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首页 11 物质中的磁场.pdf

11 物质中的磁场.pdf

11 物质中的磁场.pdf

zenger2017 2018-03-04 评分 0 浏览量 0 0 0 0 暂无简介 简介 举报

简介:本文档为《11 物质中的磁场pdf》,可适用于自然科学领域,主题内容包含TheMagneticFieldinMatterUptonow,wehavestartedalwaysfrompreciselyknowncurre符等。

TheMagneticFieldinMatterUptonow,wehavestartedalwaysfrompreciselyknowncurrentdensitydistribution()jrWhenweintendtocalculatethemagneticfieldinspacefilledwithmatterwecannolongerassumethisTherearemolecularcurrents,andmagneticmomentsofatomsandionsthatarenotknownindetail,andwhoseaveragevaluesareofinterestonlyinmacroscopicconsiderationsSo,wewillproceedasinthetreatmentoftheelectrostaticfieldinmatterThetotalcurrentdensityissplitintoapartarisingfromthemacroscopicchargetransportandapartthattakesintoaccountthecircularcurrentsofelectronsinatomsThemicroscopicvectorpotentialdependingonallcurrentsanddescribingexactlyalsotheatomicregionismacrmolAAAor,expressedintermsofthecurrentdensitydistribution,mol()()||||dVdVccjrjrArrrr()wherethesecondintegralrepresentsthecontributionoftheatomiccircularcurrentsIndependentofj,thevectorpotentialatthepositionr,belongingtoamoleculeatthepointir,canbeapproximatedby()()molmol()()||iiiimrrArrrwheremolmisthetotalmagneticmomentofthemoleculeNow,thetotalvectorpotentialis()mol()()||||iiiidVcmrrjrArrrr()AnaveragevaluemolmisassumedformolmIfNisthenumberofmoleculespervolume,thenwiththemagneticdipoledensitymolNMmwecangofromthesumtotheintegraloverthevolumeThequantityMiscalledthe(macroscopic)magnetizationCorrespondingtothepolarizationintheelectrostaticcaseitisadensityofmagneticdipolesHence,weobtainforthevectorpotential()()()||||dVdVcjrMrrrArrrr()Thisequationcanberewritteninthefollowingway:()()||||dVdVcjrAMrrrrr()Now,()()||||dVdVMrMrrrrr()()||||dVdVMrMrrrrrThefirstintegralontherighthandsidemaybetransformedintoasurfaceintegralwhichbecomeszeroundertheassumptionthatMisboundedinspace:()()||||ddVMraMrrrrr(seethemathematicalconsiderationattheendofthissection)Therefore,()()()()||||||dVcdVdVccjrMrjrMrArrrrrr()Themacroscopicmagnetizationcorrespondstoacurrent,thesocalledmagnetizingcurrent()curl()McjrMrSo,theeffectivecurrentisthesumoftheconductioncurrentjandthemagnetizingcurrentMjReally,because()()BAAAA,weobtain()()curl()()||||MMdVdVccjrjrBjrjrrrrr()()()()()MMdVccjrjrrrjrjr()Aconversionyieldscurl()cBMj()ThequantityHBM()isdenotedthemagneticfieldintensityIthastoberegardedinanalogytotheelectricfieldintensity,forwhichinadielectricEDPForthemagneticfieldintensity,therelationcurlcHj()holdsHence,itdoesnotdependonthemoleculardipolesInthepresenceofmatterthemagneticfieldintensityHreplacesthemagneticinductionBInvacuum,thesefieldquantitiesareequaltoeachother,HBMathematicalconsiderationForavectorfield()Br,wecanprovegenerallydVdaBnB()byformingthevectorfieldABCwithanarbitraryconstantvectorCFromGauss#theorem,()()dVdaBCBCnSince()()()()BCCBBCCB,weobtain()()()dVdadaCBBCnCnBorsinceCisconstantdVdaCBCnBBecauseCisanarbitraryconstantvector,dVdadBnBaBSusceptibilityandpermeabilityNow,therelationshipbetweenMandHhastobeinvestigatedExperienceshowsthatinmanycasestheansatzmMH,constm()istrueThefactormisdenotedthemagneticsusceptibilityBut,insomecasestobediscussedbelow,mdependsonHifMandHarenotparalleltoeachother,thenmisatensorTheansatzgivenaboveyieldsfortheinductionB,()mBHMHH()withthepermeability:m()Dependingonthevalueofmorthefollowingmaterialsaredistinguished:Diamagnetics:,mParamagnetics:,mFerromagnetics:,()HThevaluesofthesusceptibilityforsomematerialscanbetakenfromthefollowinglist:H:HO:Diamagnetics(C)N:Ag:mmmmTO:Pt:Paramagnetics(C)Al:mmmTFe:Co:FerromagneticsNi:mmmParamagnetismoccursformaterialswhoseatomsormoleculeshaveamagneticmomentThismagneticmomentoriginatesfromelectronslyingoutsideclosedelectronshellsThemagneticmomentsarecompletelydisorderedduetothethermalmotionApplyinganexternalmagneticfield,theaverageanglebetweenthemomentsandthefielddirectiondependsontheratioofthepotentialenergyofthemomentsinthefieldandthethermalenergy(analogoustotheconsiderationsfordielectrics)Thus,thesusceptibilityistemperaturedependent,anditisgivenbyCurie#slaw:mcT()whereisthedensityofthematerialandTistheabsolutetemperatureThestateinwhichallmomentspossessthesmallestpossibleanglewithrespecttothefielddirectioniscalledsaturationmagnetizationForparamagneticsthemomentsaresosmallthatfornormaltemperaturessaturationcannotbereachedForferromagnetics,thesaturationmagnetizationoccursalwaysinsidemicroscopicdomains,calledWeissdomainsButinthenormalcasethedirectionsofthemagnetizationintheWeissdomainsarestatisticallydistributed,sothatnomagnetizationappearsoutsideHowever,thedirectionsoftheWeissdomainswillalignthemselveswithrespecttoanappliedfieldFigureshowstheWeissdomainsinanonmagnetizedferromagnetForferromagneticsthereisnolinearrelationbetweenMandHSomematerialsevenexhibitthephenomenonthatMisnotauniquefunctionofHAchangeinMdependsonachangeinHaswellasonthemagnetizationalreadypresentThiseffectiscalledhysteresisIfonestartsfromahighmagneticfieldHanddecreasesitcontinuously,thenevenforHaremanentmagnetizationisleftAmaterialiscalledmagneticallyhardormagneticallysoftdependingonwhetheraremanenceoccurs(Figure)DecreasingthefieldintensityH,thecoerciveforce,whichdirectedoppositetoM,isneededtomakethemagnetizationMvanishAsshowninFigure,formagneticallyhardferromagnetsapartofthemagnetizationislefteveniftheexternalforceiszeroThesematerialsaresuitedforpermanentmagnetsAboveacertaintemperature,thesocalledCurietemperaturecT,ferromagnetslosetheirferromagneticpropertiesandbehavelikeparamagneticsInthiscase,theirbehaviorisdescribedalsobyCurie#slaw,buttheCurietemperaturehastobetakenintoaccount:mccTT,cTT()ExampleofCurietemperaturesarethefollowing:Fe:CCo:CNi:CcccTTTFordiamagneticmaterialsthemagnetizationMisdirectedoppositetothefieldHThiscanbeexplainedinthefollowingway:inswitchingonamagneticfield,circularcurrentsareinducedintheindividualatomswhosemagneticmomentsareorientedoppositetothefieldTherefore,atomsandmoleculeswithoutamagneticmomentarediamagneticAsalreadystated,thereasonforthemagnetizationismerelytheoccurrenceofcircularcurrentsoftheatomicelectronsInparticular,atomsandionshavingclosedraregaslikeelectronshells,eg,theionsinsalts,havenoownmomentDiamagnetismisindependentofthetemperature,asisunderstandablefromthismodelThebehaviorofBandHatboundarysurfacesInFigure,twomaterialswithpermeabilityand,respectively,aredisplayed,separatedbytheboundarysurfaceSInthefollowingwewillinvestigatehowtheBfieldchangesinthetransitionfromonematerialtotheotherTosolvethisproblemwestartfromtheequationdivBThisequationimpliesthatthefluxintothevolumeelementisequaltothefluxoutofthevolumeelementRewritingdivdVBwiththehelpofGauss#theoremasdaBnweobtainforaflatvolumewhosesidefacesmaybeneglected:()()nnBBaSincethisequationisvalidforarbitrarya,weobtainthatthenormalcomponentsareequalonbothsides()()nnBB()AcorrespondingconditioncanbederivedforHfromtheequationcurlcHjChoosingtheintegrationpathinsuchaway(Figure)thatthenormalvectorntoaistangentialtotheboundarysurfacetheequationcanbewritteninintegralform(curl)dadacHnjnThisequationcanberewrittenaccordingtotheStokestheorem(curl)ddadaKccHsHnjn()Thelefthandsideoftheequationisequalto()()ttHHLTherighthandsiderepresentsasurfacecurrentofmagnitudeKNormally,KTherefore,thetangentialcomponentofHisconservedinthetransitionfromonemediumtoanotherone()()ttHH()Superconductors,forexample,inwhichnomagneticfieldsmayexistsincetheyarecanceledbysurfacecurrents,maybeviewedasanexceptionalcaseIftherearesurfacecurrentsKtheboundarylayercondition()cLcKnHHk()isvalidinsteadofequation()Here,LkKhasthedimensioncurrentperunitoflengthhence,itrepresentsthedensityofthesurfacecurrentListhelengthoftheconsideredareaelementaLxFromtheseconsiderationsoneobtains,altogether,thefollowingconditions:(a)ThenormalcomponentofBiscontinuousontheboundarysurface(b)ThetangentialcomponentofHisingeneralcontinuousontheboundarysurfaceBecauseBH,wehavefromBnBnthatHnHn,orforthenormalcomponent()()nnHH()Example:ThemagneticfieldofauniformlymagnetizedsphereAsphereisuniformlymagnetized(Figure)inthedirectionofthezaxisifMMeFortheexternalfield,wehavedivaB,curlaBTherefore,theexternalfieldmayberepresentedasthegradientofamagneticfieldgradaMBwhereMisvalidintheexteriorregionThesolutionofLaplace#sequationisknownalreadyfromelectrostaticsExpressedintermsofLegendrepolynomialsitreads:(cos)MllllPr()Insidethesphere,HBMwherewehaveassumedthatBisparalleltoM:iBBe()iBMHee()Accordingtotheconditionsderivedabove,wehaveattheboundarysurface:irarBeBe,iaaHeHeBeSubstitutingforgradaMBandexpressingthefirstcomponentinsphericalcoordinates,thefollowingrelationshipbetweenMandBisobtained:cosMraBrTakingtheexpansion()forM,then,with(cos)cosP:(cos)(cos)()llllraPBPlrCorrespondingly,wehaveforHattheboundarysurface:(cos)()sinllMllraradPBMardWiththerelationsin()(cos)ddPacomparisonofcoefficientsyieldstheequationsBaandBMBaWeobtainBM,MaSubstitutingthisresultintoequations()and()foriBandiH,theniMBeM,iMHeM()ThefieldintheexteriorregionbecomescosgradgradaMrBcossinrrrrreeeForthevariouscomponents,weobtaincoscosrBMarr,sinsinBMarr,Borwritteninvectorform,cossin()rrraMMaarreeMeeMBThisisjustthefieldofamagneticdipole(comparethisresulttoequation()andExercise),withamMBecausedivB,weobtaindivdiv()divHBMMThequantitydivMiscalledthemagneticchargeItrepresentsthesourcesofthemagneticfieldThefieldofthemagnetizedsphereisrepresentedinFigureFortheHfieldonehastonotethefollowing:intheinteriorthereareonlyhalfasmanyfieldlinesasfortheBfieldmoreover,theyareorientedintheoppositedirectionBecause,theexternalfieldisequaltotheBfieldThisexpressestheresult(),accordingtowhich||||iiBHandiiBH注关于()式,一般的(轴对称)势应该写成:(cos)llMllllrPr但是,考虑到在无穷远处磁场为零,故l下面的例子中,无穷远处的磁场不为零Example:MagnetizablesphereinanexternalfieldAmagnetizablesphereofpermeabilityisinanexternalfieldBThemagnetizationofthespherehastobecalculatedDuetothelinearityofthefieldequations,theinternalfieldsare,accordingtoExample:iBBM()iHBM()SupposingthatthematerialisnotferromagneticthereisalinearrelationshipbetweeniBandiH,iiBHThus,fromequations()and(),BMBMSolvingthisequationweobtainforthemagnetization:MBThepolarizationinelectrostatics(Exercise)isitsanalog:PEForferromagneticsthereisnosimpleproportionalityofBtoHRather,iBisacomplicatedfunctionofiH:()iiiBBHNow,wemaysolvetheequationsiBBM,iHBMforarelationbetweeniHandiBbyeliminatingMTheresultreadsiiBHBForvariousBthisequationcorrespondstoafamilyofstraightlineswiththeslopeThehysteresiscurveofthematerialyieldsafurtherrelationbetweeniBandiHWiththesetworelationsspecialvaluesforanyexternalfieldmaybefoundForexample,ifwewanttocalculatetheresidualmagnetizationofaferromagneticsphereafterBhasbeenincreasedfirstandthenhasbeentakenbacktozerothevaluesforiBandiHresultgraphicallyastheintersectionpointPofthehysteresiscurveandthestraightlinewithB(Figure)iBM,iHMFromtheseequations,MmaybedeterminedExercise:EnergylossandhysteresisInaferromagnettheconnectionofthefieldsBandHdependsonthehistoryofthesystem(hysteresis)SuchabodyisplacedinauniformmagneticfieldBtheintensityofwhichishighenoughsothattheinternalfieldsareuniqueNow,thepolarityoftheexternalfieldisslowlyalternated(BB)andthenbroughtbacktotheoriginalstate(BB)Provethattheenergylossisgivenby()Sdrrwhere()Sristheareaenclosedbythehysteresisloop(Figure)Note:Itwillbeshownlaterthatthechangeoftheenergydensity()wrforthetransition(BB)alongthepathCisgivenby()()CwdrHBBThisexpressionforthemagneticenergydensityiscompletelyanalogoustotheexpressionfortheelectricenergydensitywdEDSolutionThevariationoftheenergydensityisgivenby()()()wddBBBBrHBBHBB()()()dSBBHBHBBr()Ascanbeseen,theintegrandisjusttheareaenclosedbythehysteresisloopOutsidetheferromagneticvolume,HHandtherefore()SrThetotalenergychangeis()WSdrr

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