rE 0rB0rEB trB0J0E t1 Contents 1 Introduction 1 11 ChargeandCurrent 2 111 TheConservationLaw 4 12 ForcesandFields 4 121 TheMaxwellEquations 6 2 Electrostatics 8 21 Gauss ID: 262542
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LentTerm,2015ElectromagnetismUniversityofCambridgePartIBMathematicalTripos DavidTongDepartmentofAppliedMathematicsandTheoreticalPhysics,CentreforMathematicalSciences,WilberforceRoad,Cambridge,CB3OBA,UKhttp://www.damtp.cam.ac.uk/user/tong/em.html d.tong@damtp.cam.ac.uk MaxwellEquations rE= 0rB=0rE=�@B @trB=0J+0@E @t{1{ RecommendedBooksandResourcesThereismoreorlessawellestablishedroutetoteachingelectromagnetism.Anumberofgoodbooksfollowthis.DavidJ.Griths,\IntroductiontoElectrodynamics"Asuperbbook.Theexplanationsareclearandsimple.Itdoesn'tcoverquiteasmuchaswe'llneedfortheselectures,butifyou'relookingforabooktocoverthebasicsthenthisistherstonetolookat.EdwardM.PurcellandDavidJ.Morin\ElectricityandMagnetism"Anotherexcellentbooktostartwith.IthassomewhatmoredetailinplacesthanGriths,butthebeginningofthebookexplainsbothelectromagnetismandvectorcalculusinanintertwinedfashion.Ifyouneedsomehelpwithvectorcalculusbasics,thiswouldbeagoodplacetoturn.Ifnot,you'llneedtospendsometimedisentanglingthetwotopics.J.DavidJackson,\ClassicalElectrodynamics"Themostcanonicalofphysicstextbooks.Thisisprobablytheonebookyoucanndoneveryprofessionalphysicist'sshelf,whetherstringtheoristorbiophysicist.Itwillseeyouthroughthiscourseandnextyear'scourse.Theproblemsarefamouslyhard.Butitdoeshavediv,gradandcurlinpolarcoordinatesontheinsidecover.A.Zangwill,\ModernElectrodynamics"Agreatbook.ItisessentiallyamoremodernandmorefriendlyversionofJackson.Feynman,LeightonandSands,\TheFeynmanLecturesonPhysics,VolumeII"Feynman'sfamouslecturesonphysicsaresomethingofamixedbag.Someexplanationsarewonderfullyoriginal,butotherscanbealittletooslicktobehelpful.Andmuchofthematerialcomesacrossasold-fashioned.Volumetwocoverselectromagnetismand,inmyopinion,isthebestofthethree.Anumberofexcellentlecturenotes,includingtheFeynmanlectures,areavailableontheweb.Linkscanbefoundonthecoursewebpage: http://www.damtp.cam.ac.uk/user/tong/em.html {2{ Contents 1 .Introduction 1 1.1 ChargeandCurrent 2 1.1.1 TheConservationLaw 4 1.2 ForcesandFields 4 1.2.1 TheMaxwellEquations 6 2 .Electrostatics 8 2.1 Gauss'Law 8 2.1.1 TheCoulombForce 9 2.1.2 AUniformSphere 11 2.1.3 LineCharges 12 2.1.4 SurfaceChargesandDiscontinuities 13 2.2 TheElectrostaticPotential 16 2.2.1 ThePointCharge 17 2.2.2 TheDipole 19 2.2.3 GeneralChargeDistributions 20 2.2.4 FieldLines 23 2.2.5 ElectrostaticEquilibrium 24 2.3 ElectrostaticEnergy 25 2.3.1 TheEnergyofaPointParticle 27 2.3.2 TheForceBetweenElectricDipoles 29 2.4 Conductors 30 2.4.1 Capacitors 32 2.4.2 BoundaryValueProblems 33 2.4.3 MethodofImages 35 2.4.4 Manymanymoreproblems 37 2.4.5 AHistoryofElectrostatics 39 3 .Magnetostatics 41 3.1 Ampere'sLaw 42 3.1.1 ALongStraightWire 42 3.1.2 SurfaceCurrentsandDiscontinuities 43 3.2 TheVectorPotential 46 3.2.1 MagneticMonopoles 47 {3{ 3.2.2 GaugeTransformations 48 3.2.3 Biot-SavartLaw 49 3.2.4 AMathematicalDiversion:TheLinkingNumber 52 3.3 MagneticDipoles 54 3.3.1 ACurrentLoop 54 3.3.2 GeneralCurrentDistributions 56 3.4 MagneticForces 57 3.4.1 ForceBetweenCurrents 57 3.4.2 ForceandEnergyforaDipole 59 3.4.3 SoWhatisaMagnet? 62 3.5 UnitsofElectromagnetism 64 3.5.1 AHistoryofMagnetostatics 65 4 .Electrodynamics 67 4.1 Faraday'sLawofInduction 67 4.1.1 Faraday'sLawforMovingWires 69 4.1.2 InductanceandMagnetostaticEnergy 71 4.1.3 Resistance 74 4.1.4 MichaelFaraday(1791-1867) 77 4.2 OneLastThing:TheDisplacementCurrent 79 4.2.1 WhyAmpere'sLawisNotEnough 80 4.3 AndThereWasLight 82 4.3.1 SolvingtheWaveEquation 84 4.3.2 Polarisation 87 4.3.3 AnApplication:Re ectiono aConductor 89 4.3.4 JamesClerkMaxwell(1831-1879) 91 4.4 TransportofEnergy:ThePoyntingVector 92 4.4.1 TheContinuityEquationRevisited 94 5 .ElectromagnetismandRelativity 95 5.1 AReviewofSpecialRelativity 95 5.1.1 Four-Vectors 96 5.1.2 ProperTime 97 5.1.3 IndicesUp,IndicesDown 98 5.1.4 Vectors,CovectorsandTensors 99 5.2 ConservedCurrents 102 5.2.1 MagnetismandRelativity 103 5.3 GaugePotentialsandtheElectromagneticTensor 105 {4{ 5.3.1 GaugeInvarianceandRelativity 105 5.3.2 TheElectromagneticTensor 106 5.3.3 AnExample:ABoostedLineCharge 109 5.3.4 AnotherExample:ABoostedPointCharge 110 5.3.5 LorentzScalars 111 5.4 MaxwellEquations 113 5.4.1 TheLorentzForceLaw 115 5.4.2 MotioninConstantFields 117 5.5 Epilogue 118 {5{ AcknowledgementsTheselecturenotescontainmaterialcoveringtwocoursesonElectromagnetism.InCambridge,thesecoursesarecalledPartIBElectromagnetismandPartIIElectro-dynamics.Thenotesoweadebttothepreviouslecturersofthesecourses,includingNatashaBerlo ,JohnPapaloizouandespeciallyAnthonyChallinor.ThenotesassumeafamiliaritywithNewtonianmechanicsandspecialrelativity,ascoveredintheDynamicsandRelativitynotes.Theyalsoassumeaknowledgeofvectorcalculus.Thenotesdonotcovertheclassicaleldtheory(LagrangianandHamiltonian)sectionofthePartIIcourse.{6{ 1.IntroductionThereare,tothebestofourknowledge,fourforcesatplayintheUniverse.Attheverylargestscales|thoseofplanetsorstarsorgalaxies|theforceofgravitydominates.Attheverysmallestdistances,thetwonuclearforcesholdsway.Foreverythinginbetween,itisforceofelectromagnetismthatrules.Attheatomicscale,electromagnetism(admittedlyinconjunctionwithsomebasicquantume ects)governstheinteractionsbetweenatomsandmolecules.Itistheforcethatunderliestheperiodictableofelements,givingrisetoallofchemistryand,throughthis,muchofbiology.Itistheforcewhichbindsatomstogetherintosolidsandliquids.Anditistheforcewhichisresponsiblefortheincrediblerangeofpropertiesthatdi erentmaterialsexhibit.Atthemacroscopicscale,electromagnetismmanifestsitselfinthefamiliarphenom-enathatgivetheforceitsname.Inthecaseofelectricity,thismeanseverythingfromrubbingaballoononyourheadandstickingitonthewall,throughtothefactthatyoucanpluganyapplianceintothewallandbeprettycondentthatitwillwork.Formag-netism,thismeanseverythingfromtheshoppingliststucktoyourfridgedoor,throughtotrainsinJapanwhichlevitateabovetherail.Harnessingthesepowersthroughtheinventionoftheelectricdynamoandmotorhastransformedtheplanetandourlivesonit.Asifthiswasn'tenough,thereismuchmoretotheforceofelectromagnetismforitis,quiteliterally,responsibleforeverythingyou'veeverseen.Itistheforcethatgivesrisetolightitself.Ratherremarkably,afulldescriptionoftheforceofelectromagnetismiscontainedinfoursimpleandelegantequations.TheseareknownastheMaxwellequations.Therearefewplacesinphysics,orindeedinanyothersubject,wheresucharichlydiversesetofphenomena owsfromsolittle.ThepurposeofthiscourseistointroducetheMaxwellequationsandtoextractsomeofthemanystoriestheycontain.However,thereisalsoasecondthemethatrunsthroughthiscourse.Theforceofelectromagnetismturnsouttobeablueprintforalltheotherforces.TherearevariousmathematicalsymmetriesandstructureslurkingwithintheMaxwellequations,struc-tureswhichNaturethenrepeatsinothercontexts.Understandingthemathematicalbeautyoftheequationswillallowustoseesomeoftheprinciplesthatunderlythelawsofphysics,layingthegroundworkforfuturestudyoftheotherforces.{1{ 1.1ChargeandCurrentEachparticleintheUniversecarrieswithitanumberofproperties.Thesedeterminehowtheparticleinteractswitheachofthefourforces.Fortheforceofgravity,thispropertyismass.Fortheforceofelectromagnetism,thepropertyiscalledelectriccharge.Forthepurposesofthiscourse,wecanthinkofelectricchargeasarealnumber,q2R.Importantly,chargecanbepositiveornegative.Itcanalsobezero,inwhichcasetheparticleisuna ectedbytheforceofelectromagnetism.TheSIunitofchargeistheCoulomb,denotedbyC.Itis,likeallSIunits,aparochialmeasure,convenientforhumanactivityratherthaninformedbytheunderlyinglawsofthephysics.(We'lllearnmoreabouthowtheCoulombisdenedinSection 3.5 ).Atafundamentallevel,Natureprovidesuswithabetterunitofcharge.Thisfollowsfromthefactthatchargeisquantised:thechargeofanyparticleisanintegermultipleofthechargecarriedbytheelectronwhichwedenotedas�e,withe=1:6021765710�19CAmuchmorenaturalunitwouldbetosimplycountchargeasq=newithn2Z.Thenelectronshavecharge�1whileprotonshavecharge+1andneutronshavecharge0.Nonetheless,inthiscourse,wewillbowtoconventionandstickwithSIunits.(Anaside:thechargeofquarksisactuallyq=�e=3andq=2e=3.Thisdoesn'tchangethespiritoftheabovediscussionsincewecouldjustchangethebasicunit.But,apartfrominextremecircumstances,quarksareconnedinsideprotonsandneutronssowerarelyhavetoworryaboutthis).Oneofthekeygoalsofthiscourseistomovebeyondthedynamicsofpointparticlesandontothedynamicsofcontinuousobjectsknownaselds.Toaidinthis,it'susefultoconsiderthechargedensity,(x;t)denedaschargeperunitvolume.ThetotalchargeQinagivenregionVissimplyQ=RVd3x(x;t).Inmostsituations,wewillconsidersmoothchargedensities,whichcanbethoughtofasarisingfromaveragingovermanypoint-likeparticles.But,onoccasion,wewillreturntotheideaofasingleparticleofchargeq,movingonsometrajectoryr(t),bywriting=q(x�r(t))wherethedelta-functionensuresthatallthechargesitsatapoint.{2{ Moregenerally,wewillneedtodescribethemovementofchargefromoneplacetoanother.ThisiscapturedbyaquantityknownasthecurrentdensityJ(x;t),denedasfollows:foreverysurfaceS,theintegralI=ZSJdScountsthechargeperunittimepassingthroughS.(HeredSistheunitnormaltoS).ThequantityIiscalledthecurrent.Inthissense,thecurrentdensityisthecurrent-per-unit-area.Theaboveisaratherindirectdenitionofthecur- Figure1:Current uxrentdensity.Togetamoreintuitivepicture,consideracontinuouschargedistributioninwhichthevelocityofasmallvolume,atpointx,isgivenbyv(x;t).Then,ne-glectingrelativistice ects,thecurrentdensityisJ=vInparticular,ifasingleparticleismovingwithvelocityv=_r(t),thecurrentdensitywillbeJ=qv3(x�r(t)).Thisisillustratedinthegure,wheretheunderlyingchargedparticlesareshownasredballs,movingthroughthebluesurfaceS.Asasimpleexample,considerelectronsmov- Figure2:Thewireingalongawire.Wemodelthewireasalongcylinderofcross-sectionalareaAasshownbe-low.Theelectronsmovewithvelocityv,paral-leltotheaxisofthewire.(Inreality,theelec-tronswillhavesomedistributionofspeeds;wetakevtobetheiraveragevelocity).Iftherearenelectronsperunitvolume,eachwithchargeq,thenthechargedensityis=nqandthecurrentdensityisJ=nqv.ThecurrentitselfisI=jJjA.Throughoutthiscourse,thecurrentdensityJplaysamuchmoreprominentrolethanthecurrentI.Forthisreason,wewilloftenrefertoJsimplyasthe\current"althoughwe'llbemorecarefulwiththeterminologywhenthereisanypossibilityforconfusion.{3{ 1.1.1TheConservationLawThemostimportantpropertyofelectricchargeisthatit'sconserved.This,ofcourse,meansthatthetotalchargeinasystemcan'tchange.Butitmeansmuchmorethanthatbecauseelectricchargeisconservedlocally.Anelectricchargecan'tjustvanishfromonepartoftheUniverseandturnupsomewhereelse.Itcanonlyleaveonepointinspacebymovingtoaneighbouringpoint.Thepropertyoflocalconservationmeansthatcanchangeintimeonlyifthereisacompensatingcurrent owingintooroutofthatregion.Weexpressthisinthecontinuityequation,@ @t+rJ=0(1.1)Thisisanimportantequation.Itarisesinanysituationwherethereissomequantitythatislocallyconserved.Toseewhythecontinuityequationcapturestherightphysics,it'sbesttoconsiderthechangeinthetotalchargeQcontainedinsomeregionV.dQ dt=ZVd3x@ @t=�ZVd3xrJ=�ZSJdSFromourpreviousdiscussion,RSJdSisthetotalcurrent owingoutthroughtheboundarySoftheregionV.(Itisthetotalcharge owingout,ratherthanin,becausedSistheoutwardnormaltotheregionV).Theminussignistheretoensurethatifthenet owofcurrentisoutwards,thenthetotalchargedecreases.Ifthereisnocurrent owingoutoftheregion,thendQ=dt=0.Thisisthestatementof(global)conservationofcharge.InmanyapplicationswewilltakeVtobeallofspace,R3,withbothchargesandcurrentslocalisedinsomecompactregion.Thisensuresthatthetotalchargeremainsconstant.1.2ForcesandFieldsAnyparticlethatcarrieselectricchargeexperiencestheforceofelectromagnetism.Buttheforcedoesnotactdirectlybetweenparticles.Instead,Naturechosetointroduceintermediaries.Theseareelds.Inphysics,a\eld"isadynamicalquantitywhichtakesavalueateverypointinspaceandtime.Todescribetheforceofelectromagnetism,weneedtointroducetwo{4{ elds,eachofwhichisathree-dimensionalvector.TheyarecalledtheelectriceldEandthemagneticeldB,E(x;t)andB(x;t)Whenwetalkabouta\force"inmodernphysics,wereallymeananintricateinterplaybetweenparticlesandelds.Therearetwoaspectstothis.First,thechargedparticlescreatebothelectricandmagneticelds.Second,theelectricandmagneticeldsguidethechargedparticles,tellingthemhowtomove.Thismotion,inturn,changestheeldsthattheparticlescreate.We'releftwithabeautifuldancewiththeparticlesandeldsastwopartners,eachdictatingthemovesoftheother.ThisdancebetweenparticlesandeldsprovidesaparadigmwhichallotherforcesinNaturefollow.ItfeelslikethereshouldbeadeepreasonthatNaturechosetointroduceeldsassociatedtoalltheforces.And,indeed,thisapproachdoesprovideoneover-ridingadvantage:allinteractionsarelocal.Anyobject|whetherparticleoreld|a ectsthingsonlyinitsimmediateneighbourhood.Thisin uencecanthenpropagatethroughtheeldtoreachanotherpointinspace,butitdoesnotdosoinstantaneously.Ittakestimeforaparticleinonepartofspacetoin uenceaparticleelsewhere.Thislackofinstantaneousinteractionallowsustointroduceforceswhicharecompatiblewiththetheoryofspecialrelativity,somethingthatwewillexploreinmoredetailinSection 5 Thepurposeofthiscourseistoprovideamathematicaldescriptionoftheinterplaybetweenparticlesandelectromagneticelds.Infact,you'vealreadymetonesideofthisdance:thepositionr(t)ofaparticleofchargeqisdictatedbytheelectricandmagneticeldsthroughtheLorentzforcelaw,F=q(E+_rB)(1.2)ThemotionoftheparticlecanthenbedeterminedthroughNewton'sequationF=mr.WeexploredvarioussolutionstothisintheDynamicsandRelativitycourse.Roughlyspeaking,anelectriceldacceleratesaparticleinthedirectionE,whileamagneticeldcausesaparticletomoveincirclesintheplaneperpendiculartoB.WecanalsowritetheLorentzforcelawintermsofthechargedistribution(x;t)andthecurrentdensityJ(x;t).Nowwetalkintermsoftheforcedensityf(x;t),whichistheforceactingonasmallvolumeatpointx.NowtheLorentzforcelawreadsf=E+JB(1.3){5{ 1.2.1TheMaxwellEquationsInthiscourse,mostofourattentionwillfocusontheothersideofthedance:thewayinwhichelectricandmagneticeldsarecreatedbychargedparticles.Thisisdescribedbyasetoffourequations,knowncollectivelyastheMaxwellequations.Theyare:rE= 0(1.4)rB=0(1.5)rE+@B @t=0(1.6)rB�00@E @t=0J(1.7)Theequationsinvolvetwoconstants.Therstistheelectricconstant(knownalso,inslightlyold-fashionedterminology,asthepermittivityoffreespace),08:8510�12m�3Kg�1s2C2Itcanbethoughtofascharacterisingthestrengthoftheelectricinteractions.Theotheristhemagneticconstant(orpermeabilityoffreespace),0=410�7mKgC�21:2510�6mKgC�2Thepresenceof4inthisformulaisn'ttellingusanythingdeepaboutNature.It'smoreare ectionofthedenitionoftheCoulombastheunitofcharge.(WewillexplainthisinmoredetailinSection 3.5 ).Nonetheless,thiscanbethoughtofascharacterisingthestrengthofmagneticinteractions(inunitsofCoulombs).TheMaxwellequations( 1.4 ),( 1.5 ),( 1.6 )and( 1.7 )willoccupyusfortherestofthecourse.Ratherthantryingtounderstandalltheequationsatonce,we'llproceedbitbybit,lookingatsituationswhereonlysomeoftheequationsareimportant.Bytheendofthelectures,wewillunderstandthephysicscapturedbyeachoftheseequationsandhowtheyttogether.{6{ However,equallyimportantly,wewillalsoexplorethemathematicalstructureoftheMaxwellequations.Atrstglance,theylookjustlikefourrandomequationsfromvectorcalculus.Yetthiscouldn'tbefurtherfromthetruth.TheMaxwellequationsarespecialand,whenviewedintherightway,aretheessentiallyuniqueequationsthatcandescribetheforceofelectromagnetism.Thefullstoryofwhythesearetheuniqueequationsinvolvesbothquantummechanicsandrelativityandwillonlybetoldinlatercourses.Butwewillstartthatjourneyhere.ThegoalisthatbytheendoftheselecturesyouwillbeconvincedoftheimportanceoftheMaxwellequationsonbothexperimentalandaestheticgrounds.{7{ 2.ElectrostaticsInthissection,wewillbeinterestedinelectricchargesatrest.Thismeansthatthereexistsaframeofreferenceinwhichtherearenocurrents;onlystationarycharges.Ofcourse,therewillbeforcesbetweenthesechargesbutwewillassumethatthechargesarepinnedinplaceandcannotmove.Thequestionthatwewanttoansweris:whatistheelectriceldgeneratedbythesecharges?Sincenothingmoves,wearelookingfortimeindependentsolutionstoMaxwell'sequationswithJ=0.ThismeansthatwecanconsistentlysetB=0andwe'releftwithtwoofMaxwell'sequationstosolve.TheyarerE= 0(2.1)andrE=0(2.2)Ifyouxthechargedistribution,equations( 2.1 )and( 2.2 )haveauniquesolution.Ourgoalinthissectionistondit.2.1Gauss'LawBeforeweproceed,let'srstpresentequation( 2.1 )inaslightlydi erentformthatwillshedsomelightonitsmeaning.ConsidersomeclosedregionVR3ofspace.We'lldenotetheboundaryofVbyS=@V.Wenowintegratebothsidesof( 2.1 )overV.Sincetheleft-handsideisatotalderivative,wecanusethedivergencetheoremtoconvertthistoanintegraloverthesurfaceS.WehaveZVd3xrE=ZSEdS=1 0ZVd3xTheintegralofthechargedensityoverVissimplythetotalchargecontainedintheregion.We'llcallitQ=Rd3x.Meanwhile,theintegraloftheelectriceldoverSiscalledthe uxthroughS.WelearnthatthetwoarerelatedbyZSEdS=Q 0(2.3)ThisisGauss'slaw.However,becausethetwoareentirelyequivalent,wealsorefertotheoriginal( 2.1 )asGauss'slaw.{8{ Figure3:The uxthroughSandS0isthesame.Figure4:The uxthroughSvanishes.Noticethatitdoesn'tmatterwhatshapethesurfaceStakes.AslongasitsurroundsatotalchargeQ,the uxthroughthesurfacewillalwaysbeQ=0.Thisisshown,forexample,intheleft-handgureabove.ThechoiceofSiscalledtheGaussiansurface;oftenthere'sasmartchoicethatmakesaparticularproblemsimple.OnlychargesthatlieinsideVcontributetothe ux.AnychargesthatlieoutsidewillproduceanelectriceldthatpenetratesthroughSatsomepoint,givingnegative ux,butleavesthroughtheothersideofS,depositingpositive ux.ThetotalcontributionfromthesechargesthatlieoutsideofViszero,asillustratedintheright-handgureabove.Forageneralchargedistribution,we'llneedtousebothGauss'law( 2.1 )andtheextraequation( 2.2 ).However,forratherspecialchargedistributions{typicallythosewithlotsofsymmetry{itturnsouttobesucienttosolvetheintegralformofGauss'law( 2.3 )alone,withthesymmetryensuringthat( 2.2 )isautomaticallysatised.Westartbydescribingtheserathersimplesolutions.We'llthenreturntothegeneralcaseinSection 2.2 .2.1.1TheCoulombForceWe'llstartbyshowingthatGauss'law( 2.3 )reproducesthemorefamiliarCoulombforcelawthatweallknowandlove.Todothis,takeasphericallysymmetricchargedistribution,centeredattheorigin,containedwithinsomeradiusR.Thiswillbeourmodelforaparticle.Wewon'tneedtomakeanyassumptionaboutthenatureofthedistributionotherthanitssymmetryandthefactthatthetotalchargeisQ.{9{ Wewanttoknowtheelectriceldatsomeradiusr� Figure5:R.WetakeourGaussiansurfaceStobeasphereofradiusrasshowninthegure.Gauss'lawstatesZSEdS=Q 0Atthispointwemakeuseofthesphericalsymmetryoftheproblem.Thistellsusthattheelectriceldmustpointra-diallyoutwards:E(x)=E(r)^r.And,sincetheintegralisonlyovertheangularcoordinatesofthesphere,wecanpullthefunctionE(r)outside.WehaveZSEdS=E(r)ZS^rdS=E(r)4r2=Q 0wherethefactorof4r2hasarisensimplybecauseit'stheareaoftheGaussiansphere.WelearnthattheelectriceldoutsideasphericallysymmetricdistributionofchargeQisE(x)=Q 40r2^r(2.4)That'snice.Thisisthefamiliarresultthatwe'veseenbefore.(See,forexample,thenoteson DynamicsandRelativity ).TheLorentzforcelaw( 1.2 )thentellsusthatatestchargeqmovingintheregionr�RexperiencesaforceF=Qq 40r2^rThis,ofcourse,istheCoulombforcebetweentwostaticchargedparticles.Noticethat,aspromised,1=0characterisesthestrengthoftheforce.Ifthetwochargeshavethesamesign,sothatQq�0,theforceisrepulsive,pushingthetestchargeawayfromtheorigin.Ifthechargeshaveoppositesigns,Qq0,theforceisattractive,pointingtowardstheorigin.WeseethatGauss'slaw( 2.1 )reproducesthissimpleresultthatweknowaboutcharges.Finally,notethattheassumptionofsymmetrywascrucialinouraboveanalysis.Withoutit,theelectriceldE(x)wouldhavedependedontheangularcoordinatesofthesphereSandsobeenstuckinsidetheintegral.Insituationswithoutsymmetry,Gauss'lawaloneisnotenoughtodeterminetheelectriceldandweneedtoalsouserE=0.We'llseehowtodothisinSection 2.2 .Ifyou'reworried,however,it'ssimpletocheckthatournalexpressionfortheelectriceld( 2.4 )doesindeedsolverE=0.{10{ CoulombvsNewtonTheinverse-squareformoftheforceiscommontobothelectrostaticsandgravity.It'sworthcomparingtherelativestrengthsofthetwoforces.Forexample,wecanlookattherelativestrengthsofNewtonianattractionandCoulombrepulsionbetweentwoelectrons.Thesearepointparticleswithmassmeandcharge�egivenbye1:610�19Coulombsandme9:110�31KgRegardlessoftheseparation,wehaveFCoulomb FNewton=e2 401 Gm2eThestrengthofgravityisdeterminedbyNewton'sconstantG6:710�11m3Kg�1s2.Plugginginthenumbersrevealssomethingextraordinary:FCoulomb FNewton1042Gravityispuny.Electromagnetismrules.Infactyouknewthisalready.ThemereactofliftingupyouarmispitchingafewelectricalimpulsesupagainstthegravitationalmightoftheentireEarth.Yettheelectricalimpulseswin.However,gravityhasatrickupitssleeve.Whileelectricchargescomewithbothpositiveandnegativesigns,massisonlypositive.Itmeansthatbythetimewegettomacroscopicallylargeobjects|stars,planets,cats|themassaccumulateswhilethechargescanceltogoodapproximation.Thiscompensatesthefactorof10�42suppressionuntil,atlargedistancescales,gravitywinsafterall.Thefactthattheforceofgravityissoridiculouslytinyattheleveloffundamentalparticleshasconsequence.Itmeansthatwecanneglectgravitywheneverwetalkabouttheverysmall.(Andindeed,weshallneglectgravityfortherestofthiscourse).However,italsomeansthatifwewouldliketounderstandgravitybetterontheseverytinydistances{forexample,todevelopaquantumtheoryofgravity|thenit'sgoingtobetrickytogetmuchguidancefromexperiment.2.1.2AUniformSphereTheelectriceldoutsideasphericallysymmetricchargedistributionisalwaysgivenby( 2.4 ).Whataboutinside?Thisdependsonthedistributioninquestion.ThesimplestisasphereofradiusRwithuniformchargedistribution.ThetotalchargeisQ=4 3R3{11{ Let'spickourGaussiansurfacetobeasphere,centeredat Figure6:theorigin,ofradiusrR.Thechargecontainedwithinthissphereis4r3=3=Qr3=R3,soGauss'lawgivesZSEdS=Qr3 0R3Again,usingthesymmetryargumentwecanwriteE(r)=E(r)^randcomputeZSEdS=E(r)ZS^rdS=E(r)4r2=Qr3 0R3ThistellsusthattheelectriceldgrowslinearlyinsidethesphereE(x)=Qr 40R3^rrR(2.5)Outsidethespherewereverttotheinverse-square Figure7:form( 2.4 ).Atthesurfaceofthesphere,r=R,theelectriceldiscontinuousbutthederivative,dE=dr,isnot.Thisisshowninthegraph.2.1.3LineChargesConsider,next,achargesmearedoutalongalinewhich Figure8:we'lltaketobethez-axis.We'lltakeuniformchargeden-sityperunitlength.(Ifyoulikeyoucouldconsiderasolidcylinderwithuniformchargedensityandthensendtheradiustozero).Wewanttoknowtheelectriceldduetothislineofcharge.Ourset-upnowhascylindricalsymmetry.WetaketheGaussiansurfacetobeacylinderoflengthLandradiusr.WehaveZSEdS=L 0Again,bysymmetry,theelectriceldpointsintheradialdirection,awayfromtheline.We'lldenotethisvectorincylindricalpolarcoordinatesas^rsothatE=E(r)^r.ThesymmetrymeansthatthetwoendcapsoftheGaussian{12{ surfacedon'tcontributetotheintegralbecausetheirnormalpointsinthe^zdirectionand^z^r=0.We'releftonlywithacontributionfromthecurvedsideofthecylinder,ZSEdS=E(r)2rL=L 0SothattheelectriceldisE(r)= 20r^r(2.6)Notethat,whiletheelectriceldforapointchargedropso as1=r2(withrtheradialdistance),theelectriceldforalinechargedropso moreslowlyas1=r.(Ofcourse,theradialdistancermeansslightlydi erentthingsinthetwocases:itisr=p x2+y2+z2forthepointparticle,butisr=p x2+y2fortheline).2.1.4SurfaceChargesandDiscontinuitiesNowconsideraninniteplane,whichwe Figure9:taketobez=0,carryinguniformchargeperunitarea,.WeagaintakeourGaus-siansurfacetobeacylinder,thistimewithitsaxisperpendiculartotheplaneasshowninthegure.Inthiscontext,thecylin-derissometimesreferredtoasaGaussian\pillbox"(onaccountofGauss'wellknownfondnessforaspirin).Onsymmetrygrounds,wehaveE=E(z)^zMoreover,theelectriceldintheupperplane,z�0,mustpointintheoppositedirectionfromthelowerplane,z0,sothatE(z)=�E(�z).Thesurfaceintegralnowvanishesoverthecurvedsideofthecylinderandweonlygetcontributionsfromtheendcaps,whichwetaketohaveareaA.ThisgivesZSEdS=E(z)A�E(�z)A=2E(z)A=A 0TheelectriceldaboveaninniteplaneofchargeisthereforeE(z)= 20(2.7)Notethattheelectriceldisindependentofthedistancefromtheplane!Thisisbecausetheplaneisinniteinextent:thefurtheryoumovefromit,themorecomesintoview.{13{ Figure10:ThenormalcomponentoftheelectriceldisdiscontinuousFigure11:Thetangentialcomponentoftheelectriceldiscontinuous.Thereisanotherimportantpointtotakeawayfromthisanalysis.Theelectriceldisnotcontinuousoneithersideofasurfaceofconstantchargedensity.WehaveE(z!0+)�E(z!0�)= 0(2.8)Forthistohold,itisnotimportantthattheplanestretchestoinnity.It'ssimpletoredotheaboveanalysisforanyarbitrarysurfacewithchargedensity.Thereisnoneedfortobeuniformand,correspondingly,thereisnoneedforEatagivenpointtobeparalleltothenormaltothesurface^n.Atanypointofthesurface,wecantakeaGaussiancylinder,asshownintheleft-handgureabove,whoseaxisisnormaltothesurfaceatthatpoint.Itscross-sectionalareaAcanbearbitrarilysmall(since,aswesaw,itdropsoutofthenalanswer).IfEdenotestheelectriceldoneithersideofthesurface,then^nEj+�^nEj�= 0(2.9)Incontrast,theelectriceldtangenttothesurfaceiscontinuous.Toseethis,weneedtodoaslightlydi erentcalculation.Consider,again,anarbitrarysurfacewithsurfacecharge.NowweconsideraloopCwithalengthLwhichliesparalleltothesurfaceandalengthawhichisperpendiculartothesurface.We'vedrawnthisloopintheright-handgureabove,wherethesurfaceisnowshownside-on.WeintegrateEaroundtheloop.UsingStoke'stheorem,wehaveICEdr=ZrEdSwhereSisthesurfaceboundedbyC.Inthelimita!0,thesurfaceSshrinkstozerosizesothisintegralgiveszero.Thismeansthatthecontributiontolineintegralmustalsovanish,leavinguswith^nE+�^nE�=0Thisisthestatementthattheelectriceldtangentialtothesurfaceiscontinuous.{14{ APairofPlanes Figure12:Asasimplegeneralisation,considerapairofin-niteplanesatz=0andz=a,carryinguniformsurfacechargedensityrespectivelyasshowninthegure.Tocomputetheelectriceldweneedonlyaddtheeldsforarisingfromtwoplanes,eachofwhichtakestheform( 2.7 ).WendthattheelectriceldbetweenthetwoplanesisE= 0^z0za(2.10)whileE=0outsidetheplanesAPlaneSlabWecanrederivethediscontinuity( 2.9 )intheelectriceldbyconsideringaninniteslabofthickness2dandchargedensityperunitvolume.WhenourGaussianpillboxliesinsidetheslab,withzd,wehave2AE(z)=2zA 0)E(z)=z 0Meanwhile,forz�dwegetourearlierresult( 2.7 ).Theelectriceldisnowcontinuousasshowninthegure.Takingthelimitd!0and!1suchthatthesurfacecharge=dremainsconstantreproducesthediscontinuity( 2.8 ). Figure13:TheGaussiansurfaceforaplaneslabFigure14:Theresultingelectriceld{15{ ASphericalShellLet'sgiveonelastexamplethatinvolvessurfacechargeand Figure15:theassociateddiscontinuityoftheelectriceld.We'llcon-siderasphericalshellofradiusR,centeredattheorigin,withuniformsurfacechargedensity.ThetotalchargeisQ=4R2Wealreadyknowthatoutsidetheshell,r�R,theelectriceldtakesthestandardinverse-squareform( 2.4 ).Whataboutinside?Well,sinceanysurfacewithrRdoesn'tsurroundacharge,Gauss'lawtellsusthatwenecessarilyhaveE=0inside.Thatmeansthatthereisadiscontinuityatthesurfacer=R,E^rj+�E^rj�=Q 4R20= 0inaccordwiththeexpectation( 2.9 ).2.2TheElectrostaticPotentialForalltheexamplesinthelastsection,symmetryconsiderationsmeantthatweonlyneededtoconsiderGauss'law.However,forgeneralchargedistributionsGauss'lawisnotsucient.Wealsoneedtoinvokethesecondequation,rE=0.Infact,thissecondequationiseasilydispatchedsincerE=0impliesthattheelectriceldcanbewrittenasthegradientofsomefunction,E=�r (2.11)Thescalar iscalledtheelectrostaticpotentialorscalarpotential(or,sometimes,justthepotential).Toproceed,wereverttotheoriginaldi erentialformofGauss'law( 2.1 ).ThisnowtakestheformofthePoissonequationrE= 0)r2 =� 0(2.12)Inregionsofspacewherethechargedensityvanishes,we'releftsolvingtheLaplaceequationr2 =0(2.13)SolutionstotheLaplaceequationaresaidtobeharmonicfunctions.{16{ Afewcomments:Thepotential isonlydeneduptotheadditionofsomeconstant.Thisseem-inglytrivialpointisactuallythebeginningofalonganddeepstoryintheoreticalphysicsknownasgaugeinvariance.We'llcomebacktoitinSection 5.3.1 .Fornow,we'lleliminatethisredundancybyrequiringthat (r)!0asr!1.WeknowfromourstudyofNewtonianmechanicsthattheelectrostaticpotentialisproportionaltothepotentialenergyexperiencedbyatestparticle.(SeeSection2.2ofthe DynamicsandRelativity lecturenotes).Specically,atestparticleofmassm,positionr(t)andchargeqmovinginabackgroundelectriceldhasconservedenergyE=1 2m_r_r+q (r)ThePoissonequationislinearinboth and.Thismeansthatifweknowthepotential 1forsomechargedistribution1andthepotential 2foranotherchargedistribution2,thenthepotentialfor1+2issimply 1+ 2.Whatthisreallymeansisthattheelectriceldforabunchofchargesisjustthesumoftheeldsgeneratedbyeachcharge.Thisiscalledtheprincipleofsuperpositionforcharges.ThislinearityoftheequationsiswhatmakeselectromagnetismeasycomparedtootherforcesofNature.WestatedabovethatrE=0isequivalenttowritingE=�r .ThisistruewhenspaceisR3or,infact,ifwetakespacetobeanyopenballinR3.ButifourbackgroundspacehasasuitablycomplicatedtopologythentherearesolutionstorE=0whichcannotbewrittenintheformE=�r .ThisistiedultimatelytothebeautifulmathematicaltheoryofdeRhamcohomology.Needlesstosay,inthisstartercoursewe'renotgoingtoworryabouttheseissues.We'llalwaystakespacetimetohavetopologyR4and,correspondingly,anyspatialhypersurfacetobeR3.2.2.1ThePointChargeLet'sstartbyderivingtheCoulombforcelawyetagain.We'lltakeaparticleofchargeQandplaceitattheorigin.Thistime,however,we'llassumethattheparticlereallyisapointcharge.Thismeansthatthechargedensitytakestheformofadelta-function,(x)=Q3(x).Weneedtosolvetheequationr2 =�Q 03(x)(2.14){17{ You'vesolvedproblemsofthiskindinyourMethodscourse.ThesolutionisessentiallytheGreen'sfunctionfortheLaplacianr2,aninterpretationthatwe'llreturntoinSection 2.2.3 .Let'srecallhowwendthissolution.Werstlookawayfromtheorigin,r6=0,wherethere'snofunnybusinessgoingonwithdelta-function.Here,we'relookingforthesphericallysymmetricsolutiontotheLaplaceequation.Thisis = rforsomeconstant .ToseewhythissolvestheLaplaceequation,weneedtousetheresultrr=^r(2.15)where^ristheunitradialvectorinsphericalpolarcoordinates,sox=r^r.Usingthechainrule,thismeansthatr(1=r)=�^r=r2=�x=r3.Thisgivesusr =� r3x)r2 =� rx r3�3xx r5Butrx=3andwendthatr2 =0asrequired.Itremainstogureoutwhattodoattheoriginwherethedelta-functionlives.Thisiswhatdeterminestheoverallnormalization ofthesolution.Atthispoint,it'ssimplesttousetheintegralformofGauss'lawtotransfertheproblemfromtheorigintothefar ungreachesofspace.Todothis,weintegrate( 2.14 )oversomeregionVwhichincludestheorigin.Integratingthechargedensitygives(x)=Q3(x))ZVd3x=QSo,usingGauss'law( 2.3 ),werequireZSr dS=�Q 0Butthisisexactlythekindofsurfaceintegralthatweweredoinginthelastsection.Substituting = =rintotheaboveequation,andchoosingStobeasphereofradiusr,tellsusthatwemusthave =Q=40,or =Q 40r(2.16)Takingthegradientofthisusing( 2.15 )givesusCoulomb'slawE(x)=�r =Q 40r2^r{18{ ThederivationofCoulomb'slawusingthepotentialwassomewhatmoreinvolvedthanthetechniqueusingGauss'lawalonethatwesawinthelastsection.However,aswe'llnowsee,introducingthepotentialallowsustowritedownthesolutiontoessentiallyanyproblem.ANoteonNotationThroughouttheselectures,wewillusexandrinterchangeablytodenotepositioninspace.Forexample,sometimeswe'llwriteintegrationoveravolumeasRd3xandsometimesasRd3r.Theadvantageofthernotationisthatitlooksmorenaturalwhenworkinginsphericalpolarcoordinates.Forexample,wehavejrj=rwhichisnice.Thedisadvantageisthatitcanleadtoconfusionwhenworkinginothercoordinatesystems,inparticularcylindricalpolar.Forthisreason,we'llalternatebetweenthetwonotations,adoptingtheattitudethatclarityismoreimportantthanconsistency.2.2.2TheDipoleAdipoleconsistsoftwopointcharges,Qand�Q,adistancedapart.Weplacetherstchargeattheoriginandthesecondatr=�d.Thepotentialissimplythesumofthepotentialforeachcharge, =1 40Q r�Q jr+djSimilarly,theelectriceldisjustthesumoftheelectriceldsmadebythetwopointcharges.Thisfollowsfromthelinearityoftheequationsandisasimpleapplicationoftheprincipleofsuperpositionthatwementionedearlier.Itwillprovefruitfultoaskwhatthedipolelookslikefarfromthetwopointcharges,atadistancer jdj.WeneedtoTaylorexpandthesecondtermabove.ThevectorversionoftheTaylorexpansionforageneralfunctionf(r)isgivenbyf(r+d)f(r)+drf(r)+1 2(dr)2f(r)+:::(2.17)Applyingthistothefunction1=jr+djgives1 jr+dj1 r+dr1 r+1 2(dr)21 r+:::=1 r�dr r3�1 2dd r3�3(dr)2 r5+:::{19{ (Toderivethelastterm,itmightbeeasiesttouseindexnotationfordr=di@i).Forourdipole,we'llonlyneedthersttwotermsinthisexpansion.Theygivethepotential Q 401 r�1 r�dr1 r+:::=Q 40dr r3+:::(2.18)Weseethatthepotentialforadipolefallso as1=r2.Correspondingly,theelectricelddropso as1=r3;bothareonepowerhigherthantheeldsforapointcharge.Theelectriceldisnotsphericallysymmetric.Theleadingordercontributionisgovernedbythecombinationp=QdThisiscalledtheelectricdipolemoment.Byconvention,itpointsfromthenegativechargetothepositive.ThedipoleelectriceldisE=�r =1 403(p^r)^r�p r3+:::(2.19)Noticethatthesignoftheelectricelddependsonwhereyousitinspace.Insomeparts,theforcewillbeattractive;inotherpartsrepulsive.It'ssometimesusefultoconsiderthelimitd!0andQ!1suchthatp=Qdremainsxed.Inthislimit,allthe:::termsin( 2.18 )and( 2.19 )disappearsincetheycontainhigherpowersofd.Oftenwhenpeopletalkaboutthe\dipole",theyimplicitlymeantakingthislimit.2.2.3GeneralChargeDistributionsOurderivationofthepotentialduetoapointcharge( 2.16 ),togetherwiththeprincipleofsuperposition,isactuallyenoughtosolve{atleastformally{thepotentialduetoanychargedistribution.ThisisbecausethesolutionforapointchargeisnothingotherthantheGreen'sfunctionfortheLaplacian.TheGreen'sfunctionisdenedtobethesolutiontotheequationr2G(r;r0)=3(r�r0)which,fromourdiscussionofthepointcharge,wenowknowtobeG(r;r0)=�1 41 jr�r0j(2.20){20{ WecannowapplyourusualGreen'sfunctionmethodstothegeneralPoissonequation( 2.12 ).Inwhatfollows,we'lltake(r)6=0onlyinsomecompactregion,V,ofspace.ThesolutiontothePoissonequationisgivenby (r)=�1 0ZVd3r0G(r;r0)(r0)=1 40ZVd3r0(r0) jr�r0j(2.21)(Tocheckthis,youjusthavetokeepyourheadandrememberwhethertheoperatorsarehittingrorr0.TheLaplacianactsonrso,ifwecomputer2 ,itpassesthroughtheintegralintheaboveexpressionandhitsG(r;r0),leavingbehindadelta-functionwhichsubsequentlykillstheintegral).Similarly,theelectriceldarisingfromageneralchargedistributionisE(r)=�r (r)=�1 40ZVd3r0(r0)r1 jr�r0j=1 40ZVd3r0(r0)r�r0 jr�r0j3Givenaverycomplicatedchargedistribution(r),thisequationwillgivebackanequallycomplicatedelectriceldE(r).Butifwesitalongwayfromthechargedistribution,there'sarathernicesimplicationthathappens...LongDistanceBehaviourSupposenowthatyouwanttoknowwhattheelectriceld Figure16:lookslikefarfromtheregionV.Thismeansthatwe'reinter-estedintheelectriceldatrwithjrj jr0jforallr02V.WecanapplythesameTaylorexpansion( 2.17 ),nowreplacingdwith�r0foreachr0inthechargedregion.Thismeanswecanwrite1 jr�r0j=1 r�r0r1 r+1 2(r0r)21 r+:::=1 r+rr0 r3+1 23(rr0)2 r5�r0r0 r3+:::(2.22)andourpotentialbecomes (r)=1 40ZVd3r0(r0)1 r+rr0 r3+:::Theleadingtermisjust (r)=Q 40r+:::{21{ whereQ=RVd3r0(r0)isthetotalchargecontainedwithinV.So,toleadingorder,ifyou'refarenoughawaythenyoucan'tdistinguishageneralchargedistributionfromapointchargelocalisedattheorigin.Butifyou'recarefulwithexperiments,youcantellthedi erence.Therstcorrectiontakestheformofadipole, (r)=1 40Q r+p^r r2+:::wherep=ZVd3r0r0(r0)isthedipolemomentofthedistribution.OneparticularlyimportantsituationiswhenwehaveaneutralobjectwithQ=0.Inthiscase,thedipoleisthedominantcontri-butiontothepotential.Weseethatanarbitrarilycomplicated,localisedchargedistributioncanbechar-acterisedbyafewsimplequantities,ofdecreasingimportance.FirstcomesthetotalchargeQ.Nextthedipolemomentpwhichcontainssomebasicinformationabouthowthechargesaredistributed.Butwecankeepgoing.Thenextcorrectioniscalledthequadrupoleandisgivenby =1 21 40rirjQij r5whereQijisasymmetrictracelesstensorknownasthequadrupolemoment,givenbyQij=ZVd3r0(r0)�3r0ir0j�ijr02Itcontainssomemorerenedinformationabouthowthechargesaredistributed.Afterthiscomestheoctopoleandsoon.Thegeneralnamegiventothisapproachisthemul-tipoleexpansion.Itinvolvesexpandingthefunction intermsofsphericalharmonics.Asystematictreatmentcanbefound,forexample,inthebookbyJackson.ACommentonInniteChargeDistributionsIntheabove,weassumedforsimplicitythatthechargedistributionwasrestrictedtosomecompactregionofspace,V.TheGreen'sfunctionapproachstillworksifthechargedistributionstretchestoinnity.However,forsuchdistributionsit'snotalwayspossibletopick (r)!0asr!1.Infact,wesawanexampleofthisearlier.Foraninnitelinecharge,wecomputedtheelectriceldin( 2.6 ).ItgoesasE(r)= 2r^r{22{ wherenowr2=x2+y2isthecylindricalradialcoordinateperpendiculartotheline.Thepotential whichgivesrisetothisis (r)=� 20logr r0Becauseofthelogfunction,wenecessarilyhave (r)!1asr!1.Instead,weneedtopickanarbitrary,butnitedistance,r0atwhichthepotentialvanishes.2.2.4FieldLinesTheusualwayofdepictingavectoristodrawanarrowwhoselengthisproportionaltothemagnitude.Fortheelectriceld,there'saslightlydi erent,moreusefulwaytoshowwhat'sgoingon.Wedrawcontinuouslines,tangenttotheelectriceldE,withthedensityoflinesproportionaltothemagnitudeofE.Thisinnovation,duetoFaraday,iscalledtheeldline.(Theyarewhatwehavebeensecretlydrawingthroughoutthesenotes).Fieldlinesarecontinuous.Theybeginandendonlyatcharges.Theycannevercross.Theeldlinesforpositiveandnegativepointchargesare: Byconvention,thepositivechargesactassourcesforthelines,withthearrowsemerg-ing.Thenegativechargesactassinks,withthearrowsapproaching.It'salsoeasytodrawtheequipotentials|surfacesofconstant |onthissamegure.Thesearethesurfacesalongwhichyoucanmoveachargewithoutdoinganywork.TherelationshipE=�r ensuresthattheequipotentialscuttheeldlinesatrightangles.Weusuallydrawthemasdottedlines: {23{ Meanwhile,wecan(very)roughlysketchtheeldlinesandequipotentialsforthedipole(ontheleft)andforapairofchargesofthesamesign(ontheright): 2.2.5ElectrostaticEquilibriumHere'sasimplequestion:canyoutrapanelectricchargeusingonlyothercharges?Inotherwords,canyoundsomearrangementsofchargessuchthatatestchargesitsinstableequilibrium,trappedbytheeldsoftheothers.There'satrivialwaytodothis:justallowanegativechargetositdirectlyontopofapositivecharge.Butlet'sthrowoutthispossibility.We'llaskthattheequilibriumpointliesawayfromalltheothercharges.Therearesomesimpleset-upsthatspringtomindthatmightachievethis.Maybeyoucouldplacefourpositivechargesattheverticesofapyramid;orperhaps8positivechargesatthecornersofacube.Isitpossiblethatatestpositivechargetrappedinthemiddlewillbestable?It'scertainlyrepelledfromallthecorners,soitmightseemplausible.Theanswer,however,isno.Thereisnoelectrostaticequilibrium.Youcannottrapanelectricchargeusingonlyotherstationaryelectriccharges,atleastnotinastablemanner.Sincethepotentialenergyoftheparticleisproportionalto ,mathematically,thisisthestatementthataharmonicfunction,obeyingr2 =0,canhavenominimumormaximum.Toprovethattherecanbenoelectrostaticequilibrium,let'ssupposetheopposite:thatthereissomepointinemptyspacer?thatisstableforaparticleofchargeq�0.By\emptyspace",wemeanthat(r)=0inaneighbourhoodofr?.Becausethepointisstable,iftheparticlemovesawayfromthispointthenitmustalwaysbepushedback.This,inturn,meansthattheelectriceldmustalwayspointinwardstowardsthepointr?;neveraway.Wecouldthensurroundr?byasmallsurfaceSandcomputeZSEdS0{24{ But,byGauss'law,theright-handsidemustbethechargecontainedwithinSwhich,byassumption,iszero.Thisisourcontradiction:electrostaticequilibriumdoesnotexist.Ofcourse,ifyou'rewillingtousesomethingotherthanelectrostaticforcesthenyoucanconstructequilibriumsituations.Forexample,ifyourestrictthetestparticletolieonaplanethenit'ssimpletocheckthatequalchargesplacedatthecornersofapolygonwillresultinastableequilibriumpointinthemiddle.Buttodothisyouneedtouseotherforcestokeeptheparticleintheplaneintherstplace.2.3ElectrostaticEnergyThereisenergystoredintheelectriceld.Inthissection,wecalculatehowmuch.Let'sstartbyrecallingafactfromourrstcourseonclassicalmechanics1.Supposewehavesometestchargeqmovinginabackgroundelectrostaticpotential .We'lldenotethepotentialenergyoftheparticleasU(r).(WeusedthenotationV(r)inthe DynamicsandRelativity coursebutwe'llneedtoreserveVforthevoltagelater).ThepotentialU(r)oftheparticlecanbethoughtofastheworkdonebringingtheparticleinfrominnity;U(r)=�Zr1Fdr=+qZr1r dr=q (r)wherewe'veassumedourstandardnormalizationof (r)!0asr!1.Consideradistributionofchargeswhich,fornow,we'lltaketobemadeofpointchargesqiatpositionsri.Theelectrostaticpotentialenergystoredinthiscongurationisthesameastheworkrequiredtoassemblethecongurationintherstplace.(Thisisbecauseifyouletthechargesgo,thisishowmuchkineticenergytheywillpickup).Sohowmuchworkdoesittaketoassembleacollectionofcharges?Well,therstchargeisfree.Intheabsenceofanyelectriceld,youcanjustputitwhereyoulike|say,r1.TheworkrequiredisW1=0.Toplacethesecondchargeatr2takesworkW2=q1q2 401 jr1�r2jNotethatifthetwochargeshavethesamesign,soq1q2�0,thenW2�0whichistellingusthatweneedtoputworkintomakethemapproach.Ifq1q20thenW20wherethenegativeworkmeansthattheparticleswantedtobedrawncloserbytheirmutualattraction. 1SeeSection2.2ofthelecturenoteson DynamicsandRelativity .{25{ Thethirdchargehastobattleagainsttheelectriceldduetobothq1andq2.TheworkrequiredisW3=q3 40q2 jr2�r3j+q1 jr1�r3jandsoon.Thetotalworkneededtoassembleallthechargesisthepotentialenergystoredintheconguration,U=NXi=1Wi=1 40Xiqiqj jri�rjj(2.23)wherePimeansthatwesumovereachpairofparticlesonce.Infact,youprobablycouldhavejustwrittendown( 2.23 )asthepotentialenergystoredintheconguration.Thewholepurposeoftheaboveargumentwasreallyjusttonaildownafactorof1=2:dowesumoverallpairsofparticlesPiorallparticlesPi6=j?Theanswer,aswehaveseen,isallpairs.Wecanmakethatfactorof1=2evenmoreexplicitbywritingU=1 21 40XiXj6=iqiqj jri�rjj(2.24)wherenowwesumovereachpairtwice.Thereisaslickerwayofwriting( 2.24 ).Thepotentialatriduetoalltheotherchargesqj,j6=iis (ri)=1 40Xj6=iqj jri�rjjwhichmeansthatwecanwritethepotentialenergyasU=1 2NXi=1qi (ri)(2.25)Thisisthepotentialenergyforasetofpointcharges.Butthereisanobviousgeneral-izationtochargedistributions(r).We'llagainassumethat(r)hascompactsupportsothatthechargeislocalisedinsomeregionofspace.ThepotentialenergyassociatedtosuchachargedistributionshouldbeU=1 2Zd3r(r) (r)(2.26)wherewecanquitehappilytaketheintegraloverallofR3,safeintheknowledgethatanywherethatdoesn'tcontainchargehas(r)=0andsowon'tcontribute.{26{ Nowthisisinaformthatwecanstarttoplaywith.WeuseGauss'lawtorewriteitasU=0 2Zd3r(rE) =0 2Zd3r[r(E )�Er ]Butthersttermisatotalderivative.Andsincewe'retakingtheintegraloverallofspaceand (r)!0asr!1,thistermjustvanishes.Inthesecondtermwecanreplacer =�E.Wendthatthepotentialenergystoredinachargedistributionhasanelegantexpressionsolelyintermsoftheelectriceldthatitcreates,U=0 2Zd3rEE(2.27)Isn'tthatnice!2.3.1TheEnergyofaPointParticleThereisasubtletyintheabovederivation.Infact,Itotallytriedtopullthewooloveryoureyes.Hereit'stimetoownup.First,letmesaythatthenalresult( 2.27 )isright:thisistheenergystoredintheelectriceld.Butthederivationabovewasdodgy.Onereasontobedissatisedisthatwecomputedtheenergyintheelectriceldbyequatingittothepotentialenergystoredinachargedistributionthatcreatesthiselectriceld.Buttheendresultdoesn'tdependonthechargedistribution.Thissuggeststhatthereshouldbeamoredirectwaytoarriveat( 2.27 )thatonlytalksabouteldsanddoesn'tneedcharges.Andthereis.Wewillseeitlater.Butthereisalsoanother,moreworryingproblemwiththederivationabove.Toillustratethis,let'sjustlookatthesimplestsituationofapointparticle.ThishaselectriceldE=q 40r2^r(2.28)So,by( 2.27 ),theassociatedelectriceldshouldcarryenergy.Butwestartedourderivationabovebyassumingthatasingleparticledidn'tcarryanyenergysinceitdidn'ttakeanyworktoputtheparticlethereintherstplace.What'sgoingon?Well,therewassomethingofasleightofhandinthederivationabove.Thisoccurswhenwewentfromtheexpressionq in( 2.25 )to in( 2.26 ).Theformeromitsthe\self-energy"terms;thereisnocontributionarisingfromqi (ri).However,thelatterincludesthem.Thetwoexpressionsarenotquitethesame.Thisisalsothereasonthatournalexpressionfortheenergy( 2.27 )ismanifestlypositive,whileq canbepositiveornegative.{27{ Sowhichisright?Well,whichformoftheenergyyouuseratherdependsonthecontext.Itistruethat( 2.27 )isthecorrectexpressionfortheenergystoredintheelectriceld.Butitisalsotruethatyoudon'thavetodoanyworktoputtherstchargeinplacesincewe'reobviouslynotghtingagainstanything.Instead,the\self-energy"contributioncomingfromEEin( 2.28 )shouldsimplybethoughtof|usingE=mc2|asacontributiontothemassoftheparticle.Wecaneasilycomputethiscontributionfor,say,anelectronwithchargeq=�e.Let'scalltheradiusoftheelectrona.ThentheenergystoredinitselectriceldisEnergy=0 2Zd3rEE=e2 320Z1adr4r2 r4=e2 801 aWeseethat,atleastasfarastheenergyisconcerned,we'dbetternottreattheelectronasapointparticlewitha!0oritwillenduphavinginnitemass.Andthatwillmakeitreallyhardtomove.Sowhatistheradiusofanelectron?Fortheabovecalculationtobeconsistent,theenergyintheelectriceldcan'tbegreaterthantheobservedmassoftheelectronme.Inotherwords,we'dbetterhavemec2�e2 801 a)a�e2 801 mec2(2.29)That,atleast,putsaboundontheradiusoftheelectron,whichisthebestwecandousingclassicalphysicsalone.Togiveamoreprecisestatementoftheradiusoftheelectron,weneedtoturntoquantummechanics.AQuickForayintoQuantumElectrodynamicsToassignameaningof\radius"toseeminglypoint-likeparticles,wereallyneedthemachineryofquantumeldtheory.Inthatcontext,thesizeoftheelectroniscalleditsComptonwavelength.Thisisthedistancescaleatwhichtheelectrongetssurroundedbyaswarmofelectron-positronpairswhich,roughlyspeaking,smearsoutthechargedistribution.Thisdistancescaleisa=~ mecWeseethattheinequality( 2.29 )translatesintoaninequalityonabunchoffundamentalconstants.Forthewholestorytohangtogether,werequiree2 80~c1{28{ Thisisanalmostfamouscombinationofconstants.It'smoreusualtodenethecombination =e2 40~cThisisknownasthenestructureconstant.Itisdimensionlessandtakesthevalue 1 137Ourdiscussionaboverequires 2.WeseethatNaturehappilymeetsthisrequire-ment.2.3.2TheForceBetweenElectricDipolesAsanapplicationofourformulaforelectrostaticenergy,wecancomputetheforcebetweentwo,farseparateddipoles.Weplacetherstdipole,p1,attheorigin.Itgivesrisetoapotential (r)=1 40p1r r3Now,atsomedistanceaway,weplaceaseconddipole.We'lltakethistoconsistofachargeQatpositionrandacharge�Qatpositionr�d,withd r.Theresultingdipolemomentisp2=Qd.We'renotinterestedintheenergystoredineachindividualdipole;onlyinthepotentialenergyneededtobringthetwodipolestogether.Thisisgivenby( 2.23 ),U=Q (r)� (r�d)=Q 40p1r r3�p1(r�d) jr�dj3=Q 40p1r r3�p1(r�d)1 r3+3dr r5+:::=Q 40p1d r3�3(p1r)(dr) r5where,togettothesecondline,we'veTaylorexpandedthedenominatorofthesecondterm.Thisnalexpressioncanbewrittenintermsoftheseconddipolemoment.Wendthenice,symmetricexpressionforthepotentialenergyoftwodipolesseparatedbydistancer,U=1 40p1p2 r3�3(p1r)(p2r) r5{29{ But,weknowfromourrstcourseondynamicsthattheforcebetweentwoobjectsisjustgivenbyF=�rU.WelearnthattheforcebetweentwodipolesisgivenbyF=1 40r3(p1r)(p2r) r5�p1p2 r3(2.30)Thestrengthoftheforce,andevenitssign,dependsontheorientationofthetwodipoles.Ifp1andp2lieparalleltoeachotherandtorthentheresultingforceisattractive.Ifp1andp2pointinoppositedirections,andlieparalleltor,thentheforceisrepulsive.Theexpressionaboveallowsustocomputethegeneralforce.2.4ConductorsLet'snowthrowsomethingnewintothemix.Aconductorisaregionofspacewhichcontainschargesthatarefreetomove.Physically,think\metal".Wewanttoaskwhathappenstothestoryofelectrostaticsinthepresenceofaconductor.Thereareanumberofthingsthatwecansaystraightaway:InsideaconductorwemusthaveE=0.Ifthisisn'tthecase,thechargeswouldmove.Butwe'reinterestedinelectrostaticsituationswherenothingmoves.SinceE=0insideaconductor,theelectrostaticpotential mustbeconstantthroughouttheconductor.SinceE=0andrE==0,wemustalsohave=0.Thismeansthattheinterioroftheconductorcan'tcarryanycharge.Conductorscanbeneutral,carryingbothpositiveandnegativechargeswhichbalanceout.Alternatively,conductorscanhavenetcharge.Inthiscase,anynetchargemustresideatthesurfaceoftheconductor.Since isconstant,thesurfaceoftheconductormustbeanequipotential.ThismeansthatanyE=�r isperpendiculartothesurface.Thisalsotsnicelywiththediscussionabovesinceanycomponentoftheelectriceldthatliestan-gentialtothesurfacewouldmakethesurfacechargesmove.Ifthereissurfacechargeanywhereintheconductorthen,byourpreviousdiscontinuityresult( 2.9 ),togetherwiththefactthatE=0inside,theelectriceldjustoutsidetheconductormustbeE= 0^n(2.31){30{ Problemsinvolvingconductorsareofaslightlydi erentnaturethanthosewe'vediscusseduptonow.Thereasonisthatwedon'tknowfromthestartwherethechargesare,sowedon'tknowwhatchargedistributionthatweshouldbesolvingfor.Instead,theelectriceldsfromothersourceswillcausethechargesinsidetheconductortoshiftarounduntiltheyreachequilibriuminsuchawaythatE=0insidetheconductor.Ingeneral,thiswillmeanthatevenneutralconductorsendupwithsomesurfacecharge,negativeinsomeareas,positiveinothers,justenoughtogenerateanelectriceldinsidetheconductorthatpreciselycancelsthatduetoexternalsources.AnExample:AConductingSphereToillustratethekindofproblemthatwehavetodealwith,it'sprobablybestjusttogiveanexample.Consideraconstantbackgroundelectriceld.(Itcould,forexample,begeneratedbytwochargedplatesofthekindwelookedatinSection 2.1.4 ).Nowplaceaneutral,sphericalconductorinsidethiseld.Whathappens?Weknowthattheconductorcan'tsu eranelectriceldinsideit.Instead,themobilechargesintheconductorwillmove:thenegativeonestooneside;thepositiveonestotheother.Thespherenowbecomespolarised.ThesechargescounteractthebackgroundelectriceldsuchthatE=0insidetheconductor,whiletheelectriceldoutsideimpingesonthesphereatright-angles.Theendresultmustlookqualitativelylikethis: We'dliketounderstandhowtocomputetheelectriceldinthis,andrelated,situations.We'llgivetheanswerinSection 2.4.4 .AnApplication:FaradayCageConsidersomeregionofspacethatdoesn'tcontainanycharges,surroundedbyacon-ductor.Theconductorsitsatconstant = 0while,sincetherearenochargesinside,wemusthaver2 =0.Butthismeansthat = 0everywhere.Thisisbecause,ifitdidn'tthentherewouldbeamaximumorminimumof somewhereinside.AndweknowfromthediscussioninSection 2.2.5 thatthiscan'thappen.Therefore,insidearegionsurroundedbyaconductor,wemusthaveE=0.{31{ Thisisaveryusefulresultifyouwanttoshieldaregionfromelectricelds.Inthiscontext,thesurroundingconductoriscalledaFaradaycage.Asanapplication,ifyou'reworriedthatthey'retryingtoreadyourmindwithelectromagneticwaves,thenyouneedonlywrapyourheadintinfoilandallconcernsshouldbealleviated.2.4.1CapacitorsLet'snowsolvefortheelectriceldinsomeconductorproblems. Figure17:Thesimplestexamplesarecapacitors.Theseareapairofcon-ductors,onecarryingchargeQ,theothercharge�Q.ParallelPlateCapacitorTostart,we'lltaketheconductorstohave at,parallelsurfacesasshowninthegure.Weusuallyassumethatthedistancedbetweenthesurfacesismuchsmallerthanp A,whereAistheareaofthesurface.Thismeansthatwecanneglectthee ectsthatarisearoundtheedgeofplatesandwe'rejustiedinassumingthattheelectriceldbetweenthetwoplatesisthesameasitwouldbeiftheplateswereinniteinextent.TheproblemreducestothesameonethatweconsideredinSection 2.1.4 .Theelectriceldnecessarilyvanishesinsidetheconductorwhile,betweentheplateswehavetheresult( 2.10 ),E= 0^zwhere=Q=Aandwehaveassumedtheplatesareseparatedinthez-direction.WedenethecapacitanceCtobeC=Q VwhereVisthevoltageorpotentialdi erencewhichis,asthenamesuggests,thedif-ferenceinthepotential onthetwoconductors.SinceE=�d =dzisconstant,wemusthave =�Ez+c)V= (0)� (d)=Ed=Qd A0andthecapacitanceforparallelplatesofareaA,separatedbydistanced,isC=A0 dBecauseVwasproportionaltoQ,thechargehasdroppedoutofourexpressionforthecapacitance.Instead,Cdependsonlyonthegeometryoftheset-up.Thisisageneralproperty;wewillseeanotherexamplebelow.{32{ Capacitorsareusuallyemployedasamethodtostoreelectricalenergy.Wecanseehowmuch.Usingourresult( 2.27 ),wehaveU=0 2Zd3xEE=A0 2Zd0dz 02=Q2 2CThisistheenergystoredinaparallelplatecapacitor.ConcentricSphereCapacitorConsiderasphericalconductorofradiusR1.Aroundthiswe Figure18:placeanotherconductorintheshapeofasphericalshellwithinnersurfacelyingatradiusR2.Weaddcharge+Qtothesphereand�Qtotheshell.Fromourearlierdiscussionofchargedspheresandshells,weknowthattheelectriceldbe-tweenthetwoconductorsmustbeE=Q 40r2^rR1rR2Correspondingly,thepotentialis =Q 40rR1rR2andthecapacitanceisgivenbyC=40R1R2=(R2�R1).2.4.2BoundaryValueProblemsUntilnow,we'vethoughtofconductorsascarryingsomexedchargeQ.Thesecon-ductorsthensitatsomeconstantpotential .Ifthereareotherconductorsinthevicinitythatcarryadi erentchargethen,aswe'veseenabove,therewillbesomexedpotentialdi erence,V= betweenthem.However,wecanalsothinkofasubtlydi erentscenario.Supposethatweinsteadxthepotential inaconductor.Thismeansthat,whateverelsehappens,whateverotherchargesaredoingallaround,theconductorremainsataxed .Itneverdeviatesfromthisvalue.Now,thissoundsabitstrange.We'veseenabovethattheelectricpotentialofaconductordependsonthedistancetootherconductorsandalsoonthechargeitcarries.If remainsconstant,regardlessofwhatobjectsarearoundit,thenitmustmeanthatthechargeontheconductorisnotxed.Andthat'sindeedwhathappens.{33{ Havingconductorsatxed meansthatchargecan owinandoutoftheconductor.Weimplicitlyassumethatthereissomebackgroundreservoirofchargewhichtheconductorcandipinto,takingandgivingchargesothat remainsconstant.Wecanthinkofthisreservoirofchargeasfollows:supposethat,somewhereinthebackground,thereisahugeconductorwithsomechargeQwhichsitsatsomepotential .Toxthepotentialofanyotherconductor,wesimplyattachittooneofthisbigreservoir-conductor.Ingeneral,someamountofchargewill owbetweenthem.Thebigconductordoesn'tmissit,whilethesmallconductormakesuseofittokeepitselfatconstant .ThesimplestexampleofthesituationabovearisesifyouconnectyourconductortotheplanetEarth.Byconvention,thisistakentohave =0anditensuresthatyourconductoralsositsat =0.Suchconductorsaresaidtobegrounded.Inpractice,onemaygroundaconductorinsideachipinyourcellphonebyattachingitthemetalcasing.Mathematically,wecanconsiderthefollowingproblem.Takesomenumberofob-jects,Si.Someoftheobjectswillbeconductorsataxedvalueof i.OtherswillcarrysomexedchargeQi.ThiswillrearrangeitselfintoasurfacechargeisuchthatE=0insidewhile,outsidetheconductor,E=4^n.Ourgoalistounderstandtheelectriceldthatthreadsthespacebetweenalloftheseobjects.Sincethereisnochargesittinginthisspace,weneedtosolvetheLaplaceequationr2 =0subjecttooneoftwoboundaryconditionsDirichletBoundaryConditions:Thevalueof isxedonagivensurfaceSiNeumannBoundaryConditions:Thevalueofr ^nisxedperpendiculartoagivensurfaceSiNoticethat,foreachSi,weneedtodecidewhichofthetwoboundaryconditionswewant.Wedon'tgettochosebothofthem.Wethenhavethefollowingtheorem.Theorem:WitheitherDirichletorNeumannboundaryconditionschosenoneachsurfaceSi,theLaplaceequationhasauniquesolution.{34{ Proof:Supposethattherearetwosolutions, 1and 2withthesamespeciedbound-aryconditions.Let'sdenef= 1� 2.WecanlookatthefollowingexpressionZVd3rr(frf)=ZVd3rrfrf(2.32)wherether2ftermvanishesbytheLaplaceequation.But,bythedivergencetheorem,weknowthatZVd3rr(frf)=XiZSifrfdSHowever,ifwe'vepickedDirichletboundaryconditionsthenf=0ontheboundary,whileNeumannboundaryconditionsensurethatrf=0ontheboundary.Thismeansthattheintegralvanishesand,from( 2.32 ),wemusthaverf=0throughoutspace.ButifwehaveimposedDirichletboundaryconditionssomewhere,thenf=0onthatboundaryandsof=0everywhere.Alternatively,ifwehaveNeumannboundaryconditionsonallsurfacesthanrf=0everywhereandthetwosolutions 1and 2candi eronlybyaconstant.But,asdiscussedinSection 2.2 ,thisconstanthasnophysicalmeaning.2.4.3MethodofImagesForparticularlysimplesituations,thereisarathercutemethodthatwecanusetosolveproblemsinvolvingconductors.Althoughthistechniqueissomewhatlimited,itdoesgiveussomegoodintuitionforwhat'sgoingon.It'scalledthemethodofimages.AchargedparticlenearaconductingplaneConsideraconductorwhichllsallofspacex0.We'llgroundthisconductorsothat =0forx0.Then,atsomepointx=d&#x]TJ/;༕ ;.9;Ւ ;&#xTf 2;.39; 0 ;&#xTd [;0,weplaceachargeq.Whathappens?We'relookingforasolutiontothePoissonequationwithadelta-functionsourceatx=d=(d;0;0),togetherwiththerequirementthat =0ontheplanex=0.Fromourdiscussionintheprevioussection,there'sauniquesolutiontothiskindofproblem.Wejusthavetondit.Here'stheclevertrick.Forgetthatthere'saconductoratx0.Instead,supposethatthere'sacharge�qplacedoppositetherealchargeatx=�d.Thisiscalledtheimagecharge.Thepotentialforthispairofchargesisjustthepotential =1 40 q p (x�d)2+y2+z2�q p (x+d)2+y2+z2!(2.33){35{ Figure19:Aparticlenearaconductingplane...Figure20:...lookslikeadipoleByconstruction,thishasthepropertythat =0forx=0andithasthecorrectsourceatx=(d;0;0).Therefore,thismustbetherightsolutionwhenx0.Acartoonofthisisshowninthegures.Ofcourse,it'sthewrongsolutioninsidetheconductorwheretheelectriceldvanishes.Butthat'strivialtox:wejustreplaceitwith =0forx0.Withthesolution( 2.33 )inhand,wecannowdispensewiththeimagechargeandexplorewhat'sreallygoingon.Wecaneasilycomputetheelectriceldfrom( 2.33 ).Ifwefocusontheelectriceldinthexdirection,itisEx=�@ @x=q 40x�d jr�dj3�x+d jr+dj3x0Meanwhile,Ex=0forx0.ThediscontinuityofExatthesurfaceoftheconductordeterminestheinducedsurfacecharge( 2.31 ).Itis=Ex0jx=0=�q 2d (d2+y2+z2)3=2Weseethatthesurfacechargeismostlyconcentratedontheplaneatthepointclosesttotherealcharge.Asyoumoveaway,itfallso as1=(y2+z2)3=2.Wecancomputethetotalinducedsurfacechargebydoingasimpleintegral,qinduced=Zdydz=�qThechargeinducedontheconductorisactuallyequaltotheimagecharge.Thisisalwaystruewhenweusetheimagechargetechnique.Finally,asfarastherealcharge+qisconcerned,aslongasitsitsatx�0,itfeelsanelectriceldwhichisidenticalinallrespectstotheeldduetoanimagecharge�qembeddedintheconductor.Thismeans,inparticular,thatitwillexperienceaforceF=�q2 160d2^xThisforceisattractive,pullingthechargetowardstheconductor.{36{ AchargedparticlenearaconductingsphereWecanplayasimilargameforaparticlenearagrounded,conductingsphere.Thedetailsareonlyslightlymorecomplicated.We'lltakethespheretositattheoriginandhaveradiusR.Theparticlehaschargeqandsitsatx=d=(d;0;0),withd�R.Ourgoalistoplaceanimagechargeq0somewhereinsidethespheresothat =0onthesurface.Thereisawaytoderivetheanswerusingconformaltransformations.However,herewe'lljuststateit.Youshouldchooseaparticleofchargeq0=�qR=d,placedatx=R2=dand,bysymmetry,y=z=0.Acartoonofthisisshowninthegure. Figure21:Aparticlenearaconductingsphere...Figure22:...lookslikeaslightlydi erentdipoleTheresultingpotentialis =q 40 1 p (x�d)2+y2+z2�R d1 p (x�R2=d)2+y2+z2!Withalittlealgebra,youcancheckthat =0wheneverx2+y2+z2=R2.Withalittlemorealgebra,youcaneasilydeterminetheinducedsurfacechargeandcheckthat,whenintegratedoverthesphere,weindeedhaveqinduced=q0.Onceagain,ourchargeexperiencesaforcetowardstheconductor.Abovewe'veseenhowtotreatagroundedsphere.Butwhatifweinsteadhaveanisolatedconductorwithsomexedcharge,Q?It'seasytoadapttheproblemabove.WesimplyaddthenecessaryexcesschargeQ�q0asanimagethatsitsattheoriginofthesphere.Thiswillinduceanelectriceldwhichemergesradiallyfromthesphere.Becauseoftheprincipleofsuperposition,wejustaddthistothepreviouselectriceldandseethatitdoesn'tmessupthefactthattheelectriceldisperpendiculartothesurface.Thisisnowoursolution.2.4.4ManymanymoreproblemsTherearemanymoreproblemsthatyoucancookupinvolvingconductors,chargesandelectrostatics.Veryfewofthemcanbesolvedbytheimagechargemethod.Instead,you{37{ needtodevelopanumberofbasictoolsofmathematicalphysics.Afairlycomprehensivetreatmentofthiscanbefoundintherst100orsopagesofJackson.Fornow,Iwouldjustliketoleaveyouwiththesolutiontotheexamplethatkickedo thissection:whathappensifyoutakeaconductingsphereandplaceitinaconstantelectriceld?Thisproblemisn'tquitesolvedbytheimagechargemethod.Butit'ssolvedbysomethingsimilar:animagedipole.We'llworkinsphericalpolarcoordinatesandchosetheoriginal,constantelectriceldtopointinthe^zdirection,E0=E0^z) 0=�E0z=�E0rcosTaketheconductingspheretohaveradiusRandbecenteredonthetheorigin.Let'saddtothisanimagedipolewithpotential( 2.18 ).We'llplacethedipoleattheorigin,andorientitalongthezaxislikeso: Figure23:Aconductingspherebetweenchargedplates...Figure24:...lookslikeadipolebetweentheplatesTheresultingpotentialis =�E0r�R3 r2cosSincewe'veaddedadipoleterm,wecanbesurethatthisstillsolvestheLaplaceequationoutsidetheconductor.Moreover,byconstruction, =0whenr=R.Thisisallwewantedfromoursolution.Theinducedsurfacechargecanagainbecomputedbyevaluatingtheelectriceldjustoutsidetheconductor.Itis=�0@ @r=0E01+2R3 r3r=Rcos=30E0cosWeseethatthesurfacechargeispositiveinonehemisphereandnegativeintheother.Thetotalinducedchargeaveragestozero.{38{ 2.4.5AHistoryofElectrostaticsPerhapsthesimplestdemonstrationoftheattractivepropertiesofelectricchargecomesfromrubbingaballoononyourheadandstickingittothewall.Thisphenomenonwasknown,atleastinspirit,totheancientGreeksandiscreditedtoThalesofMiletusaround600BC.Although,intheabsenceofanyancientballoons,hehadtomakedowithpolishingpiecesofamberandwatchingitattractsmallobjects.Asystematic,scienticapproachtoelectrostaticsstartswithWilliamGilbert,physi-cist,physicianandone-timebursarofStJohnsCollege,Cambridge.(Rumourhasitthathe'dratherhavebeenatOxford.)Hismostimportantwork,DeMagnete,pub-lishedin1600showed,amongotherthings,thatmanymaterials,notjustamber,couldbeelectried.Withduedeference,hereferredtotheseas\electrics",derivedfromtheGreek\ o"(electron)meaning\amber".Thesearematerialsthatwenowcall\insulators".Therewasslowprogressoverthenext150years,muchofitdevotedtobuildingma-chineswhichcouldstoreelectricity.Anotablebreakthroughcamefromtheexperimentsofthelittle-knownEnglishscientistStephenGrey,whowasthersttoappreciatethatthedicultyinelectrifyingcertainobjectsisbecausetheyareconductors,withanychargequickly owingthroughthemandaway.Greyspentmostofhislifeasanam-ateurastronomer,althoughhisamateurstatusappearstobeinlargepartbecausehefellfoulofIsaacNewtonwhobarredhisentryintomoreprofessionalscienticcircles.Heperformedhisexperimentsonconductorsinthe1720s,lateinlifewhenthelackofanyincomelefthimdestituteandpensionedtoChaterhouse(whichwas,perhaps,theworld'sfanciestpoorhouse).UponNewton'sdeath,thescienticcommunityclamouredtomakeamends.GreywasawardedtheRoyalSociety'srstCopleymedal.Then,pre-sumablybecausetheyfeltguilty,hewasalsoawardedthesecond.Grey'sexperimentswerelaterreproducedbytheFrenchchemistCharlesFrancoisdeCisternayDuFay,whocametothewonderfulconclusionthatallobjectscanbeelectriedbyrubbingapartfrom\metals,liquidsandanimals".Hedoesnot,tomyknowledge,statehowmuchrubbingofanimalshetriedbeforegivingup.Hewasalsothersttonoticethatstaticelectricitycangiverisetobothattractiveandrepulsiveforces.Bythe1750s,thereweremanyexperimentsonelectricity,butlittletheorytoexplainthem.Mostideasrestedona uiddescriptionofelectricity,butargumentsragedoverwhetherasingle uidortwo uidswereresponsible.Theideathattherewerebothpositiveandnegativecharges,thenthoughtofasasurplusanddecitof uid,wasintroducedindependentlybythebotanistWilliamWatsonandtheUSfoundingfather{39{ BenjaminFranklin.Franklinisarguablythersttosuggestthatchargeisconservedalthoughhisstatementwasn'tquiteasconciseasthecontinuityequation:Itisnowdiscoveredanddemonstrated,bothhereandinEurope,thattheElectricalFireisarealElement,orSpeciesofMatter,notcreatedbytheFriction,butcollectedonly.BenjaminFranklin,1747Still,it'snicetoknowthatchargeisconservedbothintheUSandinEurope.Aquantitativeunderstandingofthetheoryofelectrostaticscameonlyinthe1760s.Anumberofpeoplesuggestedthattheelectrostaticforcefollowsaninverse-squarelaw,prominentamongthemJosephPriestlywhoisbetterknownforthediscoveryofOxygenand,ofatleastequalimportance,theinventionofsodawater.In1769,theScottishphysicistJohnRobisonannouncedthathehadmeasuredtheforcetofallo as1=r2:06.Thiswasbeforetheinventionoferrorbarsandheseemstoreceivelittlecredit.Aroundthesametime,theEnglishscientistHenryCavendish,discoverofHydrogenandweigheroftheEarth,performedanumberofexperimentstodemonstratetheinverse-squarelawbut,aswithhismanyofhisotherelectromagneticdiscoveries,hechosenottopublish.ItwaslefttoFrenchphysicistCharlesAugustindeCoulombtocleanup,publishingtheresultsofhisdenitiveexperimentsin1785ontheforcethatnowcarrieshisname.Initsnalform,Coulomb'slawbecomestransmutedintoGauss'law.Foronce,thiswasdonebythepersonafterwhomit'snamed.Gaussderivedthisresultin1835,althoughitwasn'tpublisheduntil1867.{40{ 3.MagnetostaticsChargesgiverisetoelectricelds.Currentgiverisetomagneticelds.Inthissection,wewillstudythemagneticeldsinducedbysteadycurrents.ThismeansthatweareagainlookingfortimeindependentsolutionstotheMaxwellequations.Wewillalsorestricttosituationsinwhichthechargedensityvanishes,so=0.WecanthensetE=0andfocusourattentiononlyonthemagneticeld.We'releftwithtwoMaxwellequationstosolve:rB=0J(3.1)andrB=0(3.2)IfyouxthecurrentdensityJ,theseequationshaveauniquesolution.Ourgoalinthissectionistondit.SteadyCurrentsBeforewesolve( 3.1 )and( 3.2 ),let'spausetothinkaboutthekindofcurrentsthatwe'reconsideringinthissection.Because=0,therecan'tbeanynetcharge.But,ofcourse,westillwantchargetobemoving!Thismeansthatwenecessarilyhavebothpositiveandnegativechargeswhichbalanceoutatallpointsinspace.Nonetheless,thesechargescanmovesothereisacurrenteventhoughthereisnonetchargetransport.Thismaysoundarticial,butinfactit'sexactlywhathappensinatypicalwire.Inthatcase,thereisbackgroundofpositivechargeduetothelatticeofionsinthemetal.Meanwhile,theelectronsarefreetomove.Buttheyallmovetogethersothatateachpointwestillhave=0.Thecontinuityequation,whichcapturestheconservationofelectriccharge,is@ @t+rJ=0Sincethechargedensityisunchanging(and,indeed,vanishing),wehaverJ=0Mathematically,thisisjustsayingthatifacurrent owsintosomeregionofspace,anequalcurrentmust owouttoavoidthebuildupofcharge.Notethatthisisconsistentwith( 3.1 )since,foranyvectoreld,r(rB)=0.{41{ 3.1Ampere'sLawTherstequationofmagnetostatics,rB=0J(3.3)isknownasAmpere'slaw.Aswithmanyofthesevectordif- Figure25:ferentialequations,thereisanequivalentformintermsofinte-grals.Inthiscase,wechoosesomeopensurfaceSwithboundaryC=@S.Integrating( 3.3 )overthesurface,wecanuseStokes'theoremtoturntheintegralofrBintoalineintegralovertheboundaryC,ZSrBdS=ICBdr=0ZSJdSRecallthatthere'sanimplicitorientationintheseequations.ThesurfaceScomeswithanormalvector^nwhichpointsawayfromSinonedirection.Thelineintegralaroundtheboundaryisthendoneintheright-handedsense,meaningthatifyoustickthethumbofyourrighthandinthedirection^nthenyourngerscurlinthedirectionofthelineintegral.TheintegralofthecurrentdensityoverthesurfaceSisthesamethingasthetotalcurrentIthatpassesthroughS.Ampere'slawinintegralformthenreadsICBdr=0I(3.4)Formostexamples,thisisn'tsucienttodeterminetheformofthemagneticeld;we'llusuallyneedtoinvoke( 3.2 )aswell.However,thereisonesimpleexamplewheresymmetryconsiderationsmeanthat( 3.4 )isallweneed...3.1.1ALongStraightWireConsideraninnite,straightwirecarryingcurrentI.We'lltakeittopointinthe^zdirection.Thesymmetryoftheproblemisjumpingupanddowntellingusthatweneedtousecylindricalpolarcoordinates,(r;';z),wherer=p x2+y2istheradialdistanceawayfromthewire.WetaketheopensurfaceStolieinthex�yplane,centeredonthewire.Forthelineintegralin( 3.4 )togivesomethingthatdoesn'tvanish,it'sclearthatthemagneticeldhastohavesomecomponentthatliesalongthecircumferenceofthedisc.{42{ But,bythesymmetryoftheproblem,that'sactuallythe Figure26:onlycomponentthatBcanhave:itmustbeoftheformB=B(r)^'.(Ifthiswasabittooquick,we'llderivethismorecarefullybelow).AnymagneticeldofthisformautomaticallysatisesthesecondMaxwellequationrB=0.WeneedonlyworryaboutAmpere'slawwhichtellsusICBdr=B(r)Z20rd'=2rB(r)=0IWeseethatthestrengthofthemagneticeldisB=0I 2r^'(3.5)Themagneticeldcirclesthewireusingthe"right-handrule":stickthethumbofyourrighthandinthedirectionofthecurrentandyourngerscurlinthedirectionofthemagneticeld.Notethatthesimplestexampleofamagneticeldfallso as1=r.Incontrast,thesimplestexampleofanelectriceld{thepointcharge{fallsofas1=r2.Youcantracethisdi erencebacktothegeometryofthetwosituations.Becausemagneticeldsaresourcedbycurrents,thesimplestexampleisastraightlineandthe1=rfall-o isbecausetherearetwotransversedirectionstothewire.Indeed,wesawinSection 2.1.3 thatwhenwelookatalineofcharge,theelectriceldalsodropso as1=r.3.1.2SurfaceCurrentsandDiscontinuitiesConsiderthe atplanelyingatz=0withasurfacecurrentdensitythatwe'llcallK.NotethatKisthecurrentperunitlength,asopposedtoJwhichisthecurrentperunitarea.Youcanthinkofthesurfacecurrentasabunchofwires,alllyingparalleltoeachother.We'lltakethecurrenttolieinthex-direction:K=K^xasshownbelow. Fromourpreviousresult,weknowthattheBeldshouldcurlaroundthecurrentintheright-handedsense.But,withaninnitenumberofwires,thiscanonlymeanthat{43{ Bisorientedalongtheydirection.Infact,fromthesymmetryoftheproblem,itmustlooklike withBpointinginthe�^ydirectionwhenz�0andinthe+^ydirectionwhenz0.WewriteB=�B(z)^ywithB(z)=�B(�z).WeinvokeAmpere'slawusingthefollowingopensurface: withlengthLintheydirectionandextendingtoz.WehaveICBdr=LB(z)�LB(�z)=2LB(z)=0KLsowendthatthemagneticeldisconstantaboveaninniteplaneofsurfacecurrentB(z)=0K 2z�0Thisisrathersimilartothecaseoftheelectriceldinthepresenceofaninniteplaneofsurfacecharge.Theanalogywithelectrostaticscontinues.Themagneticeldisnotcontinuousacrossaplaneofsurfacecurrent.WehaveB(z!0+)�B(z!0�)=0KInfact,thisisageneralresultthatholdsforanysurfacecurrentK.WecanprovethisstatementbyusingthesamecurvethatweusedintheFigureaboveandshrinkingit{44{ untilitbarelytouchesthesurfaceonbothsides.Ifthenormaltothesurfaceis^nandBdenotesthemagneticeldoneithersideofthesurface,then^nBj+�^nBj�=0K(3.6)Meanwhile,themagneticeldnormaltothesurfaceiscontinuous.(Toseethis,youcanuseaGaussianpillbox,togetherwiththeotherMaxwellequationrB=0).Whenwelookedatelectricelds,wesawthatthenormalcomponentwasdiscontinu-ousinthepresenceofsurfacecharge( 2.9 )whilethetangentialcomponentiscontinuous.Formagneticelds,it'stheotherwayaround:thetangentialcomponentisdiscontin-uousinthepresenceofsurfacecurrents.ASolenoidAsolenoidconsistsofasurfacecurrentthattravelsaroundacylin- Figure27:der.It'ssimplesttothinkofasinglecurrent-carryingwirewindingmanytimesaroundtheoutsideofthecylinder.(Strictlyspeaking,thecross-sectionalshapeofthesolenoiddoesn'thavetobeacircle{itcanbeanything.Butwe'llstickwithacirclehereforsimplicity).Tomakelifeeasy,we'llassumethatthecylinderisinnitelylong.Thisjustmeansthatwecanneglecte ectsduetotheends.We'llagainusecylindricalpolarcoordinates,(r;';z),withtheaxisofthecylinderalong^z.Bysymmetry,weknowthatBwillpointalongthez-axis.Itsmagnitudecandependonlyontheradialdistance:B=B(r)^z.Onceagain,anymagneticeldofthisformimmediatelysatisesrB=0.WesolveAmpere'slawindi erentialform.Anywhereotherthan Figure28:thesurfaceofthesolenoid,wehaveJ=0andrB=0)dB dr=0)B(r)=constantOutsidethesolenoid,wemusthaveB(r)=0sinceB(r)isconstantandweknowB(r)!0asr!1.Togureoutthemagneticeldinsidethesolenoid,weturntotheintegralformofAmpere'slawandconsiderthesurfaceS,boundedbythecurveCshowninthegure.Onlythelinethatrunsinsidethesolenoidcontributestothelineintegral.WehaveICBdr=BL=0INL{45{ whereNisthenumberofwindingsofwireperunitlength.Welearnthatinsidethesolenoid,theconstantmagneticeldisgivenbyB=0IN^z(3.7)Notethat,sinceK=IN,thisisconsistentwithourgeneralformulaforthedisconti-nuityofthemagneticeldinthepresenceofsurfacecurrents( 3.6 ).3.2TheVectorPotentialForthesimplecurrentdistributionsofthelastsection,symmetryconsiderationswereenoughtoleadustoamagneticeldwhichautomaticallysatisedrB=0(3.8)But,formoregeneralcurrents,thiswon'tbethecase.InsteadwehavetoensurethatthesecondmagnetostaticMaxwellequationisalsosatised.Infact,thisissimpletodo.WeareguaranteedasolutiontorB=0ifwewritethemagneticeldasthecurlofsomevectoreld,B=rA(3.9)HereAiscalledthevectorpotential.Whilemagneticeldsthatcanbewrittenintheform( 3.9 )certainlysatisfyrB=0,theconverseisalsotrue;anydivergence-freemagneticeldcanbewrittenas( 3.9 )forsomeA.(Actually,thisprevioussentenceisonlytrueifourspacehasasuitablysimpletopology.SincewenearlyalwaysthinkofspaceasR3orsomeopenballonR3,werarelyrunintosubtleties.ButifspacebecomesmoreinterestingthenthepossiblesolutionstorB=0alsobecomemoreinteresting.Thisisanalogoustothestoryoftheelectrostaticpotentialthatwementionedbrie yinSection 2.2 ).Usingtheexpression( 3.9 ),Ampere'slawbecomesrB=�r2A+r(rA)=0J(3.10)where,intherstequality,we'veusedastandardidentityfromvectorcalculus.ThisistheequationthatwehavetosolvetodetermineAand,throughthat,B.{46{ 3.2.1MagneticMonopolesAbove,wedispatchedwiththeMaxwellequationrB=0fairlyquicklybywritingB=rA.Butweneverpausedtothinkaboutwhatthisequationisactuallytellingus.Infact,ithasaverysimpleinterpretation:itsaysthattherearenomagneticcharges.Apoint-likemagneticchargegwouldsourcethemagneticeld,givingrisea1=r2fall-o B=g^r 4r2Anobjectwiththisbehaviourisusuallycalledamagneticmonopole.Maxwell'sequa-tionssaysthattheydon'texist.AndwehaveneverfoundoneinNature.However,wecouldask:howrobustisthisconclusion?Arewesurethatmagneticmonopolesdon'texist?Afterall,it'seasytoadaptMaxwell'sequationstoallowforpresenceofmagneticcharges:wesimplyneedtochange( 3.8 )toreadrB=mwheremisthemagneticchargedistribution.Ofcourse,thismeansthatwenolongergettousethevectorpotentialA.Butisthatsuchabigdeal?Thetwistcomeswhenweturntoquantummechanics.Becauseinquantummechan-icswe'reobligedtousethevectorpotentialA.NotonlyisthewholeframeworkofelectromagnetisminquantummechanicsbasedonwritingthingsusingA,butitturnsoutthatthereareexperimentsthatactuallydetectcertainpropertiesofAthatarelostwhenwecomputeB=rA.Iwon'texplainthedetailshere,butifyou'reinterestedthenlookupthe\Aharonov-Bohme ect".MonopolesAfterAll?Tosummarise,magneticmonopoleshaveneverbeenobserved.Wehavealawofphysics( 3.8 )whichsaysthattheydon'texist.AndwhenweturntoquantummechanicsweneedtousethevectorpotentialAwhichautomaticallymeansthat( 3.8 )istrue.Itsoundslikeweshouldprettymuchforgetaboutmagneticmonopoles,right?Well,no.Thereareactuallyverygoodreasonstosuspectthatmagneticmonopolesdoexist.ThemostimportantpartofthestoryisduetoDirac.HegaveabeautifulargumentwhichshowedthatitisinfactpossibletointroduceavectorpotentialAwhichallowsforthepresenceofmagneticcharge,butonlyifthemagneticchargegisrelatedtothechargeoftheelectronebyge=2~nn2Z(3.11)ThisisknownastheDiracquantizationcondition.{47{ Moreover,followingworkinthe1970sby'tHooftandPolyakov,wenowrealisethatmagneticmonopolesareubiquitousintheoriesofparticlephysics.Ourbestcurrenttheory{theStandardModel{doesnotpredictmagneticmonopoles.ButeverytheorythattriestogobeyondtheStandardModel,whetherGrandUniedTheories,orStringTheoryorwhatever,alwaysendsuppredictingthatmagneticmonopolesshouldexist.They'reoneofthefewpredictionsfornewphysicsthatnearlyalltheoriesagreeupon.Thesedaysmosttheoreticalphysiciststhinkthatmagneticmonopolesprobablyexistandtherehavebeenanumberofexperimentsaroundtheworlddesignedtodetectthem.However,whiletheoreticallymonopolesseemlikeagoodbet,theirfutureobservationalstatusisfarfromcertain.Wedon'tknowhowheavymagneticmonopoleswillbe,butallevidencesuggeststhatproducingmonopolesisbeyondthecapabilitiesofourcurrent(or,indeed,future)particleaccelerators.OuronlyhopeistodiscoversomethatNaturemadeforus,presumablywhentheUniversewasmuchyounger.Unfortunately,heretoothingsseemagainstus.Ourbesttheoriesofcosmology,inparticularin ation,suggestthatanymonopolesthatwerecreatedbackintheBigBanghavelongagobeendiluted.Ataguess,thereareprobablyonlyafew oatingaroundourentireobservableUniverse.Thechancesofonefallingintoourlapsseemslim.ButIhopeI'mwrong.3.2.2GaugeTransformationsThechoiceofAin( 3.9 )isfarfromunique:therearelotsofdi erentvectorpotentialsAthatallgiverisetothesamemagneticeldB.Thisisbecausethecurlofagradientisautomaticallyzero.Thismeansthatwecanalwaysaddanyvectorpotentialoftheformr forsomefunction andthemagneticeldremainsthesame,A0=A+r )rA0=rASuchachangeofAiscalledagaugetransformation.AswewillseeinSection 5.3.1 ,itiscloselytiedtothepossibleshiftsoftheelectrostaticpotential .Ultimately,suchgaugetransformationsplayakeyroleintheoreticalphysics.But,fornow,we'resimplygoingtousethistoouradvantage.Because,bypickingacunningchoiceof ,it'spossibletosimplifyourquestforthemagneticeld.Claim:Wecanalwaysndagaugetransformation suchthatA0satisesrA0=0.MakingthischoiceisusuallyreferredtoasCoulombgauge.Proof:Supposethatwe'vefoundsomeAwhichgivesusthemagneticeldthatwewant,sorA=B,butwhenwetakethedivergencewegetsomefunctionrA= (x).WeinsteadchooseA0=A+r whichnowhasdivergencerA0=rA+r2 = +r2 {48{ SoifwewantrA0=0,wejusthavetopickourgaugetransformation toobeyr2 =� ButthisisjustthePoissonequationagain.AndweknowfromourdiscussioninSection 2 thatthereisalwaysasolution.(Forexample,wecanwriteitdowninintegralformusingtheGreen'sfunction).SomethingaLittleMisleading:TheMagneticScalarPotentialThereisanotherquantitythatissometimesusedcalledthemagneticscalarpotential, .Theideabehindthispotentialisthatyoumightbeinterestedincomputingthemagneticeldinaregionwheretherearenocurrentsandtheelectriceldisnotchangingwithtime.Inthiscase,youneedtosolverB=0,whichyoucandobywritingB=�r Nowcalculationsinvolvingthemagneticeldreallydolookidenticaltothoseinvolvingtheelectriceld.However,youshouldbewaryofwritingthemagneticeldinthisway.Aswe'llseeinmoredetailinSection 5.3.1 ,wecanalwayssolvetwoofMaxwell'sequationsbywritingEandBintermsoftheelectricpotential andvectorpotentialAandthisformulationbecomesimportantaswemoveontomoreadvancedareasofphysics.Incontrast,writingB=�r isonlyusefulinalimitednumberofsituations.Thereasonforthisreallygetstotheheartofthedi erencebetweenelectricandmagneticelds:electricchargesexist;magneticchargesdon't!3.2.3Biot-SavartLawWe'renowgoingtousethevectorpotentialtosolveforthemagneticeldBinthepresenceofageneralcurrentdistribution.Fromnow,we'llalwaysassumethatwe'reworkinginCoulombgaugeandourvectorpotentialobeysrA=0.ThenAmpere'slaw( 3.10 )becomesawholeloteasier:wejusthavetosolver2A=�0J(3.12)Butthisisjustsomethingthatwe'veseenalready.Toseewhy,it'sperhapsbesttowriteitoutinCartesiancoordinates.Thisthenbecomesthreeequations,r2Ai=�0Ji(i=1;2;3)(3.13)andeachoftheseisthePoissonequation.{49{ It'sworthgivingawordofwarningatthispoint:theexpressionr2AissimpleinCartesiancoordinateswhere,aswe'veseenabove,itreducestotheLaplacianoneachcomponent.But,inothercoordinatesystems,thisisnolongertrue.TheLaplaciannowalsoactsonthebasisvectorssuchas^rand^'.Sointheseothercoordinatesystems,r2Aisalittlemoreofamess.(Youshouldprobablyusetheidentityr2A=�r(rA)+r(rA)ifyoureallywanttocomputeintheseothercoordinatesystems).Anyway,ifwesticktoCartesiancoordinatestheneverythingissimple.Infact,theresultingequations( 3.13 )areofexactlythesameformthatwehadtosolveinelectrostatics.And,inanalogyto( 2.21 ),weknowhowtowritedownthemostgeneralsolutionusingGreen'sfunctions.ItisAi(x)=0 4ZVd3x0Ji(x0) jx�x0jOr,ifyou'refeelingbold,youcanrevertbacktovectornotationandwriteA(x)=0 4ZVd3x0J(x0) jx�x0j(3.14)whereyou'vejustgottorememberthatthevectorindexonAlinksupwiththatonJ(andnotonxorx0).CheckingCoulombGaugeWe'vederivedasolutionto( 3.12 ),butthisisonlyasolutiontoAmpere'sequation( 3.10 )iftheresultingAobeystheCoulombgaugecondition,rA=0.Let'snowcheckthatitdoes.WehaverA(x)=0 4ZVd3x0rJ(x0) jx�x0jwhereyouneedtorememberthattheindexofrisdottedwiththeindexofJ,butthederivativeinrisactingonx,notonx0.WecanwriterA(x)=0 4ZVd3x0J(x0)r1 jx�x0j=�0 4ZVd3x0J(x0)r01 jx�x0jHerewe'vedonesomethingclever.Nowourr0isdi erentiatingwithrespecttox0.Togetthis,we'veusedthefactthatifyoudi erentiate1=jx�x0jwithrespecttoxthen{50{ yougetthenegativeoftheresultfromdi erentiatingwithrespecttox0.Butsincer0sitsinsideanRd3x0integral,it'sripeforintegratingbyparts.ThisgivesrA(x)=�0 4ZVd3x0r0J(x0) jx�x0j�r0J(x0)1 jx�x0jThesecondtermvanishesbecausewe'redealingwithsteadycurrentsobeyingrJ=0.Thersttermalsovanishesifwetakethecurrenttobelocalisedinsomeregionofspace,^VVsothatJ(x)=0ontheboundary@V.We'llassumethatthisisthecase.WeconcludethatrA=0and( 3.14 )isindeedthegeneralsolutiontotheMaxwellequations( 3.1 )and( 3.2 )aswe'dhoped.TheMagneticFieldFromthesolution( 3.14 ),itissimpletocomputethemagneticeldB=rA.Again,weneedtorememberthattheractsonthexin( 3.14 )ratherthanthex0.WendB(x)=0 4ZVd3x0J(x0)(x�x0) jx�x0j3(3.15)ThisisknownastheBiot-Savartlaw.Itdescribesthemagneticeldduetoageneralcurrentdensity.Thereisaslightvariationon( 3.15 )whichmoreoftengoesbythenameoftheBiot-Savartlaw.ThisarisesifthecurrentisrestrictedtoathinwirewhichtracesoutacurveC.Then,foracurrentdensityJpassingthroughasmallvolumeV,wewriteJV=(JA)xwhereAisthecross-sectionalareaofthewireandxliestangenttoC.Assumingthatthecross-sectionalareaisconstantthroughoutthewire,thecurrentI=JAisalsoconstant.TheBiot-SavartlawbecomesB(x)=0I 4ZCdx0(x�x0) jx�x0j3(3.16)ThisdescribesthemagneticeldduetothecurrentIinthewire.AnExample:TheStraightWireRevisitedOfcourse,wealreadyderivedtheanswerforastraightwirein( 3.5 )withoutusingthisfancyvectorpotentialtechnology.Beforeproceeding,weshouldquicklycheckthattheBiot-Savartlawreproducesourearlierresult.Asbefore,we'llworkincylindricalpolar{51{ coordinates.Wetakethewiretopointalongthe^zaxisanduse Figure29:r2=x2+y2asourradialcoordinate.Thismeansthatthelineelementalongthewireisparametrisedbydx0=^zdzand,forapointxawayfromthewire,thevectordx0(x�x0)pointsalongthetangenttothecircleofradiusr,dx0(x�x0)=r^'dzSowehaveB=0I^' 4Z+1�1dzr (r2+z2)3=2=0I 2r^'whichisthesameresultwefoundearlier( 3.5 ).3.2.4AMathematicalDiversion:TheLinkingNumberThere'sarathercuteapplicationoftheseideastopuremathematics.Considertwoclosed,non-intersectingcurves,CandC0,inR3.Foreachpairofcurves,thereisanintegern2Zcalledthelinkingnumberwhichtellsyouhowmanytimesoneofthecurveswindsaroundtheother.Forexample,herearepairsofcurveswithlinkingnumberjnj=0;1and2. Figure30:Curveswithlinkingnumbern=0,n=1andn=2.Todeterminethesignofthelinkingnumber,weneedtospecifytheorientationofeachcurve.Inthelasttwoguresabove,thelinkingnumbersarenegative,ifwetraversebothredandbluecurvesinthesamedirection.Thelinkingnumbersarepositiveifwetraverseonecurveinaclockwisedirection,andtheotherinananti-clockwisedirection.Importantly,thelinkingnumberdoesn'tchangeasyoudeformeithercurve,providedthatthetwocurvesnevercross.Infancylanguage,thelinkingnumberisanexampleofatopologicalinvariant.{52{ Thereisanintegralexpressionforthelinkingnumber,rstwrittendownbyGaussduringhisexplorationofelectromagnetism.TheBiot-Savartformula( 3.16 )o ersasimplephysicsderivationofGauss'expression.SupposethatthecurveCcarriesacurrentI.Thissetsusamagneticeldeverywhereinspace.WewillthencomputeHC0Bdx0aroundanothercurveC.(IfyouwantajusticationforcomputingHC0Bdx0thenyoucanthinkofitastheworkdonewhentransportingamagneticmonopoleofunitchargearoundC,butthisinterpretationisn'tnecessaryforwhatfollows.)TheBiot-SavartformulagivesIC0B(x0)dx0=0I 4IC0dx0ICdx(x0�x) jx�x0j3wherewe'vechangedourconventionssomewhatfrom( 3.16 ):nowxlabelscoordinatesonCwhilex0labelscoordinatesonC0.Meanwhile,wecanalsouseStokes'theorem,followedbyAmpere'slaw,towriteIC0B(x0)dx0=ZS0(rB)dS=0ZS0JdSwhereS0isasurfaceboundedbyC0.Thecurrentiscarriedbytheothercurve,C,whichpiercesS0preciselyntimes,sothatIC0B(x0)dx0=0ZS0JdS=n0IComparingthetwoequationsabove,wearriveatGauss'double-lineintegralexpressionforthelinkingnumbern,n=1 4IC0dx0ICdx(x0�x) jx�x0j3(3.17)NotethatournalexpressionissymmetricinCandC0,eventhoughthesetwocurvesplayedaratherdi erentphysicalroleintheoriginaldenition,withCcarryingacurrent,andC0thepathtracedbysomehypotheticalmonopole.Toseethattheexpressionisindeedsymmetric,notethatthetripleproductcanbethoughtofasthedeterminantdet(x0;x;x0�x).Swappingxandx0changestheorderofthersttwovectorsandchangesthesignofthethird,leavingthedeterminantuna ected.Theformula( 3.17 )isratherpretty.It'snotatallobviousthattheright-hand-sidedoesn'tchangeunder(non-crossing)deformationsofCandC0;norisitobviousthattheright-hand-sidemustgiveaninteger.Yetbotharetrue,asthederivationaboveshows.Thisisthersttimethatideasoftopologysneakintophysics.It'snotthelast.{53{ 3.3MagneticDipolesWe'veseenthattheMaxwellequationsforbidmagneticmonopoleswithalong-rangeB1=r2fall-o ( 3.11 ).Sowhatisthegenericfall-o forsomedistributionofcurrentswhicharelocalisedinaregionofspace?Inthissectionwewillseethat,ifyou'restandingsuitablyfarfromthecurrents,you'lltypicallyobserveadipole-likemagneticeld.3.3.1ACurrentLoopWestartwithaspecic,simpleexample.Consider Figure31:acircularloopofwireCofradiusRcarryingacurrentI.Wecanguesswhatthemagneticeldlookslikesimplybypatchingtogetherourresultforstraightwires:itmustroughlytaketheshapeshowninthegureHowever,wecanbemoreac-curate.Herewerestrictourselvesonlytothemag-neticeldfarfromtheloop.Tocomputethemagneticeldfaraway,wewon'tstartwiththeBiot-SavartlawbutinsteadreturntotheoriginalexpressionforAgivenin( 3.14 ).We'regoingtoreturntothenotationinwhichapointinspaceislabelledasrratherthanx.(Thisismoreappropriateforlong-distancedistanceeldswhichareessentiallyanexpansioninr=jrj).ThevectorpotentialisthengivenbyA(r)=0 4ZVd3r0J(r0) jr�r0jWritingthisintermsofthecurrentI(ratherthanthecurrentdensityJ),wehaveA(r)=0I 4ICdr0 jr�r0jWewanttoaskwhatthislookslikefarfromtheloop.Justaswedidfortheelectrostaticpotential,wecanTaylorexpandtheintegrandusing( 2.22 ),1 jr�r0j=1 r+rr0 r3+:::SothatA(r)=0I 4ICdr01 r+rr0 r3+:::(3.18){54{ Therstterminthisexpansionvanishesbecausewe'reintegratingaroundacircle.Thisisjustare ectionofthefactthattherearenomagneticmonopoles.Forthesecondterm,there'sawaytowriteitinslightlymoremanageableform.Toseethis,let'sintroduceanarbitraryconstantvectorgandusethistolookatICdr0g(rr0)Recallthat,fromthepointofviewofthisintegral,bothgandrareconstantvectors;it'sthevectorr0thatwe'reintegratingover.ThisisnowthekindoflineintegralofavectorthatallowsustouseStokes'theorem.WehaveICdr0g(rr0)=ZSdSr(g(rr0))=ZSdSiijk@0j(gkrlr0l)where,inthenalequality,we'veresortedtoindexnotationtohelpusrememberwhat'sconnectedtowhat.Nowthederivative@0actsonlyonther0andwegetICdr0g(rr0)=ZSdSiijkgkrj=gZSdSrButthisistrueforallconstantvectorsgwhichmeansthatitmustalsoholdasavectoridentityoncewestripawayg.WehaveICdr0(rr0)=Srwherewe'veintroducedthevectorareaSofthesurfaceSboundedbyC,denedasS=ZSdSIftheboundaryCliesinaplane{asitdoesforus{thenthevectorSpointsoutoftheplane.Nowlet'sapplythisresulttoourvectorpotential( 3.18 ).Withtheintegraloverr0,wecantreatrastheconstantvectorgthatweintroducedinthelemma.Withthersttermvanishing,we'releftwithA(r)=0 4mr r3(3.19)wherewe'veintroducedthemagneticdipolemomentm=IS{55{ Thisisournal,simple,answerforthelong-rangebehaviourofthevectorpotentialduetoacurrentloop.Itremainsonlytocomputethemagneticeld.AlittlealgebragivesB(r)=0 43(m^r)^r�m r3(3.20)Nowweseewhymiscalledthemagneticdipole;thisformofthemagneticeldisexactlythesameasthedipoleelectriceld( 2.19 ).IstressthattheBeldduetoacurrentloopandEeldduetotwochargesdon'tlookthesamecloseup.Buttheyhaveidentical\dipole"long-rangefall-o s.3.3.2GeneralCurrentDistributionsWecannowperformthesamekindofexpansionforageneralcurrentdistributionJlocalisedwithinsomeregionofspace.WeusetheTaylorexpansion( 2.22 )inthegeneralformofthevectorpotential( 3.14 ),Ai(r)=0 4Zd3r0Ji(r0) jr�r0j=0 4Zd3r0Ji(r0) r+Ji(r0)(rr0) r3+:::(3.21)wherewe'reusingacombinationofvectorandindexnotationtohelprememberhowtheindicesontheleftandright-handsidesmatchup.Thersttermabovevanishes.Heuristically,thisisbecausecurrentscan'tstopandend,theyhavetogoaroundinloops.Thismeansthatthecontributionfromonepartmustbecancelledbythecurrentsomewhereelse.Toseethismathematically,weusetheslightlyoddidentity@j(Jjri)=(@jJj)ri+Ji=Ji(3.22)wherethelastequalityfollowsfromthecontinuityconditionrJ=0.Usingthis,weseethatthersttermin( 3.21 )isatotalderivative(of@=@r0iratherthan@=@ri)whichvanishesifwetaketheintegraloverR3andkeepthecurrentlocalisedwithinsomeinteriorregion.Forthesecondtermin( 3.21 )weuseasimilartrick,nowwiththeidentity@j(Jjrirk)=(@jJj)rirk+Jirk+Jkri=Jirk+JkriBecauseJin( 3.21 )isafunctionofr0,weactuallyneedtoapplythistricktotheJir0jtermsintheexpression.Weonceagainabandontheboundarytermtoinnity.{56{ DroppingtheargumentofJ,wecanusetheidentityabovetowritetherelevantpieceofthesecondtermasZd3r0Jirjr0j=Zd3r0rj 2(Jir0j�Jjr0i)=Zd3r01 2(Ji(rr0)�r0i(Jr))Butnowthisisinaformthatisripeforthevectorproductidentitya(bc)=b(ac)�c(ab).ThismeansthatwecanrewritethistermasZd3r0J(rr0)=1 2rZd3r0Jr0(3.23)Withthisinhand,weseethatthelongdistancefall-o ofanycurrentdistributionagaintakesthedipoleform( 3.19 )A(r)=0 4mr r3nowwiththemagneticdipolemomentgivenbytheintegral,m=1 2Zd3r0r0J(r0)(3.24)Justasintheelectriccase,themultipoleexpansioncontinuestohigherterms.Thistimeyouneedtousevectorsphericalharmonics.Justasintheelectriccase,ifyouwantfurtherdetailsthenlookinJackson.3.4MagneticForcesWe'veseenthatacurrentproducesamagneticeld.Butacurrentissimplymovingcharge.AndweknowfromtheLorentzforcelawthatachargeqmovingwithvelocityvwillexperienceaforceF=qvBThismeansthatifasecondcurrentisplacedsomewhereintheneighbourhoodoftherst,thentheywillexertaforceononeanother.Ourgoalinthissectionistogureoutthisforce.3.4.1ForceBetweenCurrentsLet'sstartsimple.TaketwoparallelwirescarryingcurrentsI1andI2respectively.We'llplacethemadistancedapartinthexdirection.{57{ Thecurrentintherstwiresetsupamagneticeld Figure32:( 3.5 ).Soifthechargesinthesecondwirearemovingwithvelocityv,theywilleachexperienceaforceF=qvB=qv0I1 2d^ywhere^yisthedirectionofthemagneticeldexperiencedbythesecondwireasshownintheFigure.ThenextstepistowritethevelocityvintermsofthecurrentI2inthesecondwire.WedidthisinSection 1.1 whenwerstintroducedtheideaofcurrents:ifthere'sadensitynoftheseparticlesandeachcarrieschargeq,thenthecurrentdensityisJ2=nqvForawirewithcross-sectionalareaA,thetotalcurrentisjustI2=J2A.Forourset-up,J2=J2^z.Finally,wewanttocomputetheforceonthewireperunitlength,f.SincethenumberofchargesperunitlengthisnAandFistheforceoneachcharge,wehavef=nAF=0I1I2 2d^z^y=�0I1I2 2d^x(3.25)Thisisouranswerfortheforcebetweentwoparallelwires.Ifthetwocurrentsareinthesamedirection,sothatI1I2�0,theoverallminussignmeansthattheforcebetweentwowiresisattractive.Forcurrentsinoppositedirections,withI1I20,theforceisrepulsive.TheGeneralForceBetweenCurrentsWecanextendourdiscussiontotheforceexperiencedbetweentwocurrentdistributionsJ1andJ2.WestartbyconsideringthemagneticeldB(r)duetotherstcurrentJ1.Aswe'veseen,theBiot-Savartlaw( 3.15 )tellsusthatthiscanbewrittenasB(r)=0 4Zd3r0J1(r0)(r�r0) jr�r0j3IfthecurrentJ1islocalisedonacurveC1,thenwecanreplacethisvolumeintegralwiththelineintegral( 3.16 )B(r)=0I1 4IC1dr1(r�r1) jr�r1j3{58{ NowweplaceasecondcurrentdistributionJ2inthismagneticeld.Itexperiencesaforceperunitareagivenby( 1.3 ),sothetotalforceisF=Zd3rJ2(r)B(r)(3.26)Again,ifthecurrentJ2isrestrictedtolieonacurveC2,thenthisvolumeintegralcanbereplacedbythelineintegralF=I2IC2drB(r)andtheforcecannowbeexpressedasadoublelineintegral,F=0 4I1I2IC1IC2dr2dr1r2�r1 jr2�r1j3Ingeneral,thisintegralwillbequitetrickytoperform.However,ifthecurrentsarelocalised,andwell-separated,thereisasomewhatbetterapproachwheretheforcecanbeexpressedpurelyintermsofthedipolemomentofthecurrent.3.4.2ForceandEnergyforaDipoleWestartbyaskingaslightlydi erentquestion.We'llforgetaboutthesecondcurrentandjustfocusontherst:callitJ(r).We'llplacethiscurrentdistributioninamagneticeldB(r)andask:whatforcedoesitfeel?Ingeneral,therewillbetwokindsofforces.Therewillbeaforceonthecentreofmassofthecurrentdistribution,whichwillmakeitmove.Therewillalsobeatorqueonthecurrentdistribution,whichwillwanttomakeitre-orientitselfwithrespecttothemagneticeld.Herewe'regoingtofocusontheformer.Ratherremarkably,we'llseethatwegettheanswertothelatterforfree!TheLorentzforceexperiencedbythecurrentdistributionisF=ZVd3rJ(r)B(r)We'regoingtoassumethatthecurrentislocalisedinsomesmallregionr=RandthatthemagneticeldBvariesonlyslowlyinthisregion.ThisallowsustoTaylorexpandB(r)=B(R)+(rr)B(R)+:::{59{ WethengettheexpressionfortheforceF=�B(R)ZVd3rJ(r)+ZVd3rJ(r)[(rr)B(R)]+:::Thersttermvanishesbecausethecurrentshavetogoaroundinloops;we'vealreadyseenaproofofthisfollowingequation( 3.21 ).We'regoingtodosomeddlymanipula-tionswiththesecondterm.TohelpusrememberthatthederivativerisactingonB,whichisthenevaluatedatR,we'llintroduceadummyvariabler0andwritetheforceasF=ZVd3rJ(r)[(rr0)B(r0)]r0=R(3.27)Nowwewanttoplayaroundwiththis.First,usingthefactthatrB=0inthevicinityofthesecondcurrent,we'regoingtoshow,thatwecanrewritetheintegrandasJ(r)[(rr0)B(r0)]=�r0[(rB(r0))J(r)]Toseewhythisistrue,it'ssimplesttorewriteitinindexnotation.Aftershuingacoupleofindices,whatwewanttoshowis:ijkJj(r)rl@0lBk(r0)=ijkJj(r)rl@0kBl(r0)Or,subtractingonefromtheother,ijkJj(r)rl(@0lBk(r0)�@0kBl(r0))=0ButthetermsinthebracketsarethecomponentsofrBandsovanish.Soourresultistrueandwecanrewritetheforce( 3.27 )asF=�r0ZVd3r(rB(r0))J(r)r0=RNowweneedtomanipulatethisalittlemore.Wemakeuseoftheidentity( 3.23 )wherewereplacetheconstantvectorbyB.Thus,uptosomerelabelling,( 3.23 )isthesameasZVd3r(Br)J=1 2BZVd3rJr=�Bmwheremisthemagneticdipolemomentofthecurrentdistribution.Suddenly,ourexpressionfortheforceislookingmuchnicer:itreadsF=r(Bm){60{ wherewe'vedroppedther0=Rnotationbecause,havinglosttheintegral,there'snocauseforconfusion:themagneticdipolemisaconstant,whileBvariesinspace.Nowweinvokeastandardvectorproductidentity.UsingrB=0,thissimpliesandwe'releftwithasimpleexpressionfortheforceonadipoleF=r(Bm)(3.28)Afterallthatwork,we'releftwithsomethingremarkablysimple.Moreover,likemanyforcesinNewtonianmechanics,itcanbewrittenasthegradientofafunction.Thisfunction,ofcourse,istheenergyUofthedipoleinthemagneticeld,U=�Bm(3.29)ThisisanimportantexpressionthatwillplayaroleinlatercoursesinQuantumMechanicsandStatisticalPhysics.Fornow,we'lljusthighlightsomethingclever:wederived( 3.29 )byconsideringtheforceonthecentreofmassofthecurrent.ThisisrelatedtohowUdependsonr.Butournalexpressionalsotellsushowtheenergydependsontheorientationofthedipolematxedposition.Thisisrelatedtothetorque.Computingtheforcegivesusthetorqueforfree.Thisisbecause,ultimately,bothquantitiesarederivedfromtheunderlyingenergy.TheForceBetweenDipolesAsaparticularexampleoftheforce( 3.28 ),considerthecasewherethemagneticeldissetupbyadipolem1.Weknowthattheresultinglong-distancemagneticeldis( 3.24 ),B(r)=0 43(m1^r)^r�m1 r3(3.30)Nowwe'llconsiderhowthisa ectstheseconddipolem=m2.From( 3.28 ),wehaveF=0 4r3(m1^r)(m2^r)�m1m2 r3whereristhevectorfromm1tom2.Notethatthestructureoftheforceisidenticaltothatbetweentwoelectricdipolesin( 2.30 ).Thisisparticularlypleasingbecauseweusedtworatherdi erentmethodstocalculatetheseforces.Ifweactwiththederivative,wehaveF=30 4r4(m1^r)m2+(m2^r)m1+(m1m2)^r�5(m1^r)(m2^r)^r(3.31){61{ Firstnotethatifweswapm1andm2,sothatwealsosendr!�r,thentheforceswapssign.ThisisamanifestationofNewton'sthirdlaw:everyactionhasanequalandoppositereaction.Recallfrom DynamicsandRelativity lecturesthatweneededNewton'sthirdlawtoprovetheconservationofmomentumofacollectionofparticles.Weseethatthisholdsforabunchofdipolesinamagneticeld.ButtherewasalsoasecondparttoNewton'sthirdlaw:toprovetheconservationofangularmomentumofacollectionofparticles,weneededtheforcetolieparalleltotheseparationofthetwoparticles.Andthisisnottruefortheforce( 3.31 ).Ifyousetupacollectionofdipoles,theywillstartspinning,seeminglyincontradictionoftheconservationofangularmomentum.What'sgoingon?!Well,angularmomentumisconserved,butyouhavetolookelsewheretoseeit.Theangularmomentumcarriedbythedipolesiscompensatedbytheangularmomentumcarriedbythemagneticelditself.Finally,afewbasiccomments:thedipoleforcedropso as1=r4,quickerthantheCoulombforce.Correspondingly,itgrowsquickerthantheCoulombforceatshortdistances.Ifm1andm2pointinthesamedirectionandlieparalleltotheseparationR,thentheforceisattractive.Ifm1andm2pointinoppositedirectionsandlieparalleltotheseparationbetweenthem,thentheforceisrepulsive.Theexpression( 3.31 )tellsusthegeneralresult.3.4.3SoWhatisaMagnet?Untilnow,we'vebeentalkingaboutthemagneticeld Figure33:associatedtoelectriccurrents.Butwhenaskedtoen-visageamagnet,mostpeoplewouldthinkifapieceofmetal,possiblystucktotheirfridge,possiblyintheformofabarmagnetliketheoneshowninthepicture.Howaretheserelatedtoourdiscussionabove?Thesemetalsarepermanentmagnets.Theyoftenin-volveiron.Theycanbethoughtofascontainingmanymicroscopicmagneticdipoles,whichaligntoformalargemagneticdipoleM.Inabarmagnet,thedipoleMpointsbetweenthetwopoles.TheironlingsinthepicturetraceoutthemagneticeldwhichtakesthesameformthatwesawforthecurrentloopinSection 3.3 .Thismeansthattheleadingforcebetweentwomagnetsisdescribedbyourresult( 3.31 ).SupposethatM1,M2andtheseparationRallliealongaline.IfM1andM2{62{ pointinthesamedirection,thentheNorthpoleofonemagnetfacestheSouthpoleofanotherand( 3.31 )tellsusthattheforceisattractive.Alternatively,ifM1andM2pointinoppositedirectionsthentwopolesofthesametypefaceeachotherandtheforceisrepulsive.This,ofcourse,iswhatwealllearnedaskids.Theonlyremainingquestionis:wheredothemicroscopicdipolemomentsmcomefrom?Youmightthinkthattheseareduetotinyelectricatomiccurrentsbutthisisn'tquiteright.Instead,theyhaveamorefundamentalorigin.Theelectriccharges|whichareelectrons|possessaninherentangularmomentumcalledspin.RoughlyyoucanthinkoftheelectronasspinningarounditsownaxisinmuchthesamewayastheEarthspins.But,ultimately,spinisaquantummechanicalphenomenonandthisclassicalanalogybreaksdownwhenpushedtoofar.Themagnitudeofthespinis:s=1 2~where,recall,~hasthesamedimensionsasangularmomentum.Wecanpushtheclassicalanalogyofspinjustalittlefurther.Classically,anelectri-callychargedspinningballwouldgiverisetoamagneticdipolemoment.Soonemaywonderifthespinningelectronalsogivesrisetoamagneticdipole.Theanswerisyes.Itisgivenbym=ge 2mswhereeisthechargeoftheelectronandmisitsmass.Thenumbergisdimensionlessandcalled,ratheruninspiringly,theg-factor.Ithasbeenoneofthemostimportantnumbersinthehistoryoftheoreticalphysics,withseveralNobelprizesawardedtopeopleforcorrectlycalculatingit!Theclassicalpictureofaspinningelectronsuggestsg=1.Butthisiswrong.Therstcorrectprediction(and,correspondingly,rstNobelprize)wasbyDirac.Hisfamousrelativisticequationfortheelectrongivesg=2SubsequentlyitwasobservedthatDirac'spredictionisnotquiteright.Thevalueofgreceivescorrections.Thebestcurrentexperimentalvalueisg=2:00231930419922(1:510�12)Ratherastonishingly,thissamevaluecanbecomputedtheoreticallyusingtheframe-workofquantumeldtheory(specically,quantumelectrodynamics).Intermsofprecision,thisisoneofthegreattriumphsoftheoreticalphysics.{63{ Thereismuchmuchmoretothestoryofmagnetism,notleastwhatcausesthemagneticdipolesmtoalignthemselvesinamaterial.Thedetailsinvolvequantummechanicsandarebeyondthescopeofthiscourse.3.5UnitsofElectromagnetismMorethananyothersubject,electromagnetismisawashwithdi erentunits.Inlargepartthisisbecauseelectromagnetismhassuchdiverseapplicationsandeveryonefromastronomers,toelectricalengineers,toparticlephysicistsneedstouseit.Butit'sstillannoying.HereweexplainthebasicsofSIunits.TheSIunitofchargeistheCoulomb.Asof20192,theCoulombisdenedintermsofthecharge�ecarriedbytheelectron.Thisistakentobeexactlye=1:60217663410�19CIfyourubaballoononyoursweater,itpicksupachargeofaround10�6Corso.Aboltoflighteningdepositsachargeofabout15C.ThetotalchargethatpassesthroughanAAbatteryinitslifetimeisabout5000C.TheSIunitofcurrentistheAmpere,denotedA.ItisdenedasoneCoulombofchargepassingeverysecond.Thecurrentthatrunsthroughsingleionchannelsincellmembranesisabout10�12A.Thecurrentthatpowersyourtoasterisaround1Ato10A.ThereisacurrentintheEarth'satmosphere,knownastheBirkelandcurrent,whichcreatestheauroraandvariesbetween105Aand106A.Galacticsizecurrentsinso-calledSeyfertgalaxies(particularlyactivegalaxies)havebeenmeasuredatawhopping1018A.TheelectriceldismeasuredinunitsofNC�1.Theelectrostaticpotential hasunitsofVolts,denotedV,wherethe1Voltisthepotentialdi erencebetweentwoinnite,parallelplates,separatedby1m,whichcreateanelectriceldof1NC�1. 2Priorto2019,areluctancetorelyonfundamentalphysicsmeantthatthedenitionswerealittlemoretortuous.TheAmperewastakentobethebaseunit,andtheCoulombwasdenedastheamountofchargetransportedbyacurrentof1Ainasecond.TheAmpere,inturn,wasdenedtobethecurrentcarriedbytwostraight,parallelwireswhenseparatedbyadistanceof1m,inordertoexperienceanattractiveforce-per-unit-lengthof210�7Nm�1.(RecallthataNewtonistheunitofforceneededtoaccelerate1Kgat1ms�1.)Fromourresult( 3.25 ),weseethatifwepluginI1=I2=1Aandd=1mthenthisforceisf=0=2A2m�1.Thisdenitionisthereasonthat0hasthestrange-lookingvalue0=410�7mKgC�2.ThenewdenitionsofSIunitsmeansthatwecannolongersaywithcertaintythat0=410�7mKgC�2,butthisonlyholdsuptotheexperimentalaccuracyofadozensignicantguresorso.Forourpurposes,themainlessontodrawfromthisisthat,fromtheperspectiveoffundamentalphysics,SIunitsarearbitraryandalittledaft.{64{ Anervecellsitsataround10�2V.AnAAbatterysitsat1:5V.Thelargestman-madevoltageis107VproducedinavanderGraafgenerator.Thisdoesn'tcompetewellwithwhatNatureiscapableof.Thepotentialdi erencebetweentheendsofalighteningboltcanbe108V.Thevoltagearoundapulsar(aspinningneutronstar)canbe1015V.TheunitofamagneticeldistheTesla,denotedT.Aparticleofcharge1C,passingthroughamagneticeldof1Tat1ms�1willexperienceaforceof1N.Fromtheexamplesthatwe'veseenaboveit'sclearthat1Cisalotofcharge.Correspondingly,1Tisabigmagneticeld.Ourbestinstruments(SQUIDs)candetectchangesinmagneticeldsof10�18T.Themagneticeldinyourbrainis10�12T.ThestrengthoftheEarth'smagneticeldisaround10�5Twhileamagnetstucktoyourfridgehasabout10�3T.ThestrongestmagneticeldwecancreateonEarthisaround100T.Again,Naturebeatsusquiteconsiderably.Themagneticeldaroundneutronstarscanbebetween106Tand109T.(Thereisanexceptionhere:in\heavyioncollisions",inwhichgoldorleadnucleiaresmashedtogetherinparticlecolliders,itisthoughtthatmagneticeldscomparabletothoseofneutronstarsarecreated.However,thesemagneticeldsare eetingandsmall.Theyarestretchoverthesizeofanucleusandlastforamillionthofasecondorso).Astheabovediscussionamplydemonstrates,SIunitsarebasedentirelyonhistoricalconventionratherthananydeepunderlyingphysics.Amuchbetterchoiceistopickunitsofchargesuchthatwecandiscard0and0.Therearetwocommonlyusedframeworksthatdothis,calledLorentz-HeavisideunitsandGaussianunits.IshouldwarnyouthattheMaxwellequationstakeaslightlydi erentformineach.Tofullyembracenaturalunits,weshouldalsosetthespeedoflightc=1.(Seetherantinthe DynamicsandRelativity lectures).Howeverwecan'tseteverythingtoone.ThereisonecombinationofthefundamentalconstantsofNaturewhichisdimensionless.Itisknownasthenestructureconstant, =e2 40~candtakesvalue 1=137.Ultimately,thisisthecorrectmeasureofthestrengthoftheelectromagneticforce.Ittellsusthat,inunitswith0=~=c=1,thenatural,dimensionlessvalueofthechargeoftheelectronise0:3.3.5.1AHistoryofMagnetostaticsThehistoryofmagnetostatics,likeelectrostatics,startswiththeGreeks.Thefactthatmagneticironore,sometimesknownas\lodestone",canattractpiecesofironwas{65{ apparentlyknowntoThales.Hethoughtthathehadfoundthesoulinthestone.Theword\magnetism"comesfromtheGreektownMagnesia,whichissituatedinanarearichinlodestone.Ittookover1500yearstoturnThales'observationintosomethinguseful.Inthe11thcentury,theChinesescientistShenKuorealisedthatmagneticneedlescouldbeusedtobuildacompass,greatlyimprovingnavigation.Themodernstoryofmagnetismbegins,aswithelectrostatics,withWilliamGilbert.FromthetimeofThales,ithadbeenthoughtthatelectricandmagneticphenomenonarerelated.OneofGilbert'simportantdiscoverieswas,ironically,toshowthatthisisnotthecase:theelectrostaticforcesandmagnetostaticforcesaredi erent.Yetoverthenexttwocenturies,suspicionsremained.Severalpeoplesuggestedthatelectricandmagneticphenomenaareintertwined,althoughnocredibleargumentsweregiven.Thetwojustsmelledalike.ThefollowingunisightfulquotefromHenryElles,writtenin1757totheRoyalSociety,prettymuchsumsupthesituation:\Therearesomethingsinthepowerofmagnetismverysimilartothoseofelectricity.ButIdonotbyanymeansthinkthemthesame".Anumberofspecicrelationshipsbetweenelectricityandmagnetismweresuggestedandallsubsequentlyrefutedbyexperiment.Whenthebreakthroughnallycame,ittookeveryonebysurprise.In1820,theDan-ishscientistHansChristian rstednoticedthattheneedleonamagnetwasde ectedwhenacurrentwasturnedonoro .Afterthat,progresswasrapid.Withinmonths, rstedwasabletoshowthatasteadycurrentproducesthecircularmagneticeldaroundawirethatwehaveseenintheselectures.InSeptemberthatyear, rsted'sexperimentswerereproducedinfrontoftheFrenchAcademybyFrancoisArago,atalkwhichseemedtomobilisethecountry'sentirescienticcommunity.FirstoutoftheblockswereJean-BaptisteBiotandFelixSavartwhoquicklydeterminedthestrengthofthemagneticeldaroundalongwireandthemathematicallawwhichbearstheirname.OfthoseinspiredbyArago'stalk,themostimportantwasAndre-MarieAmpere.Skilledinbothexperimentalandtheoreticalphysics,Amperedeterminedtheforcesthatarisebetweencurrentcarryingwiresandderivedthemathematicallawwhichnowbearshisname:HBdr=0I.Hewasalsothersttopostulatethatthereexistsanatomofelectricity,whatwewouldnowcalltheelectron.Ampere'sworkwaspublishedin1827abookwiththecatchytitle\MemoirontheMathematicalTheoryofElectrodynamicPhenomena,UniquelyDeducedfromExperience".Itisnowviewedasthebeginningofthesubjectofelectrodynamics.{66{ 4.ElectrodynamicsForstaticsituations,Maxwell'sequationssplitintotheequationsofelectrostatics,( 2.1 )and( 2.2 ),andtheequationsofmagnetostatics,( 3.1 )and( 3.2 ).Theonlyhintthatthereisarelationshipbetweenelectricandmagneticeldscomesfromthefactthattheyarebothsourcedbycharge:electriceldsbystationarycharge;magneticeldsbymovingcharge.Inthissectionwewillseethattheconnectionbecomesmoredirectwhenthingschangewithtime.4.1Faraday'sLawofInduction\IwasatrstalmostfrightenedwhenIsawsuchmathematicalforcemadetobearuponthesubject,andthenwonderedtoseethatthesubjectstooditsowell."FaradaytoMaxwell,1857OneoftheMaxwellequationsrelatestimevaryingmagneticeldstoelectricelds,rE+@B @t=0(4.1)Thisequationtellsusthatifyouchangeamagneticeld,you'llcreateanelectriceld.Inturn,thiselectriceldcanbeusedtoacceleratechargeswhich,inthiscontext,isusuallythoughtofascreatingacurrentinwire.Theprocessofcreatingacurrentthroughchangingmagneticeldsiscalledinduction.We'llconsiderawiretobeaconductor,stretchedalong Figure34:astationary,closedcurve,C,asshowninthegure.Wewillrefertoclosedwiresofthistypeasa\circuit".Weintegratebothsidesof( 4.1 )overasurfaceSwhichisboundedbyC,ZS(rE)dS=�ZS@B @tdSByStokestheorem,wecanwritethisasZCEdr=�ZS@B @tdS=�d dtZSBdSRecallthatthelineintegralaroundCshouldbeintheright-handedsense;ifthengersonyourright-handcurlaroundCthenyourthumbpointsinthedirectionofdS.(ThismeansthatintheguredSpointsinthesamedirectionasB).Togetthelastequalityabove,weneedtousethefactthatneitherCnorSchangewithtime.Bothsides{67{ ofthisequationareusuallygivennames.TheintegraloftheelectriceldaroundthecurveCiscalledtheelectromotiveforce,E,oremfforshort,E=ZCEdrIt'snotagreatnamebecausetheelectromotiveforceisnotreallyaforce.Insteadit'sthetangentialcomponentoftheforceperunitcharge,integratedalongthewire.AnotherwaytothinkaboutitisastheworkdoneonaunitchargemovingaroundthecurveC.Ifthereisanon-zeroemfpresentthenthechargeswillbeacceleratedaroundthewire,givingrisetoacurrent.TheintegralofthemagneticeldoverthesurfaceSiscalledthemagnetic uxthroughS,=ZSBdSTheMaxwellequation( 4.1 )canbewrittenasE=�d dt(4.2)Inthisform,theequationisusuallycalledFaraday'sLaw.Sometimesitiscalledthe uxrule.Faraday'slawtellsusthatifyouchangethemagnetic uxthroughSthenacurrentwill ow.Thereareanumberofwaystochangethemagneticeld.Youcouldsimplymoveabarmagnetinthepresenceofcircuit,passingitthroughthesurfaceS;oryoucouldreplacethebarmagnetwithsomeothercurrentdensity,restrictedtoasecondwireC0,andmovethat;oryoucouldkeepthesecondwireC0xedandvarythecurrentinit,perhapsturningitonando .AllofthesewillinduceacurrentinC.However,thereisthenasecondarye ect.Whenacurrent owsinC,itwillcreateitsownmagneticeld.We'veseenhowthisworksforsteadycurrentsinSection 3 .Thisinducedmagneticeldwillalwaysbeinthedirectionthatopposesthechange.ThisiscalledLenz'slaw.Ifyoulike,\Lenz'slaw"isreallyjusttheminussigninFaraday'slaw( 4.2 ).{68{ Wecanillustratethiswithasimpleexample.Con- Figure35:Lenz'slawsiderthecasewhereCisacircle,lyinginaplane.We'llplaceitinauniformBeldandthenmakeBsmallerovertime,so_0.ByFaraday'slaw,E&#x]TJ/;༕ ;.9;Ւ ;&#xTf 1;.68; 0 ;&#xTd [;0andthecurrentwill owintheright-handeddirectionaroundCasshown.Butnowyoucanwrapyourright-handinadi erentway:pointyourthumbinthedirectionofthecurrentandletyourngerscurltoshowyouthedirectionoftheinducedmagneticeld.Thesearethecirclesdrawninthegure.YouseethattheinducedcurrentcausesBtoincreaseinsidetheloop,counteractingtheoriginaldecrease.Lenz'slawisratherlikealawofinertiaformagneticelds.Itisnecessarythatitworksthiswaysimplytoensureenergyconservation:iftheinducedmagneticeldaidedtheprocess,we'dgetanunstablerunawaysituationinwhichbothcurrentsandmagneticeldswereincreasingforever.4.1.1Faraday'sLawforMovingWiresThereisanother,relatedwaytoinducecur- Figure36:Movingcircuitrentsinthepresenceofamagneticeld:youcankeeptheeldxed,butmovethewire.Perhapsthesimplestexampleisshowninthegure:it'sarectangularcircuit,butwhereoneofthewiresisametalbarthatcanslidebackwardsandfor-wards.Thiswholeset-upisthenplacedinamagneticeld,whichpassesup,perpendicularthroughthecircuit.Slidethebartotheleftwithspeedv.EachchargeqinthebarexperiencesaLorentzforceqvB,pushingitintheydirection.Thisresultsinanemfwhich,now,isdenedastheintegratedforcepercharge.Inthiscase,theresultingemfisE=vBdwheredisthelengthofthemovingbar.But,becausetheareainsidethecircuitisgettingsmaller,the uxthroughCisalsodecreasing.Inthiscase,it'ssimpleto{69{ computethechangeof ux:itisd dt=�vBdWeseethatonceagainthechangeof uxisrelatedtotheemfthroughthe uxruleE=�d dtNotethatthisisthesameformula( 4.2 )thatwederivedpreviously,butthephysicsbehinditlookssomewhatdi erent.Inparticular,weusedtheLorentzforcelawanddidn'tneedtheMaxwellequations.Asinourpreviousexample,theemfwilldriveacurrentaroundtheloopC.And,justasinthepreviousexample,thiscurrentwillopposethemotionofthebar.Inthiscase,itisbecausethecurrentinvolveschargesmovingwithsomespeeduaroundthecircuit.ThesetoofeelaLorentzforcelaw,nowpushingthebarbacktotheright.Thismeansthatifyouletthebargo,itwillnotcontinuewithconstantspeed,eveniftheconnectionisfrictionless.Insteaditwillslowdown.ThisistheanalogofLenz'slawinthepresentcase.We'llreturntothisexampleinSection 4.1.3 andcomputethebar'ssubsequentmotion. Figure37:MovingCircuitsTheGeneralCaseThereisanicewaytoincludeboththee ectsoftime-dependentmagneticeldsandthepossibilitythatthecircuitCchangeswithtime.WeconsiderthemovingloopC(t),asshowninthegure.Nowthechangein uxthroughasurfaceShastwoterms:onebecauseBmaybechanging,andonebecauseCischanging.Inasmalltimet,wehave=(t+t)�(t)=ZS(t+t)B(t+t)dS�ZS(t)B(t)dS=ZS(t)@B @ttdS+ZS(t+t)�ZS(t)B(t)dS+O(t2)Wecandosomethingwiththemiddleterms.ConsidertheclosedsurfacecreatedbyS(t)andS(t+t),togetherwiththecylindricalregionsweptoutbyC(t)whichwecallSc.BecauserB=0,theintegralofB(t)overanyclosedsurfacevanishes.But{70{ RS(t+t)�RS(t)isthetopandbottompartoftheclosedsurface,withtheminussignjustensuringthattheintegraloverthebottompartS(t)isintheoutwarddirection.ThismeansthatwemusthaveZS(t+t)�ZS(t)B(t)dS=�ZScB(t)dSFortheintegraloverSc,wecanwritethesurfaceelementasdS=(drv)twheredristhelineelementalongC(t)andvisthevelocityofapointonC.Wendthattheexpressionforthechangein uxcanbewrittenasd dt=limt!0 t=ZS(t)@B @tdS�ZC(t)(vB)drwherewe'vetakenthelibertyofrewriting(drv)B=dr(vB).NowweusetheMaxwellequation( 4.1 )torewritethe@B=@tintermsoftheelectriceld.Thisgivesusournalexpressiond dt=�ZC(E+vB)drwheretheright-handsidenowincludestheforcetangentialtothewirefrombothelectriceldsandalsofromthemotionofthewireinthepresenceofmagneticelds.Theelectromotiveforceshouldbedenedtoincludebothofthesecontributions,E=ZC(E+vB)drandweonceagaingetthe uxruleE=�d=dt.4.1.2InductanceandMagnetostaticEnergyInSection 2.3 ,wecomputedtheenergystoredintheelectriceldbyconsideringtheworkdoneinbuildingupacollectionofcharges.Butwedidn'trepeatthiscalculationforthemagneticeldinSection 3 .Thereasonisthatweneedtheconceptofemftodescribetheworkdoneinbuildingupacollectionofcurrents.SupposethataconstantcurrentI owsalongsomecurveC.FromtheresultsofSection 3 weknowthatthisgivesrisetoamagneticeldandhencea ux=RSBdSthroughthesurfaceSboundedbyC.NowincreasethecurrentI.Thiswillincreasethe ux.Butwe'vejustlearnedthattheincreasein uxwill,inturn,induceanemfaroundthecurveC.TheminussignofLenz'slawensuresthatthisactstoresistthechangeofcurrent.Theworkneededtobuildupacurrentiswhat'sneededtoovercomethisemf.{71{ InductanceIfacurrentI owingaroundacurveCgivesrisetoa ux=RSBdSthentheinductanceLofthecircuitisdenedtobeL= ITheinductanceisapropertyonlyofourchoiceofcurveC.AnExample:TheSolenoidAsolenoidconsistsofacylinderoflengthlandcross-sectionalareaA. Figure38:Wetakel p Asothatanyend-e ectscanbeneglected.AwirewrappedaroundthecylindercarriescurrentIandwindsNtimesperunitlength.Wepreviouslycomputedthemagneticeldthroughthecentreofthesolenoidtobe( 3.7 )B=0INThismeansthata uxthroughasingleturnis0=0INA.ThesolenoidconsistsofNlturnsofwire,sothetotal uxis=0IN2Al=0IN2VwithV=Althevolumeinsidethesolenoid.TheinductanceofthesolenoidisthereforeL=0N2VMagnetostaticEnergyThedenitionofinductanceisusefultoderivetheenergystoredinthemagneticeld.Let'stakeourcircuitCwithcurrentI.We'lltrytoincreasethecurrent.TheinducedemfisE=�d dt=�LdI dtAswementionedabove,theinducedemfcanbethoughtofastheworkdoneinmovingaunitchargearoundthecircuit.ButwehavecurrentI owingwhichmeansthat,intimet,achargeItmovesaroundthecircuitandtheamountofworkdoneisW=EIt=�LIdI dtt)dW dt=�LIdI dt=�L 2dI2 dt{72{ Theworkneededtobuildupthecurrentisjusttheoppositeofthis.Integratingovertime,welearnthatthetotalworknecessarytobuildupacurrentIalongacurvewithinductanceLisW=1 2LI2=1 2IFollowingourdiscussionforelectricenergyin( 2.3 ),weidentifythiswiththeenergyUstoredinthesystem.WecanwriteitasU=1 2IZSBdS=1 2IZSrAdS=1 2IICAdr=1 2Zd3xJAwhere,inthelaststep,we'veusedthefactthatthecurrentdensityJislocalisedonthecurveCtoturntheintegralintooneoverallofspace.AtthispointweturntotheMaxwellequationrB=0JtowritetheenergyasU=1 20Zd3x(rB)A=1 20Zd3x[r(BA)+B(rA)]WeassumethatBandAfallo fastenoughatinnitysothatthersttermvanishes.We'releftwiththesimpleexpressionU=1 20Zd3xBBCombiningthiswithourpreviousresult( 2.27 )fortheelectriceld,wehavetheenergystoredintheelectricandmagneticelds,U=Zd3x0 2EE+1 20BB(4.3)Thisisaniceresult.Butthere'ssomethingalittleunsatisfactorybehindourderivationof( 4.3 ).First,wereiterateacomplaintfromSection 2.3 :wehadtoapproachtheenergyinboththeelectricandmagneticeldsinaratherindirectmanner,byfocussingnotontheeldsbutontheworkdonetoassemblethenecessarychargesandcurrents.There'snothingwrongwiththis,butit'snotaveryelegantapproachanditwouldbenicetounderstandtheenergydirectlyfromtheeldsthemselves.OnecandobetterbyusingtheLagrangianapproachtoMaxwell'sequations.Second,wecomputedtheenergyfortheelectriceldsandmagneticeldsaloneandthensimplyaddedthem.Wecan'tbesure,atthispoint,thatthereisn'tsomemixedcontributiontotheenergysuchasEB.Itturnsoutthattherearenosuchterms.Again,we'llpostponeaproofofthisuntilthenextcourse.{73{ 4.1.3ResistanceYoumayhavenoticedthatourdiscussionabovehasbeenalittlequalitative.Ifthe uxchanges,wehavegivenexpressionsfortheinducedemfEbutwehavenotgivenanexplicitexpressionfortheresultingcurrent.Andthere'sagoodreasonforthis:it'scomplicated.Thepresenceofanemfmeansthatthereisaforceonthechargesinthewire.AndweknowfromNewtonianmechanicsthataforcewillcausethechargestoaccelerate.Thisiswherethingsstarttogetcomplicated.Acceleratingchargeswillemitwavesofelectromagneticradiation,aprocessthatyouwillexplorelater.Relatedly,therewillbeanoppositiontotheformationofthecurrentthroughtheprocessthatwe'vecalledLenz'slaw.Sothingsaretricky.What'smore,inrealwiresandmaterialsthereisyetanothercomplication:friction.Throughouttheselectureswehavemodelledourchargesasiftheyaremovingunimpeded,whetherthroughthevacuumofspaceorthroughacon-ductor.Butthat'snotthecasewhenelectronsmoveinrealmaterials.Instead,there'sstu thatgetsintheirway:variousmessyimpuritiesinthematerial,orsoundwaves(usuallycalledphononsinthiscontext)whichknockthemo -course,orevenotherelectrons.Allthesee ectscontributetoafrictionforcethatactsonthemovingelec-trons.Theupshotofthisisthattheelectronsdonotaccelerateforever.Infact,theydonotaccelerateforverylongatall.Instead,theyveryquicklyreachanequilibriumspeed,analogoustothe\terminalvelocity"thatparticlesreachwhenfallingingrav-itationaleldwhileexperiencingairresistance.Inmanycircumstances,theresultingcurrentIisproportionaltotheappliedemf.ThisrelationshipiscalledOhm'slaw.ItisE=IR(4.4)TheconstantofproportionalityRiscalledtheresistance.TheemfisE=REdx.IfwewriteE=�r ,thenE=V,thepotentialdi erencebetweentwoendsofthewire.ThisgivesustheversionofOhm'slawthatisfamiliarfromschool:V=IR.TheresistanceRdependsonthesizeandshapeofthewire.IfthewirehaslengthLandcross-sectionalareaA,wedenetheresistivityas=AR=L.(It'sthesameGreekletterthatweearlierusedtodenotechargedensity.They'renotthesamething.Sorryforanyconfusion!)Theresistivityhastheadvantagethatit'sapropertyofthematerialonly,notitsdimensions.Alternatively,wetalkabouttheconductivity=1=.(ThisisthesameGreekletterthatwepreviouslyusedtodenotesurface{74{ chargedensity.They'renotthesamethingeither.)ThegeneralformofOhm'slawisthenJ=EUnliketheMaxwellequations,Ohm'slawdoesnotrepresentafundamentallawofNature.Itistrueinmany,perhapsmost,materials.Butnotall.Thereisaverysimpleclassicalmodel,knownastheDrudemodel,whichtreatselectronsasbilliardballsexperiencinglineardragwhichgivesrisetoOhm'slaw..ButaproperderivationofOhm'slawneedsquantummechanicsandamoremicroscopicunderstandingofwhat'shappeninginmaterials.Needlesstosay,thisis(way)beyondthescopeofthiscourse.So,atleastinthissmallsection,wewilltakeOhm'slaw( 4.4 )asanextrainputinourtheory.WhenOhm'slawholds,thephysicsisverydi erent.Nowtheappliedforce(or,inthiscase,theemf)isproportionaltothevelocityoftheparticlesratherthantheacceleration.It'slikelivingintheworldthatAristotleenvisagedratherthantheoneGalileounderstood.Butitalsomeansthattheresultingcalculationstypicallybecomemuchsimpler.AnExampleLet'sreturntoourpreviousexampleofaslid- Figure39:ingbaroflengthdandmassmwhichformsacircuit,sittinginamagneticeldB=B^z.Butnowwewilltakeintoaccountthee ectofelec-tricalresistance.WetaketheresistanceoftheslidingbartobeR.Butwe'llmakelifeeasyforourselvesandassumethattheresistanceoftherestofthecircuitisnegligible.Therearetwodynamicaldegreesoffreedominourproblem:thepositionxoftheslidingbarandthecurrentIthat owsaroundthecircuit.WetakeI�0ifthecurrent owsalongthebarinthepositive^ydirection.TheLorentzforcelawtellsusthattheforceonasmallvolumeofthebarisF=IB^y^z.TheforceonthewholebaristhereforeF=IBd^xTheequationofmotionforthepositionofthewireisthenmx=IBd{75{ NowweneedanequationthatgovernsthecurrentI(t).Ifthetotalemfaroundthecircuitcomesfromtheinducedemf,wehaveE=�d dt=�Bd_xOhm'slawtellsusthatE=IR.Combiningthese,wegetasimpledi erentialequationforthepositionofthebarmx=�B2d2 R_xwhichwecansolvetoseethatanyinitialvelocityofthebar,v,decaysexponentially:_x(t)=�ve�B2d2t=mRNotethat,inthiscalculationweneglectedthemagneticeldcreatedbythecurrent.It'ssimpletoseethequalitativee ectofthis.Ifthebarmovestotheleft,so_x0,thenthe uxthroughthecircuitdecreases.TheinducedcurrentisI&#x]TJ/;༕ ;.9;Ւ ;&#xTf 2;.39; 0 ;&#xTd [;0whichincreasesBinsidethecircuitwhich,inaccordwithLenz'slaw,attemptstocounteractthereduced ux.Intheabovederivation,weassumedthatthetotalemfaroundthecircuitwaspro-videdbytheinducedemf.Thisistantamounttosayingthatnocurrent owswhenthebarisstationary.ButwecanalsorelaxthisassumptionandincludeinouranalysisanemfE0acrossthecircuit(provided,forexample,byabattery)whichinducesacurrentI0=E0d=R.NowthetotalemfisE=E0+Einduced=E0�Bd_xThetotalcurrentisagaingivenbyOhmslawI=E=R.Thepositionofthebarisnowgovernedbytheequationmx=�Bd R(E0�Bd_x)Again,it'ssimpletosolvethisequation.JouleHeatingInSection 4.1.2 ,wecomputedtheworkdoneinchangingthecurrentinacircuitC.Thisignoredthee ectofresistance.Infact,ifweincludetheresistanceofawirethenweneedtodoworkjusttokeepaconstantcurrent.Thisshouldbeunsurprising.It'sthesamestatementthat,inthepresenceoffriction,weneedtodoworktokeepanobjectmovingataconstantspeed.{76{ Let'sreturntoaxedcircuitC.Aswementionedabove,ifabatteryprovidesanemfE0,theresultingcurrentisI=E0=R.Wecannowrunthroughargumentssimilartothosethatwesawwhencomputingthemagnetostaticenergy.TheworkdoneinmovingaunitchargearoundCisE0whichmeansthatamountofworknecessarytokeepacurrentImovingfortimetisW=E0It=I2RtWelearnthatthepower(workperunittime)dissipatedbyacurrentpassingthroughacircuitofresistanceRisdW=dt=I2R.Thisisnotenergythatcanbeusefullystoredlikethemagneticandelectricenergy( 4.3 );insteaditislosttofrictionwhichiswhatwecallheat.(Thedi erencebetweenheatandotherformsofenergyisexplainedintheThermodynamicssectionintheStatisticalPhysicsnotes).TheproductionofheatbyacurrentiscalledJouleheatingor,sometimes,Ohmicheating.4.1.4MichaelFaraday(1791-1867)\Theword\physicist"isbothtomymouthandearssoawkwardthatIthinkIshallneverbeabletouseit.Theequivalentofthreeseparatesoundsof\s"inonewordistoomuch."FaradayinalettertoWilliamWhewell3MichaelFaraday'srouteintosciencewasfarfromthestandardone.Thesonofablacksmith,hehadlittleschoolingand,attheageof14,wasapprenticedtoabookbinder.Thereheremaineduntiltheageof20whenFaradayattendedaseriesofpopularlecturesattheRoyalInstitutionbythechemistSirHumphryDavy.Inspired,Faradaywroteuptheselectures,lovinglyboundthemandpresentedthemtoDavyasagift.Davywasimpressedandsomemonthslater,aftersu eringaneyeinjuryinanexplosion,turnedtoFaradaytoactashisassistant.Notlongafter,Davydecidedtoretireandtakeatwo-yearleisurelytourofEurope,meetingmanyofthecontinent'stopscientistsalongtheway.HeaskedFaradaytojoinhimandhiswife,halfasassistant,halfasvalet.Thesciencepartofthiswasasuccess;thevaletpartlessso.ButFaradaydutifullyplayedhisroles,emptyinghismaster'schamberpoteachmorning,whileaidinginanumberofimportantscienticdiscoveriesalongtheway,includingawonderfulcaperinFlorencewhereDavyandFaradayusedGalileo'soldlenstoburnadiamond,reducingit,forthersttime,toCarbon. 3Accordingtotherestoftheinternet,Faradaycomplainsaboutthreeseparatesoundsof\i".Therestoftheinternetiswrongandcan'treadFaraday'swriting.TheoriginalletterisintheWrenlibraryinTrinityCollegeandisshownonthenextpage.I'mgratefultoFrankJames,editorofFaraday'scorrespondence,forhelpwiththis.{77{ BackinEngland,Faradaystartedworkatthe Figure40:RoyalInstitution.Hewouldremainthereforover45years.AnearlyattempttostudyelectricityandmagnetismwasabandonedafteraprioritydisputewithhisformermentorDavyanditwasonlyafterDavy'sdeathin1829thatFaradayturnedhisat-tentionsfullytothesubject.Hemadehisdiscoveryofinductionon28thOctober,1831.Theinitialex-perimentinvolvedtwo,separatedcoilsofwire,bothwrappedaroundthesamemagnet.Turningonacurrentinonewireinducesamomentarycurrentinthesecond.Soonafter,hefoundthatacurrentisalsoinducedbypassingaloopofwireoveramag-net.Thediscoveryofinductionunderliestheelec-tricaldynamoandmotor,whichconvertmechanicalenergyintoelectricalenergyandvice-versa.Faradaywasnotagreattheoristandthemathe-maticalexpressionthatwehavecalledFaraday'slawisduetoMaxwell.YetFaraday'sintuitionledhimtomakeoneofthemostimportantcontributionsofalltimetotheoreticalphysics:hewasthersttoproposetheideaoftheeld.AsFaraday'sresearchintoelectromagnetismincreased,hefoundhimselflackingthevocabularyneededtodescribethephenomenahewasseeing.Sincehedidn'texactlyreceiveaclassicaleducation,heturnedtoWilliamWhewell,thenMasterofTrinity,forsomeadvice.Betweenthem,theycookedupthewords`anode',`cathode',`ion',`dielectric',`diamagnetic'and`paramagnetic'.Theyalsosuggestedtheelectricchargeberenamed`Franklinic'inhonourofBenjaminFranklin.Thatonedidn'tstick.ThelastyearsofFaraday'slifewerespentinthesamewayasEinstein:seekingauniedtheoryofgravityandelectromagnetism.Thefollowingquotedescribeswhatis,perhaps,therstgenuineattemptatunication:Gravity:SurelyhisforcemustbecapableofanexperimentalrelationtoElectricity,Magnetismandtheotherforces,soastobinditupwiththeminreciprocalactionandequivalente ect.Considerforamomenthowtosetabouttouchingthismatterbyfactsandtrial...Faraday,19thMarch,1849.{78{ Asthisquotemakesclear,Faraday'sapproachtothisproblemincludessomethingthatEinstein'sdidnot:experiment.Ultimately,neitherofthemfoundaconnectionbetweenelectromagnetismandgravity.ButitcouldbearguedthatFaradaymadethemoreimportantcontribution:whileanulltheoryisuseless,anullexperimenttellsyousomethingaboutNature.4.2OneLastThing:TheDisplacementCurrentWe'venowworkedourwaythroughmostoftheMaxwellequations.We'velookedatGauss'law(whichisreallyequivalenttoCoulomb'slaw)rE= 0(4.5)andthelawthatsaystherearenomagneticmonopolesrB=0(4.6)andAmpere'slawrB=0J(4.7)andnowalsoFaraday'slawrE+@B @t=0(4.8)Infact,there'sonlyonetermlefttodiscuss.Wheneldschangewithtime,thereisanextratermthatappearsinAmpere'slaw,whichreadsinfull:rB=0J+0@E @t(4.9)Thisextratermiscalledthedisplacementcurrent.It'snotagreatnamebecauseit'snotacurrent.Nonetheless,asyoucansee,itsitsintheequationinthesameplaceasthecurrentwhichiswherethenamecomesfrom.Sowhatdoesthisextratermdo?Well,somethingquiteremarkable.Butbeforewegettothis,there'sastorytotellyou.Therstfourequationsabove( 4.5 ),( 4.6 ),( 4.7 )and( 4.8 )|whichincludeAmpere'slawinunmodiedform|werearrivedatthroughmanydecadesofpainstakingex-perimentalworktotrytounderstandthephenomenaofelectricityandmagnetism.Ofcourse,ittooktheoreticalphysicistsandmathematicianstoexpresstheselawsintheelegantlanguageofvectorcalculus.Butallthehardworktouncoverthelawscamefromexperiment.{79{ Thedisplacementcurrenttermisdi erent.Thiswasarrivedatbypurethoughtalone.ThisisoneofMaxwell'scontributionstothesubjectand,inpart,whyhisnamenowlordsoverallfourequations.Herealisedthatthelawsofelectromagnetismcapturedby( 4.5 )to( 4.8 )arenotinternallyconsistent:thedisplacementcurrenttermhastobethere.Moreover,onceyouaddit,thereareastonishingconsequences.4.2.1WhyAmpere'sLawisNotEnoughWe'lllookattheconsequencesinthenextsection.Butfornow,let'sjustseewhytheunmodiedAmperelaw( 4.7 )isinconsistent.Wesimplyneedtotakethedivergencetond0rJ=r(rB)=0Thismeansthatanycurrentthat owsintoagivenvolumehastoalso owout.Butweknowthat'snotalwaysthecase.Togiveasimpleexample,wecanimagineputtinglotsofchargeinasmallregionandwatchingitdisperse.Sincethechargeisleavingthecentralregion,thecurrentdoesnotobeyrJ=0,seeminglyinviolationofAmpere'slaw.Thereisastandardthoughtexperimentinvolvingcir- Figure41:cuitswhichisusuallyinvokedtodemonstratetheneedtoamendAmpere'slaw.Thisisshowninthegure.Theideaistocookupasituationwherecurrentsarechangingovertime.Todothis,wehookituptoacapacitor|whichcanbethoughtofastwoconductingplateswithagapbetweenthem|toacircuitofresistanceR.Thecircuitincludesaswitch.Whentheswitchisclosed,thecurrentwill owoutofthecapacitorandthroughthecircuit,ultimatelyheatinguptheresistor.Sowhat'stheproblemhere?Let'strytocomputethemagneticeldcreatedbythecurrentatsomepointalongthecircuitusingAmpere'slaw.WecantakeacurveCthatsurroundsthewireandsurfaceSwithboundaryC.IfwechoseStobetheobviouschoice,cuttingthroughthewire,thenthecalculationisthesameaswesawinSection 3.1 .WehaveZCBdr=0I(4.10)whereIisthecurrentthroughthewirewhich,inthiscase,ischangingwithtime.{80{ Figure42:Thischoiceofsurfacesug-geststhereisamagneticeldFigure43:Thischoiceofsurfacesug-geststhereisnone.Suppose,however,thatweinsteaddecidedtoboundthecurveCwiththesurfaceS0,whichnowsneaksthroughthegapbetweenthecapacitorplates.NowthereisnocurrentpassingthroughS0,soifweweretouseAmpere'slaw,wewouldconcludethatthereisnomagneticeldZCBdr=0(4.11)Thisisincontradictiontoourrstcalculation( 4.10 ).Sowhat'sgoingonhere?Well,Ampere'slawonlyholdsforsteadycurrentsthatarenotchangingwithtime.Andwe'vedeliberatelyputtogetherasituationwhereIistimedependenttoseethelimitationsofthelaw.AddingtheDisplacementCurrentLet'snowseehowaddingthedisplacementcurrent( 4.9 )xesthesituation.We'llrstlookattheabstractissuethatAmpere'slawrequiresrJ=0.Ifweaddthedisplacementcurrent,thentakingthedivergenceof( 4.9 )gives0rJ+0r@E @t=r(rB)=0But,usingGauss'slaw,wecanwrite0rE=,sotheequationabovebecomesrJ+@ @t=0whichisthecontinuityequationthattellsusthatelectricchargeislocallyconserved.It'sonlywiththeadditionofthedisplacementcurrentthatMaxwell'sequationsbecomeconsistentwiththeconservationofcharge.{81{ Nowlet'sreturntoourpuzzleofthecircuitandcapacitor.WithoutthedisplacementcurrentwefoundthatB=0whenwechosethesurfaceS0whichpassesbetweenthecapacitorplates.Butthedisplacementcurrenttellsusthatwemissedsomething,becausethebuildupofchargeonthecapacitorplatesleadstoatime-dependentelectriceldbetweentheplates.Forstaticsituations,wecomputedthisin( 2.10 ):itisE=Q 0AwhereAistheareaofeachplateandQisthechargethatsitsoneachplate,andweareignoringtheedgee ectswhichisacceptableaslongasthesizeoftheplatesismuchbiggerthanthegapbetweenthem.SinceQisincreasingovertime,theelectriceldisalsoincreasing@E @t=1 0AdQ dt=1 0AI(t)SonowifwerepeatthecalculationofBusingthesurfaceS0,wendanextratermfrom( 4.9 )whichgivesZCBdr=ZS000@E @t=0IThisisthesameanswer( 4.10 )thatwefoundusingAmpere'slawappliedtothesurfaceS.Great.SoweseewhytheMaxwellequationsneedtheextratermknownasthedisplacementcurrent.Nowtheimportantthingis:whatdowedowithit?Aswe'llnowsee,theadditionofthedisplacementcurrentleadstooneofthemostwonderfuldiscoveriesinphysics:theexplanationforlight.4.3AndThereWasLightTheemergenceoflightcomesfromlookingforsolutionsofMaxwell'sequationsinwhichtheelectricandmagneticeldschangewithtime,evenintheabsenceofanyexternalchargesorcurrents.Thismeansthatwe'redealingwiththeMaxwellequationsinvacuum:rE=0andrB=00@E @trB=0andrE=�@B @tTheessenceofthephysicsliesinthetwoMaxwellequationsontheright:iftheelectriceldshakes,itcausesthemagneticeldtoshakewhich,inturn,causestheelectric{82{ eldtoshake,andsoon.Toderivetheequationsgoverningtheseoscillations,westartbycomputingthesecondtimederivativeoftheelectriceld,00@2E @t2=@ @t(rB)=r@B @t=�r(rE)(4.12)Tocompletethederivation,weneedtheidentityr(rE)=r(rE)�r2EBut,therstofMaxwellequationstellsusthatrE=0invacuum,sothersttermabovevanishes.Wendthateachcomponentoftheelectriceldsatises,1 c2@2E @t2�r2E=0(4.13)Thisisthewaveequation.Thespeedofthewaves,c,isgivenbyc=r 1 00Identicalmanipulationsholdforthemagneticeld.Wehave@2B @t2=�@ @t(rE)=�r@E @t=�1 00r(rB)=1 00r2Bwhere,inthelastequality,wehavemadeuseofthevectoridentity( 4.12 ),nowappliedtothemagneticeldB,togetherwiththeMaxwellequationrB=0.Weagainndthateachcomponentofthemagneticeldsatisesthewaveequation,1 c2@2B @t2�r2B=0(4.14)Thewavesofthemagneticeldtravelatthesamespeedcasthoseoftheelectriceld.Whatisthisspeed?Attheverybeginningoftheselecturesweprovidedthenumericalvaluesoftheelectricconstant0=8:85418781710�12m�3Kg�1s2C2andthemagneticconstant,0=410�7mKgC�2Plugginginthesenumbersgivesthespeedofelectricandmagneticwavestobec=299792458ms�1Butthisissomethingthatwe'veseenbefore.It'sthespeedoflight!This,ofcourse,isbecausetheseelectromagneticwavesarelight.Inthewordsofthemanhimself{83{ \Thevelocityoftransverseundulationsinourhypotheticalmedium,calcu-latedfromtheelectro-magneticexperimentsofMM.KohlrauschandWeber,agreessoexactlywiththevelocityoflightcalculatedfromtheopticalex-perimentsofM.Fizeau,thatwecanscarcelyavoidtheinferencethatlightconsistsinthetransverseundulationsofthesamemediumwhichisthecauseofelectricandmagneticphenomena"JamesClerkMaxwellThesimplecalculationthatwehavejustseenrepresentsoneofthemostimportantmomentsinphysics.NotonlyareelectricandmagneticphenomenauniedintheMaxwellequations,butnowoptics{oneoftheoldesteldsinscience{isseentobecapturedbytheseequationsaswell.4.3.1SolvingtheWaveEquationWe'vederivedtwowaveequations,oneforEandoneforB.Wecansolvetheseindependently,butit'simportanttokeepinourmindthatthesolutionsmustalsoobeytheoriginalMaxwellequations.ThiswillthengiverisetoarelationshipbetweenEandB.Let'sseehowthisworks.We'llstartbylookingforaspecialclassofsolutionsinwhichwavespropagateinthex-directionanddonotdependonyandz.Thesearecalledplane-wavesbecause,byconstruction,theeldsEandBwillbeconstantinthe(y;z)planeforxedxandt.TheMaxwellequationrE=0tellsusthatwemusthaveExconstantinthiscase.AnyconstantelectriceldcanalwaysbeaddedasasolutiontotheMaxwellequationsso,withoutlossofgenerality,we'llchoosethisconstanttovanish.WelookforsolutionsoftheformE=(0;E(x;t);0)whereEsatisesthewaveequation( 4.13 )whichisnow1 c2@2E @t2�r2E=0ThemostgeneralsolutiontothewaveequationtakestheformE(x;t)=f(x�ct)+g(x+ct)Heref(x�ct)describesawaveprolewhichmovestotherightwithspeedc.(Because,astincreases,xalsohastoincreasetokeepfconstant).Meanwhile,g(x+ct)describesawaveprolemovingtotheleftwiththespeedc.{84{ Themostimportantclassofsolutionsofthiskindarethosewhichoscillatewithasinglefrequency!.Suchwavesarecalledmonochromatic.Fornow,we'llfocusontheright-movingwavesandtaketheproletobethesinefunction.(We'lllookattheoptiontotakecosinewavesorothershiftsofphaseinamomentwhenwediscusspolarisation).WehaveE=E0sinh!x c�tiWeusuallywritethisasE=E0sin(kx�!t)(4.15)wherekisthewavenumber.Thewaveequation( 4.13 )requiresthatitisrelatedtothefrequencyby!2=c2k2Equationsofthiskind,expressingfrequencyintermsofwavenumber,arecalleddis-persionrelations.Becausewavesaresoimportantinphysics,there'sawholebunchofassociatedquantitieswhichwecandene.Theyare:Thequantity!ismoreproperlycalledtheangularfrequencyandistakentobepositive.Theactualfrequencyf=!=2measureshowoftenawavepeakpassesyouby.Butbecausewewillonlytalkabout!,wewillbelazyandjustrefertothisasfrequency.TheperiodofoscillationisT=2=!.Thewavelengthofthewaveis=2=k.Thisisthepropertyofwavesthatyourstlearnaboutinkindergarten.Thewavelengthofvisiblelightisbetween3:910�7mand710�7m.Atoneendofthespectrum,gammarayshavewavelength10�12mandX-raysaround10�10to10�8m.Attheotherend,radiowaveshave1cmto10km.Ofcourse,theelectromagneticspectrumdoesn'tstopatthesetwoends.Solutionsexistforall.Althoughwegrowupthinkingaboutwavelength,movingforwardthewavenum-berkwillturnouttobeamoreusefuldescriptionofthewave.E0istheamplitudeofthewave.{85{ Sofarwehaveonlysolvedfortheelectriceld.Todeterminethemagneticeld,weuserB=0totellusthatBxisconstantandweagainsetBx=0.WeknowthattheothercomponentsByandBzmustobeythewaveequation( 4.14 ).ButtheirbehaviourisdictatedbywhattheelectriceldisdoingthroughtheMaxwellequationrE=�@B=@t.ThistellsusthatB=(0;0;B)with@B @t=�@E @x=�kE0cos(kx�!t)WendB=E0 csin(kx�!t)(4.16)WeseethattheelectricEandmagneticBeldsoscillateinphase,butinorthogonaldirections.Andbothoscillateindirectionswhichareorthogonaltothedirectioninwhichthewavetravels. BecausetheMaxwellequationsarelinear,we'reallowedtoaddanynumberofsolu-tionsoftheform( 4.15 )and( 4.16 )andwewillstillhaveasolution.Thissometimesgoesbythenameoftheprincipleofsuperposition.(Wementioneditearlierwhendiscussingelectrostatics).Thisisaparticularlyimportantpropertyinthecontextoflight,becauseit'swhatallowlightraystravellingindi erentdirectionstopassthrougheachother.Inotherwords,it'swhywecanseeanythingatall.ThelinearityoftheMaxwellequationsalsoencouragesustointroducesomenewnotationwhich,atrstsight,looksratherstrange.Wewilloftenwritethesolutions( 4.15 )and( 4.16 )incomplexnotation,E=E0^yei(kx�!t);B=E0 c^zei(kx�!t)(4.17){86{ Thisisstrangebecausethephysicalelectricandmagneticeldsshouldcertainlyberealobjects.Youshouldthinkofthemassimplytherealpartsoftheexpressionsabove.ButthelinearityoftheMaxwellequationsmeansbothrealandimaginarypartsofEandBsolvetheMaxwellequations.And,moreimportantly,ifwestartaddingcomplexEandBsolutions,thentheresultingrealandimaginarypieceswillalsosolvetheMaxwellequations.Theadvantageofthisnotationissimplythatit'stypicallyeasiertomanipulatecomplexnumbersthanlotsofcosandsinformulae.However,youshouldbeawarethatthisnotationcomeswithsomedanger:wheneveryoucomputesomethingwhichisn'tlinearinEandB|forexample,theenergystoredintheelds,whichisaquadraticquantity|youcan'tusethecomplexnotationabove;youneedtotaketherealpartrst.4.3.2PolarisationAbovewehavepresentedaparticularsolutiontothewaveequation.Let'snowlookatthemostgeneralsolutionwithaxedfrequency!.Thismeansthatwelookforsolutionswithintheansatz,E=E0ei(kx�!t)andB=B0ei(kx�!t)(4.18)where,fornow,bothE0andB0couldbecomplex-valuedvectors.(Again,weonlygetthephysicalelectricandmagneticeldsbytakingtherealpartoftheseequations).Thevectorkiscalledthewavevector.Itsmagnitude,jkj=k,isthewavenumberandthedirectionofkpointsinthedirectionofpropagationofthewave.Theexpressions( 4.18 )alreadysatisfythewaveequations( 4.13 )and( 4.14 )if!andkobeythedispersionrelation!2=c2k2.WegetfurtherconstraintsonE0,B0andkfromtheoriginalMaxwellequations.ThesearerE=0)ikE0=0rB=0)ikB0=0rE=�@B @t)ikE0=i!B0Let'snowinterprettheseequations:LinearPolarisationSupposethatwetakeE0andB0tobereal.ThersttwoequationsabovesaythatbothE0andB0areorthogonaltothedirectionofpropagation.Thelastoftheequations{87{ abovesaysthatE0andB0arealsoorthogonaltoeachother.YoucancheckthatthefourthMaxwellequationdoesn'tleadtoanyfurtherconstraints.Usingthedispersionrelation!=ck,thelastconstraintabovecanbewrittenas^k(E0=c)=B0Thismeansthatthethreevectors^k,E0=candB0formaright-handedorthogonaltriad.Wavesofthisformaresaidtobelinearlypolarised.Theelectricandmagneticeldsoscillateinxeddirections,bothofwhicharetransversetothedirectionofpropagation.CircularandEllipticPolarisationSupposethatwenowtakeE0andB0tobecomplex.Theactualelectricandmagneticeldsarejusttherealpartsof( 4.18 ),butnowthepolarisationdoesnotpointinaxeddirection.Toseethis,writeE0= �iTherealpartoftheelectriceldisthenE= cos(kx�!t)+sin(kx�!t)withMaxwellequationsensuringthat k=k=0.IfwelookatthedirectionofEatsomexedpointinspace,saytheoriginx=0,weseethatitdoesn'tpointinaxeddirection.Instead,itrotatesovertimewithintheplanespannedby and(whichistheplaneperpendiculartok).AspecialcaseariseswhenthephaseofE0isei=4,sothatj j=jj,withthefurtherrestrictionthat =0.ThenthedirectionofEtracesoutacircleovertimeintheplaneperpendiculartok.Thisiscalledcircularpolarisation.Thepolarisationissaidtoberight-handedif=^k andleft-handedif=�^k .Ingeneral,thedirectionofEatsomepointinspacewilltraceoutanellipseintheplaneperpendiculartothedirectionofpropagationk.Unsurprisingly,suchlightissaidtohaveellipticpolarisation.GeneralWaveAgeneralsolutiontothewaveequationconsistsofcombinationsofwavesofdi erentwavenumbersandpolarisations.ItisnaturallyexpressedasaFourierdecompositionbysummingoversolutionswithdi erentwavevectors,E(x;t)=Zd3k (2)3E(k)ei(kx�!t)Here,thefrequencyofeachwavedependsonthewavevectorbythenow-familiardis-persionrelation!=ck.{88{ 4.3.3AnApplication:Re ectiono aConductorTherearelotsofthingstoexplorewithelectromagneticwavesandwewillseemanyexampleslaterinthecourse.Fornow,welookatasimpleapplication:wewillre ectwaveso aconductor.Weallknowfromexperiencethatconductors,likemetals,lookshiny.Herewe'llseewhy.Supposethattheconductoroccupiesthehalfofspace Figure44:x�0.Westartbyshiningthelighthead-onontothesurface.Thismeansanincidentplanewave,travellinginthex-direction,Einc=E0^yei(kx�!t)where,asbefore,!=ck.Insidetheconductor,weknowthatwemusthaveE=0.ButthecomponentE^yliestangentialtothesurfaceandso,bycontinuity,mustalsovanishjustoutsideatx=0�.Weachievethisbyaddingare ectedwave,travellingintheoppositedirectionEref=�E0^yei(�kx�!t)SothatthecombinationE=Einc+ErefsatisesE(x=0)=0asitmust.Thisisillustratedinthegure.(Note,however,thatthegureisalittlebitmisleading:thetwowavesareshowndisplacedbut,inreality,bothllallofspaceandshouldbesuperposedontopofeachother).We'vealreadyseenabovethatthecorrespondingmagneticeldcanbedeterminedbyrE=�@B=@t.ItisgivenbyB=Binc+Bref,withBinc=E0 c^zei(kx�!t)andBref=E0 c^zei(�kx�!t)(4.19)ThisobeysBn=0,asitshouldbycontinuity.Butthetangentialcomponentdoesn'tvanishatthesurface.Instead,wehaveB^zjx=0�=2E0 ce�i!tSincethemagneticeldvanishesinsidetheconductor,wehaveadiscontinuity.Butthere'snomysteryhere.Weknowfromourpreviousdiscussion( 3.6 )thatthiscorre-spondstoasurfacecurrentKinducedbythewaveK=2E0 c0^ye�i!tWeseethatthesurfacecurrentoscillateswiththefrequencyofthere ectedwave.{89{ Re ectionatanAngle Figure45:Let'snowtrysomethingalittlemorecomplicated:we'llsendintheoriginalrayatanangle,,tothenormalasshowninthegure.OurincidentelectriceldisEinc=E0^yei(kx�!t)wherek=kcos^x+ksin^zNoticethatwe'vemadeaspecicchoiceforthepolarisationoftheelectriceld:itisoutofthepageinthegure,tangentialtothesurface.Nowwehavetwocontinuityconditionstoworryabout.Wewanttoaddare ectedwave,Eref=�E0^ei(k0x�!0t)wherewe'veallowedforthepossibilitythatthepolarisation^,thewavevectork0andfrequency!0arealldi erentfromtheincidentwave.Werequiretwocontinuitycondi-tionsontheelectriceld(Einc+Eref)^n=0and(Einc+Eref)^n=0where,forthisset-up,thenormalvectoris^n=�^x.Thisisachievedbytaking!0=!and=^y,sothatthere ectedwavechangesneitherfrequencynorpolarisation.There ectedwavevectorisk0=�kcos^x+ksin^zWecanalsocheckwhatbecomesofthemagneticeld.ItisB=Binc+Bref,withBinc=E0 c(^k^y)ei(kx�!0t)andBref=�E0 c(^k0^y)ei(k0x�!0t)Notethat,incontrastto( 4.19 ),thereisnowaminussigninthere ectedBref,butthisissimplytoabsorbasecondminussigncomingfromtheappearanceof^k0inthepolarisationvector.ItissimpletocheckthatthenormalcomponentB^nvanishesattheinterface,asitmust.Meanwhile,thetangentialcomponentagaingivesrisetoasurfacecurrent.{90{ Themainupshotofallofthisdiscussionisrelationshipbetweenkandk0whichtellsussomethingthatweknewwhenwewereve:theangleofincidenceisequaltotheangleofre ection.Onlynowwe'vederivedthisfromtheMaxwellequations.Ifthisisalittleunderwhelming,we'llderivemanymorepropertiesofwaveslater.4.3.4JamesClerkMaxwell(1831-1879)Stillthosepaperslaybeforeme,Problemsmadeexpresstoboreme,Whenasilentchangecameo'erme,Inmyharduneasychair.Fireandfog,andcandlefaded,Spectralformstheroominvaded,Littlecreatures,thatparadedOntheproblemslyingthere.JamesClerkMaxwell,\AVisionofaWrangler,ofaUniversity,ofPedantry,andofPhilosophy"JamesClerkMaxwellwasaverysmartman.BorninEdinburgh,hewasastudent,rstinhishometown,andlaterinCambridge,atPeterhouseandthenatTrinity.HeheldfacultypositionsattheUniversityofAberdeen(wheretheyredhim)andKingsCollegeLondonbeforereturningtoCambridgeastherstCavendishprofessorofphysics.PerhapstherstverysmartthingthatMaxwelldid Figure46:Maxwell'svor-ticeswastodeterminethecompositionofSaturn'srings.Hedidn'tdothisusingatelescope.Hediditusingmathe-matics!Heshowedthatneitherasolidnora uidringcouldbestable.Suchringscouldonlybemadeofmanysmallparticles.ForthishewasawardedtheAdamsPrize.(Thesedaysyoucanwinthisprizeformuchmuchless!)Maxwell'sgreatworkonelectromagnetismwasaccom-plishedbetween1861and1862.Hestartedbyconstruct-inganelaboratemechanicalmodelofelectricityandmag-netisminwhichspaceislledbyvorticesofanincom-pressible uid,separatedbytinyrotatingparticlesthatgiverisetoelectricity.Oneofhisillustrationsisshownabove.Needlesstosay,wedon'tteachthispictureofspace{91{ anymore.Fromthis,hemanagedtodistilleverythingthatwasknownaboutelectro-magnetisminto20coupledequationsin20variables.Thiswastheframeworkinwhichhediscoveredthedisplacementcurrentanditsconsequencesforlight.YoumightthinkthattheworldchangedwhenMaxwellpublishedhiswork.Infact,noonecared.Theequationsweretoohardforphysicists,thephysicstoohardformathematicians.Thingsimprovedmarginallyin1873whenMaxwellreducedhisequationstojustfour,albeitwritteninquaternionnotation.ThemodernversionofMaxwellequations,writteninvectorcalculusnotation,isduetoOliverHeavisidein1881.Inall,ittookalmost30yearsforpeopletoappreciatethesignicanceofMaxwell'sacheivement.Maxwellmadeanumberofotherimportantcontributionstoscience,includingthersttheoryofcolourvisionandthetheoryofcolourphotography.Hisworkonther-modynamicsandstatisticalmechanicsdeservesatleastequalstatuswithhisworkonelectromagnetism.Hewasthersttounderstandthedistributionofvelocitiesofmoleculesinagas,thersttoextractanexperimentalpredictionfromthetheoryofatomsand,remarkably,therst(withthehelpofhiswife)tobuildtheexperimentanddothemeasurement,conrminghisowntheory.4.4TransportofEnergy:ThePoyntingVectorElectromagneticwavescarryenergy.Thisisanimportantfact:wegetmostofourenergyfromthelightoftheSun.Herewe'dliketounderstandhowtocalculatethisenergy.Ourstartingpointistheexpression( 4.3 )fortheenergystoredinelectricandmag-neticelds,U=ZVd3x0 2EE+1 20BBTheexpressioninbracketsistheenergydensity.HerewehaveintegratedthisonlyoversomenitevolumeVratherthanoverallofspace.Thisisbecausewewanttounderstandthewayinwhichenergycanleavethisvolume.WedothisbycalculatingdU dt=ZVd3x0E@E @t+1 0B@B @t=ZVd3x1 0E(rB)�EJ�1 0B(rE){92{ wherewe'veusedthetwoMaxwellequations.NowweusetheidentityE(rB)�B(rE)=�r(EB)andwritedU dt=�ZVd3xJE�1 0ZS(EB)dS(4.20)wherewe'veusedthedivergencetheoremtowritethelastterm.ThisequationissometimescalledthePoyntingtheorem.Thersttermontheright-handsideisrelatedtosomethingthatwe'vealreadyseeninthecontextofNewtonianmechanics.TheworkdoneonaparticleofchargeqmovingwithvelocityvfortimetinanelectriceldisW=qvEt.TheintegralRVd3xJEaboveissimplythegeneralisationofthistocurrents:itshouldbethoughtofastherateofgainofenergyoftheparticlesintheregionV.Sinceitappearswithaminussignin( 4.20 ),itistherateoflossofenergyoftheparticles.Nowwecaninterpret( 4.20 ).IfwewriteitasdU dt+ZVd3xJE=�1 0ZS(EB)dSthentheleft-handsideisthecombinedchangeinenergyofbotheldsandparticlesinregionV.Sinceenergyisconserved,theright-handsidemustdescribetheenergythatescapesthroughthesurfaceSofregionV.WedenethePoyntingvectorS=1 0EBThisisavectoreld.Ittellsusthemagnitudeanddirectionofthe owofenergyinanypointinspace.(ItisunfortunatethatthecanonicalnameforthePoyntingvectorisSbecauseitmakesitnotationallydiculttointegrateoverasurfacewhichweusuallyalsoliketocallS.Needlesstosay,thesetwothingsarenotthesameandhopefullynoconfusionwillarise).Let'snowlookattheenergycarriedinelectromagneticwaves.BecausethePoyntingvectorisquadraticinEandB,we'renotallowedtousethecomplexformofthewaves.Weneedtoreverttotherealform.Forlinearpolarisation,wewritethesolutionsintheform( 4.17 ),butwitharbitrarywavevectork,E=E0sin(kx�!t)andB=1 c(^kE0)sin(kx�!t){93{ ThePoyntingvectoristhenS=E20 c0^ksin2(kx�!t)Averagingoveraperiod,T=2=!,wehaveS=E20 2c0^kWelearnthattheelectromagneticwavedoesindeedtransportenergyinitsdirectionofpropagation^k.It'sinstructivetocomparethistotheenergydensityoftheeld( 4.3 ).Evaluatedontheelectromagneticwave,theenergydensityisu=0 2EE+1 20BB=0E20sin2(kx�!t)AveragedoveraperiodT=2=!,thisisu=0E20 2Then,usingc2=1=00,wecanwriteS=cu^kTheinterpretationissimplythattheenergySisequaltotheenergydensityinthewaveutimesthespeedofthewave,c.4.4.1TheContinuityEquationRevisitedRecallthat,waybackinSection 1 ,weintroducedthecontinuityequationforelectriccharge,@ @t+rJ=0Thisequationisnotspecialtoelectriccharge.Itmustholdforanyquantitythatislocallyconserved.Nowwehaveencounteredanotherquantitythatislocallyconserved:energy.InthecontextofNewtonianmechanics,weareusedtothinkingofenergyasasinglenumber.Now,ineldtheory,itisbettertothinkofenergydensityE(x;t).Thisincludestheenergyinbotheldsandtheenergyinparticles.Thinkinginthisway,wenoticethat( 4.20 )issimplytheintegratedversionofacontinuityequationforenergy.Wecouldequallywellwriteitas@E @t+rS=0WeseethatthePoyntingvectorSistoenergywhatthecurrentJistocharge.{94{ 5.ElectromagnetismandRelativityWe'veseenthatMaxwell'sequationshavewavesolutionswhichtravelatthespeedoflight.Butthere'sanotherplaceinphysicswherethespeedoflightplaysapromi-nentrole:thetheoryofspecialrelativity.Howdoeselectromagnetismtwithspecialrelativity?Historically,theMaxwellequationswerediscoveredbeforethetheoryofspecialrel-ativity.Itwasthoughtthatthelightwaveswederivedabovemustbeoscillationsofsomesubstancewhichllsallofspace.Thiswasdubbedtheaether.TheideawasthatMaxwell'sequationsonlyholdintheframeinwhichtheaetherisatrest;lightshouldthentravelatspeedcrelativetotheaether.Wenowknowthattheconceptoftheaetherisunnecessarybaggage.Instead,Maxwell'sequationsholdinallinertialframesandaretherstequationsofphysicswhichareconsistentwiththelawsofspecialrelativity.Ultimately,itwasbystudyingtheMaxwellequationsthatLorentzwasabletodeterminetheformoftheLorentztransformationswhichsubsequentlylaidthefoundationforEinstein'svisionofspaceandtime.Ourgoalinthissectionistoviewelectromagnetismthroughthelensofrelativity.Wewillndthatobserversindi erentframeswilldisagreeonwhattheycallelectriceldsandwhattheycallmagneticelds.Theywillobservedi erentchargedensitiesanddi erentcurrents.ButallwillagreethatthesequantitiesarerelatedbythesameMaxwellequations.Moreover,thereisapay-o tothis.It'sonlywhenweformulatetheMaxwellequationsinawaywhichismanifestlyconsistentwithrelativitythatweseetheirtruebeauty.Theslightlycumbersomevectorcalculusequationsthatwe'vebeenplayingwiththroughouttheselectureswillbereplacedbyamuchmoreelegantandsimple-lookingsetofequations.5.1AReviewofSpecialRelativityWestartwithaveryquickreviewoftherelevantconceptsofspecialrelativity.(Formoredetailsseethelecturenoteson DynamicsandRelativity ).Thebasicpostulateofrelativityisthatthelawsofphysicsarethesameinallinertialreferenceframes.Thegutsofthetheorytellushowthingslooktoobserverswhoaremovingrelativetoeachother.TherstobserversitsinaninertialframeSwithspacetimecoordinates(ct;x;y;z)thesecondobserversitsinaninertialframeS0withspacetimecoordinates(ct0;x0;y0;z0).{95{ IfwetakeS0tobemovingwithspeedvinthex-directionrelativetoSthenthecoordinatesystemsarerelatedbytheLorentzboostx0= x�v cctandct0= ct�v cx(5.1)whiley0=yandz0=z.Herecisthespeedoflightwhichhasthevalue,c=299792458ms�1Meanwhile istheubiquitousfactor =s 1 1�v2=c2(5.2)TheLorentztransformation( 5.1 )encodeswithinitallofthefunideasoftimedilationandlengthcontractionthatwesawinourrstcourseonrelativity.5.1.1Four-VectorsIt'sextremelyusefultopackagethesespacetimecoordinatesin4-vectors,withindicesrunningfrom=0to=3X=(ct;x;y;z)=0;1;2;3Notethattheindexisasuperscriptratherthansubscript.Thiswillbeimportantshortly.AgeneralLorentztransformationisalinearmapfromXtoX0oftheform(X0)=XHereisa44matrixwhichobeysthematrixequationT=,=(5.3)withtheMinkowskimetric=diag(+1;�1;�1;�1)Thesolutionsto( 5.3 )fallintotwoclasses.Therstclassissimplyrotations.Givena33rotationmatrixRobeyingRTR=1,wecanconstructaLorentztransformationobeying( 5.3 )byembeddingRinthespatialpart,=0BBBB@1 000 0 0 R0 1CCCCA(5.4)Thesetransformationsdescribehowtorelatethecoordinatesoftwoobserverswhoarerotatedwithrespecttoeachother.{96{ Theotherclassofsolutionsto( 5.3 )aretheLorentzboosts.Thesearethetransfor-mationsappropriateforobserversmovingrelativetoeachother.TheLorentztransfor-mation( 5.1 )isequivalentto=0BBBB@ � v=c 00� v=c 00 00 1000 011CCCCA(5.5)Therearesimilarsolutionsassociatedtoboostsalongtheyandzaxes.Thebeautyof4-vectorsisthatit'sextremelyeasytowritedowninvariantquantities.Thesearethingswhichallobservers,nomatterwhichtheirreferenceframe,canagreeon.Toconstructthesewetaketheinnerproductoftwo4-vectors.ThetrickisthatthisinnerproductusestheMinkowskimetricandsocomeswithsomeminussigns.Forexample,thesquareofthedistancefromtheorigintosomepointinspacetimelabelledbyXisXX=XX=c2t2�x2�y2�z2whichistheinvariantinterval.Similarly,ifwe'regiventwofour-vectorsXandYthentheinnerproductXY=XYisalsoaLorentzinvariant.5.1.2ProperTimeThekeytobuildingrelativistictheoriesofNatureistondthevariablesthathavenicepropertiesunderLorentztransformations.The4-vectorsX,labellingspacetimepoints,areagoodstart.Butweneedmore.Herewereviewhowtheotherkinematicalvariablesofvelocity,momentumandaccelerationtinto4-vectors.Supposethat,insomeframe,theparticletracesoutaworldline.Theclevertrickistondawaytoparameterisethispathinawaythatallobserversagreeupon.Thenaturalchoiceisthepropertime ,thedurationoftimeexperiencedbytheparticleitself.Ifyou'resittinginsomeframe,watchingsomeparticlemovewithanold-fashionedNewtonian3-velocityu(t),thenit'ssimpletoshowthattherelationshipbetweenyourtimetandthepropertimeoftheparticle isgivenbydt d = (u){97{ Thepropertimeallowsustodenethe4-velocityandthe4-momentum.SupposethattheparticletracesoutapathX( )insomeframe.Thenthe4-velocityisU=dX d = cu!Similarly,the4-momentumisP=mUwheremistherestmassoftheparticle.WewriteP= E=cp!(5.6)whereE=m c2istheenergyoftheparticleandp= muisthe3-momentuminspecialrelativity.TheimportanceofUandPisthattheytooare4-vectors.Becauseallobserversagreeon ,thetransformationlawofUandPareinheritedfromX.ThismeansthatunderaLorentztransformation,theytoochangeasU!UandP!P.AnditmeansthatinnerproductsofUandPareguaranteedtobeLorentzinvariant.5.1.3IndicesUp,IndicesDownBeforewemoveon,wedoneedtointroduceoneextranotationalnovelty.TheminussignsintheMinkowskimetricmeansthatit'susefultointroduceaslighttwisttotheusualsummationconventionofrepeatedindices.Forallthe4-vectorsthatweintroducedabove,wealwaysplacethespacetimeindex=0;1;2;3asasuperscript(i.e.up)ratherthanasubscript.X= ctx!Thisisbecausethesameobjectwithanindexdown,X,willmeansomethingsubtlydi erent.WedeneX= ct�x!Withthisconvention,theMinkowskiinnerproductcanbewrittenusingtheusualconventionofsummingoverrepeatedindicesasXX=c2t2�xxIncontrast,XX=c2t2+x2isadumbthingtowriteinthecontextofspecialrelativitysinceitlooksverydi erenttoobserversindi erentinertialframes.Infact,wewillshortlydeclareitillegaltowritethingslikeXX.{98{ ThereisanaturalwaytothinkofXintermsofXusingtheMinkowskimetric=diag(+1;�1;�1;�1).Thefollowingequationistriviallytrue:X=XThismeansthatwecanthinkoftheMinkowskimetricasallowingustolowerindices.Toraiseindicesbackup,weneedtheinverseofwhich,fortunately,isthesamematrix:=diag(+1;�1;�1;�1)whichmeanswehave=andwecanwriteX=XFromnowon,we'regoingtoretainthisdistinctionbetweenallupperandlowerindices.Allthefour-vectorsthatwe'vemetsofarhaveupperindices.Butallcanbeloweredinthesameway.Forexample,wehaveU= c�u!(5.7)ThistrickofdistinguishingbetweenindicesupandindicesdownprovidesasimpleformalismtoensurethatallobjectshavenicetransformationpropertiesundertheLorentzgroup.Weinsistthat,justasintheusualsummationconvention,repeatedindicesonlyeverappearinpairs.Butnowwefurtherinsistthatpairsalwaysappearwithoneindexupandtheotherdown.TheresultwillbeanobjectwhichisinvariantunderLorentztransformations.5.1.4Vectors,CovectorsandTensorsInfuturecourses,youwilllearnthatthereissomewhatdeepermathematicslyingbe-hinddistinguishingXandX:formally,theseobjectsliveindi erentspaces(some-timescalleddualspaces).We'llcontinuetorefertoXasvectors,buttodistinguishthem,we'llcallXcovectors.(Inslightlyfancierlanguage,thecomponentsofthevec-torXaresometimessaidtobecontravariantwhilethecomponentsofthecovectorXaresaidtobecovariant).Fornow,theprimarydi erencebetweenavectorandcovectorishowtheytransformunderrotationsandboosts.Weknowthat,underaLorentztransformation,any4-vectorchangesasX!X0=X{99{ Fromthis,weseethatacovectorshouldtransformasX!X0=X0=X=XUsingourruleforraisingandloweringindices,nowappliedtotheLorentztransforma-tion,wecanalsowritethisasX!Xwhereournotationisnowgettingdangerouslysubtle:youhavetostaretoseewhethertheupperorlowerindexontheLorentztransformationcomesrst.Thereisasenseinwhichcanbethoughtofathecomponentsoftheinversematrix�1.Toseethis,wegobacktothedenitionoftheLorentztransformation( 5.3 ),andstarttouseournewrulesforraisingandloweringindices=)=)=)=Inthelastlineabove,we'vesimplyreversedtheorderofthetwotermsontheleft.(Whenwritteninindexnotation,thesearejusttheentriesofthematrixsothere'snoproblemwithcommutingthem).Nowwecomparethistotheformulafortheinverseofamatrix,(�1)=)(�1)=(5.8)Notethatyouneedtobecarefulwhereyouplacetheindicesinequationslikethis.Theresult( 5.8 )isanalogoustothestatementthattheinverseofarotationmatrixisthetransposematrix.ForgeneralLorentztransformations,welearnthattheinverseissortofthetransposewhere\sortof"meansthatthereareminussignsfromraisingandlowering.Theplacementofindicesin( 5.8 )tellsuswherethoseminussignsgo.Theupshotof( 5.8 )isthatifwewanttoabandonindexnotationalltogetherthenvectorstransformasX!Xwhilecovectors{which,forthepurposeofthissentence,we'llcall~X{transformas~X!�1~X.However,inwhatfollows,wehavenointentionofabandoningindexnotation.Instead,wewillembraceit.ItwillbeourfriendandourguideinshowingthattheMaxwellequationsareconsistentwithspecialrelativity.{100{ Aparticularlyusefulexampleofacovectoristhefour-derivative.Thisistherela-tivisticgeneralisationofr,denedby@=@ @X=1 c@ @t;rNoticethatthesuperscriptonthespacetime4-vectorXhasmigratedtoasubscriptonthederivative@.Forthistomakenotationalsense,weshouldcheckthat@doesindeedtransformascovector.Thisisasimpleapplicationofthechainrule.UnderaLorentztransformation,X!X0=X,sowehave@=@ @X!@ @X0=@X @X0@ @X=(�1)@=@whichisindeedthetransformationofaco-vector.TensorsVectorsandcovectorsarethesimplestexamplesofobjectswhichhavenicetransfor-mationpropertiesundertheLorentzgroup.Buttherearemanymoreexamples.Themostgeneralobjectcanhaveabunchofupperindicesandabunchoflowerindices,T1:::n1:::m.Theseobjectsarealsocalledtensorsoftype(n;m).Inordertoqualifyasatensor,theymusttransformunderaLorentztransformationasT01:::n1:::m=11:::nn11:::mmT1:::n1:::m(5.9)YoucanalwaysusetheMinkowskimetrictoraiseandlowerindicesontensors,changingthetypeoftensorbutkeepingthetotalnumberofindicesn+mxed.Tensorsofthiskindarethebuildingblocksofallourtheories.Thisisbecauseifyoubuildequationsonlyoutoftensorswhichtransforminthismannerthen,aslongasthe;;:::indicesmatchuponbothsidesoftheequation,you'reguaranteedtohaveanequationthatlooksthesameinallinertialframes.Suchequationsaresaidtobecovariant.You'llseemoreofthiskindofthingincoursesonGeneralRelativityandDi erentialGeometry.Insomesense,thisindexnotationistoogood.Rememberallthosewonderfulthingsthatyourstlearnedaboutinspecialrelativity:timedilationandlengthcontractionandtwinsandspaceshipssoon.You'llneverhavetoworryaboutthoseagain.Fromnowon,youcanguaranteethatyou'reworkingwithatheoryconsistentwithrelativitybyensuringtwosimplethingsThatyouonlydealwithtensors.Thattheindicesmatchuponbothsidesoftheequation.It'ssad,buttrue.It'sallpartofgrowingupandnothavingfunanymore.{101{ 5.2ConservedCurrentsWestartedtheselecturesbydiscussingthechargedensity(x;t),thecurrentdensityJ(x;t)andtheirrelationthroughthecontinuityequation,@ @t+rJ=0whichtellsusthatchargeislocallyconserved.Thecontinuityequationisalreadyfullyconsistentwithrelativity.Toseethis,werstneedtoappreciatethatthechargeandcurrentdensitiessitnicelytogetherina4-vector,J= cJ!Ofcourse,placingobjectsinafour-vectorhasconsequence:ittellsushowtheseobjectslooktodi erentobservers.Let'squicklyconvinceourselvesthatitmakessensethatchargedensityandcurrentdoindeedtransforminthisway.Wecanstartbyconsideringasituationwherethereareonlystaticchargeswithdensity0andnocurrent.SoJ=(0;0).Now,inaframethatisboostedbyvelocityv,thecurrentwillappearasJ0=JwiththeLorentztransformationgivenby( 5.5 ).Thenewchargedensityandcurrentarethen0= 0;J0=� vTherstoftheseequationstellsusthatdi erentobserversseedi erentchargedensities.ThisisbecauseofLorentzcontraction:chargedensitymeanschargeperunitvolume.AndthevolumegetssqueezedbecauselengthsparalleltothemotionundergoLorentzcontraction.That'sthereasonforthefactorof intheobservedchargedensity.Meanwhile,thesecondoftheseequationsisjusttherelativisticextensionoftheformulaJ=vthatwerstsawintheintroduction.(Theextraminussignisbecausevheredenotesthevelocityoftheboostedobserver;thechargeisthereforemovingwithrelativevelocity�v).Inournew,relativistic,notation,thecontinuityequationtakestheparticularlysim-pleform@J=0(5.10)ThisequationisLorentzinvariant.Thisfollowssimplybecausetheindicesarecon-tractedintherightway:oneup,andonedown.{102{ 5.2.1MagnetismandRelativityWe'velearnedsomethingunsurprising:boostedchargegivesrisetoacurrent.But,combinedwithourpreviousknowledge,thistellsussomethingnewandimportant:boostedelectriceldsmustgiverisetomagneticelds.Therestofthischapterwillbedevotedtounderstandingthedetailsofhowthishappens.Butrst,we'regoingtolookatasimpleexamplewherewecanre-derivethemagneticforcepurelyfromtheCoulombforceandadoseofLorentzcontraction.Tostart,considerabunchofpositivecharges Figure47:+qmovingalongalinewithspeed+vandabunchofnegativecharges�qmovingintheoppositedirectionwithspeed�vasshowninthegure.Ifthereisequaldensity,n,ofpositiveandnegativechargesthenthechargedensityvanisheswhilethecurrentisI=2nAqvwhereAisthecross-sectionalareaofthewire.Nowconsideratestparticle,alsocarryingchargeq,whichismovingparalleltothewirewithsomespeedu.Itdoesn'tfeelanyelectricforcebecausethewireisneutral,butweknowitexperiencesamagneticforce.Herewewillshowhowtondanexpressionforthisforcewithouteverinvokingthephenomenonofmagnetism.Thetrickistomovetotherestframeofthetestparticle.Thismeanswehavetoboostbyspeedu.Theusualadditionformulatellsusthatthevelocitiesofthepositiveandnegativechargesnowdi er,givenbyv=vu 1uv=c2ButwiththeboostcomesaLorentzcontractionwhichmeansthatthechargedensitychanges.Moreover,becausethevelocitiesofpositiveandnegativechargesarenowdi erent,thiswillmeanthat,viewedfromtherestframeofourparticle,thewireisnolongerneutral.Let'sseehowthisworks.First,we'llintroducen0,thedensityofchargeswhentheparticlesinthewireareatrest.Thenthedensityofthe+qchargesintheoriginalframeis=qn= (v)qn0Thechargedensityofthe�qparticlesisthesame,butwithoppositesign,sothatintheoriginalframethewireisneutral.However,inournewframe,thechargedensities{103{ are=qn=q (v)n0=1uv c2 (u) (v)qn0whereyou'vegottodoalittlebitofalgebratogettothelastresult.Sincev��v+,wehaven��n+andthewirecarriesnegativecharge.Theoverallnetchargedensityinthenewframeis0=qn0=q(n+�n�)=�2uv c2 (u)qnButweknowthatalineofelectricchargecreatesanelectriceld;wecalculateditin( 2.6 );itisE(r)=�2uv c2 (u)qnA 20r^rwhereristheradialdirectionawayfromthewire.Thismeansthat,initsrestframe,theparticleexperiencesaforceF0=�u (u)nAq2v 0c2rwheretheminussigntellsusthattheforceistowardsthewireforu�0.Butifthere'saforceinoneframe,theremustalsobeaforceinanother.Transformingbacktowherewecamefrom,weconcludethatevenwhenthewireisneutraltherehastobeaforceF=F0 (u)=�unq2Av 0c2r=�uq0I 2r(5.11)ButthispreciselyagreeswiththeLorentzforcelaw,withthemagneticeldgivenbytheexpression( 3.5 )thatwecomputedforastraightwire.Noticethatifu�0thenthetestparticle{whichhaschargeq{ismovinginthesamedirectionastheparticlesinthewirewhichhavechargeqandtheforceisattractive.Ifu0thenitmovesintheoppositedirectionandtheforceisrepulsive.Thisanalysisprovidesanexplicitdemonstrationofhowanelectricforceinoneframeofreferenceisinterpretedasamagneticforceinanother.There'salsosomethingrathersurprisingabouttheresult.We'reusedtothinkingoflengthcontractionasanexoticresultwhichisonlyimportantwhenweapproachthespeedoflight.Yettheelectronsinawirecrawlalong.Theytakearoundanhourtotravelameter!Nonetheless,wecaneasilydetectthemagneticforcebetweentwowireswhich,aswe'veseenabove,canbedirectlyattributedtothelengthcontractionintheelectrondensity.{104{ Thediscussionaboveneedsaminoralterationforactualwires.Intherestframeofthewirethepositivecharges{whichareions,atomsstrippedofsomeoftheirelectrons{arestationarywhiletheelectronsmove.Followingtheexplanationabove,youmightthinkthatthereisanimbalanceofchargedensityalreadyinthisframe.Butthat'snotcorrect.Thecurrentisduetosomebatteryfeedingelectronsintothewireandtakingthemouttheotherend.Andthisisdoneinsuchawaythatthewireisneutralintherestframe,withtheelectrondensityexactlycompensatingtheiondensity.Incontrast,ifwemovedtoaframeinwhichtheionsandelectronshadequalandoppositespeeds,thewirewouldappearcharged.Althoughthestartingpointisslightlydi erent,theendresultremains.5.3GaugePotentialsandtheElectromagneticTensorUnderLorentztransformations,electricandmagneticeldswilltransformintoeachother.Inthissection,wewanttounderstandmorepreciselyhowthishappens.Atrstsight,itlooksasifit'sgoingtobetricky.SofartheobjectswhichhavenicetransformationpropertiesunderLorentztransformationsare4-vectors.Butherewe'vegottwo3-vectors,EandB.Howdowemakethosetransformintoeachother?5.3.1GaugeInvarianceandRelativityTogetanideaforhowthishappens,werstturntosomeobjectsthatwemetprevi-ously:thescalarandvectorpotentials andA.Recallthatweintroducedthesetosolvesomeoftheequationsofelectrostaticsandmagnetostatics,rE=0)E=�r rB=0)B=rAHowever,ingeneraltheseexpressionscan'tbecorrect.WeknowthatwhenBandEchangewithtime,thetwosource-freeMaxwellequationsarerE+@B @t=0andrB=0Nonetheless,it'sstillpossibletousethescalarandvectorpotentialstosolvebothoftheseequations.ThesolutionsareE=�r �@A @tandB=rAwherenow = (x;t)andA=A(x;t).{105{ Justaswesawbefore,thereisnouniquechoiceof andA.WecanalwaysshiftA!A+r andBremainsunchanged.However,nowthisrequiresacompensatingshiftof . ! �@ @tandA!A+r (5.12)with = (x;t).Thesearegaugetransformations.TheyreproduceourearliergaugetransformationforA,whilealsoencompassingconstantshiftsin .Howdoesthishelpwithourattempttoreformulateelectromagnetisminawaycompatiblewithspecialrelativity?Well,nowwehaveascalar,anda3-vector:theseareripetoplaceina4-vector.WedeneA= =cA!Or,equivalently,A=( =c;�A).Inthislanguage,thegaugetransformations( 5.12 )takeaparticularlyniceform,A!A�@ (5.13)where isanyfunctionofspaceandtime5.3.2TheElectromagneticTensorWenowhavealltheingredientsnecessarytodeterminehowtheelectricandmagneticeldstransform.Fromthe4-derivative@=(@=@(ct);r)andthe4-vectorA=( =c;�A),wecanformtheanti-symmetrictensorF=@A�@AThisisconstructedtobeinvariantundergaugetransformations( 5.13 ).WehaveF!F+@@ �@@ =FThisalreadysuggeststhatthecomponentsinvolvetheEandBelds.Tocheckthatthisisindeedthecase,wecandoafewsmallcomputations,F01=1 c@(�Ax) @t�@( =c) @x=Ex candF12=@(�Ay) @x�@(�Ax) @y=�Bz{106{ Similarcomputationsforallotherentriesgiveusamatrixofelectricandmagneticelds,F=0BBBB@0Ex=cEy=cEz=c�Ex=c0�BzBy�Ey=cBz0�Bx�Ez=c�ByBx01CCCCA(5.14)This,then,istheanswertoouroriginalquestion.YoucanmakeaLorentzcovariantobjectconsistingoftwo3-vectorsbyarrangingtheminananti-symmetrictensor.Fiscalledtheelectromagnetictensor.Equivalently,wecanraisebothindicesusingtheMinkowskimetrictogetF=F=0BBBB@0�Ex=c�Ey=c�Ez=cEx=c0�BzByEy=cBz0�BxEz=c�ByBx01CCCCABothFandFaretensors.Theyaretensorsbecausethey'reconstructedoutofobjects,A,@and,whichthemselvestransformnicelyundertheLorentzgroup.ThismeansthattheeldstrengthmusttransformasF0=F(5.15)Alternatively,ifyouwanttogetridoftheindices,thisreadsF0=FT.TheobserverinanewframeseeselectricandmagneticeldsE0andB0thatdi erfromtheoriginalobserver.Thetwoarerelatedby( 5.15 ).Let'slookatwhatthismeansinacoupleofillustrativeexamples.RotationsTocomputethetransformation( 5.15 ),it'sprobablysimplesttojustdothesumsthatareimplicitintherepeatedandlabels.Alternatively,ifyouwanttoreverttomatrixmultiplicationthenthisisthesameasF0=FT.Eitherway,wegetthesameresult.Forarotation,the33matrixRisembeddedinthelower-righthandblockofasshownin( 5.4 ).Aquickcalculationshowsthatthetransformationoftheelectricandmagneticeldsin( 5.15 )isasexpected,E0=REandB0=RB{107{ BoostsThingsaremoreinterestingforboosts.Let'sconsideraboostvinthex-direction,withgivenby( 5.5 ).Again,youneedtodoafewshortcalculations.Forexample,wehave�E0x c=F001=01F=0011F01+0110F10= 2v2 c2Ex c� 2Ex c=�Ex cand�E0y c=F002=02F=0022F02+0122F12=� Ey c+ v cBz=� c(Ey�vBz)and�B0z=F012=12F=1022F02+1122F12= v c2Ey� Bz=� (Bz�vEy=c2)Thenalresultforthetransformationoftheelectriceldafteraboostinthex-directionisE0x=ExE0y= (Ey�vBz)(5.16)E0z= (Ez+vBy)and,forthemagneticeld,B0x=BxB0y= By+v c2Ez(5.17)B0z= Bz�v c2EyAsweanticipatedabove,whatappearstobeamagneticeldtooneobserverlookslikeanelectriceldtoanother,andviceversa.{108{ Notethatinthelimitv c,wehaveE0=E+vBandB0=B.ThiscanbethoughtofastheGalileanboostofelectricandmagneticelds.WerecogniseE+vBasthecombinationthatappearsintheLorentzforcelaw.We'llreturntothisforceinSection 5.4.1 wherewe'llseehowit'scompatiblewithspecialrelativity.5.3.3AnExample:ABoostedLineChargeInSection 2.1.3 ,wecomputedtheelectriceldduetoalinewithuniformchargedensityperunitlength.Ifwetakethelinetoliealongthex-axis,wehave( 2.6 )E= 20(y2+z2) 0yz!(5.18)Meanwhile,themagneticeldvanishesforstaticelectriccharges:B=0.Let'sseewhatthislookslikefromtheperspectiveofanobservermovingwithspeedvinthex-direction,paralleltothewire.Inthemovingframetheelectricandmagneticeldsaregivenby( 5.16 )and( 5.17 ).ThesereadE0= 20(y2+z2) 0yz!= 20(y02+z02) 0y0z0!B0= v 20c2(y2+z2) 0z�y!= v 20c2(y02+z02) 0z0�y0!(5.19)Inthesecondequality,we'verewrittentheexpressionintermsofthecoordinatesofS0which,becausetheboostisinthex-direction,aretrivial:y=y0andz=z0.FromtheperspectiveofanobserverinframeS0,thechargedensityinthewireis0= ,wherethefactorof comesfromLorentzcontraction.Thiscanbeseenintheexpressionabovefortheelectriceld.Sincethechargedensityisnowmoving,theobserverinframeS0seesacurrentI0=� v.Thenwecanrewrite( 5.19 )asB0=0I0 2p y02+z02^'0(5.20)Butthisissomethingthatwe'veseenbefore.It'sthemagneticeldduetoacurrentinawire( 3.5 ).WecomputedthisinSection 3.1.1 usingAmpere'slaw.Butherewe'vere-derivedthesameresultwithoutevermentioningAmpere'slaw!Instead,ourstartingpoint( 5.18 )neededGauss'lawandwethenusedonlytheLorentztransformationofelectricandmagneticelds.Wecanonlyconcludethat,underaLorentztransforma-tion,Gauss'lawmustberelatedtoAmpere'slaw.Indeed,we'llshortlyseeexplicitlythatthisisthecase.Fornow,it'sworthrepeatingthelessonthatwelearnedinSection 5.2.1 :themagneticeldcanbeviewedasarelativistice ect.{109{ 5.3.4AnotherExample:ABoostedPointChargeConsiderapointchargeQ,stationaryinaninertialframeS.Weknowthatit'selectriceldisgivenbyE=Q 40r2^r=Q 40[x2+y2+z2]3=2 xyz!whileitsmagneticeldvanishes.Nowlet'slookatthissameparticlefromtheframeS0,movingwithvelocityv=(v;0;0)withrespecttoS.TheLorentzboostwhichrelatesthetwoisgivenby( 5.5 )andsothenewelectriceldaregivenby( 5.16 ),E0=Q 40[x2+y2+z2]3=2 x y z!Butthisisstillexpressedintermsoftheoriginalcoordinates.WeshouldnowrewritethisintermsofthecoordinatesofS0,whicharex0= (x�vt)andy0=yandz0=z.Invertingthese,wehaveE0=Q 40[ 2(x0+vt0)2+y02+z02]3=2 x0+vt0y0z0!(5.21)IntheframeS0,theparticlesitsatx0=(�vt0;0;0),soweseethattheelectriceldemanatesfromthepositionofthecharge,asitshould.Fornow,let'slookattheelectriceldwhent0=0sothattheparticlesitsattheorigininthenewframe.Theelectriceldpointsoutwardsradially,alongthedirectionr0= x0y0z0!However,theelectriceldisnotisotropic.Thisarisesfromthedenominatorof( 5.21 )whichisnotproportionaltor03becausethere'sanextrafactorof 2infrontofthex0component.Instead,att0=0,thedenominatorinvolvesthecombination 2x02+y02+z02=( 2�1)x02+r02=v2 2 c2x02+r02=v2 2 c2cos2+1r02= 21�v2 c2sin2r02{110{ Figure48:TheisotropiceldlinesofastaticchargeFigure49:Thesqueezedeldlinesofamovingchargewheretheistheanglebetweenr0andthex0-axisand,inthelastline,we'vejustusedsomesimpletrigandthedenitionof 2=1=(1�v2=c2).ThismeansthatwecanwritetheelectriceldinframeS0asE0=1 2(1�v2sin2=c2)3=2Q 40r02^r0Thepre-factorisresponsibleforthefactthattheelectriceldisnotisotropic.Weseethatitreducestheelectriceldalongthex0-axis(i.ewhen=0)andincreasestheeldalongtheperpendiculary0andz0axes(i.e.when==2).ThiscanbethoughtofasaconsequenceofLorentzcontraction,squeezingtheelectriceldlinesinthedirectionoftravel.Themovingparticlealsogivesrisetoamagneticeld.ThisiseasilycomputedusingtheLorentztransformations( 5.17 ).ItisB=0Q v 4[ 2(x0+vt0)2+y02+z02]3=2 0z0�y0!5.3.5LorentzScalarsWecannowaskafamiliarquestion:isthereanycombinationoftheelectricandmagneticeldsthatallobserversagreeupon?Nowwehavethepowerofindexnotationatourdisposal,thisiseasytoanswer.Wejustneedtowritedownanobjectthatdoesn'thaveany oatingorindices.Unfortunately,wedon'tgettousetheobviouschoiceofFbecausethisvanishesonaccountoftheanti-symmetryofF.Thesimplestthingwecanwritedownis1 2FF=�E2 c2+B2NotetherelativeminussignbetweenEandB,mirroringasimilarminussigninthespacetimeinterval.{111{ However,thisisn'ttheonlyLorentzscalarthatwecanconstructfromEandB.Thereisanother,somewhatmoresubtle,object.Tobuildthis,weneedtoappreciatethatMinkowskispacetimecomesequippedwithanothernaturaltensorobject,beyondthefamiliarmetric.Thisisthefullyanti-symmetricobjectknownasthealternatingtensor,=(+1ifisanevenpermutationof0123�1ifisanoddpermutationof0123while=0ifthereareanyrepeatedindices.ToseewhythisisanaturalobjectinMinkowskispace,let'slookathowitchangesunderLorentztransformations.Theusualtensortransformationis0= It'ssimpletocheckthat0isalsofullanti-symmetric;itinheritsthispropertyfrom ontheright-handside.Butthismeansthat0mustbeproportionalto.Weonlyneedtodeterminetheconstantofproportionality.Todothis,wecanlookat00123=012 3 =det()NowanyLorentztransformationshavedet()=1.Thosewithdet()=1makeupthe\properLorentzgroup"SO(1;3).(ThiswascoveredintheDynamicsandRelativitynotes).TheseproperLorentztransformationsdonotincludere ectionsortimereversal.WelearnthatthealternatingtensorisinvariantunderproperLorentztransformations.Whatit'sreallytellingusisthatMinkowskispacecomeswithanorientedorthonormalbasis.ByloweringindiceswiththeMinkowskimetric,wecanalsoconstructthetensorwhichhas0123=�1.Thealternatingtensorallowsustoconstructasecondtensoreld,sometimescalledthedualelectromagnetictensor(although\dual"isperhapsthemostoverusedwordinphysics),~F=1 2F=0BBBB@0�Bx�By�BzBx0Ez=c�Ey=cBy�Ez=c0Ex=cBzEy=c�Ex=c01CCCCA(5.22)~Fissometimesalsowrittenas?F.WeseethatthisislooksjustlikeFbutwiththeelectricandmagneticeldsswappedaround.Actually,lookingcloselyyou'llseethatthere'saminussigndi erenceaswell:~FarisesfromFbythesubstitutionE!cBandB!�E=c.{112{ Thestatementthat~FisatensormeansthatittoohasnicepropertiesunderLorentztransformations,~F0=~FandwecanusethistobuildnewLorentzinvariantquantities.Takingtheobvioussquareof~Fdoesn'tgiveusanythingnew,since~F~F=�FFButbycontracting~FwiththeoriginalFwedondanewLorentzinvariant1 4~FF=1 cEBThistellsusthattheinner-productofEandBisthesameviewedinallframes.5.4MaxwellEquationsWenowhavethemachinerytowritetheMaxwellequationsinawaywhichismanifestlycompatiblewithspecialrelativity.Theytakeaparticularlysimpleform:@F=0Jand@~F=0(5.23)Prettyaren'tthey!TheMaxwellequationsarenotinvariantunderLorentztransformations.Thisisbecausethereisthedanglingindexonbothsides.However,becausetheequationsarebuiltoutofobjectswhichtransformnicely{F,~F,Jand@{theequationsthemselvesalsotransformnicely.Forexample,wewillseeshortlythatGauss'lawtransformsintoAmpere'slawunderaLorentzboost,somethingweanticipatedinSection 5.3.3 .WesaythattheequationsarecovariantunderLorentztransformations.Thismeansthatanobserverinadi erentframewillmixeverythingup:spaceandtime,chargesandcurrents,andelectricandmagneticelds.Althoughobserversdisagreeonwhatthesethingsare,theyallagreeonhowtheyttogether.Thisiswhatitmeansforanequationtobecovariant:theingredientschange,buttherelationshipbetweenthemstaysthesame.Allobserversagreethat,intheirframe,theelectricandmagneticeldsaregovernedbythesameMaxwellequations.GiventheobjectsF,~F,Jand@,theMaxwellequationsarenottheonlythingyoucouldwritedowncompatiblewithLorentzinvariance.Buttheyarebyfarthesimplest.Anyotherequationwouldbenon-linearinFor~Forcontainmorederivativetermsorsomesuchthing.Ofcourse,simplicityisnoguaranteethatequationsarecorrect.Forthisweneedexperiment.Butsurprisinglyofteninphysicswendthatthesimplestequationsarealsotherightones.{113{ UnpackingtheMaxwellEquationsLet'snowcheckthattheMaxwellequations( 5.23 )inrelativisticformdoindeedcoincidewiththevectorcalculusequationsthatwe'vebeenstudyinginthiscourse.Wejustneedtoexpandthedi erentpartsoftheequation.ThecomponentsoftherstMaxwellequationgive@iFi0=0J0)rE= 0@Fi=0Ji)�1 c2@E @t+rB=0JIntherstequation,whicharisesfrom=0,wesumonlyoverspatialindicesi=1;2;3becauseF00=0.MeanwhilethecomponentsofthesecondMaxwellequationgive@i~Fi0=0)rB=0@~Fi=0)@B @t+rE=0These,ofcourse,arethefamiliarequationsthatwe'veallgrowntoloveoverthiscourse.Hereafewfurther,simplecommentsabouttheadvantagesofwritingtheMaxwellequationsinrelativisticform.First,theMaxwellequationsimplythatcurrentiscon-served.ThisfollowsbecauseFisanti-symmetric,so@@F=0automatically,simplybecause@@issymmetric.TherstoftheMaxwellequations( 5.23 )thenrequiresthatthecontinuityequationholds@J=0ThisisthesamecalculationthatwedidinvectornotationinSection 4.2.1 .Notethatit'smarginallyeasierintherelativisticframework.ThesecondMaxwellequationcanbewritteninanumberofdi erentways.Itisequivalentto:@~F=0,@F=0,@F+@F+@F=0wherethelastoftheseequalitiesfollowsbecausetheequationisconstructedsothatitisfullyanti-symmetricwithrespecttoexchanginganyoftheindices,and.(Justexpandoutforafewexamplestoseethis).{114{ ThegaugepotentialAwasoriginallyintroducedtosolvethetwoMaxwellequationswhicharecontainedin@~F=0.Again,thisismarginallyeasiertoseeinrelativisticnotation.IfwewriteF=@A�@Athen@~F=1 2@F=1 2@(@A�@A)=0wherethenalequalityholdsbecauseofthesymmetryofthetwoderivatives,combinedwiththeanti-symmetryofthe-tensor.ThismeansthatwecouldequallywellwritetheMaxwellequationsas@F=0JwhereF=@A�@ATherstofthesecoincideswiththerstequationin( 5.23 );thesecondisanalternativewayofwritingthesecondequationin( 5.23 ).Inmoreadvancedformulationsofelec-tromagnetism(forexample,intheLagrangianformulation),thisistheforminwhichtheMaxwellequationsarise.5.4.1TheLorentzForceLawThere'sonelastaspectofelectromagnetismthatweneedtoshowiscompatiblewithrelativity:theLorentzforcelaw.IntheNewtonianworld,theequationofmotionforaparticlemovingwithvelocityuandmomentump=muisdp dt=q(E+uB)(5.24)Wewanttowritethisequationin4-vectornotationinawaythatmakesitclearhowalltheobjectschangeunderLorentztransformations.Bynowitshouldbeintuitivelyclearhowthisisgoingtowork.Amovingparticleexperiencesthemagneticforce.Butifweboosttoitsrestframe,thereisnomagneticforce.Instead,themagneticeldtransformsintoanelectriceldandwendthesameforce,nowinterpretedasanelectricforce.Therelativisticversionof( 5.24 )involvesthe4-momentumP,denedin( 5.6 ),thepropertime ,reviewedinSection 5.1.2 ,andournewfriendtheelectromagnetictensorF.TheelectromagneticforceactingonapointparticleofchargeqcanthenbewrittenasdP d =qFU(5.25){115{ wherethe4-velocityisU=dX d = cu!(5.26)andthe4-momentumisP=mU.Again,weseethattherelativisticformoftheequation( 5.25 )issomewhatprettierthantheoriginalequation( 5.24 ).UnpackingtheLorentzForceLawLet'schecktoseethattherelativisticequation( 5.25 )isgivingustherightphysics.Itis,ofcourse,fourequations:oneforeach=0;1;2;3.It'ssimpletomultiplyouttheright-handside,rememberingthatUcomeswithanextraminussigninthespatialcomponentsrelativeto( 5.26 ).Wendthatthe=1;2;3componentsof( 5.25 )arrangethemselvesintoafamiliarvectorequation,dp d =q (E+uB))dp dt=q(E+uB)(5.27)wherewe'veusedtherelationshipdt=d = .WendthatwerecovertheLorentzforcelaw.Actually,there'saslightdi erencefromtheusualNewtonianforcelaw( 5.24 ),althoughthedi erenceisburiedinournotation.IntheNewtoniansetting,themomentumisp=mu.However,intherelativisticsettingabove,themomentumisp=m u.Needlesstosay,therelativisticversioniscorrect,althoughthedi erenceonlyshowsupathighspeeds.TherelativisticformulationoftheLorentzforce( 5.25 )alsocontainsanextraequationcomingfrom=0.ThisreadsdP0 d =q c Eu(5.28)Recallthatthetemporalcomponentofthefour-momentumistheenergyP0=E=c.HeretheenergyisE=m c2whichincludesboththerest-massoftheparticleanditskineticenergy.Theextraequationin( 5.25 )issimplytellingusthatthekineticenergyincreaseswhenworkisdonebyanelectriceldd(Energy) dt=qEuwhereI'vewrittenenergyasawordratherthanasEtoavoidconfusingitwiththeelectriceldE.{116{ 5.4.2MotioninConstantFieldsWealreadyknowhowelectricandmagneticeldsactonparticlesinaNewtonianworld.Electriceldsaccelerateparticlesinstraightlines;magneticeldsmakeparticlesgoincircles.Herewe'regoingtoredothisanalysisintherelativisticframework.TheLorentzforcelawremainsthesame.Theonlydi erenceisthatmomentumisnowp=m u.We'llseehowthischangesthings.ConstantElectricFieldConsideravanishingmagneticeldandconstantelectriceldE=(E;0;0).(NotethatEheredenoteselectriceld,notenergy!).Theequationofmotion( 5.27 )forachargedparticlewithvelocityu=(u;0;0)ismd( u) dt=qE)m u=qEtwherewe'veimplicitlyassumedthattheparticlestartsfromrestatt=0.Rearranging,wegetu=dx dt=qEt p m2+q2E2t2=c2Reassuringly,thespeedneverexceedsthespeedoflight.Instead,u!cast!1asonewouldexpect.It'ssimpletointegratethisoncemore.Iftheparticlestartsfromtheorigin,wehavex=mc2 qE r 1+q2E2t2 m2c2�1!Forearlytimes,whenthespeedsarenottoohigh,thisreducestomx1 2qEt2+:::whichistheusualnon-relativisticresultforparticlesundergoingconstantaccelerationinastraightline.ConstantMagneticFieldNowlet'sturntheelectriceldo andlookatthecaseofconstantmagneticeldB=(0;0;B).Inthenon-relativisticworld,weknowthatparticlesturncircleswithfrequency!=qB=m.Let'sseehowrelativitychangesthings.{117{ Westartbylookingatthezerothcomponentoftheforceequation( 5.28 )which,intheabsenceofanelectriceld,readsdP0 d =0Thistellsusthatmagneticeldsdonowork.WeknewthisfromourcourseonNewtonianphysics,butitremainstrueintherelativisticcontext.Soweknowthatenergy,E=m c2,isconstant.Butthistellsusthatthespeed(i.e.themagnitudeofthevelocity)remainsconstant.Inotherwords,thevelocity,andhencetheposition,onceagainturncircles.Theequationofmotionisnowmd( u) dt=quBSince isconstant,theequationtakesthesameformasinthenon-relativisticcaseandthesolutionsarecircles(orhelicesiftheparticlealsomovesinthez-direction).Theonlydi erenceisthatthefrequencywithwhichtheparticlemovesinacirclenowdependsonhowfasttheparticleismoving,!=qB m Ifyouwanted,youcouldinterpretthisasduetotherelativisticincreaseinthemassofamovingparticle.Naturally,forsmallspeeds 1andwereproducethemorefamiliarcyclotronfrequency!qB=m.SofarwehavelookedatsituationsinwhichE=0andinwhichB=0.Butwe'veseenthatEB=0andE2�B2arebothLorentzinvariantquantities.Thismeansthatthesolutionswe'vedescribedabovecanbeboostedtoapplytoanysituationwhereEB=0andE2�B2iseither�0or0.Inthegeneralsituation,bothelectricandmagneticeldsareturnedonsoEB6=0andwehavethreepossibilitiestoconsiderdependingonwhetherE2�B2is&#x]TJ/;༕ ;.9;Ւ ;&#xTf 1;.42; 0 ;&#xTd [;0or0or=0.5.5EpilogueThisbringustotheendofourrstcourseonelectromagnetism.WehavelearnedhowtheMaxwellequationscapturetheelectricandmagneticforces,therelationshipbe-tweenthem,andtheexistenceofelectromagneticwaves.However,therearemanymorephenomenastilltodiscoverintheseequations.Prominentamongthemisthewayinwhichacceleratedchargesemitlight,andadescriptionofhowelectromagnetismworksinsidedi erentmaterials.Bothofthesewillbecoveredinnextyear'sElectrodynamicscourse.{118{ Thesecondstrandrunningthroughtheselecturesisthat,despitethecomplexityofphenomenathattheycontain,theMaxwellequationsthemselvesareextraordinarilysimpleandelegant.Indeed,whenviewedintherightway,intermsoftheelectromag-netictensorF=@A�@A,theequationsare@F=0JThisformalismprovidesablueprintfortheotherforcesinNature.Likeelectromag-netism,allforcesaredescribedintermsofeldsinteractingwithparticleswhichcarryatypeofconservedcharge.Ratherremarkably,theequationsgoverningtheweakandstrongnuclearforceareessentiallyidenticaltotheMaxwellequationsabove;allthatchangesisthemeaningofthetensorF(and,forwhatit'sworth,thederivativeisalsoreplacedbysomethingcalleda\covariantderivative").TheresultingequationsarecalledtheYang-MillsequationsandwilldescribedinsomedetailincoursesinPartIII.Gravityissomewhatdi erent,buteventheretheingredientsthatgointotheequa-tionsareverysimilartothoseseenhere.And,aswithelectromagnetism,onceyouhavetheseingredientstheequationsthatgovernourUniverseturnouttobethesimplestonesthatyoucanwritedown.Indeed,thisisperhapsthemostimportantlessontotakefromthiscourse:thelawsofphysicsaregloriouslysimple.Youshouldlearnthem.{119{