Lecture Notes in Physics Arnab Rai Choudhuri Advanced Electromagnetic Theory Lecture Notes in Physics FoundingEditors WolfBeiglböck JürgenEhlers KlausHepp Hans-ArwedWeidenmüller Volume 1009 SeriesEditors RobertaCitro,Salerno,Italy PeterHänggi,Augsburg,Germany MortenHjorth-Jensen,Oslo,Norway MaciejLewenstein,Barcelona,Spain AngelRubio,Hamburg,Germany WolfgangSchleich,Ulm,Germany StefanTheisen,Potsdam,Germany JamesD.Wells,AnnArbor,MI,USA GaryP.Zank,Huntsville,AL,USA The series Lecture Notes in Physics (LNP), founded in 1969, reports new developmentsinphysicsresearchandteaching-quicklyandinformally,butwitha highqualityandtheexplicitaimtosummarizeandcommunicatecurrentknowledge in an accessible way. 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Proposals should be sent to a member of the Editorial Board, or directly to the responsibleeditoratSpringer: DrLisaScalone SpringerNature Physics Tiergartenstrasse17 69121Heidelberg,Germany [email protected] 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) https://doi.org/10.1007/978-981-19-5944-8 ©SpringerNatureSingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore 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