13 麦克斯韦方程

113Maxwell'sEquationsMaxwell'sequationsarebasedonthefollowingempiricalfacts:(1)TheelectricchargesarethesourcesandsinksofthevectorfieldofthedielectricdisplacementdensityD.Hence,forthefluxofthedielectricdisplacementthroughasurfaceenclosingthechargewehave44daQdVDn.Here,nistheoutwardpointingunitnormalvector.ThisrelationcanbederivedfromCoulomb'sforcelaw.(2)Faraday'sinductionlaw:1dVdcdtErwithdaBn.(3)Thefactthattherearenoisolatedmonopolesimplies0daBnthatis,themagneticinductionissource-free;itsfieldlinesareclosedcurves.(4)Oersted'sorAmpere'slaw:44dIdaccHrjn.Fromtheseobservations(1)-(4),Maxwell'sequationsinintegralrepresentationcanbeinferreddirectly:4dadVDn(13.1)1dddacdtErBn(13.2)0daBn(13.3)41414ddddadadadaccdtcdtHrjnDnjnDn(13.4)wherethelastterminequation(13.4)hasthedimensionofacurrent(compareequa-tion(13.1))introducedbyMaxwellasthedisplacementcurrent.ThisadditionaltermisnecessarytofulfillthecontinuityequationddVdadtjn.(13.5)Thisrelationimpliestheconservationoftheelectriccharge:TheelectricchargeinaregionofspaceVcanchangeintimeonlyifacurrentflowsthroughitssurface.Amoredetailedjustificationofthedisplacementcurrentmaybegivenaftertrans-formingtheequations(13.1)-(13.5)intotheassociateddifferentialform,whichcanbedonebymeansofGauss'andStokes'theorems.Asanexample,weconsiderequation(13.1):div4dadVdVDnD.Immediately,div4D.Analogously,wetransformtheremainingequations(13.2)-(13.5).Then,weobtainMaxwell'sequationsindifferentialform:div4D(13.6)1curlctBE(13.7)div0B(13.8)4141curl4cctctDDHjj(13.9)2aswellasthecontinuityequationdiv0tj.(13.10a)Inaddition,thereistheforcelaw1cfEjB.(13.10b)Theforcelawlinkselectrodynamicsandmechanics.Atthispoint,Ampere'slaw,givenin(11.9),reads(compareequation(13.9))4curlcHj.(13.11)Inthisform,itholdsonlyforstationarycurrentdistributions.Undertheassumptionthatthesameassignmentexistsalsofortime-varyingcurrents,equation(13.11)requiresacrucialcompletioninthecasethattheconductioncurrentisinterrupted,e.g.,byacapacitor(Figure13.1):Attheplatesofthecapacitor,thedivergenceofjisnotequaltozero,whiletheleft-handsideofequation(13.11)isalwayssource-free(divcurl0H).Afterintroducingthedisplacementcurrentatthevector144tDZjisalwayssource-freeduetotheequations(13.6)and(13.10a):11div4divdiv4divdiv4div044tttDZjjDjthatis,afterintroducingMaxwell'sdisplacementcurrentthereareonlyclosedcur-rents.Withoutthedisplacementcurrent(14)tDinequation(13.9),thesystemofMaxwell'sequations(13.6)-(13.9)wouldnotbeconsistentifthecontinuityequa-tion(13.10a)wastakenintoaccount.Onlytheequationscompletedinthiswayareconsistent.Maxwell'sequations(13.6)-(13.9)arecompletedbytheso-calledconnectingequa-tionsrelatingthevectorsDandBtothefieldintensitiesEandH:4DEP4BHM(13.12)wherePisthevectoroftheelectricpolarization,andMisthevectorofthemagnetization.Thelinktomechanicsisrealizedbytheforceequation(13.10b),whichdescribestheLorentzforce.Insteadyfields,forisotropicmaterialswhichmaybepolarizedandmagnetizedinnormalmanner,thereareconstitutiveequations(aspecialcaseof(13.12)):4(14)eDEPEE4(14)mBHMHH.(13.13)3TheconnectionofjandEyieldsthegeneralizedOhm'slaw:jE(13.14)whereistheelectricconductivity.Innonlinearmedia,,,andaretensorsdependingonthefieldintensities;moreover,intime-varyingfields,,andarefrequencydependent.Toexpressequation(13.9)intermsofthequantitiesEandB,weusethecon-nectingequations(13.12):41curlcctDHj(13.15)41curl(4)(4)cctBMjEP(13.16)41curlcurlcctctPEBjM.(13.17)Obviously,thecurloftheB-fielddependsonthedensitiesoftheconductingcurrentj,themagnetizationcurrentcurlcMjM,andthepolarizationcurrenttPjP.Sinceinvacuumthecurrentdensities,Mjj,andPjvanish,onlytheterm(1)ctEinconnectionwithequation(13.7)mayexplainthepropagationofelectromagneticradiationinvacuum.(SeeChapter14.)Maxwell'sequations(13.6)-(13.9)arepartial,linear,coupleddifferentialequationsofthefirstorder.Duetolinearitytheprincipleofsuperpositionisvalid.Hence,theinterferencefringesofelectromagneticwaves,inparticular,andalsothoseoflightasspecialelectromagneticwavesareguaranteed.WehaveinferredMaxwell'sequations(13.6)-(13.10b)includingtheconnectingrelations(13.13)and(13.14),stepbystepfromempiricalobservations:equation(13.6)follows,e.g.,fromtheCoulombinteractionofcharges,equation(13.7)fromFaraday'sinductionlaw,andequation(13.8)fromthefactthattherearenomagneticmonopolecharges.Withoutthedisplacementcurrent,equation(13.9)representsthesecondAmpere'slawbeingagainequivalenttotheBiot-Savartlaw.Theconservationofchargeisrelatedto(13.10a)andthefirstAmpere'slaw(theLorentzforce)isfixedin(13.10b).So,wehaveorientedourselvesaccordingtotheresultsofexperimentstofind,finally,thefundamentalequationsofelectrodynamics,Maxwell'sequations.Wefoundalsonew,unexpectedthingslikethedisplacementcurrent,whichwasnotcontainedintheexperimentalphenomenabywhichwehavebeenguided.Now,wewillacceptMaxwell'sequationsasfundamental;wewillusethemasaxiomsfromwhichtodeduceeverythingthatfollows.Thesefundamentalequationsarefurthersupportedbythesuccessofthesedeductions.Inmechanicseverythingwasmuchsimpler:Newton'sforcelawpFcouldbegraspedrapidly;soonwecouldplaceitatthetopofourtheory.Thefundamentalelectromagneticequations(theMaxwellequations)aremuchmorecomplicated.Therefore,thewayhasbeenmuchmoredifficult.1Exercise13.1:Ohm'slawWewanttodemonstratethatfromOhm'slawinitsdifferentialformjEundercertainassumptionstheknownintegralform1212IRcanbederived.Theconstantofproportionalityistheconductivityofthematerial,12isthedifferenceofthevoltagebetweentwopointsoftheconductor,12Risthecorrespondingohmicresistance,andIisthecurrent.SolutionWerestrictourselvestofieldsEthatareconstantintimeandtouniformchargedistributionssothatthecurrentdistributionsarestationary.Now,weconsideracylindricalconductorbetweentwocross-sectionalareas1aand2ahavingthepotentials1()aand2()a,respectively(equipotentialsurfaces).SeeFigure13.2.Inthiscase,theelectricvoltage12betweenthetwosurfacesis21222212121111()()IdsaaddddajsEss.Hence,theintegralformofOhm'slawforthevoltageis1212VIRwith2121dsRa.Duetothespecificresistance1,theresistancedependsonthepropertiesofthematerial,andduetosanda,itdependsonthespatialdimensionsoftheconductor.Foracylindricconductorcomposedofahomogeneousmaterialwithlengthlandtheconstantcrosssectionalareaa,theohmicresistanceis21211lRdsaa.1Theenergyconservationlawofelectrodynamics:ThePoyntingvectorThePoyntingtheoremrepresentstheenergyconservationlawofelectrodynamics;itcanbederivedwiththehelpofMaxwell'sequations.Forthispurpose,weperformthescalarproductofequation(13.7)andH,andofequation(13.9)andE,andthenweaddbothequations.Weobtain41curlcurlccttDBEHHEjEEH.(13.18)Withtheidentity()()()EHEHHE.Theequation(13.18)canberewritten41()ccttDBEHjEEH.Multiplyingthisequationby4c,thenitcanbebroughttotheform1()44cttDBEHjEEH.(13.19)Now,wemustinterpretthevariousterms.Atfirst,basedonourexperiences,e.g.,theelectrostatics(seeequation(7.3)),werecognizethefollowing:theright-handsideofequation(13.19)istherateofchangeofthetotalenergydensityoftheelectro-magneticfield:elm1()4wwtttDBEH.(13.20a)Thiscanbeseenbyintroducingtheenergydensityoftheelectricfield,22el01114888wdDEDDEED(13.20b)andtheenergydensityofthemagneticfield22m01114888wdBHBBHHB(13.20c)bysubstitutingtheseexpressionsforelwandmwintoequation(13.20a)andper-formingthepartialdifferentiationwithrespecttothetime.Here,wehaveassumedthatDEandBH,andhence,alsothatttEDDE,ttHBBH.Wealreadyhaveencounteredtherelations(13.20a)-(13.20c)fortheenergydensityoftheelectricandthemagneticfieldinspecialcasesinelectrostaticsandmagnetostatics,respectively.Now,theexpressions(13.20a)-(13.20c)followfromthefundamentalequations.So,inthefirstform(13.20a)theyarevalidgenerally.Then,wecanwritetheequation(13.19)inthefirstformelm()()4cwwtEHjE.(13.21a)Equation(13.21a)describesthetransferofenergyduringadecreaseofthetotalenergydensityoftheelectrodynamicsfieldintime.ThetermjErepresentstheworkdonebythefieldontheelectriccurrentdensity.TheenergyputintothecurrentinthiswaycanbefoundastheJouleheatgeneratedpersecondperunitvolume;itisthethermalpowerofthefield.Forexample,fortheohmicwirecorrespondingtoExercise13.1wehave222222VVdVEadsEalalIRlRjE.2Thisistheohmicpowerofaresistor.Afurtherreasonforthedecreaseoftheenergyofthefieldisthedivergenceofthevector4cSEH(13.21b)denotedthePoyntingvector.Thisvectordescribestheintensityoftheenergyflowperunittimethroughaunitareaperpendiculartothedirectionoftheflow.Strictly,onlythesurfaceintegralextendedoveraclosedsurfacedaSnhasaphysicalmeaning.Thiscanbeseenbyintegratingequation(13.21a)overanarbitraryvolumeandapplyingGauss'theoremelm()dwwdVdVdadtjESn.Thisequationrepresentstheenergyconservationlawofelectrodynamics(Poyntingtheorem);thetermdaSngivestheenergyflowcrossingtheareaelementdapersecondindirectionoftheouternormaln.Aswewillseelateroninequation(15.30),foraplaneelectromagneticwavethePoyntingvectorpointsindirectionofthepropagationoftheenergy.Thequantity2cSrepresentsthemomentumdensitytransferredbyapropagatingelectromagneticwavethatleadstoaradiationpressureonthematerialboundarysurfaces.(Theseresultswillbederivedinthenextsection.)1TheMaxwellianstresstensor:TheconservationofthelinearmomentumintheelectromagneticfieldThePoyntingtheoremintheformelm()dwwdVdVdadtjESnoffersthepossibilitytoderivethemomentumoftheelectromagneticfield.TheworkdonebytheelectricfieldEperunitvolumeandtime,jE,impliestheconversionofelectromagneticenergyintomechanicalenergyorthermalenergy:mechddVEdtjEthere,mechEisthetotalmechanicalenergyoftheparticlesinsidethevolumeV.Theenergyconservationinaclosedsystemmaythenberepresentedasmechfield()dEdEEdadtdtSn.fieldEisthetotalfieldenergypervolume,whichisgivenbytheequationfield1()8EdVEDHB.Nowthatwehavefoundaformulationoftheconservationofenergyintheelectro-magneticfieldinformofthePoyntingtheoremwewanttosetupacorrespondingconservationlawforthelinearmomentum.Forthispurpose,weconsiderthecasethat,apartfromtheelectromagneticfield,therearemovingchargesinthewholespace.ThetotalforceonthesechargesfollowsfromtheLorentz-forcedensity1cfEjBbyvolumeintegration:meth1()ddVdVdtcPFfrEjB(13.22)wheremethPmeansthetotalmechanical(linear)momentumoftheparticle.Thegreatimportanceofequation(13.22)liesinthefactthatitlinkselectrodynamicstome-chanicsIthastobeviewedasthemechanicalsupplementaryequationtoMaxwell'sequation.Now,withthehelpofMaxwell'sequations(13.6)and(13.9)weexpressjandbythefields.Weget1div4D,1curl4cctDjH.Andequation(13.22)becomesmeth11()()4ddVdtctDPDEHBB.Addingfurthertheterms()0BH(Maxwellequation(13.8):div0B)andutilizingtherelation:()tttDBDBBDourequationbecomesmeth1[()()()4ddtPDEBHHB11()dVctctBDBD.(13.23)2SubstitutingtheMaxwellequation(13.7)intoequation(13.22),weobtainmeth1[()()()4ddtPDEBHHB1()()dVctDBDE,(13.24)meth11[()()44ddVdtcPDBDEBH()()]dVHBED.(13.25)Tosimplifytheremainingderivation,weconsidertheelectromagneticfieldinvacuum,with1;thatis,EDandBH.Then,equation(13.25)readsmeth11[()()44ddVdtcPEBEEBB()()]dVBBEE.(13.26a)Thevolumeintegralontheleft-handsideofequation(13.26a)maybeinterpretedastheelectromagneticfieldmomentuminthevolumeV.Theexpression2114ccEBS(13.26b)hastoberegardedasthespatialdensityofthefieldmomentumfieldP.Itispropor-tionaltothedensityoftheenergyflow,thePoyntingvectorS,withthefactor21c,field2114dVdVccPEBS.SubstitutingfieldPintoequation(13.26a)methfield1()[()()()()]4ddVdtPPEEBBEEBB.(13.27)Theintegrandofequation(13.27)maybeexpressedasthedivergenceofatensor,theMaxwellianstresstensor.Todemonstratethis,fortherepresentationofdivergenceandcurlweuseEinstein'ssummationconvention,accordingtowhichonesumsautomaticallyoverindicesappearingtwice.Then,thesummationsignissuppressed.Forexample,inCartesiancoordinates:divjijjijiiFFxxFeeee,curljijjijiiFFxxFeeeewhereFisanydifferentiablevectorfield.Wenowrewritethepartoftheintegrandofequation(13.27)thatdependsontheE-field:()()jjijkkijkkiiEEEExxEEEEeeeeeejjjijkkkkjiikkjiiiEEEEEExxxeeeeeeejjikkjiijiiiEEEEEExxxeeejjiijjijjiEEEEEExxxebecausewecanrewritetheindicesinthevarioustermsduetothesummationoverallindices.3Inthefollowing,weconsideronlythethicomponentofourexpressioninordertorewritefurther1[()()]()()2jjiiijjjijjjjijjEEEEEEEEEExxxxxEEEE1()2jiijkkjEEEEx.Acompletelyanalogousresultisobtainedforthepartoftheintegrandofequation(13.27)thatdependsonlyonB:1[()()]()2ijiijkkjBBBBxBBBB.(13.28)Withtheserelations,theintegrandofequation(13.27)maybeexpressedasthediver-genceofatensormethfield11()()42iijijijkkkkjdEEBBEEBBdVdtxPP.(13.29)Thisisthedivergence(jx)oftheMaxwellianstresstensorwiththecomponents:11()42ijijijijkkkkTEEBBEEBB(,,1,2,3ijk).(13.30)Obviously,theMaxwellianstresstensorisasymmetrictensor,thatisijjiTT.Wewriteitinmatrixformusingthecomponents,xy,andzasindices:2222221()4xxxyxzxxxyxyxzxzijyxyyyzxyxyyyyzyzzxzyzzxzxzyzyzzzTTTEBWEEBBEEBBTTTTEEBBEBWEEBBTTTEEBBEEBBEBW(13.31)2222222211()()22xyzxyzWEEEBBBEB.(13.32)Consideringtheelectromagneticfieldinspacefilledwithmatter,wemayobtainthecomponentsoftheMaxwellianstresstensorinananalogousmannerfromequation(13.26a)insteadofequation(13.27):()xxxyxzijyxyyyzzxzyzzTTTTTTTTTT14xxxxxyxyxzxzxyxyyyyyyzyzxzxzyzyzzzzzEDBHWEDBHEDBHEDBHEDBHWEDBHEDBHEDBHEDBHW(13.33)11()()22xxyyzzxxyyzzWEDEDEDBHBHBHEDBH.(13.34)ThetraceoftheMaxwellianstresstensorhasthevalueelm11Tr()()()48ijiiTTWwwEDBH.(13.35)WiththehelpoftheMaxwellianstresstensor,themomentumequation(13.27)canbewrittenasmethfield1()4ijijTddVdtxPP.(13.36)4UsingGauss'theorem,thevolumeintegralisrewrittenformallyintoasurfaceintegral,sothatijijjjTdVTndax(13.37)wherejnmeansthethjcomponentoftheunitnormalvector(directioncosine)perpendiculartotheareada.Thevalidityof(13.37)canbeseenatoncebytheremarkthat123(,,)iiiTTTrepresentthecomponentsofthethirowvectoroftheMaxwelliantensor;soequation(13.37)isjusttheusualGauss'theoremforthethirowvector.Withthistransformation,weobtaintheconservationlawofthelinearmomentum:methfield()0iijjdTndadtPP(13.38)(forthethicomponent).Ifthearealiesinaregionshieldedfromthefieldoratinfinity,thenthesurfaceintegralvanishes.From(13.38)welearnthatonlythetotalmomentummechanicalmomentumplusfieldmomentumisconserved.ItisclearthatthequantityijjTndoesnotvanishingeneral.Obviously,itrepresentsthethicomponentofthemomentumflowthroughtheunitofarea.Westressthetermmomentumflow,thatis,momentumpersecondthroughunitarea,thusarateofchangeofthemomentumperunitofarea.Then,theexpressionijjTnamaybeunderstoodasthethicomponentofaforceiijjFTna(13.39)ontheareaelement.Iftheradiationis,e.g.,absorbedbyablackareaelementaan,thentheforceiiijijFnTnnaFnisexertedontheareaabytheradiation.Accordingtoequation(13.38),thetotalsystemconsistingofmechanicalforces,fieldforces,andforcesexertedbythefieldonitsboundaryisatequilibrium,asithastobe.Takingtheforceperareaunit,thenwearedealingobviouslywithapressure,theradiationpressure(Figure13.3):radiationijijpTnnaFn.(13.40)Thismeansthat,besidesenergy,momentumalsoistransferredfromtheradiationsourcetoanabsorberbyelectromagneticradiation.Therateofchangeofthismomen-tumperunitareacanbemeasuredontheabsorber.Todetecttheradiationpressure,LebedevandHullusedatorsionbalance.Themetalplatesattachedtotheendofabarareexposedalternatingtoradiationwiththefrequencyofthenormalmode,andwellobservabledeflectionsareobtainedinresonance.5Theelectromagneticradiationpressurereachesextremelyhighvaluesintheinteriorofthestars.Forthetemperaturesofmanymillionsofdegreesprevailingthere,themomentumflowisratherremarkableandplaysanimportantroleforthestabilityofthestarsbutalsoforexplosions(e.g.,supernova).Oneofthemoststrikingeffectsontheradiationpressureisfoundintheobservationofcomets:thetailsofthecometsarealwaysdirectedawayfromthesun(Figure13.4).Becausethegasofthecometshasaverylowdensityandbecausetheradiationactsonthecometforalongtime,theradiationpressureoriginatingfromthesunissufficienttocreatethetailsandtoorientthem.1Exercise13.2:EnergytransportinaconductingwireTheimportanceofthePoyntingvectormaybeillustratedbytheexampleofenergytransportinawireofradiusacarryingadirectcurrentI.CalculatetheJouleheatandthePoyntingenergyflux.SolutionInawire,theJouleheat2IVIRarises.TheJouleheatIVlIEperunitlengthisspent.Here,Visthevoltage,Ristheresistance,andlisthelengthofthewire.Buthowdidtheenergycometothepositionl?TheanswerisgivenbythePoyntingvectorS.Inthiscase,Eliesalongtheaxisofthewire,andHEwindsaroundthewire(seeFigure13.5).Therefore,atthesurfaceofthewireEHandthus(4)cSEHaredirectedintothewire.Further,4244242ccIcIEISEHEEcacaa.Theenergyflowintothewireperunitlengthis21SaEIand,hence,quantitativelyequaltotheheatreleased,EIperunitlengthofthewire.Atthepositionswheretheenergyisspent,theenergydoesnotflowduetoelectrontransportbutfromthefieldintothewire,wherethedensityoftheenergyflowisgivenbythePoyntingvectorS.Really,itseemstoberatherunlikelyfromtheverybeginningthatthisenergyistransportedtherebytheconductionelectronsinthewire,becauseduetotheirlargenumbertheymoveonlywitharelativelylowaveragevelocity(somecmsec)throughoutthewireevenforthelargestcurrents.Correspon-dingly,forexample,foratransmittingantennatheenergyflowsfromthewiretotheradiatedfield.注.由于是稳恒电流,电场与磁场均不变,由坡印廷定理,elm()0ddVdawwdVdtjESn即dVdajESn.1Example13.3:MagneticfieldenergyofacoilFromMaxwell'sequations,oneobtainsthemagneticfieldenergymag14UdHBormag14dUdHB.Thisresultmaybederivedinanotherwayandthuscheckedandexplained:letthewirebewounduniformlyonarodwithnturnspercm,crosssectionalareaa,andlengthl.Abatteryofvoltage()eVsuppliesthecurrentI.(SeeFigure13.6).Inthetimedt,thework()edAIVdtisdonebythebattery.Thepoweris()edAIVdt.Accordingtotheinductionlawwehave()()11eeddBIRVVnlacdtcdt.Themagneticfieldinsidethecoilisgivenby4HnIc.Theworkdoneis()214edBaldAIVdtIIRnladtIRdtHdBcdt.Sincealisequaltothevolumeofthesolenoid,besidestheJouleheat2IR,wefindtheenergymag4HdBdUdeliveredtotheunitvolumeofthefield.Thebatterysuppliestheohmicresistanceaswellasthemagneticfieldofthecoil.1Exercise13.4:ContinuityequationandMaxwell'sequationsDerivethecontinuityequationtjfromMaxwell'sequations.SolutionTheinhomogenousmacroscopicMaxwellequationsare4D,14ctcDHj.Differentiatethefirstequationwithrespecttotandtakethedivergenceofthesecondone,then4ttD,14()ctcDHj.Duetothevectoridentity()0Aaddingtheseequationstogetheryieldstheassertion0tj.1Exercise13.5:Magneticfieldenergyofsteadycurrents(a)Showthattheenergyofthemagneticfieldofastationarycurrentdistributionjinvacuumisdeterminedby3321()()2||Wdrdrcjrjrrr.(b)Theenergyofasystemofncurrent-carryingconductors(currentiI)canbedescribedbythequadraticform2111,12nnniiijijiijjiWLILII.Findtheexpressionfortheself-inductioncoefficientsiLandthemutualinductioncoefficientsijL.(c)Calculatetheself-inductioncoefficientdefinedin(b)foralongcurrent-carryingcoaxialline(Figure13.7).Lettheinnerconductor(radiusa)havetheperme-ability0.Thespacebetweentheconductorsshouldbefilledwithamaterialofpermeability.Solution(a)TheenergycontentofavolumecontainingtheelectricfieldsEandDis3el18WdrED.AsshownaboveforthemagneticfieldsHandB,wehave3mag18WdrHB.UsingBAandtheidentity()()()HAAHHAtheintegralmaybetransformed(magWissimplycalledW)333111()()()888W