11 物质中的磁场

111TheMagneticFieldinMatterUptonow,wehavestartedalwaysfrompreciselyknowncurrentdensitydistribution()jr.Whenweintendtocalculatethemagneticfieldinspacefilledwithmatterwecannolongerassumethis.Therearemolecularcurrents,andmagneticmomentsofatomsandionsthatarenotknownindetail,andwhoseaveragevaluesareofinterestonlyinmacroscopicconsiderations.So,wewillproceedasinthetreatmentoftheelectrostaticfieldinmatter.Thetotalcurrentdensityissplitintoapartarisingfromthemacroscopicchargetransportandapartthattakesintoaccountthecircularcur-rentsofelectronsinatoms.ThemicroscopicvectorpotentialdependingonallcurrentsanddescribingexactlyalsotheatomicregionismacrmolAAAor,expressedintermsofthecurrentdensitydistribution,mol()1()1||||dVdVccjrjrArrrr(11.1)wherethesecondintegralrepresentsthecontributionoftheatomiccircularcurrents.Independentofj,thevectorpotentialatthepositionr,belongingtoamoleculeatthepointir,canbeapproximatedby()()molmol3()()||iiiimrrArrrwheremolmisthetotalmagneticmomentofthemolecule.Now,thetotalvectorpotentialis()mol3()1()||||iiiidVcmrrjrArrrr.(11.2)Anaveragevaluemolmisassumedformolm.IfNisthenumberofmoleculespervolume,thenwiththemagneticdipoledensitymolNMmwecangofromthesumtotheintegraloverthevolume.ThequantityMiscalledthe(macroscopic)magnetization.Correspondingtothepolarizationintheelectrostaticcaseitisadensityofmagneticdipoles.Hence,weobtainforthevectorpotential31()()()||||dVdVcjrMrrrArrrr.(11.3)Thisequationcanberewritteninthefollowingway:1()1()||||dVdVcjrAMrrrrr.(11.4)Now,11()()||||dVdVMrMrrrrr()()||||dVdVMrMrrrrr.Thefirstintegralontheright-handsidemaybetransformedintoasurfaceintegralwhichbecomeszeroundertheassumptionthatMisboundedinspace:()()||||ddVMraMrrrrr(seethemathematicalconsiderationattheendofthissection).2Therefore,1()()1()()||||||dVcdVdVccjrMrjrMrArrrrrr.(11.5)Themacroscopicmagnetizationcorrespondstoacurrent,theso-calledmagnetizingcurrent()curl()McjrMr.So,theeffectivecurrentisthesumoftheconductioncurrentjandthemagnetizingcurrentMj.Really,because()()BAAAA,weobtain()()111curl[()()]||||MMdVdVccjrjrBjrjrrrrr44[()()]()[()()]MMdVccjrjrrrjrjr.(11.6)Aconversionyields4curl(4)cBMj.(11.7)Thequantity4HBM(11.8)isdenotedthemagneticfieldintensity.Ithastoberegardedinanalogytotheelectricfieldintensity,forwhichinadielectric4EDP.Forthemagneticfieldintensity,therelation4curlcHj.(11.9)holds.Hence,itdoesnotdependonthemoleculardipoles.InthepresenceofmatterthemagneticfieldintensityHreplacesthemagneticinductionB.Invacuum,thesefieldquantitiesareequaltoeachother,HB.MathematicalconsiderationForavectorfield()Br,wecanprovegenerallydVdaBnB(11.10)byformingthevectorfieldABCwithanarbitraryconstantvectorC.FromGauss'theorem,()()dVdaBCBCn.Since()()()()BCCBBCCB,weobtain()()()dVdadaCBBCnCnBorsinceCisconstantdVdaCBCnB.BecauseCisanarbitraryconstantvector,dVdadBnBaB.1SusceptibilityandpermeabilityNow,therelationshipbetweenMandHhastobeinvestigated.ExperienceshowsthatinmanycasestheansatzmMH,constm(11.11)istrue.Thefactormisdenotedthemagneticsusceptibility.But,insomecasestobediscussedbelow,mdependsonH;ifMandHarenotparalleltoeachother,thenmisatensor.TheansatzgivenaboveyieldsfortheinductionB,4(14)mBHMHH(11.12)withthepermeability:14m.(11.13)Dependingonthevalueofmorthefollowingmaterialsaredistinguished:Diamagnetics:1,0mParamagnetics:1,0mFerromagnetics:1,()H.Thevaluesofthesusceptibilityforsomematerialscanbetakenfromthefollowinglist:9252825H:2.310HO:1.210Diamagnetics(0C)N:0.710Ag:2.510mmmmT8255O:1.810Pt:2.710Paramagnetics(20C)Al:2.110mmmT666Fe:10Co:10FerromagneticsNi:10mmm.Paramagnetismoccursformaterialswhoseatomsormoleculeshaveamagneticmo-ment.Thismagneticmomentoriginatesfromelectronslyingoutsideclosedelectronshells.Themagneticmomentsarecompletelydisorderedduetothethermalmotion.Applyinganexternalmagneticfield,theaverageanglebetweenthemomentsandthefielddirectiondependsontheratioofthepotentialenergyofthemomentsinthefieldandthethermalenergy(analogoustotheconsiderationsfordielectrics).Thus,thesusceptibilityistemperature-dependent,anditisgivenbyCurie'slaw:mcT(11.14)whereisthedensityofthematerialandTistheabsolutetemperature.Thestateinwhichallmomentspossessthesmallestpossibleanglewithrespecttothefielddirectioniscalledsaturationmagnetization.Forparamagneticsthemomentsaresosmallthatfornormaltemperaturessaturationcannotbereached.Forferromagnetics,thesaturationmagnetizationoccursalwaysinsidemicroscopicdomains,calledWeissdomains.Butinthenormalcasethedirectionsofthemagne-tizationintheWeissdomainsarestatisticallydistributed,sothatnomagnetizationappearsoutside.However,thedirectionsoftheWeissdomainswillalignthemselveswithrespecttoanappliedfield.2Figure11.1showstheWeissdomainsinanonmagnetizedferromagnet.Forferromag-neticsthereisnolinearrelationbetweenMandH.SomematerialsevenexhibitthephenomenonthatMisnotauniquefunctionofH.AchangeinMdependsonachangeinHaswellasonthemagnetizationalreadypresent.Thiseffectiscalledhysteresis.IfonestartsfromahighmagneticfieldHanddecreasesitcontin-uously,thenevenfor0Haremanentmagnetizationisleft.Amaterialiscalledmagneticallyhardormagneticallysoftdependingonwhetheraremanenceoccurs(Figure11.2).DecreasingthefieldintensityH,thecoerciveforce,whichdirectedoppositetoM,isneededtomakethemagnetizationMvanish.AsshowninFigure11.2,formagneticallyhardferromagnetsapartofthemagne-tizationislefteveniftheexternalforceiszero.Thesematerialsaresuitedforperma-nentmagnets.Aboveacertaintemperature,theso-calledCurietemperaturecT,ferromagnetslosetheirferromagneticpropertiesandbehavelikeparamagnetics.Inthiscase,theirbehaviorisdescribedalsobyCurie'slaw,buttheCurietemperaturehastobetakenintoaccount:mccTT,cTT.(11.15)ExampleofCurietemperaturesarethefollowing:Fe:774CCo:1131CNi:372C.cccTTTFordiamagneticmaterialsthemagnetizationMisdirectedoppositetothefieldH.Thiscanbeexplainedinthefollowingway:inswitchingonamagneticfield,circularcurrentsareinducedintheindividualatomswhosemagneticmomentsareorientedoppositetothefield.Therefore,atomsandmoleculeswithoutamagneticmomentarediamagnetic.Asalreadystated,thereasonforthemagnetizationismerelytheoccurrenceofcircularcurrentsoftheatomicelectrons.Inparticular,atomsandionshavingclosedraregas-likeelectronshells,e.g.,theionsinsalts,havenoownmo-ment.Diamagnetismisindependentofthetemperature,asisunderstandablefromthismodel.1ThebehaviorofBandHatboundarysurfacesInFigure11.3,twomaterialswithpermeability1and2,respectively,aredis-played,separatedbytheboundarysurfaceS.InthefollowingwewillinvestigatehowtheB-fieldchangesinthetransitionfromonematerialtotheother.Tosolvethisproblemwestartfromtheequationdiv0B.Thisequationimpliesthatthefluxintothevolumeelementisequaltothefluxoutofthevolumeelement.Rewritingdiv0dVBwiththehelpofGauss'theoremas0daBnweobtainforaflatvolumewhosesidefacesmaybeneglected:12[()()]0nnBBa.Sincethisequationisvalidforarbitrarya,weobtainthatthenormalcomponentsareequalonbothsides12()()nnBB.(11.16)AcorrespondingconditioncanbederivedforHfromtheequation4curlcHj.Choosingtheintegrationpathinsuchaway(Figure11.4)thatthenormalvectorntoaistangentialtotheboundarysurfacetheequationcanbewritteninintegralform4(curl)dadacHnjn.ThisequationcanberewrittenaccordingtotheStokestheorem44(curl)ddadaKccHsHnjn.(11.17)Theleft-handsideoftheequationisequalto12[()()]ttHHL.Theright-handsiderepresentsasurfacecurrentofmagnitudeK.Normally,0K.Therefore,thetangentialcomponentofHisconservedinthetransitionfromonemediumtoanotherone.12()()ttHH.(11.18)2Superconductors,forexample,inwhichnomagneticfieldsmayexistsincetheyarecanceledbysurfacecurrents,maybeviewedasanexceptionalcase.IftherearesurfacecurrentsKtheboundary-layercondition2144()cLcKnHHk(11.19)isvalidinsteadofequation(11.18).Here,LkKhasthedimensioncurrentperunitoflength;hence,itrepresentsthedensityofthesurfacecurrent.ListhelengthoftheconsideredareaelementaLx.Fromtheseconsiderationsoneobtains,altogether,thefollowingconditions:(a)ThenormalcomponentofBiscontinuousontheboundarysurface.(b)ThetangentialcomponentofHisingeneralcontinuousontheboundarysurface.BecauseBH,wehavefrom12BnBnthat1122HnHn,orforthenormalcomponent2121()()nnHH.(11.20)1Example11.1:ThemagneticfieldofauniformlymagnetizedsphereAsphereisuniformlymagnetized(Figure11.5)inthedirectionofthez-axisif03MMe.Fortheexternalfield,wehavediv0aB,curl0aB.Therefore,theexternalfieldmayberepresentedasthegradientofamagneticfieldgradaMBwhere0Misvalidintheexteriorregion.ThesolutionofLaplace'sequationisknownalreadyfromelectrostatics.ExpressedintermsofLegendrepolynomialsitreads:101(cos)MllllPr.(11.21)Insidethesphere,4HBMwherewehaveassumedthatBisparalleltoM:03iBBe(11.22)03034iBMHee.(11.23)Accordingtotheconditionsderivedabove,wehaveattheboundarysurface:irarBeBe,iaaHeHeBe.SubstitutingforgradaMBandexpressingthefirstcomponentinsphericalcoordinates,thefollowingrelationshipbetweenMand0Bisobtained:0cosMraBr.Takingtheexpansion(11.21)forM,then,with1(cos)cosP:0120(cos)(cos)(1)llllraPBPlr.Correspondingly,wehaveforHattheboundarysurface:0020(cos)1(4)sinllMllraradPBMard.Withtherelation1sin()(cos)ddPacomparisonofcoefficientsyieldstheequations1032Baand10003142BMBa.Weobtain0083BM,31043Ma.Substitutingthisresultintoequations(11.22)and(11.23)foriBandiH,then038833iMBeM,034433iMHeM.(11.24)Thefieldintheexteriorregionbecomes12cosgradgradaMrB1211cossinrrrrreee.2Forthevariouscomponents,weobtain31033cos8cos23rBMarr,31033sin4sin3BMarr,0Borwritteninvectorform,3300332cossin3()4433rrraMMaarreeMeeMB.Thisisjustthefieldofamagneticdipole(comparethisresulttoequation(8.1)andExercise1.4),with343amM.Becausediv0B,weobtaindivdiv(4)4divHBMM.ThequantitydivMiscalledthemagneticcharge.Itrepresentsthesourcesofthemagneticfield.ThefieldofthemagnetizedsphereisrepresentedinFigure11.6.FortheH-fieldonehastonotethefollowing:intheinteriorthereareonlyhalfasmanyfieldlinesasfortheB-field;moreover,theyareorientedintheoppositedirection.Because1,theexternalfieldisequaltotheB-field.Thisexpressestheresult(11.24),accordingtowhich||2||iiBHand2iiBH.注.关于(11.21)式,一般的(轴对称)势应该写成:10(cos)llMllllrPr.但是,考虑到在无穷远处磁场为零,故0l.下面的例子中,无穷远处的磁场不为零.1Example11.2:MagnetizablesphereinanexternalfieldAmagnetizablesphereofpermeabilityisinanexternalfield0B.Themagneti-zationofthespherehastobecalculated.Duetothelinearityofthefieldequations,theinternalfieldsare,accordingtoExample11.1:083iBBM(11.25)043iHBM.(11.26)SupposingthatthematerialisnotferromagneticthereisalinearrelationshipbetweeniBandiH,iiBH.Thus,fromequations(11.25)and(11.26),008433BMBM.Solvingthisequationweobtainforthemagnetization:03142MB.Thepolarizationinelectrostatics(Exercise6.2)isitsanalog:03142PE.ForferromagneticsthereisnosimpleproportionalityofBtoH.Rather,iBisacomplicatedfunctionofiH:()iiiBBH.Now,wemaysolvetheequations083iBBM,043iHBM.forarelationbetweeniHandiBbyeliminatingM.Theresultreads023iiBHB.Forvarious0Bthisequationcorrespondstoafamilyofstraightlineswiththeslope2.ThehysteresiscurveofthematerialyieldsafurtherrelationbetweeniBandiH.Withthesetworelationsspecialvaluesforanyexternalfieldmaybefound.Forexample,ifwewanttocalculatetheresidualmagnetizationofaferromagneticsphereafter0Bhasbeenincreasedfirstandthenhasbeentakenbacktozerothe2valuesforiBandiHresultgraphicallyastheintersectionpointPofthehysteresiscurveandthestraightlinewith00B(Figure11.7)83iBM,43iHM.Fromtheseequations,Mmaybedetermined.1Exercise11.3:EnergylossandhysteresisInaferromagnettheconnectionofthefieldsBandHdependsonthehistoryofthesystem(hysteresis).Suchabodyisplacedinauniformmagneticfield0Btheintensityofwhichishighenoughsothattheinternalfieldsareunique.Now,thepola-rityoftheexternalfieldisslowlyalternated(00BB)andthenbroughtbacktotheoriginalstate(00BB).Provethattheenergylossisgivenby31()4Sdrrwhere()Sristheareaenclosedbythehysteresisloop(Figure11.8).Note:Itwillbeshownlaterthatthechangeoftheenergydensity()wrforthetran-sition(12BB)alongthepathCisgivenby1()()4CwdrHBB.Thisexpressionforthemagneticenergydensityiscompletelyanalogoustotheexpre-ssionfortheelectricenergydensity14wdED.SolutionThevariationoftheenergydensityisgivenby00001211()()()44wddBBBBrHBBHBB002111[()()]()44dSBBHBHBBr.(11.27)Ascanbeseen,theintegrandisjusttheareaenclosedbythehysteresisloop.Outsidetheferromagneticvolume,12HHandtherefore()0Sr.Thetotalenergychangeis31()4WSdrr.