Table Of ContentLecture Notes in Physics
Arnab Rai Choudhuri
Advanced
Electromagnetic
Theory
Lecture Notes in Physics
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Volume 1009
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Arnab Rai Choudhuri
Advanced Electromagnetic
Theory
ArnabRaiChoudhuri
DepartmentofPhysics
IndianInstituteofScience
Bangalore,India
ISSN 0075-8450 ISSN 1616-6361 (electronic)
LectureNotesinPhysics
ISBN 978-981-19-5943-1 ISBN 978-981-19-5944-8 (eBook)
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To
OpuandTipu
Preface
Thisbookisbasedonaone-semestercourseonelectromagnetictheoryatadvanced
undergraduateorbeginninggraduatelevelwhichIhavetaughtattheIndianInstitute
ofScienceseveraltimes.Aprerequisiteforthiscourseisamoreelementarycourseat
thelevelofPurcell’sElectricityandMagnetism,teachingstudentshowtoformulate
thebasicprinciplesofelectromagnetismthroughvectorialequations.Theadvanced
course on which this book is based is the last course on classical electrodynamics
takenbyourgraduatestudentsbeforetheyembarkonresearchindifferentbranches
ofphysics.Theaimofthiscourseistoexposestudentstoalltheimportantbasicprin-
ciplesofthesubjectwhichprofessionalphysicistsworkinginanyareaoftheoretical
orexperimentalphysicsareexpectedtoknow.
Teaching a course on electromagnetic theory at the level of this book is quite
challenging due to the wide diversity of topics which are expected to be covered.
Ontheonehand,therearecertainpracticaltopicswhichhaveextensiveengineering
applications,suchasthetheoryofwaveguidesandantennae.Ontheotherhand,one
hastocoverothertopicswhicharegatewaystoadvancedtheoreticalphysics,suchas
therelationofelectromagnetictheorywithspecialrelativityandtheactionprinciple
formulation in a Lorentz-invariant manner. Additionally, there are topics like the
calculation of electromagnetic fields from the Liénard–Wiechert potentials, which
involve mathematical derivations more complicated than probably any derivations
whichstudentsatthislevelarelikelytoencounterinotherbasicphysicscourses.I
have tried my best to write a balanced book which should be suitable for students
goingintodifferentbranchesoftheoreticalorexperimentalphysics.Imaybeslightly
biasedbymypersonalresearchinterestinplasmaastrophysics,butIbelievethatI
have kept this bias within acceptable limits by adding only a short chapter on the
basicsofplasmaphysicsattheendofthisbook.Apartfromthegrowingimportance
of plasma physics, I firmly believe that even from a conceptual point of view it is
desirablethatallphysicsstudentsshouldhavesomeideaaboutthebasicsofplasma
physics—themany-bodytheoryofclassicalelectrodynamics.
Althoughmostofthebasicprinciplesofelectromagnetictheorycoveredinabook
likethiswereestablishedmorethanacenturyagoandhavenotchangedsincethen,
therehavebeenmanychangesinthelastfewdecadesinthewaytheseprinciplesare
vii
viii Preface
taughtattheadvancedundergraduateorthebeginninggraduatelevel.WhenIwas
goingtograduateschoolinthe1980s,itwascustomaryformanyuniversitiesaround
the world (especially American universities) to teach electromagnetic theory in a
leisurelymanneroverayear(intwosemestersorthreequarters).Veryoften,there
would not be a separate course on mathematical methods of physics, and students
wereexpectedtobetaughtatleastsomebasictechniquesofmathematicalmethodsin
theelectromagnetictheorycourse.Onetextbookwhichheldundisputedswayinthat
erawasJackson’sClassicalElectrodynamics.Ibelongtothegenerationofphysicists
whowere‘raised’onJackson’sbook.Withtremendousadvancesindifferentareas
ofphysicsoverthelastfewdecades,ithasnowbecomeimperativetogivestudents
some exposure to these modern topics at a stage sufficiently early in their career.
Thiscanonlybedonebycuttingoutsomepartsoftheoldercurriculum.Whileany
competentphysicistwouldagreethatitisnotpossibletobecomeagoodphysicist
withoutagoodcommandoftheolderclassicalareasofphysics,itisalsogenerally
agreedthatperhapsnoteverytopicintheseolderareasofphysicswhichusedtobe
taughttoallphysicsstudentsafewdecadesagoneedstobetaughttoday.Onecan
prunetheoldercurriculumtoretainonlytheessentialswhicheveryphysicsstudent
oughttoknow.Manyuniversitiesaroundtheworldnowofferaone-semestercourse
ofelectromagnetictheoryattheadvancedundergraduateorthebeginninggraduate
level,ratherthanayear-longsequenceofcourses.
Manyoftheoldertextbooksofelectromagnetictheorywerewrittenwiththeaimof
teachingthesubjectoverayear(thethirdeditionofJackson’sbookrunstoabout800
pages).Incontrast,myaimhasbeentowritealeanandthinadvancedelectromagnetic
theorytextbookfromamodernperspectivewhichcoversalltheimportanttopicsthat
aprofessionalphysicistneedstoknowandwhichcanbecoveredcomfortablyina
one-semestercourse.Therebeingmanytextbookswithamorecompletecoverageof
thesubject,Ihavenotfelttheneedtostriveforcompleteness.Thisbookessentially
coversthematerialswhichIwouldmanagetoteachinaone-semestercourseatthe
IndianInstituteofScience,lecturingthreehoursperweek.AlthoughIamconsidered
a reasonably fast teacher, I trust that a teacher proceeding at a moderate pace will
beabletocoveralargepartofthisbook.Whiledesigningthiscourse,Ihadtobe
ruthlesslyselectiveindecidingwhichtopicsofelectromagnetictheoryaprofessional
physicistmustknowfromthelastcourseonthesubject.Althoughnoteverybodycan
beexpectedtobein100%agreementwithme,Idobelievethatanyprofessorwhois
compelledtoteachtheessentialsofadvancedelectromagnetictheoryinonesemester
inabalancedcoursesuitableforallphysicsstudentswillhavetoendupwithacourse
notverydifferentfromwhatispresentedinthisbook.Whiledecidingwhichtopics
toincludeandwhichtopicstoleaveout,Ihadoneguidingprinciple.Igaveahigher
prioritytotopicswhichintroduceimportantnewconceptsratherthantopicswhich
are more like detailed working out of already established concepts. For example,
althoughIdiscussedtwo-dimensionalboundaryvalueproblemsinelectrostaticsin
considerable detail, I left out three-dimensional problems and thereby saved some
precioustimewhichcouldbeusedforteachingothertopicswhichIconsideredmore
important.
Preface ix
Atime-honouredpracticefollowedinmanyofthestandardtextbooksofelectro-
magnetictheoryhadbeentoprovideanextensivesetofexerciseproblems.Jackson’s
bookisfamous(ornotorious,dependingonyourpointofview!)foritscollectionof
difficultproblems.Inkeepingwiththeoverallspiritofthebook,Ihaveincludeda
limitednumberofexerciseproblemsspecificallyselectedforadvancingthestudent’s
understandingofthematerialsdiscussedinthetext.Ihaveavoidedproblemsforthe
sole purpose of testing a student’s capacity for intellectual gymnastics at its limit.
Thereisanothertrickyissueconnectedwithexerciseproblems:ascribingthecorrect
creditto thepersons who might have firstinvented some of these problems.Since
thisisimpossibletodo,IherebydeclarethatIcollectedmanyoftheproblemsfrom
differentsourcesovertheyears(Icannotevenrecallnowwhichproblemmighthave
beentakenfromwhere)andImakenoclaimofmyintellectualownershipoverthe
exerciseproblems.
Whilethethoughtofwritingthisbookhasbeenonmymindforseveralyears,Iat
lastgottheforcedopportunityofworkingonthisbookduringtheseveraldepressing
monthswhenmuchoftheworld—especiallymycountryIndia—wasunderalmost
completelockdownduetotheCOVID-19pandemic.Workingonthisbookhelped
metomaintainmymentalsanityatthisdifficulttime.Ihavenaturallybeeninfluenced
bymanyauthorswhowrotebooksonthissubjectbeforeme.Inparticular,Iexpress
myindebtednesstoFeynman,Griffiths,Jackson,LandauandLifshitz,Panofskyand
Phillips. I am grateful to many students who have taken this course from me over
theyearsandencouragedmethroughtheirquestionsandfeedback.Ihavealsohad
fruitful discussions with many departmental colleagues about the teaching of this
course.Iwouldparticularlyliketomentiontwoyoungercolleagueswhohadtaught
this course in our department many times like me and who were very enthusiastic
aboutmyideaofwritingthisbook,butwhoarenolongeramongustoseethisbook:
Vasant Natarajan and V. Venkataraman. I thank Bibhuti Kumar Jha for preparing
most of the figures in this book. Finally, the book would not have been possible
withoutthecontinuoussupportofmywifeMahua.
Bangalore,India ArnabRaiChoudhuri
Contents
1 Introduction ................................................... 1
1.1 WhyElectromagneticTheoryAgain? ........................ 1
1.2 APossibleAxiomaticFormulation ........................... 2
1.3 ElectrostaticsandMagnetostatics ............................ 5
1.4 AUsefulRepresentationoftheDiracδ-Function ............... 6
1.5 GeneralSolutionofaVectorFieldwithGivenDivergence
andCurl ................................................. 8
1.6 ConcludingRemarks ....................................... 9
Exercises ....................................................... 12
References ..................................................... 12
2 Electrostatics .................................................. 13
2.1 Coulomb’sLaw ........................................... 13
2.2 ElectrostaticPotentialasPotentialEnergy .................... 15
2.3 Poisson’sandLaplace’sEquations ........................... 16
2.4 ElectricFieldDuetoaDipoleandaSurfaceDipoleLayer ....... 16
2.5 DipolesinElectromagneticTheory .......................... 20
2.6 Gauss’sLawinElectrostaticsandApplications ................ 21
2.7 CylindricalandSphericalCoordinates ........................ 23
2.8 BoundaryValueProblemsandUniquenessTheorem ............ 25
2.9 MethodofImages ......................................... 28
2.10 BoundaryValueProblemsinTwo-DimensionalCartesian
Coordinates .............................................. 30
2.11 BoundaryValueProblemsinPolarCoordinates ................ 35
2.11.1 ConductingCylinderinaUniformElectricField ....... 36
2.11.2 Wedge-ShapedRegionBetweenConductors ........... 38
2.12 BoundaryValueProblemsinSphericalCoordinates ............ 39
2.12.1 SomePropertiesofLegendrePolynomials ............. 41
2.12.2 BoundaryValueProblemAroundaSphere ............ 41
2.13 MultipoleExpansion ....................................... 42
2.14 PolarizationinDielectricMedium ........................... 45
xi
