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首页 13 麦克斯韦方程.pdf

13 麦克斯韦方程.pdf

13 麦克斯韦方程.pdf

上传者: zenger2017 2018-03-11 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《13 麦克斯韦方程pdf》,可适用于自然科学领域,主题内容包含Maxwell#sEquationsMaxwell#sequationsarebasedonthefollowingempiricalfacts:(符等。

Maxwell#sEquationsMaxwell#sequationsarebasedonthefollowingempiricalfacts:()TheelectricchargesarethesourcesandsinksofthevectorfieldofthedielectricdisplacementdensityDHence,forthefluxofthedielectricdisplacementthroughasurfaceenclosingthechargewehavedaQdVDnHere,nistheoutwardpointingunitnormalvectorThisrelationcanbederivedfromCoulomb#sforcelaw()Faraday#sinductionlaw:dVdcdtErwithdaBn()ThefactthattherearenoisolatedmonopolesimpliesdaBnthatis,themagneticinductionissourcefreeitsfieldlinesareclosedcurves()Oersted#sorAmpere#slaw:dIdaccHrjnFromtheseobservations()(),Maxwell#sequationsinintegralrepresentationcanbeinferreddirectly:dadVDn()dddacdtErBn()daBn()ddddadadadaccdtcdtHrjnDnjnDn()wherethelastterminequation()hasthedimensionofacurrent(compareequation())introducedbyMaxwellasthedisplacementcurrentThisadditionaltermisnecessarytofulfillthecontinuityequationddVdadtjn()Thisrelationimpliestheconservationoftheelectriccharge:TheelectricchargeinaregionofspaceVcanchangeintimeonlyifacurrentflowsthroughitssurfaceAmoredetailedjustificationofthedisplacementcurrentmaybegivenaftertransformingtheequations()()intotheassociateddifferentialform,whichcanbedonebymeansofGauss#andStokes#theoremsAsanexample,weconsiderequation():divdadVdVDnDImmediately,divDAnalogously,wetransformtheremainingequations()()Then,weobtainMaxwell#sequationsindifferentialform:divD()curlctBE()divB()curlcctctDDHjj()aswellasthecontinuityequationdivtj(a)Inaddition,thereistheforcelawcfEjB(b)TheforcelawlinkselectrodynamicsandmechanicsAtthispoint,Ampere#slaw,givenin(),reads(compareequation())curlcHj()Inthisform,itholdsonlyforstationarycurrentdistributionsUndertheassumptionthatthesameassignmentexistsalsofortimevaryingcurrents,equation()requiresacrucialcompletioninthecasethattheconductioncurrentisinterrupted,eg,byacapacitor(Figure):Attheplatesofthecapacitor,thedivergenceofjisnotequaltozero,whilethelefthandsideofequation()isalwayssourcefree(divcurlH)AfterintroducingthedisplacementcurrentatthevectortDZjisalwayssourcefreeduetotheequations()and(a):divdivdivdivdivdivtttDZjjDjthatis,afterintroducingMaxwell#sdisplacementcurrentthereareonlyclosedcurrentsWithoutthedisplacementcurrent()tDinequation(),thesystemofMaxwell#sequations()()wouldnotbeconsistentifthecontinuityequation(a)wastakenintoaccountOnlytheequationscompletedinthiswayareconsistentMaxwell#sequations()()arecompletedbythesocalledconnectingequationsrelatingthevectorsDandBtothefieldintensitiesEandH:DEPBHM()wherePisthevectoroftheelectricpolarization,andMisthevectorofthemagnetizationThelinktomechanicsisrealizedbytheforceequation(b),whichdescribestheLorentzforceInsteadyfields,forisotropicmaterialswhichmaybepolarizedandmagnetizedinnormalmanner,thereareconstitutiveequations(aspecialcaseof()):()eDEPEE()mBHMHH()TheconnectionofjandEyieldsthegeneralizedOhm#slaw:jE()whereistheelectricconductivityInnonlinearmedia,,,andaretensorsdependingonthefieldintensitiesmoreover,intimevaryingfields,,andarefrequencydependentToexpressequation()intermsofthequantitiesEandB,weusetheconnectingequations():curlcctDHj()curl()()cctBMjEP()curlcurlcctctPEBjM()Obviously,thecurloftheBfielddependsonthedensitiesoftheconductingcurrentj,themagnetizationcurrentcurlcMjM,andthepolarizationcurrenttPjPSinceinvacuumthecurrentdensities,Mjj,andPjvanish,onlytheterm()ctEinconnectionwithequation()mayexplainthepropagationofelectromagneticradiationinvacuum(SeeChapter)Maxwell#sequations()()arepartial,linear,coupleddifferentialequationsofthefirstorderDuetolinearitytheprincipleofsuperpositionisvalidHence,theinterferencefringesofelectromagneticwaves,inparticular,andalsothoseoflightasspecialelectromagneticwavesareguaranteedWehaveinferredMaxwell#sequations()(b)includingtheconnectingrelations()and(),stepbystepfromempiricalobservations:equation()follows,eg,fromtheCoulombinteractionofcharges,equation()fromFaraday#sinductionlaw,andequation()fromthefactthattherearenomagneticmonopolechargesWithoutthedisplacementcurrent,equation()representsthesecondAmpere#slawbeingagainequivalenttotheBiotSavartlawTheconservationofchargeisrelatedto(a)andthefirstAmpere#slaw(theLorentzforce)isfixedin(b)So,wehaveorientedourselvesaccordingtotheresultsofexperimentstofind,finally,thefundamentalequationsofelectrodynamics,Maxwell#sequationsWefoundalsonew,unexpectedthingslikethedisplacementcurrent,whichwasnotcontainedintheexperimentalphenomenabywhichwehavebeenguidedNow,wewillacceptMaxwell#sequationsasfundamentalwewillusethemasaxiomsfromwhichtodeduceeverythingthatfollowsThesefundamentalequationsarefurthersupportedbythesuccessofthesedeductionsInmechanicseverythingwasmuchsimpler:Newton#sforcelawpFcouldbegraspedrapidlysoonwecouldplaceitatthetopofourtheoryThefundamentalelectromagneticequations(theMaxwellequations)aremuchmorecomplicatedTherefore,thewayhasbeenmuchmoredifficultExercise:Ohm#slawWewanttodemonstratethatfromOhm#slawinitsdifferentialformjEundercertainassumptionstheknownintegralformIRcanbederivedTheconstantofproportionalityistheconductivityofthematerial,isthedifferenceofthevoltagebetweentwopointsoftheconductor,Risthecorrespondingohmicresistance,andIisthecurrentSolutionWerestrictourselvestofieldsEthatareconstantintimeandtouniformchargedistributionssothatthecurrentdistributionsarestationaryNow,weconsideracylindricalconductorbetweentwocrosssectionalareasaandahavingthepotentials()aand()a,respectively(equipotentialsurfaces)SeeFigureInthiscase,theelectricvoltagebetweenthetwosurfacesis()()IdsaaddddajsEssHence,theintegralformofOhm#slawforthevoltageisVIRwithdsRaDuetothespecificresistance,theresistancedependsonthepropertiesofthematerial,andduetosanda,itdependsonthespatialdimensionsoftheconductorForacylindricconductorcomposedofahomogeneousmaterialwithlengthlandtheconstantcrosssectionalareaa,theohmicresistanceislRdsaaTheenergyconservationlawofelectrodynamics:ThePoyntingvectorThePoyntingtheoremrepresentstheenergyconservationlawofelectrodynamicsitcanbederivedwiththehelpofMaxwell#sequationsForthispurpose,weperformthescalarproductofequation()andH,andofequation()andE,andthenweaddbothequationsWeobtaincurlcurlccttDBEHHEjEEH()Withtheidentity()()()EHEHHETheequation()canberewritten()ccttDBEHjEEHMultiplyingthisequationbyc,thenitcanbebroughttotheform()cttDBEHjEEH()Now,wemustinterpretthevarioustermsAtfirst,basedonourexperiences,eg,theelectrostatics(seeequation()),werecognizethefollowing:therighthandsideofequation()istherateofchangeofthetotalenergydensityoftheelectromagneticfield:elm()wwtttDBEH(a)Thiscanbeseenbyintroducingtheenergydensityoftheelectricfield,elwdDEDDEED(b)andtheenergydensityofthemagneticfieldmwdBHBBHHB(c)bysubstitutingtheseexpressionsforelwandmwintoequation(a)andperformingthepartialdifferentiationwithrespecttothetimeHere,wehaveassumedthatDEandBH,andhence,alsothatttEDDE,ttHBBHWealreadyhaveencounteredtherelations(a)(c)fortheenergydensityoftheelectricandthemagneticfieldinspecialcasesinelectrostaticsandmagnetostatics,respectivelyNow,theexpressions(a)(c)followfromthefundamentalequationsSo,inthefirstform(a)theyarevalidgenerallyThen,wecanwritetheequation()inthefirstformelm()()cwwtEHjE(a)Equation(a)describesthetransferofenergyduringadecreaseofthetotalenergydensityoftheelectrodynamicsfieldintimeThetermjErepresentstheworkdonebythefieldontheelectriccurrentdensityTheenergyputintothecurrentinthiswaycanbefoundastheJouleheatgeneratedpersecondperunitvolumeitisthethermalpowerofthefieldForexample,fortheohmicwirecorrespondingtoExercisewehaveVVdVEadsEalalIRlRjEThisistheohmicpowerofaresistorAfurtherreasonforthedecreaseoftheenergyofthefieldisthedivergenceofthevectorcSEH(b)denotedthePoyntingvectorThisvectordescribestheintensityoftheenergyflowperunittimethroughaunitareaperpendiculartothedirectionoftheflowStrictly,onlythesurfaceintegralextendedoveraclosedsurfacedaSnhasaphysicalmeaningThiscanbeseenbyintegratingequation(a)overanarbitraryvolumeandapplyingGauss#theoremelm()dwwdVdVdadtjESnThisequationrepresentstheenergyconservationlawofelectrodynamics(Poyntingtheorem)thetermdaSngivestheenergyflowcrossingtheareaelementdapersecondindirectionoftheouternormalnAswewillseelateroninequation(),foraplaneelectromagneticwavethePoyntingvectorpointsindirectionofthepropagationoftheenergyThequantitycSrepresentsthemomentumdensitytransferredbyapropagatingelectromagneticwavethatleadstoaradiationpressureonthematerialboundarysurfaces(Theseresultswillbederivedinthenextsection)TheMaxwellianstresstensor:TheconservationofthelinearmomentumintheelectromagneticfieldThePoyntingtheoremintheformelm()dwwdVdVdadtjESnoffersthepossibilitytoderivethemomentumoftheelectromagneticfieldTheworkdonebytheelectricfieldEperunitvolumeandtime,jE,impliestheconversionofelectromagneticenergyintomechanicalenergyorthermalenergy:mechddVEdtjEthere,mechEisthetotalmechanicalenergyoftheparticlesinsidethevolumeVTheenergyconservationinaclosedsystemmaythenberepresentedasmechfield()dEdEEdadtdtSnfieldEisthetotalfieldenergypervolume,whichisgivenbytheequationfield()EdVEDHBNowthatwehavefoundaformulationoftheconservationofenergyintheelectromagneticfieldinformofthePoyntingtheoremwewanttosetupacorrespondingconservationlawforthelinearmomentumForthispurpose,weconsiderthecasethat,apartfromtheelectromagneticfield,therearemovingchargesinthewholespaceThetotalforceonthesechargesfollowsfromtheLorentzforcedensitycfEjBbyvolumeintegration:meth()ddVdVdtcPFfrEjB()wheremethPmeansthetotalmechanical(linear)momentumoftheparticleThegreatimportanceofequation()liesinthefactthatitlinkselectrodynamicstomechanicsIthastobeviewedasthemechanicalsupplementaryequationtoMaxwell#sequationNow,withthehelpofMaxwell#sequations()and()weexpressjandbythefieldsWegetdivD,curlcctDjHAndequation()becomesmeth()()ddVdtctDPDEHBBAddingfurthertheterms()BH(Maxwellequation():divB)andutilizingtherelation:()tttDBDBBDourequationbecomesmeth()()()ddtPDEBHHB()dVctctBDBD()SubstitutingtheMaxwellequation()intoequation(),weobtainmeth()()()ddtPDEBHHB()()dVctDBDE,()meth()()ddVdtcPDBDEBH()()dVHBED()Tosimplifytheremainingderivation,weconsidertheelectromagneticfieldinvacuum,withthatis,EDandBHThen,equation()readsmeth()()ddVdtcPEBEEBB()()dVBBEE(a)Thevolumeintegralonthelefthandsideofequation(a)maybeinterpretedastheelectromagneticfieldmomentuminthevolumeVTheexpressionccEBS(b)hastoberegardedasthespatialdensityofthefieldmomentumfieldPItisproportionaltothedensityoftheenergyflow,thePoyntingvectorS,withthefactorc,fielddVdVccPEBSSubstitutingfieldPintoequation(a)methfield()()()()()ddVdtPPEEBBEEBB()Theintegrandofequation()maybeexpressedasthedivergenceofatensor,theMaxwellianstresstensorTodemonstratethis,fortherepresentationofdivergenceandcurlweuseEinstein#ssummationconvention,accordingtowhichonesumsautomaticallyoverindicesappearingtwiceThen,thesummationsignissuppressedForexample,inCartesiancoordinates:divjijjijiiFFxxFeeee,curljijjijiiFFxxFeeeewhereFisanydifferentiablevectorfieldWenowrewritethepartoftheintegrandofequation()thatdependsontheEfield:()()jjijkkijkkiiEEEExxEEEEeeeeeejjjijkkkkjiikkjiiiEEEEEExxxeeeeeeejjikkjiijiiiEEEEEExxxeeejjiijjijjiEEEEEExxxebecausewecanrewritetheindicesinthevarioustermsduetothesummationoverallindicesInthefollowing,weconsideronlythethicomponentofourexpressioninordertorewritefurther()()()()jjiiijjjijjjjijjEEEEEEEEEExxxxxEEEE()jiijkkjEEEExAcompletelyanalogousresultisobtainedforthepartoftheintegrandofequation()thatdependsonlyonB:()()()ijiijkkjBBBBxBBBB()Withtheserelations,theintegrandofequation()maybeexpressedasthedivergenceofatensormethfield()()iijijijkkkkjdEEBBEEBBdVdtxPP()Thisisthedivergence(jx)oftheMaxwellianstresstensorwiththecomponents:()ijijijijkkkkTEEBBEEBB(,,,,ijk)()Obviously,theMaxwellianstresstensorisasymmetrictensor,thatisijjiTTWewriteitinmatrixformusingthecomponents,xy,andzasindices:()xxxyxzxxxyxyxzxzijyxyyyzxyxyyyyzyzzxzyzzxzxzyzyzzzTTTEBWEEBBEEBBTTTTEEBBEBWEEBBTTTEEBBEEBBEBW()()()xyzxyzWEEEBBBEB()Consideringtheelectromagneticfieldinspacefilledwithmatter,wemayobtainthecomponentsoftheMaxwellianstresstensorinananalogousmannerfromequation(a)insteadofequation():()xxxyxzijyxyyyzzxzyzzTTTTTTTTTTxxxxxyxyxzxzxyxyyyyyyzyzxzxzyzyzzzzzEDBHWEDBHEDBHEDBHEDBHWEDBHEDBHEDBHEDBHW()()()xxyyzzxxyyzzWEDEDEDBHBHBHEDBH()ThetraceoftheMaxwellianstresstensorhasthevalueelmTr()()()ijiiTTWwwEDBH()WiththehelpoftheMaxwellianstresstensor,themomentumequation()canbewrittenasmethfield()ijijTddVdtxPP()UsingGauss#theorem,thevolumeintegralisrewrittenformallyintoasurfaceintegral,sothatijijjjTdVTndax()wherejnmeansthethjcomponentoftheunitnormalvector(directioncosine)perpendiculartotheareadaThevalidityof()canbeseenatoncebytheremarkthat(,,)iiiTTTrepresentthecomponentsofthethirowvectoroftheMaxwelliantensorsoequation()isjusttheusualGauss#theoremforthethirowvectorWiththistransformation,weobtaintheconservationlawofthelinearmomentum:methfield()iijjdTndadtPP()(forthethicomponent)Ifthearealiesinaregionshieldedfromthefieldoratinfinity,thenthesurfaceintegralvanishesFrom()welearnthatonlythetotalmomentummechanicalmomentumplusfieldmomentumisconservedItisclearthatthequantityijjTndoesnotvanishingeneralObviously,itrepresentsthethicomponentofthemomentumflowthroughtheunitofareaWestressthetermmomentumflow,thatis,momentumpersecondthroughunitarea,thusarateofchangeofthemomentumperunitofareaThen,theexpressionijjTnamaybeunderstoodasthethicomponentofaforceiijjFTna()ontheareaelementIftheradiationis,eg,absorbedbyablackareaelementaan,thentheforceiiijijFnTnnaFnisexertedontheareaabytheradiationAccordingtoequation(),thetotalsystemconsistingofmechanicalforces,fieldforces,andforcesexertedbythefieldonitsboundaryisatequilibrium,asithastobeTakingtheforceperareaunit,thenwearedealingobviouslywithapressure,theradiationpressure(Figure):radiationijijpTnnaFn()Thismeansthat,besidesenergy,momentumalsoistransferredfromtheradiationsourcetoanabsorberbyelectromagneticradiationTherateofchangeofthismomentumperunitareacanbemeasuredontheabsorberTodetecttheradiationpressure,LebedevandHullusedatorsionbalanceThemetalplatesattachedtotheendofabarareexposedalternatingtoradiationwiththefrequencyofthenormalmode,andwellobservabledeflectionsareobtainedinresonanceTheelectromagneticradiationpressurereachesextremelyhighvaluesintheinteriorofthestarsForthetemperaturesofmanymillionsofdegreesprevailingthere,themomentumflowisratherremarkableandplaysanimportantroleforthestabilityofthestarsbutalsoforexplosions(eg,supernova)Oneofthemoststrikingeffectsontheradiationpressureisfoundintheobservationofcomets:thetailsofthecometsarealwaysdirectedawayfromthesun(Figure)Becausethegasofthecometshasaverylowdensityandbecausetheradiationactsonthecometforalongtime,theradiationpressureoriginatingfromthesunissufficienttocreatethetailsandtoorientthemExercise:EnergytransportinaconductingwireTheimportanceofthePoyntingvectormaybeillustratedbytheexampleofenergytransportinawireofradiusacarryingadirectcurrentICalculatetheJouleheatandthePoyntingenergyfluxSolutionInawire,theJouleheatIVIRarisesTheJouleheatIVlIEperunitlengthisspentHere,Visthevoltage,Ristheresistance,andlisthelengthofthewireButhowdidtheenergycometothepositionlTheanswerisgivenbythePoyntingvectorSInthiscase,Eliesalongtheaxisofthewire,andHEwindsaroundthewire(seeFigure)Therefore,atthesurfaceofthewireEHandthus()cSEHaredirectedintothewireFurther,ccIcIEISEHEEcacaaTheenergyflowintothewireperunitlengthisSaEIand,hence,quantitativelyequaltotheheatreleased,EIperunitlengthofthewireAtthepositionswheretheenergyisspent,theenergydoesnotflowduetoelectrontransportbromthefieldintothewire,wherethedensityoftheenergyflowisgivenbythePoyntingvectorSReally,itseemstoberatherunlikelyfromtheverybeginningthatthisenergyistransportedtherebytheconductionelectronsinthewire,becauseduetotheirlargenumbertheymoveonlywitharelativelylowaveragevelocity(somecmsec)throughoutthewireevenforthelargestcurrentsCorrespondingly,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()ddVdawwdVdtjESn即dVdajESnExample:MagneticfieldenergyofacoilFromMaxwell#sequations,oneobtainsthemagneticfieldenergymagUdHBormagdUdHBThisresultmaybederivedinanotherwayandthuscheckedandexplained:letthewirebewounduniformlyonarodwithnturnspercm,crosssectionalareaa,andlengthlAbatteryofvoltage()eVsuppliesthecurrentI(SeeFigure)Inthetimedt,thework()edAIVdtisdonebythebatteryThepoweris()edAIVdtAccordingtotheinductionlawwehave()()eeddBIRVVnlacdtcdtThemagneticfieldinsidethecoilisgivenbyHnIcTheworkdoneis()edBaldAIVdtIIRnladtIRdtHdBcdtSincealisequaltothevolumeofthesolenoid,besidestheJouleheatIR,wefindtheenergymagHdBdUdeliveredtotheunitvolumeofthefieldThebatterysuppliestheohmicresistanceaswellasthemagneticfieldofthecoilExercise:ContinuityequationandMaxwell#sequationsDerivethecontinuityequationtjfromMaxwell#sequationsSolutionTheinhomogenousmacroscopicMaxwellequationsareD,ctcDHjDifferentiatethefirstequationwithrespecttotandtakethedivergenceofthesecondone,thenttD,()ctcDHjDuetothevectoridentity()AaddingtheseequationstogetheryieldstheassertiontjExercise:Magneticfieldenergyofsteadycurrents(a)Showthattheenergyofthemagneticfieldofastationarycurrentdistributionjinvacuumisdeterminedby()()||Wdrdrcjrjrrr(b)Theenergyofasystemofncurrentcarryingconductors(currentiI)canbedescribedbythequadraticform,nnniiijijiijjiWLILIIFindtheexpressionfortheselfinductioncoefficientsiLandthemutualinductioncoefficientsijL(c)Calculatetheselfinductioncoefficientdefinedin(b)foralongcurrentcarryingcoaxialline(Figure)Lettheinnerconductor(radiusa)havethepermeabilityThespacebetweentheconductorsshouldbefilledwithamaterialofpermeabilitySolution(a)TheenergycontentofavolumecontainingtheelectricfieldsEandDiselWdrEDAsshownaboveforthemagneticfieldsHandB,wehavemagWdrHBUsingBAandtheidentity()()()HAAHHAtheintegralmaybetransformed(magWissimplycalledW)()()()W

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