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首页 10 磁矩.pdf

10 磁矩.pdf

10 磁矩.pdf

上传者: zenger2017 2018-03-04 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《10 磁矩pdf》,可适用于自然科学领域,主题内容包含MagneticMomentMagneticdipolefieldWewishtoconsideranarrowlyboundedregioninw符等。

MagneticMomentMagneticdipolefieldWewishtoconsideranarrowlyboundedregioninwhichthesteadycurrentdistribution()jrisgiven(Figure)Forthevectorpotentialgeneratedbythiscurrentdistribution,weobtain()()||dVcjrArrr()Letthepointofobservationrliefaroutsidethecurrentdistribution,sothatrrThen,thedenominator||rrcanbeexpandedinaTaylorseriesretainingonlythefirsttwoterms:||rrrrrrWepickoutoneofthe(Cartesian)componentsofthecurrentandthepotential,eg,thethione:()()()||iiijAdVjdVccrrrrrrrrr()()()iijdVjdVcrcrrrrrSincethevolumeintegralscovertheentirecurrentdistribution,thefirstintegralvanishesbecausedivjAlthoughthisisobvious,onecanseeitbythefollowingtrick:()xSxdVxdVxdVxdajdVjjjjn(a)ThesurfaceintegralvanishessinceSisfaroutsidetheregionofthecurrentHence,xjdV,andanalogouslyfortheothercomponentsSo,theclosedcurrentregionhasnomonopoleTherefore,()()()iiAjdVcrrrrrorwritteninvectorform,()()()dVcrArjrrrTheintegrandisrewrittenusingtherelation()()()rrjrjrrrjNow,wewilldemonstrate,thatapartfromasign,theintegralsoverthelefthandsideandthefirsttermontherighthandsideareequaltoeachother,sothat()()()dVdVrrjrrjrr(b)Forthispurpose,wetreatthexcomponentfirstStartingfromoneoftheterms,eg,xyyjdV,wefactorouttheprimedcomponentfromtheintegralsignandconsiderfurther(()jrj):()xyjdVyxdVjbecause()xxxxxjxjjjjjandthedivergencejAnintegrationbypartsyields()()yyxdVyxdVxydVxjdVjjjthus,xyyjdVxjdVoringeneral,ikikxjdVjxdVIntheintegrationbyparts,thefirsttermvanishesbyGauss#theoremFurthermore,gradxxxehasbeenusedSo,wehaveiikiikiixxjdVxjxdV()Thisisthethkcomponentof(b)Finally,wehave()()()dVdVjrrrrrjr()Forthevectorpotentialthisimplies()()()()dVdVcrrcrArjrrrrjrTheintegralisthemagneticmomentmwedefine()dVcmrjrhence,()rmrAr()isthevectorpotentialofamagneticdipolemThemagneticfieldcorrespondingtothisvectorpotentialresultsfromcurlBAoutsidethecurrentdistributionAfterashortcalculation,onefinds(Exercise):()()()rrrmrrmnnmmBr()wherernrisaunitvectorinrdirectionReally,with(,,)xxxrwecalculatedirectly(curl)()jjjrrrrrrAmmmjjxmrrrmNowrrrrrrandjjijijijiixxxxxrrxxrrrHence,()(curl)jijijjjjiiiiiixxxxmmmmrrrrrmrAsothat()curlrrmrrmAThisisthedesiredresultItconfirmsourassumptionforthemagneticdipolefield(asbeingcompletelyanalogoustotheelectricdipolefield)anddemonstratestheconsistencyofourconsiderationsInotherwords,theassumptionintheoriginalmeasurementofmandBthatthefieldofamagneticdipolecanbedeterminedincompleteanalogytothefieldofanelectricdipolefollowsdirectlyfromtheBiotSavartlawOntheotherhand,thelatterhasbeendeducedfromexperimentsutilizing()Foranarbitrarilyshapedconductinglooplyinginaplane(Figure)themagneticmomentisgivenby()dVIdccmrjrrsIIdaccrrn()wherenisthenormalvectortotheplaneThecrossproductgivestheareaelement||dadrrandthemagneticmomentismIacifaistheareaenclosedbytheloopasinExerciseforthecircularconductorForceandtorqueonamagneticdipoleinamagneticfieldConsideranexternalmagneticfield()Br(seeFigure)Thethicomponent()iBrmaybeexpandedinaTaylorseriesabouttheorigin:()()()()iiiBBBrrWeassumethattheBfielddependsonlyweaklyonthepositionthatis,itisessentiallyuniformThen,alltermswithhigherpowersofrcanbeneglectedIfabodywiththecurrentdensitydistribution()jrisplacedinthefield,thenaforceandatorqueactonitAccordingto()theforceonacurrentdensitydistributionis()()()()dVdVccFjrBrBrjr()SubstitutingintheTaylorexpansiongivenaboveweget()()()()()dVdVccFBjrrBjr()()()()()dVdVccBjrjrrBThevolumeintegral()dVjr(thatis,themonopoleofthecurrentdistribution)isequaltozeroagainsee(a)Thus,theforceis()()()dVcFjrrB()Notethetensorcharacteroftheterm()rBwhose,eg,xcomponentis()xxxxBBBxyzxyzrBSinceforanexternalmagneticfieldoutsidethesourcesj,()cBj,oneobtains()()()rBrBrBSubstitutingthisintotheequationfortheforce,weobtain()()()()()()dVdVccFrBjrrBjrrBjr()()()()dVdVccrBjrrBjrToconvinceourselvesthatthistransformationiscorrectwecheck,eg,thexcomponentHere,wewillrefrainfromdoingsoComparingtheintegrandwithequation(),()()()dVdVjrrrrrjrandidentifyingrwith()Br(botharenotintegrated),thentheforcecanbewrittenintheform()dVcFBrjr(writingBfor()B)Buttheintegralisjustequaltothemagneticdipolemomentmsothat()()()FBmBmmBmBBm()()mBmBandbecausedivB()()FmBmB()Here,becausecurlB,therelations()()()()()()mBmBBmmBBmmBhavebeenusedThiscanbealsocheckedbycomputing,eg,thexcomponent:()yxxxxzxxyzxyzBBBBBBmmmmmmxyzxxxmB()xmBHere,becausecurlB,therelationsyxBByxandxzBBzxhasbeenusedTheenergyofthemagneticdipolemintheexternalfieldcanbecalculatedeasilyfrom()UddFrmBmB()Thetorqueofacurrentdensitydistributioninamagneticinductionfield()BBisgivenby(())ddVcNrFrjrB()()()dVdVccrBjrrjrBInmagnetostatics,divjThesecondpartoftheintegralis(()()BrBcanbefactoredoromundertheintegralsign):()()()dVdVdVrjrrjrrjr()()dVdarjrrjrnbecausethecurrentdistributionisboundedandthesurfacecanbeplacedarbitrarilyfaroutsidethecurrentdistributionThus,weobtainforthetorque()()dVcNrBjrTakingintoaccountequation(),()dVcNBrjr()Again,theintegralgivesthemagneticmoment,andfinallywecanwriteNmB()Relations(),(),and()arevalidgenerallyinanexternalfield,andtheyareanalogoustotherelationsbetweenelectricdipolesinanelectricfieldTheyareneededfrequentlyAlso,thisresultdemonstratestheconsistencyofourconsiderations,because()isidenticalwith(),whichisoneofthestartingpointsofmagnetostatics,inadditionto()Example:IllustrationofthemagneticmomentWeconsideraplane,rectangularcurrentloop(Figuresand),whichisplacedinauniforminductionfieldBThefiguredisplaysthissituationinperspectiveaswellasinatopviewWecanimmediatelyclarifythedirectionoftheforcesdrawn(Lorentzforce!)Obviously,FFandFFTheforcesFandFtrytosquashtheloopontheotherhand,theforcesFandFexertatorqueontherectangleWedenotethetorquebyDItsmagnitudeis(comparethefigure)||||||cos||sindaaDFFFNow,||IdVIdbBcccFjBsBandthus||sin||sinIabBBcDmwheremIabcisthemagnitudeofthemagneticdipolemomentDefiningitsdirectionasindicatedinthefigure,wecanwriteinvectornotationDmBforthetorqueThisrelationistheexactanalogtothetorqueofanelectricdipolemomentinanelectricfield,DpEExercise:MagneticmomentandangularmomentumofachargedparticleDerivetherelationshipbetweenthemagneticmomentmandtheangularmomentumofachargedparticleLettheparticlemoveinacentralfieldRefertoFigureSolutionTheareaaofaparticle#sclosedorbitis||dLLadadmdtdtTmdtmmrrrrwhereTistheperiodWehaveusedthefactthatLisaconstantvector(centralforcefield)WiththechargeqthecurrentisqITForapositivechargeq,mandLhavethesamedirectionTherefore,IqLqaTcLcTmLmcLLmLInquantummechanics,Ln,whereJsh,andhisPlanck#sconstant,,,narethequantumnumbersoftheangularmomentumHence,themagneticmomentisamultipleofBmqmcThisiscalledtheBohrmagnetonfortheparticleofchargeqandmassmExercise:ForceandtorquebetweentwocircularconductorsDeterminethetorqueandtheforcebetweentwocircularconductorswithparallelaxescarryingequalandequidirectionalcurrentsIifthedistancebetweenthecentersoftheloopsislarge(LR)ExpresstheforceandthetorqueasafunctionoftheanglebetweenthetielineandtheconductorplaneRefertoFigureSolutionAtlargedistanceLRthefieldofasmallconductorloopofradiusRisgivenby(seeequation())()()()rrrrmrrmmrrmBr(rL)()whereIRcmn,IRcmn()IisthecurrentnisaunitvectorThecoordinatesystemcanalwaysbechosensuchthatthedipoleslieinthe,xyplane,andxne(Figure)Then,correspondingtoequation()theinteractionenergyofthetwomagneticdipolesisgivenby()()()(cos)mUrrrmrmrmmmBr()and()sinrUrrrFeeemBrcosrmrrree(cos)sinrmree(cos)sinrmLee()Here,wehavesetmmm,asgiven,andnotedinequation()Werecognizethatequation()iscompletelyanalogoustothecaseoftheinteractingelectricdipoles(seeequation()):()()WrrprprpppEThetorque()()xxmmrrrerrmrrmDmBmecoscoszzmmrLee()

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