l m nRadiationfield

c(8.101)

The first part of the field, the velocityfield tendsto the ordinaryCoulombfieldwhen Ú Ù 0 anddoesnot contributeto theradiation.Thesecondpartof thefield,theacceleration field, is radiatedinto the far zoneandis thereforealsocalledtheradiationfield.

In asimilar waywe cancomputethemagneticfield:

B + `4 x - ' Lba A + L - a A ' + L - a A 0 x 0 x , 9 a B ^^` , E x

A' 0 & ,4<>= 0

2 9 2

x 0 x ,@x 0 x , @ a u 0 x 0 x ,@

x 0 x , @ a B ^^` E xA (8.102)

wherewemadeuseof Equation(8.70)onpage119andFormula(8.94)onpage123.But, accordingto (8.98a),


x 0 x ,@x 0 x , @ a + L - ; ' & ,

4<K= 0 2 9 2

x 0 x ,@x 0 x , @ a u (8.103)

sothat

B + `g4 x - ' x 0 x ,@x 0 x , @ a O 0 + L - ; 0 B ^^` E x

A P' x 0 x ,@x 0 x , @ a E + `4 x - (8.104)

Theradiationpartof theelectricfield is obtainedfrom theaccelerationfield inFormula(8.101)on theprecedingpageas

Erad+ `4 x - ' lim x ~ x Ó E + `4 x -' & ,

4<>= 0 2 9 3

+ x 0 x , - aed O + x 0 x , - 0 @x 0 x , @ u P a u f' & ,

4<>= 0 2 9 3

+ x 0 x , - a q½+ x 0 x0 - a u r (8.105)

wherein thelaststepweusedFormula(8.75)onpage120.

Radiationfor smallvelocities

Whenthechargemovesat low speedsÚ _ 1 andFormula(8.71)on page119

simplifiesto9 ' x 0 x , 0 + x 0 x , - J u x 0 x , 4 Ú (8.106)

andFormula(8.75)onpage120

x 0 x0' x 0 x ,0 @

x 0 x , @ u x 0 x , 4 Ú (8.107)

sothattheradiationfield Equation(8.105)canbeapproximatedby

Erad+ `4 x - ' & ,

4<K= 0 2

@x 0 x , @ 3 + x 0 x , - a q½+ x 0 x , - a u r 4 Ú (8.108)

from whichwe obtain,with theuseof Formula(8.104)above, themagneticfield

Brad+ `4 x - ' & ,

4< 3@x 0 x , @ 2 q u a + x 0 x , -Cr 4 Ú (8.109)

It is interestingto notetheclosecorrespondencewhich existsbetweenthenon-relativistic fields(8.108)and(8.109)andtheelectricdipolefield Equations(8.49)on page114if we introduce


p1'G& , x , + `C, - (8.110)

andat thesametimemake thetransitions& , u ' p1 Ù 0 2p ó 1 (8.111a)

x 0 x , ' x 0 x0 (8.111b)

Thepower flux in thefar zoneis describedby thePoynting vectorasa functionof Erad andBrad. Weusetheclosecorrespondencewith thedipolecaseto find thatit becomes

S ' d0& , 2 + u - 2

16< 2 @ x 0 x , @ 2 sin2 Ê x 0 x ,@x 0 x , @ (8.112)

where Ê is the anglebetweenu andx 0 x0. The total radiatedpower (integratedover aclosedsphericalsurface)becomes° ' d

0& , 2 + u - 26< ' & , 2 ˙Ú 2

6<K= 0 3 (8.113)

which is theLarmor formulafor radiatedpowerfrom anacceleratedcharge.Notethatherewearetreatingachargewith Ú but otherwisetotally unspecifiedmo-tion while we comparewith formulaederived for a stationaryoscillating dipole.Theelectricandmagneticfields,Equation(8.108)ontheprecedingpageandEqua-tion (8.109)on thefacingpage,respectively, andtheexpressionsfor thePoyntingflux andpower derived from them,arehereinstantaneousvalues,dependentonthe instantaneouspositionof thechargeat x , + ` , - . Theangulardistribution is thatwhich is “frozen” to thepoint from which theenergy is radiated.

Bremsstrahlung

An importantspecialcaseof radiationis whenthevelocity u andtheaccelerationu arecollinear(parallelor antiparallel)so thatu a u ' 0. This condition(for anarbitrarymagnitudeof u) insertedinto expression(8.105)on the precedingpagefor theradiationfield, yields

Erad+ `4 x - ' & ,

4<>= 0 2 9 3

+ x 0 x , - a q½+ x 0 x , - a u r 4 u g u (8.114)

from which we obtain,with the useof Formula (8.104)on the facingpage,themagneticfield

Brad+ `4 x - ' & , @ x 0 x , @

4<K= 0 3 9 3

q u a + x 0 x , -:r 4 u g u (8.115)

The differencebetweenthis caseandthe previous caseof Ú is that the ap-proximateexpression(8.106)on the precedingpagefor 9 is not valid; we must


h5i 0 j 5kh5i 0 j 25ku

h5i 0

Figure 8.5. Polar diagram of the energy loss angular distribution factorsin2 lnm J 1 É ` coslnmpo M 5 during bremsstrahlungfor particle speedsrq 0,`sq 0 t 25o , and `bq 0 t 5o .

insteadusethecorrectexpression(8.71)on page119. Theangulardistribution ofthepowerflux (Poyntingvector)thereforebecomes

S u v 0 wyx 2 ˙z 2

16 2 | x ~ x x 2 sin2 1 ~Q cos 6

x ~ x x x ~ x x (8.116)

It is interestingto notethat themagnitudesof theelectricandmagneticfieldsarethesamewhetheru andu areparallelor antiparallel.

We mustbecarefulwhenwe computetheenergy (S integratedover time). ThePoynting vectoris relatedto thetime whenit is measuredandto a fixedsurfacein space.Theradiatedpower into a solid angleelement Ω, measuredrelative totheparticle’s retardedposition,is givenby theformula rad 8 Ω u S x ~ x x x ~ x x Ω u v 0 wnx 2 ˙z 2

16 2 | sin2 1 ~Q cos 6 Ω

(8.117)

On theotherhand,the radiationlossdueto radiationfrom thechargeat retardedtime x : rad8 x Ω u rad8 x x

Ω (8.118)

UsingFormula(8.94)onpage123,we obtain


x 1x 2 u

x x 2 < # Ω

S

Figure8.6. Locationof radiationbetweentwo spheresasthechargemoveswithvelocityu from x 1 to x 2 duringthetime interval y¡ ¢£ .¤¥

rad¤8¦-§ ¤Ω ¨ ¤¥

rad¤8¦ ©ªx « x

§ ª ¤ Ω ¨ S ¬ x « x§ ® © ¤ Ω (8.119)

InsertingEquation(8.116)on the facingpagefor S into (8.119),we obtaintheexplicit expressionfor the energy lossdue to radiationevaluatedat the retardedtime ¤¥

rad ¯ ®¤8¦°§ ¤Ω ¨ ± 0 ² § 2 ˙³ 2

16 2 µ sin2 ¯¶1 «Q· ¸ cos ¹ 5

¤Ω (8.120)

Theangularfactorsof thisexpresssion,for threedifferentparticlespeeds,is plottedin Figure8.5.

Comparingexpression(8.117)on the precedingpagewith expression(8.120),weseethatthey differ by a factor1 « ³ cos º µ whichcomesfrom theextra factor© º ª x « x

§ ªintroducedin (8.119).Let usexplain this in geometricalterms.

Duringtheinterval ¦ §B» ¦ §¼ ¤8¦ § ® andwithin thesolidangleelement¤

Ω theparticleradiatesanenergy ½ ¤¥ rad ¯ ® º ¤8¦ § ¾ ¤8¦ § . As shown in Figure8.6thisenergy is at time¦ §

locatedbetweentwospheres,oneouterwith itsorigin in x§1 ¦ § ® andoneinnerwith

its origin in x§1 ¦ §¼ ¤8¦ § ® ¨ x

§1 ¦ § ® ¼ u

¤8¦andradiusµ ½ ¦ « ¦ §¼ ¤8¦ § ® ¾ ¨ µ ¦ « ¦ § « ¤8¦ § ® .

FromFigure8.6we seethatthevolumeelementsubtendingthesolid angleele-ment ¤

Ω ¨ ¤5¿ªx « x

§2

ª 2 (8.121)

is


3À Á 5ÂÃ8Ä Á x ~ x x2 2 Ω 8Ä (8.122)

Here, 8Ä denotesthedifferentialdistancebetweenthetwo spheresandcanbeeval-uatedin thefollowing way8Ä Á x ~ x x2 ÆÅ | 8 x ~ x ~ x x2 x ~ x x2 uÇ È É Êz cos 8 x ~ x ~ x x2

Á | ~ x ~ x x2 x ~ x x2 u 8 x Á | Ë x ~ x x2 8 x (8.123)

whereFormula(8.71)on page119 wasusedin the last step. Hence,the volumeelementunderconsiderationis 3À Á 5ÂÃ8Ä Á Ë x ~ x x2 Ì Â | 8 x (8.124)

Weseethattheenergywhichis radiatedperunit solidangleduringthetimeinterval xÍ x Å 8 x is locatedin avolumeelementwhosesizeis dependent.Thisexplainsthedifferencebetweenexpression(8.117)on page128andexpression(8.120)onthepreviouspage.

Let the radiatedenergy, integratedover Ω, be denoted˜ rad. After tedious,butrelatively straightforward integrationof Formula (8.120)on the precedingpage,oneobtains ˜ rad8 x Á v 0 wyx 2 ˙z 2

6 | 1Î1 ~ 2 2 Ï 3 (8.125)

If we know u x , we can integrate this expressionover x and obtain the totalenergy radiatedduring the accelerationor decelerationof the particle. This waywe obtain a classicalpicture of bremsstrahlung (braking radiation). Often, anatomistictreatmentis requiredfor anacceptableresult.Ð BREMSSTRAHLUNG AT LOW SPEEDS AND SHORT ACCELERATION TIMESEXAMPLE 8.1Calculatethebremsstrahlungwhena chargedparticle,moving at a non-relativistic speed,is acceleratedor deceleratedduringaninfinitely shorttime interval.

Weapproximatethevelocitychangeat time ÑÒ8ÓÔÑ 0 by adeltafunction:

u ÕÑ ÒÖ Ó ∆u ×ØÕÑ Ò Ù Ñ 0 Ö (8.126)

which meansthat

∆u ÓÛÚÝÜÞ Ü u ßØÑ (8.127)


Also, weassumeà mpoâá 1 sothat,accordingto Formula(8.71)onpage119,ãåäçæ x Ù x Ò æ (8.128)

and,accordingto Formula(8.75)on page120,

x Ù x0ä x Ù x Ò (8.129)

From the generalexpression(8.104)on page126 we concludethat E è B and that itsufficesto consideréëê æErad

æ . Accordingto the “bremsstrahlungexpression”for Erad,Equation(8.114)on page127,éìÓ í Ò sin î

4ïñð 0o 2 æ x Ù x Ò æ ∆ àå×ØÕÑ Ò Ù Ñ 0 Ö (8.130)

In thissimplecaseòóê æBradæ is givenbyòóÓ é o (8.131)

Fouriertransformingexpression(8.130)above for é is trivial, yieldingéõôöÓ í Ò sin î8ï 2 ð 0

o 2 æ x Ù x Ò æ ∆ àø÷ù ôyú 0 (8.132)

We note that the magnitudeof this Fourier componentis independentof û . This is aconsequenceof the infinitely short“impulsive step” ×ØÕÑÒ Ù Ñ 0 Ö in the time domainwhichproducesaninfinite spectrumin thefrequency domain.

Thetotal radiationenergy is givenby theexpression

˜ü rad Ó Ú ß ˜ü radßØÑ Ò ß¢Ñ Ò Ó Ú ÜÞ Ü Ú¢ýöþ E ÿ B0

ß S ßØÑ ÒÓ 10Ú ýåÚ ÜÞ Ü é ò¡ß¢Ñ Ò ß 2 Ó 1

0o Ú¢ýåÚ ÜÞ Ü é 2 ßØÑ Ò ß 2Ó ð 0

o Ú ý Ú ÜÞ Ü é 2 ßØÑ Ò ß 2 (8.133)

Accordingto Parseval’s identity [cf. Equation(8.15)on page107] thefollowing equalityholds: ÚÝÜÞ Ü é 2 ßØÑ Ò Ó 4ï<ÚÝÜ

0

æ éâô æ 2 ßØû (8.134)

which meansthattheradiatedenergy in thefrequency interval Õûû ßØû Ö is

˜ü rad ßØû Ó 4ï5ð 0o þ Ú ý æ éõô æ 2 ß 2 ßØû (8.135)


For our infinite spectrum,Equation(8.132)on thepreviouspage,weobtain

˜ü rad ßØû Ó í Ò 2 Õ ∆ à Ö 216ï 3 ð 0

o 3 Ú ý sin2 îæ x Ù x Ò æ 2 ß 2 ßØûÓ í Ò 2 Õ ∆ à Ö 216ï 3 ð 0

o 3 Ú 20

ß Ú 0

sin2 î sin î ß î ßØûÓ í Ò 23ïñð 0

o þ ∆ ào 2 ßØû2ï (8.136)

We seethat theenergy spectrum˜ü rad is independentof frequency û . This meansthat ifwe integrateit overall frequenciesû 0 Ö , adivergentintegralwould result.

In reality, all spectrahave finite widths, with an uppercutoff limit set by the quantumcondition û Ó 1

2 Õ ∆ à Ö 2 (8.137)

which expressesthat the highestpossiblefrequency in the spectrumis that for which allkinetic energy differencehasgoneinto onesinglefield quantum(photon) with energy

û .If we adoptthepicturethat the total energy is quantisedin termsof ô photonsradiatedduringtheprocess,wefind that

˜ü rad ßØû û Ó ß (8.138)

or, for anelectronwhereí Ò Ó Ù æ ÷ æ , where ÷ is theelementarycharge,ß Ó ÷ 2

4ï5ð 0

o 23ï þ ∆ ào 2 ßØûû ä 1

1372

3ï þ ∆ ào 2 ßØûû (8.139)

whereweusedthevalueof thefinestructureconstant÷ 2 m Õ 4ïñð 0

o Ö ä 1m 137.

Evenif thenumberof photonsbecomesinfinitewhenû 0, thesephotonshavenegligibleenergiessothatthetotal radiatedenergy is finite.

END OF EXAMPLE 8.1 Cyclotron andsynchrotron radiation

Formula(8.104)andFormula(8.105)on page126 for themagneticfield andtheradiationpartof theelectricfield aregeneral,valid for any kind of motionof thelocalisedcharge. A very importantspecialcaseis circular motion, i.e., the caseu u.

With thechargedparticleorbitingin the À planeasin Figure8.7,anorbit radius


x

u

y

!#"%$'&)(0

* + u

",$ &%- x & ("%$ - x ( x . x &

Figure 8.7. Coordinatesystemfor a particle in circular motion with velocityu ÕÑÒ Ö alongthe tangentandconstantaccelerationu ÕÑÒ Ö toward the origin at thesourcepoint ÕÑ Ò x Ò Ö . The / axesarechosensothattherelativefield pointvectorx Ù x Ò makesanangle0 with the 1 axiswhich is normalto theplanof theorbital

motion.Theradiusof theorbit is 2 .3 , andanangularfrequency 4 0, we obtain5 x Á 4 0 x (8.140a)

x x x Á 3 6 ˆ7 cos5 x Å ˆ8 sin 5 x :9 (8.140b)

u x Á x x x Á 3 4 06 ~ ˆ7 sin 5 x Å ˆ8 cos5 x ;9 (8.140c)z Á u Á 3 4 0 (8.140d)

u x Á x x x Á ~ 3 4 206 ˆ7 cos5 x Å ˆ8 sin 5 x ;9 (8.140e)

˙z Á u Á 3 4 20 (8.140f)

Wealsoseethatwecanexpresstherelativevectorx ~ x x as

x ~ x x Á x ~ x x ˆ8 sin < Å ˆ= cos< (8.141)

where < is the anglebetweenx ~ x x and the normalplaneof the particleorbit.


Fromtheseexpressionsweobtain x ~ x x u Á x ~ x x z sin < cos5 (8.142a) x ~ x x u Á ~ x ~ x x ˙z sin < sin 5 (8.142b)

The power flux is given by the Poynting vector, which, with the help of For-mula(8.104)onpage126,canbewritten

S Á 1v 0 E > B Á 1| v 0

E 2 x ~ x x x ~ x x (8.143)

Insertingthis into Equation(8.119)on page129,weobtain rad < Í 5 8 x Á x ~ x x Ë| v 0

E 2 (8.144)

wheretheretardeddistanceË is givenby expression(8.71)on page119. With theradiationpart of the electricfield, expression(8.105)on page126, inserted,andusing(8.142a)and(8.142b)above,oneobtains,aftersomealgebra,that rad < Í 5 8 x Á v 0 w x 2 ˙z 2

16 2 | 1 ~Q sin < cos5 2 ~ Î

1 ~ 2 2 Ï sin2 < sin2 51 ~ sin < cos5 5

(8.145)

The angles and 5 vary in time during the rotation,so that refersto a movingcoordinatesystem.But we canparametrisethesolid angle Ω in theangle 5 andthe(fixed)angle< sothat Ω Á sin < < 5 . Integrationof Equation(8.145)overthis Ω, givesaftersomecumbersomealgebratheangularintegratedexpression ˜ rad8 x Á v 0 w x 2 ˙z 2

6 | 1Î1 ~Q 2 2 Ï 2 (8.146)

In Equation(8.145)above, two limits areparticularlyinteresting:

1. z ? |A@ 1 whichcorrespondsto cyclotronradiation.

2. z ? | B 1 whichcorrespondsto synchrotronradiation.


Cyclotron radiation For anon-relativistic speedz @ | , Equation(8.145)onthefacingpagereducesto rad < Í 5 8 x Á v 0 wyx 2 ˙z 2

16 2 | 1 ~ sin2 < sin2 5 (8.147)

But, accordingto Equation(8.142b)on theprecedingpage

sin2 < sin2 5 Á cos2 (8.148)

where is definedin Figure8.7. This meansthatwe canwrite rad 8 x Á v 0 wyx 2 ˙z 2

16 2 | 1 ~ cos2 Á v 0 wyx 2 ˙z 2

16 2 | sin2 (8.149)

Consequently, a fixedobserver neartheorbit planewill observe cyclotronradi-ationtwiceperrevolution in theform of two equallybroadpulsesof radiationwithalternatingpolarisation.

Synchrotron radiation Whentheparticleis relativistic, z B | , thedenominatorin Equation(8.145)on the facing pagebecomesvery small if sin < cos5DC 1,which definesthe forward directionof the particlemotion ( < C ? 2 Í 5EC 0).Equation(8.145)on theprecedingpagethenbecomes rad ? 2 Í 08 x Á v 0 w x 2 ˙z 2

16 2 | 11 ~Q 3 (8.150)

whichmeansthatanobserverneartheorbit planeseesaverystrongpulsefollowed,half anorbit periodlater, by a muchweaker pulse.

Thetwo casesrepresentedby Equation(8.149)andEquation(8.150)above arevery importantresultssincethey canbeusedto determinethecharacteristicsof theparticlemotion both in particleacceleratorsandin astrophysical objectswhereadirectmeasurementof particlevelocitiesareimpossible.

In theorbit plane( < Á ? 2), Equation(8.145)on thefacingpagegives rad ? 2 Í 5 8 x Á v 0 wyx 2 ˙z 2

16 2 | 1 ~ cos5 2 ~ Î

1 ~ 2 2 Ï sin2 51 ~Q cos5 5

(8.151)

whichvanishesfor angles5 0 which fulfil


x . x &u

y

!#"%$F&)(0

+ "%$ &%- x & (∆

∆

u

"%$ - x (

Figure8.8. Whentheobservationpoint is in theplaneof theparticleorbit, i.e.,0 ÓÔï m 2 thelobewidth is givenby ∆ î .cos5 0

Á z | (8.152a)

sin 5 0ÁHG 1 ~ z 2| 2 (8.152b)

Hence,theangle 5 0 is a measureof thesynchrotron radiation lobewidth ∆ ; seeFigure8.8. For ultra-relativistic particles,definedbyI Á 1J

1 ~ 2 2 K 1 Í G 1 ~ z 2| 2@ 1 Í (8.153)

onecanapproximate50C sin 5 0

ÁEG 1 ~ z 2| 2Á 1I (8.154)

Hence,synchrotronradiationfrom ultra-relativistic chargesis characterizedbya radiationlobewidth which is approximately

∆ C 1I (8.155)

This angularinterval is sweptby thechargeduringthetime interval


∆ x Á ∆ 4 0(8.156)

duringwhich theparticlemovesa lengthinterval

∆ L Á z ∆ x Á z ∆ 4 0(8.157)

in thedirectiontowardtheobserverwhothereforemeasuresapulsewidth of length

∆ Á ∆ x ~ ∆ L| Á ∆ x ~ z ∆ x| Á Î1 ~ z | Ï ∆ x Á Î

1 ~ z | Ï ∆ 4 0C Î1 ~ z | Ï 1I 4 0

Á 1 ~ 1 Å

1 Å z |Ç È É ÊC 2

1I 4 0

C 1 ~ z 2| 2Ç È É Ê1? I 2

12I 4 0

Á 12I 3

14 0(8.158)

As ageneralrule,thespectralwidth of apulseof length∆ is ∆ 4 B 1? ∆ . In theultra-relativistic synchrotroncaseonecanthereforeexpectfrequency componentsup to 4 max

C 1∆ Á 2I 3 4 0 (8.159)

A spectralanalysisof theradiationpulsewill exhibit FouriercomponentsM 4 0 fromM Á 1 up to M C 2I 3.When N electronsare contributing to the radiation,we can discernbetween

threesituations:

1. All electronsarevery closeto eachothersothattheindividual phasediffer-encesarenegligible. Thepowerwill bemultipliedby N 2 relative to asingleelectronandwe talk aboutcoherentradiation.

2. Theelectronsareperfectlyevenly distributedin theorbit. This is thecase,for instance,for electronsin acircularcurrentin aconductor. In thiscasetheradiationfieldscancelcompletelyandno farfieldsaregenerated.

3. Theelectronsareunevenlydistributedin theorbit. Thishappensfor anopenring currentwhich is subjectto fluctuationsof order O N as for all opensystems.As a resultwe get incoherent radiation. Examplesof this canbefoundbothin earthlylaboratoriesandundercosmicconditions.

Radiationin thegeneral case

Werecallthatthegeneralexpressionfor theradiationE field from amoving chargeconcentrationis givenby expression(8.105)onpage126.Thisexpressionin Equa-


tion (8.144)on page134yieldsthegeneralformula rad 8 x Á v 0 wyx 2 x ~ x x 16 2 | Ë 5 P x ~ x x >RQ x ~ x x ~ x ~ x x u| S > u T 2

(8.160)

Integrationover thesolidangleΩ givesthetotally radiatedpoweras ˜ rad8 x Á v 0 w x 2 ˙z 2

6 | 1 ~Q 2 2 sin2 UÎ1 ~Q 2 2 Ï 3 (8.161)

whereU is theanglebetweenu andu.In thelimit u V u, sin U Á 0, which correspondsto bremsstrahlung. For u u,

sin U Á 1, whichcorrespondsto cyclotronradiationor synchrotronradiation.

Theconvectionpotentialandtheconvectionforce

Let usconsiderin moredetailthetreatmentof theradiationfrom auniformly mov-ing rigid charge distribution. We recall from Equation(8.82) on page121 thatin this particularphysical caseof uniform velocity u, time andspacederivativesarecloselyrelatedin thefollowing way whenthey operateon a scalarfunctionof Í x : XW ~ u ZY (8.162)

If wereturnto theoriginaldefinitionof thepotentialsandtheinhomogeneouswaveequation,Formula(3.19)onpage36,for agenericpotentialcomponentΨ Í x anda genericsourcecomponent[ Í x ,\ 2Ψ Í x Á 1| 2 2 2 ~ ] 2 Ψ Í x Á [ Í x (8.163)

we find thatundertheassumptionthatu Á z ˆ7 1, this equationcanbewritten 1 ~ z 2| 2 2Ψ À 21Å 2Ψ À 2

2Å 2Ψ À 2

3

Á ~^[ x (8.164)

i.e., in a time-independentform. Transforming_1Á À

1`1 ~ z 2 ? | 2

(8.165a)_2Á À

2 (8.165b)_3Á À

3 (8.165c)

thetime-independentequation(8.164)reducesto anordinaryPoisson’sequation


] 2a Ψ _ Á ~A[ J 1 ~ z 2 ? | 2_

1 Í _ 2 Í _ 3 (8.166)

whichhasasthewell-known Coulombpotentialsolution

Ψ b Á 14 c d [ b x b ~ b x 3_ x (8.167)

After inversetransformationbackto theoriginal coordinates,thisbecomes

Ψ x Á 14 c d [ x x Ë 3À x (8.168)

where,in thedenominator,Ë Á À 1 ~ À x1 2 Å 1 ~ z 2 ? | 2 6 À 2 ~ À x2 2 Å À 3 ~ À x3 2 9 12 (8.169)

Applying this to theexplicit scalarandvectorpotentialcomponents,realisingthatfor a rigid charge distribution e moving with velocity u, the currentis given byj Á e u weobtainf Í x Á 1

4hg 0 c d e x x Ë 3À x (8.170a)

A Í x Á 14hg 0

| 2 c d u e x x Ë 3À x Á u| 2

f Í x (8.170b)

For a localisedchargewhere i e 3À x Á wnx , theseexpressionsreducetof Í x Á wnx4hg 0

Ë (8.171a)

A Í x Á wyx u4hg 0

| 2 Ë (8.171b)

which we recogniseas the Liénard-Wiechert potentials; cf. Equations(8.70) onpage119. We notice,however, that thederivationhere,basedon a mathematicaltechniquewhich in fact is a Lorentztransformation, is of more generalvaliditythantheoneleadingto Equations(8.70)on page119.

Letusnow considertheactionof thefieldsproducedfrom amoving, rigid chargedistributionrepresentedby wnx moving with velocityu, onachargedparticle w , alsomoving with velocity u. This forceis givenby theLorentzforce

F Á w E Å u > B (8.172)

TheE andB fieldsareobtainedfrom Formulae(8.80)on page121andthepoten-tialsgivenbyEquations(8.170)in thefollowingway, makinguseof Formula(8.162)


on page138:

E Á ~^Y f ~ A Á ~^Y f ~ 1| 2 uf Á ~^Y f Å u| 2 u ZY f Á Î u| jY f Ï u| ~kY f (8.173a)

B Á Yl> A Á Yl> Î u| 2

f Ï Á Y f > u| 2Á ~ u| 2 >mY fÁ u| 2 >on Î u| ZY f Ï u| ~kY fqp Á u| 2 > E (8.173b)

whereweusedthefactthatu > u r 0.This meansthatwe canrewrite expression(8.172)on theprecedingpageas

F Á w n E Å u > Î u| 2 > E Ï p Á w n Î u| ZY f Ï u| ~kY f ~ u| > Î u| >mY f Ï p(8.174)

Applying the“bac-cab”rule,Formula(F.56)on page159,on thelasttermyields

u| > Î u| >Y f Ï Á Î u| ZY f Ï u| ~ z 2| 2 Y f (8.175)

whichmeansthatwe canwrite

F Á ~ w Y U (8.176)

whereU Á 1 ~ z 2| 2 f (8.177)

Thescalarfunction U is calledtheconvectionpotentialor theHeavisidepoten-tial. Whenthe rigid charge distribution is well localisedso that we canusethepotentials(8.171)theconvectionpotentialbecomesU Á 1 ~ z 2| 2 wyx

4hg 0Ë (8.178)

Theconvectionpotentialfrom a point chargeis constanton flattenedellipsoidsofrevolution,definedthroughEquation(8.169)on theprecedingpageass À

1 ~ À x1`1 ~ z 2 ? | 2 t 2 Å À 2 ~ À x2 2 Å À 3 ~ À x3 2Á I 2 À 1 ~ À x1 2 Å À 2 ~ À x2 2 Å À 3 ~ À x3 2 Á Const (8.179)

TheseHeavisideellipsoidsareequipotentialsurfaces,andsincetheforceis propor-tional to thegradientof U , whichmeansit is perpendicularto theellipsoidsurface,


B

u uwv ˆxyz v

|~ˆ

Figure8.9. Theperpendicularfield of amoving charge.

the forcebetweenthe two chargesis in generalnot directedalongthe line whichconnectsthe charges. A consequenceof this is that a systemconsistingof twocomoving chargesconnectedwith a rigid bar, will experiencea torque.This is theideabehindtheTrouton-Nobleexperiment,aimedatmeasuringtheabsolutespeedof the earthor the galaxy. The negative outcomeof this experimentis explainedby thespecialtheoryof relativity whichpostulatesthatmechanicallawsfollow thesamerulesaselectromagneticlaws, sothata compensatingtorqueappearsduetomechanicalstresseswithin thecharge-barsystem.

Virtual photons

Accordingto Formula(8.84)onpage121andFigure8.9 Á Á w x4hg 0

Ë 3 1 ~ z 2| 2 x ~ x0 ˆ=Á w x

4hg 0 I 2 6 z 2 Å 2 ? I 2 9 3 2 (8.180)

which representsa contractedfield, approachingthe field of a planewave. Thepassageof this field “pulse” correspondsto a frequency distribution of the fieldenergy. Fouriertransforming,weobtain Á 1

2 c 8 Á w

4 2 g 0 z Q 4zqI 1 4zqI S(8.181)

Here, 1 is theKelvin function(Besselfunctionof thesecondkind with imaginaryargument)whichbehavesin suchaway for smallandlargeargumentsthat

142 CHAPTER 8. ELECTROMAGNETIC RADIATION w4 2 g 0 z Í 4 @ zI (8.182a)

0 Í 4 K zI (8.182b)

showing thatthe“pulse” lengthis of theorder ? zqI .Dueto theequipartitionof thefield energy into theelectricandmagneticfields,

thetotal field energy canbewritten Á g 0 c d 2 3À x Á g 0 c max

minc 2 z 8 2 (8.183)

wherethe volumeintegrationis over the planeperpendicularto u. With the useof Parseval’s identity for Fourier transforms,Formula(8.15)on page107,we canrewrite thisas Á c 0

Ô 4 Á 4g 0z c max

minc 0

# 2 4 2 C w 2

2 2 g 0z c 0 c min

4 (8.184)

from whichwe concludethat C w 2

2 2 g 0z ln zI min 4 (8.185)

whereanexplicit valueof min canbecalculatedin quantumtheoryonly.As in the caseof bremsstrahlung,it is intriguing to quantisethe energy into

photons[cf. Equation(8.138)onpage132]. Thenwe find thatNXö 4 C 2< ln | I min 4 44 (8.186)

where< Á 2 ? 4g 0 | C 1? 137is thefinestructureconstant.Let us considerthe interactionof two electrons,1 and 2. The result of this

interactionis that they changetheir linear momentafrom p1 to p x1 andp2 to p x2,respectively. Heisenberg’s uncertaintyprinciplegives min

? p1 ~ p x1 sothatthenumberof photonsexchangedin theprocessis of theorderN 4 C 2< ln

Î | I 4 p1 ~ p x1 Ï 44 (8.187)

Sincethischangein momentumcorrespondsto achangein energy 4 Á 1 ~ x1

and

1Á

0I | , weseethatNXö 4 C 2< ln

10| 2

| p1 ~ | p x1 1 ~ x1 44 (8.188)

a formulawhich givesa reasonableaccountof electron-andphoton-inducedpro-cesses.


8.3.4 Radiationfrom chargesmoving in matter

When electromagneticradiationis propagating throughmatter, new phenomenamayappearwhich are(at leastclassically)not presentin vacuum.As mentionedearlier, onecanundercertainsimplifying assumptionsinclude,to someextent,theinfluencefrom matteron the electromagneticfields by introducingnew, derivedfield quantitiesD andH accordingto

D Á g Í x E Á g 0E (8.189)

B Á v Í x H Á m v 0H (8.190)

Expressedin termsof thesederivedfield quantities,theMaxwell equations,oftencalledmacroscopicMaxwellequations, take theformY D Á e Í x (8.191a)Y> E Å B Á 0 (8.191b)Y B Á 0 (8.191c)Yl> H ~ D Á j Í x (8.191d)

Assumingfor simplicity that the electric permittivity g and the magneticper-meability v , andhencetherelativepermittivity andthe relativepermeability m

all have fixedvalues,independenton time andspace,for eachtypeof materialweconsider, we canderive thegeneraltelegrapher’s equation[cf. Equation(2.23)onpage25] 2E 2 ~ v E ~g v 2E 2 Á 0 (8.192)

describing(1D) wave propagationin amaterialmedium.In Chapter2 weconcludedthattheexistenceof afinite conductivity, manifesting

itself in a collisional interaction betweenthe charge carriers,causesthe wavesto decayexponentiallywith time andspace.Let us thereforeassumethat in ourmedium Á 0 sothatthewave equationsimplifiesto 2E 2 ~g v 2E 2 Á 0 (8.193)

If we introducethephasevelocityin themediumas ¡ Á 1O g v Á 1O g 0

m v 0

Á |O ¢ m(8.194)

whereaccordingto Equation(1.9) on page5 | Á 1? O g 0 v 0 is thespeedof light,i.e., thephasespeedof electromagneticwavesin vacuum,thenthegeneralsolutionto eachcomponentof Equation(8.193)

144 CHAPTER 8. ELECTROMAGNETIC RADIATION £ Á [ ~ ¡ Å¥¤ Å ¡ Í ¦ Á 1 Í 2 Í 3 (8.195)

Theratio of thephasespeedin vacuumandin themedium| ¡ Á O ¢ mÁ | O g v defr M (8.196)

is called the refractive index of the medium. In generalM is a function of bothtime andspaceasarethe quantitiesg , v , , and m themselves. If, in addition,themediumis anisotropic or birefringent, all thesequantitiesarerank-two tensorfields.Underoursimplifying assumptions,in eachmediumweconsiderM Á Constfor eachfrequency componentof thefields.

Associatedwith thephasespeedof amediumfor awaveof agivenfrequency 4we haveawavevector, definedas

kdefr¨§ ˆ© Á § ˆª ¡ Á 4 ¡ v ¡ ¡ (8.197)

As in thevacuumcasediscussedin Chapter2, assumingthatE is time-harmonic,i.e., canbe representedby a Fourier componentproportionalto exp « ~¬ 4 , thesolutionof Equation(8.193)canbewritten

E Á E0%® k ¯ x ±° (8.198)

wherenow k is thewave vector in themediumgivenby Equation(8.197)above.With thesedefinitions,thevacuumformulafor theassociatedmagneticfield,Equa-tion (2.30)on page25,

B Á O g v ˆ© > E Á 1 ¡ ˆ© > E Á 14 k > E (8.199)

is valid also in a materialmedium(assuming,as mentioned,that M hasa fixedconstantscalarvalue). A consequenceof a ³²Á 1 is that theelectricfield will, ingeneral,have a longitudinalcomponent.

It is importantto noticethatdependingon theelectricandmagneticpropertiesof amedium,and,hence,on thevalueof therefractive index M , thephasespeedinthemediumcanbesmalleror largerthanthespeedof light: ~¡ Á |M Á 4 § (8.200)

where,in thelaststep,weusedEquation(8.197)above.


If themediumhasa refractive index which,asis usuallythecase,dependentonfrequency 4 , we saythatthemediumis dispersive. Becausein this casealsok 4 and 4 k), sothatthegroupvelocity

gÁ 4 § (8.201)

hasauniquevaluefor eachfrequency component,andis differentfrom ¡ . Exceptin regionsof anomalousdispersion, ¡ is alwayssmallerthan | . In a gasof freecharges,suchasaplasma, therefractive index is givenby theexpressionM 2 ´ Á 1 ~ 4 2

p4 2 (8.202)

where4 2pÁ¶µ· N · w 2

·g 0 · (8.203)

is the plasmafrequency. Here · and N · denotethe massand numberdens-ity, respectively, of charged particle species . In an inhomogeneousplasma,N · Á N · x so that the refractive index andalso the phaseandgroupvelocit-iesarespacedependent.As canbeeasilyseen,for eachgivenfrequency, thephaseandgroupvelocitiesin a plasmaaredifferentfrom eachother. If thefrequency 4is suchthat it coincideswith 4 p at somepoint in the medium,thenat that point ¡ Wl¸ while g W 0 andthewave Fouriercomponentat 4 is reflectedthere.

Vavilov-Cerenkov radiation

As we saw in Subsection8.3.3,a chargein uniform, rectilinearmotion in vacuumdoesnot give rise to any radiation;seein particularEquation(8.84)on page121.Let usnow considerachargein uniform,rectilinearmotionin a mediumwith elec-tric propertieswhicharedifferentfrom thoseof a (classical)vacuum.Specifically,consideramediumwhereg Á Const ¹³g 0 (8.204a)v Á v 0 (8.204b)

This impliesthatin thismediumthephasespeedis ¡ Á |M Á 1O g v 0 º | (8.205)

Hence,in thisparticularmedium,thespeedof propagationof (thephaseplanesof)electromagneticwavesis lessthanthespeedof light in vacuum,whichweknow is


anabsolutelimit for themotionof anything, includingparticles.A mediumof thiskind hasthe interestingpropertythat particles,enteringinto the mediumat highspeeds u , which,of course,arebelow thephasespeedin vacuum, canexperiencethattheparticlespeedsarehigher thanthephasespeedin themedium. This is thebasisfor theVavilov-Cerenkov radiationthatwe shallnow study.

If we recall the generalderivation, in the vacuumcase,of the retarded(andadvanced)potentialsin Chapter3 and the Liénard-Wiechert potentials,Equa-tions (8.70) on page119, we realisethat we obtain the latter in the mediumbya simpleformal replacement| W | ? M in theexpression(8.71)on page119for Ë .Hence,theLiénard-Wiechertpotentialsin a mediumcharacterizedby a refractiveindex M , aref Í x Á 1

4g 0

wyx x ~ x x ~ M ® x x » ° ¯ u Á 14g 0

wyxË (8.206a)

A Í x Á 14g 0

| 2wyx u x ~ x x ~ M ® x x » ° ¯ u Á 1

4hg 0| 2

wyx uË (8.206b)

wherenowË Á x ~ x x ~ M x ~ x x u| (8.207)

Theneedfor theabsolutevalueof theexpressionfor Ë is obviousin thecasewhenz ? |½¼ 1? M becausethen the secondterm can be larger than the first term; ifz ? |³@ 1? M we recover the well-known vacuumcasebut with modified phasespeed.We alsonotethat the retardedandadvancedtimesin the mediumare[cf.Equation(3.34)onpage39] xret

Á xret Í x ~ x x Á ~ § x ~ x x 4 Á ~ x ~ x x M| (8.208a) xadvÁ xadv Í x ~ x x Á Å § x ~ x x 4 Á Å x ~ x x M| (8.208b)

so that the usualtime interval ~ó x betweenthe time measuredat the point ofobservationandtheretardedtime in a mediumbecomes ~ x Á x ~ x x M| (8.209)

For z ? |X¼ 1? M , theretardeddistanceË , andtherforethedenominatorsin Equa-tions(8.206)vanishwhen


u¾c ¿ c

Figure8.10. Instantaneouspictureof the expandingfield spheresfrom a pointchargemoving with constantspeedà mpo À 1m Á in a mediumwhere ÁÀ 1. This

generatesaVavilov-Cerenkov shockwave in theform of a cone.M x ~ x x u| Á x ~ x x M z| cos c Á x ~ x x (8.210)

or, equivalently, when

cos c Á |M z (8.211)

In the directiondefinedby this angle c, the potentialsbecomesingular. Duringthetime interval ~ x givenby expression(8.209)on theprecedingpage,thefieldexistswithin asphereof radius x ~ x x aroundtheparticlewhile theparticlemovesadistanceL Á z ~ x (8.212)

alongthedirectionof u.In thedirection c wherethepotentialsaresingular, all field spheresaretangent

to astraightconewith its apex at theinstantaneouspositionof theparticleandwiththeapex half angle< c definedaccordingto

sin < cÁ cos c Á |M z (8.213)

Thisconeof potentialsingularitiesandfield spherecircumferencespropagateswithspeed| ? M in the form of a shock front, calledVavilov-Cerenkov radiation.1 TheVavilov-Cerenkov coneis similar in natureto theMach conein acoustics.1Thefirst observationof this radiationwasmadeby P. A. Cerenkov in 1934,who wasthena post-graduatestudentin S. I. Vavilov’s researchgroupat theLebedev Institutein Moscow. Vavilov wrote


In orderto make somequantitative estimatesof this radiation,we notethatwecandescribethemotionof eachchargedparticle wnx asacurrentdensity:

j Á w x u Â x x ~ u x Á w x z Â À x ~ z x Â x Â Ã x ˆ7 (8.214)

whichhasthetrivial Fouriertransform

j Á wyx

2 Ä » Â x Â Ã x ˆ7 (8.215)

This Fouriercomponentcanbeusedin theformulaederivedfor a linearcurrentinSubsection8.3.1if only we make thereplacementsg 0 W g Á M 2 g 0 (8.216a)§ W M 4| (8.216b)

In thismanner, usingj from Equation(8.215),theresultingFouriertransformsoftheVavilov-Cerenkov magneticandelectricradiationfieldscanbecalculatedfromtheexpressions(8.4)and(8.5)onpage104,respectively.

The total energy contentis then obtainedfrom Equation(8.15) on page107(integratedover a closedsphereat largedistances).For a Fouriercomponentoneobtains[cf. Equation(8.18)onpage107] radÅ Ω C 1

4hg 0 M | c d j > k k ¯ x » 3À x 2 ΩÁ wyx 2 M 4 2

16 3 g 0| 3

c exp Q±¬ 4 À xz ~k§ À x cos S À x 2 sin2 Ω

(8.217)

where is theanglebetweenthedirectionof motion, ˆ7 x , andthedirectionto theobserver, ˆ© . Theintegral in (8.217)is singularof a “Dirac deltatype.” If we limitthespatialextentof themotionof theparticleto theclosedinterval 6 ~Æ Í Æ 9 onthe

a manuscriptwith theexperimentalfindings,put Cerenkov astheauthor, andsubmittedit to Nature.In themanuscript,Vavilov explainedthe resultsin termsof radioactive particlescreatingComptonelectronswhich gave rise to the radiation(which wasthecorrectinterpretation),but thepaperwasrejected. The paperwas then sentto PhysicalReview and was, after somecontroversy with theAmericaneditorswho claimedthe resultsto be wrong,eventuallypublishedin 1937. In the sameyear, I .E.TammandI. M. Frankpublishedthetheoryfor theeffect (“the singingelectron”).In fact,predictionsof a similar effect hadbeenmadeasearlyas1888by Heaviside,andby Sommerfeldinhis1904paper“Radiatingbodymoving with velocityof light”. OnMay 8, 1937,Sommerfeldsentaletterto Tammvia Austria,sayingthathewassurprisedthathis old 1904ideaswerenow becominginteresting.Tamm,FrankandCerenkov receivedtheNobelPrizein 1958“for thediscoveryandtheinterpretationof theCerenkov effect” [V. L. Ginzburg, privatecommunication].

Ç È axiswecanevaluatetheintegral to obtainÉradÅ#Ê Ω ËÍÌ È 2 M 4 2 sin2 Î

4Ï 3 g 0 Ð 3 sin2 ÑÓÒ 1 ÔÖÕZ×Ø cosÎ Ù Ú Û×ÝÜÑÓÒ 1 Ô½ÕZ×Ø cosÎ ÙÛ ×ÞÜ 2 Ê Ω (8.218)

which hasa maximumin the direction Î c as expected. The magnitudeof thismaximumgrows and its width narrows as Æ ß ¸ . The integrationof (8.218)over Ω thereforepicksup themaincontributionsfrom Î à Î c. Consequently, wecansetsin2 Î à sin2 Î c andtheresultof theintegrationis

˜É radá Ë 2Ï â ã0

Éradá sin Î Ê Î Ë 2Ï â ã

0

ÉradáÊ ä Ô cosÎ åæ ç è éêà Ì È 2 ë ì 2 sin2 Î c

2Ï 2 í0 Ð 3 â 1î 1

sin2 ïñð 1 ò ÕZ×ôóØõ Ú Û×Ýöï÷ð 1 ò ÕZ×ôóØ õ Û × ö 2 Ê ê (8.219)

The integral (8.219) is strongly in peaked nearê ËøÔ Ð ù ä ë ú å , or, equivalently,

nearcosÎ c Ë Ð ù ä ë ú å so we can extend the integration limits to ûXü withoutintroducingtoo mucherror. Via yet anothervariablesubstitutionwe canthereforeapproximate

sin2 cosÎ c â 1î 1

sin2 ï÷ð 1 ò ÕZ×ôóØÝõ Ú Û×Ýöïñð 1 ò ÕZ×ôóØ õ Û × ö 2 Ê ê à ý 1 Ô Ð 2ë 2 ú 2 þ Ð÷ÿì ë â î sin2 ÇÇ 2 Ê ÇË Ð÷ÿ Ïì ë ý 1 Ô Ð 2ë 2 ú 2 þ (8.220)

leadingto the final approximateresult for the total energy loss in the frequencyinterval ä ìÓì ò Ê ì å

˜É radá#Ê ì Ë Ì È 2 ÿ2Ï í 0 Ð 2 ý 1 Ô Ð 2ë 2 ú 2 þ ì Ê ì (8.221)

As mentionedearlier, therefractive index is usuallyfrequency dependent.Real-ising this, we find that theradiationenergy perfrequency unit andper unit lengthis

˜É radá#Ê ì2ÿ Ë Ì È 2 ì

4Ï í 0 Ð 2 ý 1 Ô Ð 2ë 2 ä ì å ú 2 þ Ê ì (8.222)

This resultwasderived underthe assumptionthat ú ù Ð 1ù ë ä ì å , i.e., undertheconditionthattheexpressioninsidetheparenthesesin theright handsideis posit-ive. For all mediait is true that ë ä ì å ß 1 when ì ß ü , so thereexist alwaysa

149


highestfrequency for whichwecanobtainVavilov-Cerenkov radiationfrom afastcharge in a medium.Our derivationabove for a fixedvalueof ë is valid for eachindividualFouriercomponent.

BIBLIOGRAPHY 8

[1] RichardBecker. ElectromagneticFields and Interactions. Dover Publications,Inc.,New York, NY, 1982. ISBN 0-486-64290-9.

[2] Vitaliy Lazarevich Ginzburg. Applicationsof Electrodynamicsin Theoretical PhysicsandAstrophysics. GordonandBreachSciencePublishers,New York, London,Paris,Montreux,Tokyo andMelbourne,Revisedthird edition,1989. ISBN 2-88124-719-9.


[4] JerryB. MarionandMark A. Heald.ClassicalElectromagneticRadiation. AcademicPress,Inc. (London)Ltd., Orlando,. . . , secondedition,1980. ISBN 0-12-472257-1.


[6] JackVanderlinde.ClassicalElectromagneticTheory. JohnWiley & Sons,Inc., NewYork, Chichester, Brisbane,Toronto,andSingapore,1993. ISBN 0-471-57269-1.

151

APPENDIX F

Formulae

F.1 TheElectromagneticField

F.1.1 Maxwell’sequationsD Ë (F.1)B Ë 0 (F.2)E ËoÔ B (F.3)H Ë j ò D (F.4)

Constitutiverelations

D Ë í E (F.5)

H Ë B(F.6)

j Ë E (F.7)

P Ë í 0 E (F.8)

153

154 APPENDIX F. FORMULAE

F.1.2 Fieldsandpotentials

Vectorandscalarpotentials

B Ë A (F.9)

E Ë Ô Ô A (F.10)

Lorentz’gaugeconditionin vacuumA ò 1Ð 2 0 (F.11)

F.1.3 Forceandenergy

Poynting’s vector

S

E

H (F.12)

Maxwell’s stresstensor ò Ô 12

ä "! ! ò# ! ! å (F.13)

F.2 ElectromagneticRadiation

F.2.1 Relationshipbetweenthefield vectorsin aplanewave

B ˆ$ EÐ (F.14)

F.2.2 Thefarfieldsfrom anextendedsourcedistribution

B Û&% rad ä x å Ô(' 0

4Ï )+* !-, x ,.x. â / 021 3 3 4 ) î * k 5 x 0 j Û k (F.15)

E Û&% rad ä x å '4Ï í 0 Ð )+* !-, x ,.

x. ˆ6 â / 0 1 3 3 4 ) î * k 5 x 0 j Û k (F.16)

F.2. ELECTROMAGNETIC RADIATION 155

F.2.3 Thefarfieldsfrom anelectricdipole

B Û7% rad ä x å Ô ì 0

4Ï )+* !-, x ,.x. p Û&% 1 k (F.17)

E Û7% rad ä x å Ô 14Ï í 0

)+* !-, x ,.x. ä p Û&% 1 k å k (F.18)

F.2.4 Thefarfieldsfrom amagneticdipole

B Û7% rad ä x å Ô 0

4Ï )+* !-, x ,.x. ä m Û k å k (F.19)

E Û7% rad ä x å 84Ï í 0 Ð )+* !-, x ,.

x. m Û k (F.20)

F.2.5 Thefarfieldsfrom anelectricquadrupole

B Û7% rad ä x å ' 0ì

8Ï ) * !-, x ,.x. ä k Q Û å k (F.21)

E Û7% rad ä x å '8Ï í 0

) * !-, x ,.x.:9 ä k Q Û å k ; k (F.22)

F.2.6 Thefieldsfrom apoint chargein arbitrarymotion

E ä x å <4Ï í 0 = 3 > R ? ý 1 Ô ú 2Ð 2 þ ò ä x Ô x 4 å R ? uÐ 2 @ (F.23)

B ä x å ä x Ô x 4 å E ä x åÐ . x Ô x 4 . (F.24)= .x Ô x 4 . Ô ä x Ô x 4 å uÐ (F.25)

R ? ä x Ô x 4 å Ô . x Ô x 4 . uÐ (F.26)ý 4 þx

.x Ô x 4 .= (F.27)


F.2.7 Thefieldsfrom apoint chargein uniformmotion

E ä x å <4Ï í 0 = 3 ý 1 Ô ú 2Ð 2 þ R0 (F.28)

B ä x å u

E ä x åÐ 2 (F.29)

= .R0.2 Ô ý R0

uÐ þ 2

(F.30)

R0

x Ô x0 (F.31)

F.3 SpecialRelativity

F.3.1 Metric tensorACB D EFFG 1 0 0 00 Ô 1 0 00 0 Ô 1 00 0 0 Ô 1

H IIJ (F.32)

F.3.2 Covariantandcontravariantfour-vectorsKLB ACB D K D (F.33)

F.3.3 Lorentztransformationof a four-vector3 4 B ΛB D 3 D (F.34)

ΛB D EFFG M Ô M&N 0 0Ô M&N M 0 0

0 0 1 00 0 0 1

H IIJ (F.35)

M 1O1 Ô N 2

(F.36)

N ú Ð (F.37)

F.3. SPECIAL RELATIVITY 157

F.3.4 Invariantline element1 = QP 1 M QP 1SR (F.38)

F.3.5 Four-velocityú B d3d= B ð M M uP õ (F.39)

F.3.6 Four-momentumT B QU0P 2 ú B ä P p å (F.40)

F.3.7 Four-currentdensityV B 0ú B ð uP õ (F.41)

F.3.8 Four-potentialW B ä P A å (F.42)

F.3.9 Field tensorX B D W D 3 B Ô YW B 3 D EFFG 0 Ô Z Ô [ Ô ]\ Z0 Ô P \ P [ [ P \

0 Ô P Z \ Ô P [ P Z0

H IIJ (F.43)


F.4 VectorRelationsLet x be the radius vector (coordinatevector), from the origin to the pointä 3 1

32 3

3å ^ ä 3 `_Yba å andlet

.x.denotethemagnitude(“length”) of x. Let fur-

ther c ä x å N ä x å ededed bearbitraryscalarfieldsanda ä x å b ä x å c ä x å d ä x å ededed arbitraryvectorfields.

Thedifferentialvectoroperator

is in Cartesiancoordinatesgivenby ^ 3f g1

def^ ˆ6 3 def^ih (F.44)

where ˆ6 , j 1 2 3 is the j th unit vectorand ˆ6 1^ ˆ6 , ˆ6 2

^ ˆk , and ˆ6 3^ ˆl . In

component(tensor)notation

canbewrittenm ý 31

32

33þ ý 3 _ a þ (F.45)

F.4.1 Sphericalpolarcoordinates

Basevectors

ˆn sin o cosp ˆ6 ò sin o sin p ˆk ò coso ˆl (F.46)

ˆq coso cosp ˆ6 ò coso sin p ˆksr sin o ˆl (F.47)

ˆt r sin p ˆ6 ò cosp ˆk (F.48)

Directedline element1 3 ˆ6 1 l 1Su ˆn òuv1 o ˆq ò#u 2 sin o 1p ˆt (F.49)

Solidangleelement1 Ω

sin o 1 o 1p (F.50)

Directedareaelement1 23 ˆw 1 S 1x ˆn u 2 1 Ω ˆn (F.51)

Volumeelement1 33 1y 1SuC1x u 2 1Suv1 Ω (F.52)

F.4. VECTOR RELATIONS 159

F.4.2 Vectorformulae

General relations

ab

ba ez |"Qz7

coso (F.53)

a

b r b

a !2z ~!

ˆ6 (F.54)

a ä b cå ä a b å c (F.55)

a ä b cå b ä a cå r c ä a b å (F.56)

a ä b cå ò b

ä c aå ò c ä a b å 0 (F.57)ä a b å ä c d å a

9 b ä c d å ; ä a cå ä b d å r ä a d å ä b cå (F.58)ä a b å ä c d å ä a bd å c r ä a b

cå d (F.59) ä c N å c N ò N c (F.60) ä c aå a

- c òc a (F.61) ä c aå c

a r as c (F.62) ä a b å b

ä aå r a

ä b å (F.63) ä a b å a ä

b å r b ä aå ò ä b - å a r ä a - å b (F.64) ä a b å a

ä b å ò b

ä aå ò ä b - å a ò ä a - å b (F.65)L c m 2 c (F.66)s c 0 (F.67) ä

aå 0 (F.68) ä aå ä

aå r m 2a (F.69)

Specialrelations

In thefollowing k is anarbitraryconstantvector.x

3 (F.70)x

0 (F.71) .x. x.

x. (F.72) ý 1.

x. þ r x.

x.3 (F.73) ý x.

x.3 þ r m 2 ý 1.

x. þ 4 ä x å (F.74) ý k.

x. þ k

> ý 1.x. þ @ r k

x.

x.3 (F.75) >

k ý x.

x.3 þ @ r ý k

x.

x.3 þ if

.x.

0 (F.76)m 2 ý k.x. þ k

m 2 ý 1.x. þ r 4 k

ä x å (F.77) ä k aå k ä aå ò k

ä aå r ä k aå (F.78)

Integral relations

Let y ä x å be the volume boundedby the closedsurface x ä y å . Denotethe 3-dimensionalvolumeelementby 1 33 ä ^ dy å andthesurfaceelement,directedalongtheoutwardpointingsurfacenormalunit vector ˆw , by 1 Sä ^ d23 ˆw å .â / ä

aå 1 33 1 S a (F.79)â / ä c å 1 33 1 S c (F.80)â / ä aå 1 33 1 S

a (F.81)

If x ä å is anopensurfaceboundedby thecontour ä x å , whoseline elementis1 l, then

160

F.4. VECTOR RELATIONS 161 c1 l 1 S s c (F.82)

a 1 l

1 S ä aå (F.83)

BIBLIOGRAPHY F

[1] GeorgeB. ArfkenandHansJ.Weber. MathematicalMethodsfor Physicists. AcademicPress,Inc.,SanDiego,CA . . . , fourth,internationaledition,1995.ISBN 0-12-059816-7.

[2] Philip M. Morse and HermanFeshbach. Methodsof Theoretical Physics. Part I.McGraw-Hill BookCompany, Inc.,New York, NY . . . , 1953. ISBN 07-043316-8.


163

APPENDIX M

MathematicalMethods

M.1 Scalars,VectorsandTensors

Every physical observablecanbe describedby a geometricobject. We will de-scribetheobservablesin classicalelectrodynamicsmathematicallyin termsof scal-ars,pseudoscalars,vectors,pseudovectors,tensorsor pseudotensorsandwill notexploit differentialformsto any significantdegree.

A scalar describesa scalarquantity that may or may not be constantin timeand/orspace. A vector describessomekind of physical motion due to vectionand a tensordescribesthe motion or deformationdue to someform of tension.However, generalisationsto moreabstractnotionsof thesequantitiesarecommon-place. The differencebetweena scalar, vector and tensorand a pseudoscalar,pseudovectoranda pseudotensoris that the latter behave differently undersuchcoordinatetransformationswhichcannotbereducedto purerotations.

Throughoutwe adopttheconventionthatLatin indices j V 8 ededed run over therange1 2 3 to denotevector or tensorcomponentsin the real Euclideanthree-dimensional(3D) configurationspace 3, andGreekindices

ededed , whichareusedin four-dimensional(4D) space,runover therange0 1 2 3.

M.1.1 Vectors

Radiusvector

Any vectorcanberepresentedmathematicallyin severaldifferentways.Onesuit-able representationis in termsof an ordered -tuple, or row vector, of the co-ordinates3 where is thedimensionalityof thespaceunderconsideration.Themostbasicvectoris radiusvectorwhich is thevectorfrom theorigin to thepointof interest. Its -tuple representationsimply enumeratesthe coordinateswhich

165

166 APPENDIX M. MATHEMATICAL METHODS

describethis point. In this sense,the radiusvector from the origin to a point issynonymouswith thecoordinatesof thepoint itself.

In the3D space 3, wehave 3 andtheradiusvectorcanberepresentedby

thetriplet ä 3 1 3

2 3

3å of coordinates3 , j 1 2 3. Thecoordinates3 arescalar

quantitiesthatdescribethepositionalongtheunit basevectors ˆ6 which span 3.Thereforea representationof theradiusvectorin 3 is

x 3f g

1

ˆ6 3 def^ ˆ6 3 (M.1)

wherewehave introducedEinstein’ssummationconvention(EΣ) whichstatesthata repeatedindex in a termimpliessummationover therangeof theindex in ques-tion. Whenever possibleandconvenientwe shall in the following alwaysassumeEΣ andsuppressexplicit summationin our formulae.Typographically, we repres-enta3D vectorby aboldfaceletteror symbolin aRomanfont.

Alternatively, we maydescribethe radiusvectorin componentnotationasfol-lows: 3 def^ ä 3 1

32 3

3å ^ ä 3 `_Yba å (M.2)

Thiscomponentnotationis particularlyusefulin 4D spacewherewecanrepres-enttheradiusvectoreitherin its contravariantcomponentform3 B def^ ä 3 0 3 1 3 2 3 3 å (M.3)

or its covariant componentform3 B def^ ä 3 0 3

1 3

2 3

3å (M.4)

The relationbetweenthe covariantandcontravariant forms is determinedby themetrictensor(alsoknown asthefundamentaltensor) whoseactualform is dictatedby the physics. The dual representationof vectorsin contravariantandcovariantforms is most convenientwhen we work in a non-Euclideanvector spacewithan indefinitemetric. An exampleis Lorentzspace 4 which is a 4D Riemannianspace. 4 is oftenutilisedto formulatethespecialtheoryof relativity.

We notethat for a changeof coordinates3 B ß 3 4 B 3 4 B ä 3 0 3 1 3 2 3 3 å , dueto a transformationfrom a systemΣ to anothersystemΣ 4 , the differential radiusvector 1 3 B transformsas1 3 4 B 3 4 B 3 D 1 3 D (M.5)

which follows trivially from therulesof differentiationof 3 4 B consideredasfunc-tionsof four variables3 D .

M.1. SCALARS, VECTORS AND TENSORS 167

M.1.2 Fields

A field is a physicalentity which dependson oneor morecontinuousparameters.Sucha parametercanbeviewedasa “continuousindex” thatenumeratesthe“co-ordinates”of thefield. In particular, in a field which dependson theusualradiusvectorx of 3, eachpoint in thisspacecanbeconsideredasonedegreeof freedomsothata field is a representationof a physicalentity which hasaninfinite numberof degreesof freedom.

Scalarfields

Wedenoteanarbitraryscalarfield in 3 byc ä x å c ä 3 1 3

2 3

3å def^ c ä 3 å (M.6)

This field describeshow thescalarquantity c variescontinuouslyin 3D 3 space.In 4D, a four-scalarfield is denotedc ä 3 0 3 1 3 2 3 3 å def^ c ä 3 B å (M.7)

which indicatesthat the four-scalar c dependson all four coordinatesspanningthis space.Sincea four-scalarhasthe samevalueat a given point regardlessofcoordinatesystem,it is alsocalledan invariant.

Analogousto thetransformationrule,Equation(M.5) onthefacingpage,for thedifferential 1 3 B , thetransformationrule for thedifferentialoperator

ù 3 B undera transformation3 B ß 3 4 B becomes 3 4 B 3 D 3 4 B 3 D (M.8)

which,again, follows trivially from therulesof differentiation.

Vectorfields

Wecanrepresentanarbitraryvectorfield a ä x å in 3 asfollows:

a ä x å ˆ6 z ä x å (M.9)

In componentnotationthissamevectorcanberepresentedasz ä x å ä z 1 ä x å z 2 ä x å z 3 ä x åå Qz ä 3 å (M.10)

In 4D, an arbitrary four-vector field in contravariant componentform can berepresentedas

168 APPENDIX M. MATHEMATICAL METHODSz B ä 3 D å ä z 0 ä 3 D å z 1 ä 3 D å z 2 ä 3 D å z 3 ä 3 D åå (M.11)

or, in covariant componentform, asz B ä 3 D å ä z 0 ä 3 D å z 1 ä 3 D å z 2 ä 3 D å z 3 ä 3 D åå (M.12)

where 3 D is the radiusfour-vector. Again, the relationbetweenz B

andz B is de-

terminedby themetricof thephysical4D systemunderconsideration.Whetheran arbitrary -tuple fulfils the requirementof being an ( -dimen-

sional)contravariantvectoror not,dependson its transformationpropertiesduringa changeof coordinates.For instance,in 4D anassemblage_ B ä _ 0 `_ 1 `_ 2 `_ 3 åconstitutesacontravariantfour-vector(or thecontravariantcomponentsof a four-vector)if andonly if, duringatransformationfrom asystemΣ with coordinates3 Bto a systemΣ 4 with coordinates3 4 B , it transformsto thenew systemaccordingtotherule_ 4 B 3 4 B 3 D _ D (M.13)

i.e., in thesamewayasthedifferentialcoordinateelement1 3 B transformsaccord-ing to Equation(M.5) onpage166.

Theanalogousrequirementfor acovariant four-vectoris thatit transforms,dur-ing thechangefrom Σ to Σ 4 , accordingto therule_ 4B 3 D 3 4 B _ D (M.14)

i.e., in the sameway asthe differentialoperator ù 3 B transformsaccordingto

Equation(M.8) on theprecedingpage.

Tensorfields

We denoteanarbitrarytensorfield in 3 by A ä x å . This tensorfield canberepres-entedin anumberof ways,for instancein thefollowing matrix form:

A ä x å EG W 11 ä x å W12 ä x å W

13 ä x åW21 ä x å W

22 ä x å W23 ä x åW

31 ä x å W32 ä x å W

33 ä x å HJ def^ W ä 3 ! å (M.15)

where,in thelastmember, wehaveagainusedthemorecompactcomponentnota-tion. Strictly speaking,thetensorfield describedhereis a tensorof rank two.


A particularlysimplerank-two tensorin 3 is the3D Kronecker delta symbol , with thefollowing properties: 0 if j V

1 if j V (M.16)

The3D Kroneckerdeltahasthefollowing matrix representationä å EG 1 0 00 1 00 0 1

HJ (M.17)

Anothercommonandusefultensoris the fully antisymmetrictensorof rank3,alsoknown astheLevi-Civita tensor ! 1 if j V 8 is anevenpermutationof 1,2,3

0 if at leasttwo of j V 8 areequalr 1 if j V 8 is anoddpermutationof 1,2,3(M.18)

with thefollowing furtherproperty !- b ! r ` ! (M.19)

In fact,tensorsmayhave any rank ë . In this picturea scalaris consideredto bea tensorof rank ë 0 anda vectora tensorof rank ë 1. Consequently, thenotationwherea vector(tensor)is representedin its componentform is calledthetensornotation. A tensorof rank ë 2 mayberepresentedby a two-dimensionalarrayor matrixwhereashigherranktensorsarebestrepresentedin theircomponentforms(tensornotation).

In 4D, wehave threeformsof four-tensorfieldsof rank ë . Wespeakof: A contravariantfour-tensorfield, denotedW B

1B

2 B ¡ ä 3 D å , or acovariant four-tensorfield, denotedW B

1B

2 B ¡ ä 3 D å , or amixedfour-tensorfield, denotedW B

1B

2 BC¢BC¢ £1 B ¡ ä 3 D å .

The 4D metric tensor(fundamentaltensor) mentionedabove is a particularlyimportantfour-tensorof rank 2. In covariantcomponentform we shall denoteitACB D . This metric tensordeterminestherelationbetweenanarbitrarycontravariantfour-vector

z Bandits covariantcounterpart

z B accordingto thefollowing rule:z B ä 3 ¤ å def^ A B D z D ä 3 ¤ å (M.20)

This rule is sometimescalledthe“loweringof index.”


Fromtheaboveweseethat ACB D “lowers”oneindex andthat A B D “raises”onein-dex of thetensoronwhichit operates.In particular, the“raisingof index” analogueof the“loweringof index” rule,Equation(M.20) on thepreviouspage,is:z B ä 3 ¤ å def^ A B D z D ä 3 ¤ å (M.21)

More generally, the following “lowering” and“raising” ruleshold for arbitraryrank ë mixedtensorfields:A D ¢BC¢ W B 1

B2 BC¢¥ 1

BC¢B ¢ £1B ¢ £

2 B ¡ ä 3 ¤ å W B 1B

2 BC¢b¥ 1D ¢ B ¢ £1 B ¡ ä 3 ¤ å (M.22)A D ¢ B ¢ W B 1

B2 B ¢¥ 1BC¢bBC¢ £

1 B ¡ ä 3 ¤ å W B 1B

2 B ¢¥ 1D ¢BC¢ £

1BC¢ £

2 B ¡ ä 3 ¤ å (M.23)

Successive loweringandraisingof morethanoneindex is achievedby a repeatedapplicationof this rule. For example,a dualapplicationof theloweringoperationon a rank2 tensorin contravariantform yieldsW B D A B ¤ A§¦ D W ¤ ¦ (M.24)

i.e., the samerank 2 tensorin covariantform. This operationis alsoknown asatensorcontraction.¨ TENSORS IN 3D SPACEEXAMPLE M.1

Considerthetetrahedron-likevolumeelement© indicatedin FigureM.1 onthefacingpageof a solid, fluid, or gaseousbody, whoseatomisticstructureis irrelevant for the presentanalysis.Let ª S «ª 2¬ ˆ in FigureM.1 on thenext pagebea directedsurfaceelementofthis volumeelementandlet thevectorT ˆ® ª 2¬ betheforcethatmatter, lying on thesideofª 2¬ towardwhich theunit normalvector ˆ points,actson matterthat lies on theoppositesideof ª 2¬ . This forceconceptis meaningfulonly if theforcesareshort-rangeenoughthatthey canbe assumedto act only in the surfaceproper. Accordingto Newton’s third law,this surfaceforcefulfils

T ¯ ˆ® « ° T ˆ® (M.25)

Using (M.25) andNewton’s secondlaw, we find that the matterof mass± , which at agiveninstantis locatedin © obeys theequationof motion

T ˆ® ª 2¬ ° T ˆ² 1 ª 2¬ ° T ˆ² 2 ª 2¬ ° T ˆ² 3 ª 2¬³ Fext « ± a (M.26)

whereFext is theexternalforceanda is theaccelerationof thevolumeelement.In otherwords

T ˆ® « ´ 1T ˆ² 1³ ´ 2T ˆ² 2

³ ´ 3T ˆ² 3³ ±ª 2¬µ a ° Fext± ¶ (M.27)


·1

¸¹ 2· ˆº·

3

·2

FigureM.1. Terahedron-like volumeelement» containingmatter.

Sincebotha andFext ¼ ½ remainfinite whereas½ ¼e¾ 2¿ÁÀ 0 as » À 0, onefinds that inthis limit

T ˆÂÄÃ 3Å Æ Ç1 È Æ T ˆÉLÊÌË È Æ T ˆÉCÊ (M.28)

Fromtheabove derivationit is clearthatEquation(M.28) is valid not only in equilibriumbut alsowhenthematterin » is in motion.

IntroducingthenotationÍ ÆÎ ÃÐÏ T ˆÉ ÊÒÑ Î (M.29)

for the Ó th componentof thevectorT ˆÉ Ê , wecanwrite Equation(M.28)abovein componentform asfollowsÍ

ˆÂ Î ÃÐÏ T ˆÂ Ñ Î Ã 3Å Æ Ç1 È Æ Í ÆÔÎ Ë È Æ Í ÆÎ (M.30)

UsingEquation(M.30),wefind thatthecomponentof thevectorT ˆÂ in thedirectionof an


arbitraryunit vector ˆÕ isÖˆ® ˆ× « T ˆ®Ø ˆÕ« 3Ù Ú Û

1

Öˆ® Ú ± Ú « 3Ù Ú Û

1 Ü 3Ù Ý Û1

´ Ý Ö ÝÚÞ ± Úàß ´ Ý Ö ÝÚ ± Ú « ˆ Ø T Ø ˆÕ (M.31)

Hence,the á th componentof the vectorT ˆ²Lâ , heredenotedÖ ÝÚ

, canbe interpretedastheã á th componentof a tensorT. Note thatÖ

ˆ® ˆ× is independentof theparticularcoordinatesystemusedin thederivation.

We shall now show how onecanusethe momentumlaw (force equation)to derive theequationof motion for anarbitraryelementof massin thebody. To this endwe considera part © of thebody. If theexternalforcedensity(forceperunit volume)is denotedby fandthevelocity for amasselementª ± is denotedby v, weobtainªª§äæå ç v ª ± « å ç f ª 3¬³ åéè T ˆ® ª 2¬ (M.32)

The á th componentof thisequationcanbewrittenå ç ªª§ä2ê Ú ª ± « å ç ë Ú ª 3¬³ åìè Ö ˆ® Ú ª 2¬ « å ç ë Ú ª 3¬³ åìè ´ Ý Ö ÝÔÚ ª 2¬ (M.33)

where,in thelaststep,Equation(M.30)onthepreviouspagewasused.Settingª ± «îíæª 3¬andusingthedivergencetheoremon thelastterm,wecanrewrite theresultaså ç í ªªìä2ê Ú ª 3¬ « å ç ë Ú ª 3¬³ å ç ï Ö ÝÚï ¬ Ý ª 3¬ (M.34)

Sincethis formulais valid for any arbitraryvolume,wemustrequirethatí ªªìä ê Ú ° ë Ú ° ï Ö ÝÚï ¬ Ý « 0 (M.35)

or, equivalentlyí ï ê Úï ä ³ í v Øeð êÚ ° ë Ú ° ï Ö ÝÚï ¬ Ý « 0 (M.36)

Note that ï ê Úòñ ï ä is the rateof changewith time of the velocity componentê Ú at a fixedpointx «ôó ¬ 1 õ ¬ 1 õ ¬ 3 ö .

END OF EXAMPLE M.1 ÷


M.1.3 Vectoralgebra

Scalarproduct

Thescalarproduct(dot product, inner product) of two arbitrary3D vectorsa andb in ordinary 3 spaceis thescalarnumber

a ø b ˆ6 z ø ˆ6 2ù" ˆ6 ø ˆ6 ez ù" ez ù"úz (M.37)

wherewe usedthe fact that the scalarproduct ˆ6 ø ˆ6 is a representationof theKroneckerdelta

definedin Equation(M.16) onpage169. In Russianliterature,

thescalarproductis oftendenotedû abü .In 4D spacewedefinethescalarproductof two arbitraryfour-vectors

zþýand

ÿýin thefollowing wayz ý ý ý z ý z þý z ý (M.38)

wherewemadeuseof theindex “lowering” and“raising” rules(M.20)and(M.21).Theresultis a four-scalar, i.e., aninvariantwhich is independenton in which iner-tial systemit is measured.

Thequadratic differential form 2 ý ý ý ý (M.39)

i.e., the scalarproductof the differential radiusfour-vectorwith itself, is an in-variantcalledthemetric. It is alsothesquareof the line element

which is the

distancebetweenneighbouringpointswith coordinates ý

and ý ý

.¨ SCALAR PRODUCT, NORM AND METRIC IN LORENTZ SPACE EXAMPLE M.2In 4 themetrictensorattainsasimpleform [seeEquation(5.7)onpage56for anexample]and,hence,the scalarproductin Equation(M.38) canbe evaluatedalmosttrivially andbecomes

0 õ a 0 õ b 0 0 a b (M.40)

Theimportantscalarproductof the 4 radiusfour-vectorwith itself becomes 0 õ x 0 õ x ! õ x " õ x ! 2 1 2 2 2 3 2 $# 2 (M.41)

which is theindefinite,realnormof 4. The 4 metricis thequadraticdifferentialform% # 2 % % & 2 % 2 % 1 2 % 2 2 % 3 2 (M.42)

END OF EXAMPLE M.2 ÷


' METRIC IN GENERAL RELATIVITYEXAMPLE M.3

In thegeneraltheoryof relativity, severalimportantproblemsaretreatedin a4D sphericalpolarcoordinatesystemwheretheradiusfour-vectorcanbegivenas ! "( ) (!*("+ andthemetrictensoris, .- /01 243 0 0 0

0 2 5 6 0 00 0 ) 2 00 0 0 ) 2 sin2

* 798: (M.43)

where ; ; " "( ) (<*=("+ and > > ! "( ) (!*("+ . In sucha space,themetrictakestheform% # 2 & 2 2 3 % 2 2 6 % ) 2 ) 2 % * 2 ) 2 sin2

* % + 2 (M.44)

In generalrelativity themetrictensoris notgivena priori but is determinedby theEinsteinequations.

END OF EXAMPLE M.3 ?Dyadicproduct

A tensorA û x ü cansometimesberepresentedin thedyadicform A û x ü @ a û x ü b û x ü .Thedyadicnotationwith two juxtaposedvectorsa andb is interpretedasanouterproductandthisdyadis operatedonby anothervectorc “from theright” and“fromtheleft” with ascalar(inner)productin thefollowing two ways:

A ø c def@ ab ø c def@ a û b ø cü (M.45a)

c ø A def@ c ø abdef@ û c ø aü b (M.45b)

thus producingnew vectors,proportionalto a andb. In mathematics,a dyadicproductis oftencalledtensorproductandis frequentlydenoteda A b.

Vectorproduct

Thevectorproductor crossproductof two arbitrary3D vectorsa andb in ordinaryB 3 spaceis thevector

c

a C bED F GIHKJ GML H

ˆN F (M.46)

HereD F GIH

is theLevi-Civita tensordefinedin Equation(M.18) onpage169.Some-timesthevectorproductof a andb is denoteda O b or, particularlyin theRussianliterature, P abQ .


A spatial reversalof the coordinatesystem û R1 S R2 S R3 ü û T 1 S T 2 S T 3 üchangessignof thecomponentsof thevectorsa andb sothat in thenew coordin-atesystema

R T a andbR T b, which is to saythatthedirectionof anordinary

vectoris not dependenton thechoiceof directionsof thecoordinateaxes. On theotherhand,asis seenfrom Equation(M.46) on thefacingpage,thecrossproductvectorc doesnot changesign.Thereforea (or b) is anexampleof a “true” vector,or polar vector, whereasc is anexampleof anaxial vector, or pseudovector.

A prototypefor a pseudovectoris theangularmomentumvectorandhencetheattribute “axial.” Pseudovectorstransformasordinaryvectorsundertranslationsandproperrotations,but reversetheir signrelative to ordinaryvectorsfor any co-ordinatechangeinvolving reflection.Tensors(of any rank)that transformanalog-ously to pseudovectorsarecalledpseudotensors. Scalarsaretensorsof rankzero,andzero-rankpseudotensorsarethereforealsocalledpseudoscalars, an examplebeingthepseudoscalarN F ø û ˆN G C ˆN H ü . This triple productis arepresentationof theUVXW

componentof theLevi-Civita tensorD F GIH

which is a rankthreepseudotensor.

M.1.4 Vectoranalysis

Thedeloperator

InB 3 thedeloperator is adifferential vectoroperator, denotedin Gibb’snotation

by Y anddefinedasY def@ ˆN F ZZ F def@\[ (M.47)

where ˆN F is theUth unit vectorin aCartesiancoordinatesystem.Sincetheoperator

in itself hasvectorialproperties,we denoteit with a boldfacenabla.In “compon-ent” notationwecanwriteZ F ] ZZ 1

S ZZ 2S ZZ 3 ^ (M.48)

In 4D, the contravariantcomponentrepresentationof the four-del operator isdefinedbyZ ý ] ZZ 0

S ZZ 1S ZZ 2

S ZZ 3 ^ (M.49)

whereasthecovariantfour-deloperatorisZ ý ] ZZ 0 S ZZ 1 S ZZ 2 S ZZ 3 ^ (M.50)

Wecanusethisfour-deloperatorto expressthetransformationproperties(M.13)and(M.14) onpage168as


_ R ýa` Z R ýb _ (M.51)

and _ Rý ` Z Rý b _ (M.52)

respectively.' THE FOUR-DEL OPERATOR IN LORENTZ SPACEEXAMPLE M.4In 4 thecontravariantform of thefour-del operatorcanberepresentedasc dfe 1 cc ( gih fe 1 cc ( jkh (M.53)

andthecovariantform asc fe 1 cc ( gih le 1 cc ( jkh (M.54)

Takingthescalarproductof thesetwo, oneobtainsc c 12

c 2c 2 m 2

$n2 (M.55)

which is thed’Alembertoperator, sometimesdenotedn

, andsometimesdefinedwith anoppositesignconvention.

END OF EXAMPLE M.4 ?With thehelpof thedeloperatorwecandefinethegradient,divergenceandcurl

of a tensor(in thegeneralisedsense).

Thegradient

Thegradientof anB 3 scalarfield o û x ü , denotedYpo û ü , is an

B 3 vectorfield a û x ü :Ypo û x ü [qo û x ü ˆN F Z F o û x ü a û x ü (M.56)

Fromthis we seethat theboldfacenotationfor thenablaanddel operatorsis veryhandyasit elucidatesthe3D vectorialpropertyof thegradient.

In 4D, the four-gradient is a covariantvector, formedasa derivative of a four-scalarfield o û ý ü , with thefollowing componentform:Z ý o û ü Z o û üZ ý (M.57)


' GRADIENTS OF SCALAR FUNCTIONS OF RELATIVE DISTANCES IN 3D EXAMPLE M.5

Veryoftenelectrodynamicquantitiesaredependenton therelativedistancein r 3 betweentwo vectorsx andx s , i.e., on t x x sut . In analogywith Equation(M.47) on page175,wecandefinethe“primed” del operatorin thefollowing way:j s ˆv w cc sw g s (M.58)

Usingthis, the“unprimed”version,Equation(M.47) onpage175,andelementaryrulesofdifferentiation,weobtainthefollowing two veryusefulresults:j t x x s t ˆv w c t x x sxtc w x x st x x s t ˆv w c t x x sutc sw j s t x x s t (M.59)j e 1t x x s t h x x st x x s t 3 j s e 1t x x s t h (M.60)

END OF EXAMPLE M.5 ?Thedivergence

Wedefinethe3D divergenceof avectorfield inB 3 asYø a û x ü [ ø ˆN G J G û x ü y F G Z F J G û x ü Z F J F û x ü Z J F û x üZ F o û x ü (M.61)

which, asindicatedby thenotation o û x ü , is a scalarfield inB 3. We maythink of

the divergenceasa scalarproductbetweena vectorialoperatoranda vector. Asis thecasefor any scalarproduct,the resultof a divergenceoperationis a scalar.Againweseethattheboldfacenotationfor the3D deloperatoris veryconvenient.

Thefour-divergenceof a four-vectorJ ý

is thefollowing four-scalar:Z ý Jý û ü Z ýJ ý û ü Z J ý û üZ ý (M.62)


' DIVERGENCE IN 3DEXAMPLE M.6For anarbitrary r 3 vectorfield a

x sz , thefollowing relationholds:j s e a

x st x x s t h j s a x s|t x x s t& a

x s j s e 1t x x s t h (M.63)

which demonstrateshow the “primed” divergence,definedin termsof the “primed” deloperatorin Equation(M.58) on theprecedingpage,works.

END OF EXAMPLE M.6 ?TheLaplacian

The3D Laplaceoperator or Laplaciancanbedescribedasthedivergenceof thegradientoperator:~ 2 ∆

Yø.Y ZZ F ˆN F ø ˆN GZZ G y F G Z F Z G Z 2F Z 2Z 2F @ 3 F 1

Z 2Z 2F(M.64)

Thesymbol~ 2 is sometimesreaddelsquared. If, for ascalarfield o û x ü , ~ 2 o 0

at somepoint in 3D space,it is asignof concentrationof o at thatpoint.' THE LAPLACIAN AND THE DIRAC DELTAEXAMPLE M.7A veryusefulformulain 3D r 3 isj j e 1t x x s t h m 2

e 1t x x s t h 4ï x x s (M.65)

where x x sz is the3D Diracdelta“function.”

END OF EXAMPLE M.7 ?Thecurl

InB 3 thecurl of a vectorfield a û x ü , denotedYC a û x ü , is another

B 3 vectorfieldb û x ü whichcanbedefinedin thefollowing way:YC a û x ü D F GIH ˆN F Z G JH û x ü D F GIH ˆN F Z JH û x üZ G

b û x ü (M.66)

whereusewasmadeof theLevi-Civita tensor, introducedin Equation(M.18) onpage169.

Thecovariant4D generalisationof thecurl of a four-vectorfieldJ ý û ü is the

antisymmetricfour-tensorfield ý û ü Z ý J û ü T Z J ý û ü T ý û ü (M.67)

A vectorwith vanishingcurl is saidto be irr otational.


' THE CURL OF A GRADIENT EXAMPLE M.8

Usingthedefinitionof the r 3 curl, Equation(M.66) on thefacingpage,andthegradient,Equation(M.56) onpage176,weseethatj jk x w! ˆv w c c x (M.68)

which,dueto theassumedwell-behavednessof x , vanishes: w! ˆv w c c x w! cc cc x ˆv w e c 2c 2c

3 c 2c

3c

2h x ˆv 1 e c 2c

3c

1 c 2c

1c

3h x ˆv 2 e c 2c

1c

2 c 2c

2c

1h x ˆv 3 0 (M.69)

We thusfind thatj jk x 0 (M.70)

for any arbitrary, well-behaved r 3 scalarfield x .In 4D we notethatfor any well-behavedfour-scalarfield 3 c c - c - c 3 0 (M.71)

sothatthefour-curl of a four-gradientvanishesjust asdoesacurl of agradientin r 3.

Hence,a gradientis alwaysirrotational.

END OF EXAMPLE M.8 ?' THE DIVERGENCE OF A CURL EXAMPLE M.9

With theuseof thedefinitionsof thedivergence(M.61) andthecurl, Equation(M.66) onthefacingpage,wefind thatj j a

x c w ja a

x w w! c w c x (M.72)

Usingthedefinitionfor theLevi-Civita symbol,definedby Equation(M.18) on page169,


we find that,dueto theassumedwell-behavednessof a ,c w w! c x cc w w! cc e c 2c

2c

3 c 2c

3c

2h 1

x

e c 2c 3c

1 c 2c

1c

3h 2

x

e c 2c 1c

2 c 2c

2c

1h 3

x 0 (M.73)

i.e., thatj j ax 0 (M.74)

for any arbitrary, well-behaved r 3 vectorfield ax .

In 4D, thefour-divergenceof thefour-curl is not zero,forc -M .- c c - - 3 n 2 3 0 (M.75)

END OF EXAMPLE M.9 ?Numerousvectoralgebraandvectoranalysisformulaearegiven in ChapterF.

Thosethat arenot found therecanoften be easilyderived by usingthe compon-ent formsof thevectorsandtensors,togetherwith theKronecker andLevi-Civitatensorsandtheir generalisationsto higherranks.A shortbut very usefulreferencein this respectis thearticleby A. Evett [3].

M.2 AnalyticalMechanics

M.2.1 Lagrange’sequations

As is well known from elementaryanalyticalmechanics,theLagrange functionorLagrangian is givenby û F S ˙ F S" ü ] F S F S" ^ ¢¡ T¤£ (M.76)

where F is the generalisedcoordinate,¡

the kinetic energy and £ the potentialenergyof amechanicalsystem,TheLagrangiansatisfiestheLagrangeequations

ZZ ] Z Z ˙ F ^ T Z Z F 0 (M.77)

Wedefinetheto thegeneralisedcoordinate F canonicallyconjugatemomentum¥ F accordingto¥ F Z Z ˙ F (M.78)

andnotefrom Equation(M.77) thatZ Z F ˙¥ F (M.79)

M.2.2 Hamilton’sequations

From , theHamiltonian(Hamilton function) ¦ canbedefinedvia theLegendretransformation¦ û ¥ F S F S" ü ¥ F ˙ F T û F S ˙ F S" ü (M.80)

After differentiatingtheleft andright handsidesof thisdefinitionandsettingthemequalwe obtainZ ¦Z ¥ F ¥ F Z ¦Z F F Z ¦Z ˙ F ¥ F ¥ F ˙ F T Z Z F F T Z Z ˙ F ˙ F T Z Z

(M.81)

Accordingto the definition of ¥ F , Equation(M.78) above, the secondandfourthtermson the right handsidecancel.Furthermore,noting thataccordingto Equa-tion (M.79) thethird termontheright handsideof Equation(M.81) above is equalto T ˙¥ F F andidentifying terms,we obtaintheHamiltonequations:Z ¦Z ¥ F ˙ F F (M.82a)Z ¦Z F T ˙¥ F T ¥ F (M.82b)

181

BIBLIOGRAPHY M

[1] GeorgeB. ArfkenandHansJ.Weber. MathematicalMethodsfor Physicists. AcademicPress,Inc.,SanDiego,CA . . . , fourth,internationaledition,1995.ISBN 0-12-059816-7.

[2] R. A. Dean. Elementsof Abstract Algebra. Wiley & Sons,Inc., New York, NY . . . ,1967. ISBN 0-471-20452-8.

[3] Arthur A. Evett. Permutationsymbolapproachto elementaryvectoranalysis.Amer-ican Journalof Physics, 34,1965.

[4] Philip M. Morse and HermanFeshbach. Methodsof Theoretical Physics. Part I.McGraw-Hill BookCompany, Inc.,New York, NY . . . , 1953. ISBN 07-043316-8.

[5] Barry Spain. TensorCalculus. Oliver andBoyd, Ltd., Edinburgh andLondon,thirdedition,1965. ISBN 05-001331-9.

183

INDEX

accelerationfield, 125advancedtime,38Ampère’s law, 5Ampère-turndensity, 95anisotropic,144anomalousdispersion,145antisymmetrictensor, 67associatedLegendrepolynomial,112associative,59axial gauge,36axial vector, 67,175

Biot-Savart’s law, 6birefringent,144brakingradiation,130bremsstrahlung,130,138

canonicallyconjugatefour-momentum,76

canonicallyconjugatemomentum,76,181

canonicallyconjugatemomentumdens-ity, 83

characteristicimpedance,23chargedensity, 4classicalelectrodynamics,8closedalgebraicstructure,59coherentradiation,137collisionalinteraction,143complex notation,27

componentnotation,166concentration,178conservativefield, 11conservative forces,81constitutiverelations,14contraction,56contravariantcomponentform,56,166contravariantfield tensor, 68contravariantfour-tensorfield, 169contravariantfour-vector, 168contravariantfour-vectorfield, 58contravariantvector, 56convectionpotential,140convectivederivative,12cosineintegral,110Coulombgauge,36Coulomb’s law, 2covariant,53covariantcomponentform, 166covariantfield tensor, 68covariantfour-tensorfield, 169covariantfour-vector, 168covariantfour-vectorfield, 59covariantvector, 56crossproduct,174curl, 178cutoff, 132cyclotronradiation,134,138

185

186 INDEX

d’Alembertoperator, 34,63,176del operator, 175del squared,178differentialdistance,57differentialvectoroperator, 175Dirac delta,178Dirac-Maxwellequations,15dispersive,145displacementcurrent,10divergence,177dotproduct,173duality transformation,16dummyindex, 56dyadicform, 174

E1 radiation,114E2 radiation,116Einsteinequations,174Einstein’ssummationconvention,166electricchargeconservationlaw, 9electricconductivity, 10electricdipolemoment,113electricdipolemomentvector, 91electricdipoleradiation,114electricdisplacement,14electricdisplacementvector, 93electricfield, 3electricfield energy, 96electricmonopolemoment,91electricpermittivity, 143electricpolarisation,92electricquadrupolemomenttensor, 91electricquadrupoleradiation,116electricquadrupoletensor, 116electricsusceptibility, 93electricvolumeforce,98electromagneticfield tensor, 68electromagneticpotentials,32electromagneticscalarpotential,33electromagneticvectorpotential,33

electromagnetodynamicequations,15electromagnetodynamics,16electromotive force(EMF), 11electrostaticscalarpotential,31electrostatics,1energy theoremin Maxwell’s theory,

96equationof continuity, 9, 64equationof continuity for magnetic

monopoles,15equationsof classicalelectrostatics,8equationsof classicalmagnetostatics,

8Euclideanspace,60Euclideanvectorspace,57Euler-Lagrangeequation,82Euler-Lagrangeequations,83Euler-Mascheroniconstant,110event,60

farfield, 46farzone,103Faraday’s law, 11field, 167field Lagrangedensity, 84field point,3field quantum,132finestructureconstant,132,142four-current,63four-deloperator, 175four-dimensionalHamiltonequations,

77four-dimensionalvectorspace,55four-divergence,177four-gradient,176four-Hamiltonian,76four-Lagrangian,74four-momentum,62four-potential,64four-scalar, 167

INDEX 187

four-tensorfields,169four-vector, 58,167four-velocity, 62Fouriercomponent,22Fouriertransform,36functionalderivative,82fundamentaltensor, 56,166,169

Galileo’s law, 53gaugefixing, 36gaugefunction,35gaugeinvariant,35gaugetransformation,35Gauss’s law, 4generalisefour-coordinate,76generalisedcoordinate,76,180Gibb’snotation,175gradient,176Green’s function,37grouptheory, 59groupvelocity, 145

Hamiltondensity, 83Hamiltondensityequations,84Hamiltonequations,76,181Hamiltonfunction,181Hamiltongauge,36Hamiltonian,181Heavisidepotential,140Helmholtz’ theorem,34helpvector, 111Hertz’ method,110Hertz’ vector, 111homogeneouswave equation,21,22Huygen’sprinciple,37

identityelement,59in amedium,146incoherentradiation,137indefinitenorm,57inductionfield, 46

inertial referenceframe,53inertial system,53inhomogeneousHelmholtzequation,

37inhomogeneoustime-independentwave

equation,37inhomogeneouswave equation,36innerproduct,173instantaneous,127interactionLagrangedensity, 84intermediatefield, 49invariant,167invariantline element,57inverseelement,59irrotational,4, 178

Kelvin function,141kineticenergy, 81,180Kroneckerdelta,169

Lagrangedensity, 81Lagrangeequations,180Lagrangefunction,80,180Lagrangian,80,180Laplaceoperator, 178Laplacian,178Larmor formula for radiatedpower,

127law of inertia,53Legendrepolynomial,112Legendretransformation,181Levi-Civita tensor, 169Liénard-Wiechertpotentials,67,119,

139light cone,58light-like interval, 58line element,173linearmassdensity, 81linearlypolarisedwave,25longitudinalcomponent,24

188 INDEX

Lorentzboostparameter, 61Lorentzequations,34Lorentzforce,13,97,139Lorentzgauge,36Lorentzgaugecondition,34,64Lorentzspace,57,166Lorentztransformation,55,139

M1 radiation,116Machcone,147macroscopicMaxwell equations,143magneticchargedensity, 15magneticcurrentdensity, 15magneticdipolemoment,94,115magneticdipoleradiation,116magneticfield, 6magneticfield energy, 96magneticfield intensity, 95magneticflux, 11magneticflux density, 6magneticinduction,6magneticmonopoles,15magneticpermeability, 143magneticsusceptibility, 95magnetisation,94magnetisationcurrents,94magnetisingfield, 14,95magnetostaticvectorpotential,32magnetostatics,5massivephotons,88mathematicalgroup,59matrix form, 168Maxwell stresstensor, 98Maxwell’smacroscopicequations,15,

95Maxwell’smicroscopicequations,14Maxwell-Lorentzequations,14mechanicalLagrangedensity, 84metric,166,173metrictensor, 56,166,169

Minkowski equation,76Minkowski space,60mixedfour-tensorfield, 169mixing angle,16momentumtheoremin Maxwell’sthe-

ory, 98multipoleexpansion,110,113

nearzone,49Newton’sfirst law, 53non-Euclideanspace,57non-lineareffects,10norm,57,173null vector, 58

observationpoint,3Ohm’s law, 10one-dimensionalwave equation,25outerproduct,174

Parseval’s identity, 107,131,142phasevelocity, 143photon,132physicalmeasurable,27planepolarisedwave,25plasma,145plasmafrequency, 145Poisson’sequation,138Poissons’equation,31polarvector, 67,175polarisationcharges,93polarisationcurrents,94polarisationpotential,111polarisationvector, 111positivedefinite,60positivedefinitenorm,57potentialenergy, 81,180potentialtheory, 112powerflux, 96Poyntingvector, 96Poynting’s theorem,96

INDEX 189

ProcaLagrangian,88propagator, 37propertime,58pseudoscalar, 165pseudoscalars,175pseudotensor, 165pseudotensors,175pseudovector, 67,165,175

quadraticdifferentialform, 57,173quantummechanicalnonlinearity, 3

radiationfield, 46,49,125radiationfields,103radiationgauge,36radiationresistance,109radiusfour-vector, 55radiusvector, 165rank,168rapidity, 61refractive index, 144relativeelectricpermittivity, 98relativemagneticpermeability, 98relativepermeability, 143relativepermittivity, 143Relativity principle,53relaxationtime,22restmassdensity, 84retardedCoulombfield, 49retardedpotentials,39retardedrelativedistance,119retardedtime,38Riemannianmetric,57Riemannianspace,56,166row vector, 165

scalar, 165,177scalarfield, 59,167scalarproduct,173shockfront, 147signature,56

skew-symmetric,68skindepth,27sourcepoint,3spacecomponents,57space-like interval, 58space-time,57specialtheoryof relativity, 53sphericalBesselfunction of the first

kind, 112sphericalHankel functionof thefirst

kind, 112sphericalwaves,105super-potential,111synchrotronradiation,134,138synchrotronradiationlobewidth,136

telegrapher’sequation,25,143temporaldispersivemedia,10temporalgauge,36tensor, 165tensorcontraction,170tensorfield, 168tensornotation,169tensorproduct,174three-dimensionalfunctionalderivat-

ive,83timecomponent,57time-harmonicwave,22time-independentdiffusionequation,

23time-independenttelegrapher’sequa-

tion, 26time-independentwaveequation,23time-like interval, 58total charge,91transversecomponents,24transversegauge,36

vacuumpermeability, 5vacuumpermittivity, 2

190 INDEX

vacuumpolarisationeffects,3vacuumwave number, 23Vavilov-Cerenkov radiation,146,147vector, 165vectorproduct,174velocity field, 125virtual simultaneousradiusvector, 125

wave vector, 25,144world line, 60

Young’smodulus,81Yukawa mesonfield, 88

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