第一章麦克斯韦方程english含中文

Preface引言Wecanmastertheelectromagnetismsystemicallybylearningthiscourse.Learninghowtousethiselectromagnetismtheorytosettletheactualproblems,anditcandosomehelptosomestudiesinotherrelateddomain,suchastheoryandtechnologyofthewireless,thetechnologyoflaser,plasmaphysics,astrophysics.通过对它的学习可以指导系统地掌握电磁学理论。学会用此电磁学理论解决实际问题，同时有助于对其他有关领域的学科的学习，如无线电理论与技术，激光技术，等离子体理论，天体物理。Intheelectrodynamics,wewillstudythephenomenonaboutelectricityandmagnetisminthenaturefurther,togetherwithsomeexperimentallaws,motionallawsandthefundamentaltheoriesamongtheirmutualrelation,whichmakethetheoriesintegrate.

电动力学进一步深入研究自然界中电和磁现象及其许多实验定律，运动定律及其相互联系的基本理论，使理论系统化。1方法：在学完每一章节后及时看书理解基本概念和规律、并辅以适当的练习学习电动力学的目的：掌握规律，理解电磁场的性质（时空性）

获得分析处理该类问题的能力

参考书：电动力学，场论，电磁场理论，电磁场与电磁波等F.D.JacksonClassicalElectrodynamicNewyork1975J.A.KonyTheoryofElectromagneticWave，2Theaim:Tomasterthelawsandtheory,understandthecharacteroftheelectromagnetism(itspropertyofspaceandtime)Toobtaintheabilityofanalyzingandsettlingthosekindofproblems学习电动力学的目的：掌握规律，理解电磁场的性质（时空性）

获得分析处理该类问题的能力Method:Reviewingthebasicconceptsandlawsaftertheclasses,withsomeproperexercises.方法：在学完每一章节后及时看书理解基本概念和规律、并辅以适当的练习Referencebooks：Electrodynamics,Fields,Electromagnetismtheory,ElectromagnetismfieldsandtheElectromagnetismwave(inChinese),etc.F.D.JacksonClassicalElectrodynamicsNewYork1975J.A.KonyTheoryofElectromagneticWave，参考书：电动力学，场论，电磁场理论，电磁场与电磁波等3AppendixVectorAnalysisandFields

附录：矢量代数和场论1、Vectoralgebra矢量代数Thetriplevectormixtureproductofthreevectors三矢量混合积Theproductisunchangedbyanexchangeofdotandcrossorderbyacyclicpermutionofthethreevector.

Thetriplevectorproductofthreevectors三矢量矢积42、Divergence，CurlandGradient散度、旋度、梯度(1)、Divergenceofvector散度(2)、Curlofvector旋度when回总目录5(3)、Gradientofscalarquantity标量场的梯度Thatis(4)、Integralform积分表达式BackDivergencetheorem散度公式Stokes’s

theorem斯托克斯公式6(5)、Theexpressionaboutdivergence，curlandandgradientinrectangularcoordinates直角坐标系中散度、旋度和梯度的表达式exey

ez

arethethreeunitvectorsinrectangularcoordinates,respectively.

exeyez是直角坐标系的三个单位矢量回总目录76、Del(

)operator算符回总目录Usingthisoperator,wecanave

:利用算符：8Notice：

1、Del(

)isadifferentialoperatorinthatitisusedonlyinfrontofafunction,whichitdifferentiates;itisavectorinthatitobeythelawsofvectoralgebra.soithasthetwocharactersofvectoranddifferentialduringthecalculation.注意：1、算子▽的定义，表明▽是一个矢性微分算子；因此它在计算中具有矢性和微分的双重性质。

回总目录2、Theoperationrulesofoperator

showthatwhen

worksonafunctionorvectorfunction,itcanbeexpressedas：2、算子▽的运算规则表明，作用在一个数性函数或矢性函数上时，其方式有如下三种：9Usefullformulae:有用公式回总目录Formulaeaboutoperator▽:10Formulaeaboutoperator▽:▽算符公式11公式证明示例12Laplace’soperatorandequation

拉普拉斯算符及方程LaplacianoperatorLaplacianequation13GeneralsolutionofLaplace’sequationwhenaxissymmetryGeneralsolution:Generalsolutionofsymmetryaxis14Emphases:Maxwellequations(formsofdifferentialandintegral),boundaryconditionsand

Lorentz

formulae(重点：麦克斯韦方程组、边界条件和罗仑兹力公式)chapterⅠGenerallawsofelectromagneticphenomenaDifficulty:Themediumcharactersandboundaryconditions(介质特性和边界条件)15Byanalyzingtheexperimentallaws,summarizethegenerallawsaboutelectromagnetism,setupMaxwellequationsandLorentz’s

formula.(分析实验规律，总结电磁普遍规律，建立麦克斯韦理论和罗仑兹力公式)

Asaformofmaterial,wecanuseelectricfieldE(x,y,z,t)andmagneticinductionB(x,y,z,t),thetwobasicphysicquantities,twovectorfunctionstodescribetherulesandcharactersofitsmovement

of

electromagneticfieldstate,whileusingitsdifferentialequationstodescribeitsrules.物质的一种形式，由矢量函数E、B描述。由微分方程描述规律。回总目录The(macroscopic)electricfieldistheforceperunitchargeonatestchargeembeddedinthedielectric,inthelimitwherethetestchargeissosmallthatitdoesnotitselfaffectthechargedistribution.（单位试验电荷所受的力，试验电荷足够小，不影响原电场。）16§1ElectricchargeandtheelectricfieldBystudyinganexperimentallaw,

Gauss’law

,wecananalyzeGauss’law,divergenceandcurlinelectrostaticswhicharecalledthecharactersofelectrostaticfield.分析实验定律、高斯定律，及描述静电场特征的电场散度和旋度。回总目录1、Coulomb’slaw

（库伦定律）Invacuum,thetwopointchargesexertoneachotherforcesthatactalongthelinejoiningthemandareinverselyproportionaltothesquareofthedistancebetweenthem,andtheseforcesarealsoproportionaltotheproductofthecharges,arerepulsiveforlikecharge,andattractiveforunlikecharges.Itobeythecoulomb’slaw.ThatisFieldaroundapointchargecanbeobtainas：17

Fromthesuperpositionprinciple,wecanknowthefieldofasystemofnpointcharges：由叠加原理，点电荷组的场：回总目录Whenchargeisacontinuouschargedistribution,wecanhave：连续分布P(x,y,z)andItistheintegralformofelectricfieldintheelectrostaticsfield.Thesourceofcontinuouschargedistributioncanbevolume,surface,orlineardistribution,andwecanchangetheexpressionofdQaccordingly.Theintegralformreflectselectrostaticsfield’seffectsbroughtupbychargemacroscopically,whileitsdifferentialformshowstherelationshipbetweenachargeandotherfieldnearbydirectly.连续分布有线分布、面分布、体分布，积分反映宏观，微分反映场点临域的微观。182、Gauss’lawandthedivergenceintheelectricfieldTheabilityofchargesproducingafieldcanbereflectbytheflux,whilethefluxofthefieldthroughaclosedsurfacewiththetotalchargeQenclosedbythesurfaceis通量反映电荷产生场的能力，电场对闭曲面积分。Thesurfaceintegralofthenormalcomponentofthiselectricfieldoveraclosedsurfacethatenclosetheoriginandconsequently,thechargeQ.ds

issurfaceelement.回总目录TocalculatethefluxofthechargeQ,youcanchoosesphericalsurfaceinwhichQlieoriginandradiusisrasgausssurface计算出该点电荷的电通量，可以以点电荷为球心选一半径为r的球面为高斯面（书上给出了一般闭曲面的点通量计算Page7）19Asforseveralchargesorthecontinuousdistributioncharge：多点分布电荷或连续电荷：Therightofequationincludeallchargesenclosedbytheclosedsurface.右端必须为闭曲面所全包含的电荷Whenthevolumeoftheclosedsurfacecometozero，thepreviousequationcanbywrittenas：所选闭曲面所围体积元趋于零时，上式变为ItcanbeattainedbyGauss’law：直接由高斯定理回总目录20ThedifferentialequationofGauss’lawshowsthedivergenceofacertainfieldspointinthenearareaonlyrelatesthechargedensityonthispoint,havingnomatterwithotherchargeinthedistance.Chargeonlyproducethefieldofitsnearregion,thefieldofotherregionistransferredbyitself.Thedifferentialequationofgauss’slawisuniversalrightforchargeproducingfield.高斯定理的微分形式反映确定场点的散度只与其临近区域的电荷密度有关，与远处的无关。回总目录Gauss’slawlieslargelyinprovidingaveryeasywaytocalculateelectricfieldsinsufficientlysymmetricsituations.213、

CurlinElectrostaticField（静电场的旋度）.Thecirculationproducedbypointchargeis（点电荷的环量）电场为有源场。起于点电荷，止于负电荷或无穷远，场在自由空间连续分布。静电场为无旋电场。Curloffieldreflectthecharacterofcircumfluence.(旋度反映电场的环流特性)回总目录E22sample.ChargeQisdistributinguniformlywithinaspherewhichradiusisa.Calculatetheelectricfieldofeverypoint,andits

divergence.计算电荷均匀的球内外的场Answer：Thisproblemissphericalsymmetry,wemayconstructagaussiansurfacethatisasphericalsurface,sphericalcenterisorigin,radiusisr.1、whenr≥a，chargesenclosedinGaussiansurfaceareQ,wecanhavethefollowingequationfromGauss’law:Divergenceformulainsphericalcoordinate：回总目录ra23Forthesphericalcase:

2、

whenr<a,chargesenclosedinGausssurface:FromGauss’slaw:

回总目录24Whileitsdivergence：also,Ehasthesamedirectionasradius回总目录25§2Electriccurrentandmagneticfield电流和磁场1、Chargeconversationlaw电荷守恒定律thecurrentdensity

Jisthequantityofchargesthroughperunitcrosssectionareaandpertimealongthedirectionofcurrent电流密度J为沿电流方向上单位时间垂直通过单位面积的电量

Thecurrentdistributionthroughawireisvarious,someuniformdistributingonthesectionoflead,whileotherdistributingunequally,forexampletheskineffectintheconditionofhighfrequency.

Weintroducecurrentdensity

Jtodescribethedistributionofcurrent.通过导线的电流分布是多样化的。有的均匀分布在导线截面上，有的则不均匀，如高频时的趋肤效应。为表达电流在电流在导线上的分布，引入电流密度J，导线上的任一截面的电流元ThecurrentthroughthesurfaceS,anarbitrarilyshapedsurfaceareaofmacroscopicsize,isgivenbytheintegral:

A

Currentdensity电流密度26Thecurrentofmovingcharges,supposethechargedensityρ

with

thesame

velocity

v

运动电荷的电流，isJ=ρvAsafewparticleswithcharge：

BEquationofcontinuity――ChargeconservationlawChargeconservationlaw:Thechargecanneitherbecreatednordestroyed;thechargechanginginacertainclosedsurfaceequalstothesumofchargesflowingoutandinthearea.Itcanbedescribedbycontinuitylaw:电荷既不能被创造也不能被破坏，闭曲面内的电荷变化等于流入流出该区域的总电荷。WhichdenotethatthetotalcurrentflowingoutfromtheinterfaceequaltotherateofchargesdecreasingwithinregionV.流出电荷等于区域内的减少。

回总目录27FromGauss’slaw:Hence:Thisisequationofcontinuity,anditisthedifferentialformofchargeconservationlaw.这就是连续性方程，它是电荷守恒定律的微分形式。回总目录Movetheminustotheleft,then:ThereforethetotalcurrentflowingintotheinterfaceequaltotherateofchargesincreasingwithinareaV.

（注意闭曲面的方向为外法线方向为正向）

28Discuss:

1whenVisfullspace,thennocurrentoutorin,thetotalchargeremainthesame.全空间无流入流出，so,Forsteadycurrent:

▽·J＝0whileitsdistributionisacloselinewithoutsource.292、

ThelawofBiotandSavart

Thelawdescribesthemagneticeffectsofcurrent

Itdenotethatthemagneticinductionatfieldpointxwasproducedbythesourcecurrent,thepositionvectorrdirectsfromthesourcepoint(currentelement)tofieldpoint.

表示源电流在离源r远的场点x处的所产生的磁感应强度Allcurrentlieonthelead,thenanelementofcurrent

回总目录HencetheforceofcurrentelementIdlatthispointis303、CirculationandcurlofB磁场的环量和旋度Forconvenience,themagneticinductionofainfinitecurrentlinearleadis考虑无限长直导线。Fromthesymmetry,themagneticfieldofP:回总目录ThecirculationofBis（∵B与dl同向）

ThisisAmpere’scircuitallaw.ThecirculationofBintheclosedcurveisμ0I.安培环路定律Pdlr31ForgeneralFromStockes’stheorem:4、

Thedivergenceandmagneticfluxofmagneticfield磁场的磁通量和散度Tostudyacertaindivergence,wefirstlycalculatethemagneticfluxoftheclosedsurface.choosingthegaussiansurfaceiscylinderwhichsymmetricalaxisisthelead

计算闭曲面的磁通量，以长直导线为对称轴作一高斯面。

B回总目录32Fromelectromagnetics,weknowthatthemagneticinductionlineproductbycurrentisclose.由电磁学知。电流激发的磁感应线为闭合线。故B为无源场。则B对任意闭曲面的总通量为05、证明磁场的旋度和散度公式∵▽是对场点的微分，与源无关，故后二项为0，同时积分为对源积分。33Itisvectorpotential。Fromthevectoranalysis,thedivergenceofcurliszero回总目录b.（∵▽和积分分别对场和源）34对于此积分只有上时被积分函数不为0，此时B为柱对称，代入柱坐标下的散度计算表达式，当r≠0时，回总目录被积函数35则上式可见，

和

由恒定电流下的毕奥－萨法定律导出，但前者是在任何磁场都成立的。后者仅在恒定下成立。

36例，电流I

在均匀分布于半径为a的无限长直导线内。求空间各点的磁感应强度，并计算其旋度。

解

由对称性知，B关于导线轴心轴对称，以导线轴心为圆心作一垂直导线的圆。当r>a时，圆心总电流为I，由安培环路定律得当r<a时环路内的电流回总目录37FromAmpere’scircuitallaw,

38§3Maxwell’sEquations前两节总结了恒定电磁场的基本规律（电磁场的与电荷,电流的关系，积分，微分，散度，旋度）本节研究变化的电磁场规律,建立描述电磁现象的普遍规律Maxwell’sEquations和Lorentzformula.1、电磁感应定律(LawofElectromagneticInduction)Theinductionelectromotiveforce(emf)ofacircuitisproportionaltothedecreasingrateofmagneticfluxthroughthecircuit.闭合导体回路中的感生电动势与通过以该回路为边界的任一曲面的磁通量减少率成正比。39ThenegativesigninFaraday’slawindicatethatthedirectionoftheinducedemfissuchastoopposethechangethatproduceit.ItisLenz’slaw:incaseofachangeinamagneticsystem,thatthinghappenswhichtendstoopposethechange.

(负号表示阻止改变发送，即楞次定律)Usingthecirculationofelectricfielddenotetheelectromotiveforce.

Ifthecircuitisarigidstationarycircuit,thetimederivativecanbetakeninsidetheintegral,whereitbecomesapartialtimederivative,then回总目录402、位移电流(displacementcurrent)

回总目录Fromtheequationofcontinuity,wehave：ItisthedifferentialformofFaraday’slaw,anditisanindependentexperimentallaw–itcannotbederivedfromotherexperimentallaw.Theinductionfieldisacurlfield.ItrepresentsoneofMaxwell’smajorcontributiontoelectromagnetictheory.Therehasacontradiction.41回总目录MaxwellbringDisplacementcurrentJDforwardtosolvethiscontradiction.Itis

Fromthechargeconversationlawandthedifferentialformofthegauss’slaw,wehave：

Fromthiswecanseethatdisplacementcurrentisthechangingrateofelectricfield.423、

麦克斯韦方程组Maxwell’sequations

Maxwellgivestheuniversallawoftheelectomagneticfieldbymaxwell’sequations回总目录Eachoftheseequationsrepresentsageneralizationofcertainexperimentobservation:thefirstisthedifferentialformofFaraday’slawofelectromagneticinduction;thesecondisanextentionofAmpere’slaw;thethirdisgauss’slaw,whichinturnderivesfromthecoulomb’slaw;thelastisthefactthatsinglemagneticpoleshaveneverbeenobserved.它反映了一般情况下电荷电流激发电磁场内部运动规律。变化的电场和磁场也可以相互激发。这种相互激发在空间传播形成电磁波，同时也体现了场能独立于电荷之外而存在（如空间的无限电波）可见场亦是一种物质形态。434、洛仑兹力公式Lorentz’sformulaTheelectricfieldproducebychargescanexistandpropagateinitselfform,andcanactontheothercharges.

电场由电荷产生，并能以其自身的形态存在、传播，它亦能作用于电荷、电流。TheelectromagneticforceofaunitvolumeofcontinuousdistributionchargesisLorentz’formulaforforcedensity.

若电荷为连续分布，其密度为则电荷系统单位体积所受的力密度f为洛仑兹力密度公式

若v为电荷为e的粒子速度Theelectromagneticforcetothesinglechargeoftheelectromagneticfieldis则可得单个带电粒子受到的电磁力，即洛仑兹力的表达式反映电荷与场的作用关系回总目录theelectrostaticforce:Themagneticforceofsteadycurrentelement:44前面这些讨论都是基本于真空和无界情况下进行，而实际生活中，电磁场几乎都存在于介质中。为此，完全理解，掌握和运用电磁场理论于实际生活中，必须以此为基础，结合介质特性研究介质中的电磁理论及边值关系。45§4介质的电磁性质electromagneticpropertiesofmedium

1、介质及其于电磁场的相互作用(Interactofdielectricmediumandelectromagnetic)ADielectricmediumarecomposedofmolecules,havequantitiesmovingchargedparticle,buttheoveralleffectfromthemacroscopicpointofviewisthatthereisnonetcharge.介质就是在空间一定区域中聚集的大量的运动着的带电粒子，其宏观特性就是电中性如半导体材料等物体，某些液体。

Btheelectromagneticfieldcausesaforcetobeexertedoneachchargeparticle,thepositiveandnegativeparticleofeachmoleculearedisplacedfromtheequilibriumpositioninoppositedirection,itispolarize,ormagnetize.Thesedisplacementarelimitedbystrongrestoringforceswhicharesetupbythechangingchargeconfigurationinmolecule.Boundcharge,freecharge,magnetizationcurrent.当其与电磁场作用时，就出现的电荷电流，它们也将产生新的场分布，从而影响原场的分布。此那种电荷电流分别称为束缚电荷，磁化电流。场作用主要使有极分子趋向有序化。无极分子被极化并有序取向。462、

介质的极化Polarizationofdielectricmedium回总目录ASortofMedium介质分类Adielectricmediumiscomposedofatomormolecule.Itisclassifiedaspolarornonpolar.所有介质都是由原子，分子构成。

1.

nonpolarmoleculehavethesame‘centersofthegravity’ofthepositiveandnegativechargedistribution.无极分子正负电荷中心重合。2.Apolarmoleculehavedifferent‘centerofthegravity’ofthepositiveandnegative,ithasapermanentdipolemomentelectricdipolewithanelectricdipolemoment.

有极分子正负电荷中心不重合，可看成偶极子，有一定的的电偶极距。47BPolarizingofdielectric介质极化Apolarizeddielectric,eventhoughitiselectricallyneutralontheaverage,producesanelectricfield,bothatexteriorpointandinsidethedielectricaswell.极化的电介质即使总体表现为电中性，但在介质内和外都有电场。Thepolarizationofdielectricdependsonthetotalelectricfieldinthemedium,butapartoftheelectricfieldisproducedbythedielectricitself.Furthermore,thedistantelectricfieldofthedielectricmaymodifythefreechargedistributiononconductingbodies,andthisinturnwillchangetheelectricfieldinthedielectric.介质极化依赖于介质中的总电场，其中部分为介质产生，介质远处的场分布又改变电场分布，如此相互影响。48Ifthemediumispolarized,thenaseparationofpositiveandnegativechargehavebeeneffected,anditischaracterizedbytheelectricdipolemomentperunitvolume.

介质极化，电场作用于介质，使有极分子运动有序化，无极分子亦变为有极性的并运动亦有序化的现象。

wedefineapolarizationvectorintermofthenumberofdipolemomentsperunitvolume.用矢量p描述介质的宏观电偶极距的分布，即电极化强度是矢量Isverysmallvolumeinclusionquantitydipolefromthemacroscopicviewpoint

为包含一定量的电偶极子的物理小体积。49C计算束缚电荷密度与电极化强度间的关系Relationofboundchargedensityandtheelectricpolarizationvector.回总目录Integraltheclosedsurface,wehavethetotalnegativechargequantitiesinsidetheclosedsurfacewhichisequaltothepositivechargeoutoftheclosedsurface.Thepolarizationchargedensity,δP,

ρP对闭曲面积分，则在闭曲面内的负电荷等于曲面外的正电荷。即：由高斯公式Choosingaclosedsurfaceinsidethemediumwithsurfaceelementds,Ifthepositivechargeofthedipoleisoutoftheds,theunitvolumemoleculeisn,thequantitiespositivechargeintheoutsidedsare:取一闭曲面S（在介质内），设偶极子的偶极矩Pi。在任意曲面上取一面元ds，设偶极子的正电荷在ds外，负电荷在ds内，若单位体积分子数为N。则在ds外的正电荷50DBoundcharge介质界面上的面束缚电荷Nowdiscussingtheboundchargeoftheinterfaceoftwomedium.Choosingathinlaminawhichcontaintheinterface.

现讨论面界面间的束缚电荷，在界面取一薄层，包含两界面。则Theenteringpositivechargesfrommedium1tothelaminais

由介质1进入薄层的正电荷为

Theenteringnegativechargesfrommedium2tothelaminais

由介质2进入薄层的负电荷为

Thenetchargeinsidethelaminais薄层内的净余电荷为：回总目录51E介质中的电场，电位移矢量theelectricfieldinsideadielectric,thedisplacementvectorThepolarizationofdielectricdependsonthetotalelectricfieldinthemedium,butapartoftheelectricfieldisproducedbythedielectricitself.Furthermore,thedistantelectricfieldofthedielectricmaymodifythefreechargedistributiononconductingbodies,andthisinturnwillchangetheelectricfieldinthedielectric.Thefieldinsidethedielectricisproducedbythefreechargeandthepolarizationcharge.电场使介质极化，极化电荷又产生场与原场叠加而成介质中的场，因此介质中的场是由两个源产生，即：自由电荷，束缚电荷。

即有:

Nowtherelationofthesourceandfieldis:

即:Infact，wecannotmeasuretheboundcharge,nowsupposeadisplacementvector

为引入电位移矢量,电位移矢量D是辅助场量:52回总目录

isthepermittivityofthematerial.为介质的介电常数（电容率）

为相对介电常数（相对电容率）TherelationofDandEisdifferentfromothermedium.TheDandEoflinearisotropydielectricobeythelinearlaws

D与E的关系随介质性质而各异，一般各向同性线性介质。极化强度D和E为线性关系。533、介质磁化mediummagnetization回总目录Current：aconventioncurrentandatomcurrentbothkindsofcurrentmayproducemagneticmoment,m=ia.A：磁化(magnetization)

theatomcurrentchangetosamedirectionundertheexteriormagneticfield,itbecomeamacroscopiccurrent,magnetizationcurrentIm,themagnetizationcurrentdensityJm.在磁场作用下，介质内的分子电流取向有序化，形成宏观磁化电流密度JmB：磁化强度M与磁化电流密度的关系relationofmagnetizationMandthemagnetizationcurrentdensity

ThemagnetizationcurrentofasurfaceisThemagnetizationcurrentisthesumoftheallatomcurrent.Thecurrentsinthevariousloopstendtocanceleachotherout,andthereisnotneteffectivecurrentintheinteriorofthematerial,theneteffectivecurrentistheclosedlinelinkingatomcurrent.另一方面，磁化电流由磁化分子电流总和组成，在曲面内，只有闭曲线穿过分子流线圈的分子流对电流有贡献，其余则由分子的闭合性，而贡献为0。也就是说只有闭曲线链环的分子电流有贡献。54

C极化电流PolarizationcurrentThechangingelectricfieldcausesthepolarizationvectorchanging,andproducethepolarizationcurrent.

电场变化引起P变化，从而产生极化电流。与之和为介质中的诱导电流abductioncurrent。回总目录55D介质中的磁场问题，磁场强度themagneticfieldinmedium,themagneticintensityHTheabductioncurrentproducedbymagneticfieldfolduptheoriginmagneticfield.在介质中磁场引起诱导电流，诱导电流又产生磁场为原场叠加，成介质中点的场分布。即：Infact,wecanonlymeasuretheconventioncurrent,hereweintroduceanauxiliaryvector,themagneticintensity,H.

在实际工作中只能控制自由电流，磁化电流则不然。为此引入磁场强度H这一辅助量。TheMaxwell’equationcometo回总目录56Theauxiliaryvectorsimplifytheequation.

辅助量使介质中的方程简化。但H并不代表介质内场强。只有B才是一个基本物理量。因此要找到B和H的关系。TheconstitutiverelationofMandHofthenon-ferromagneticisotropyofmaterialislinearrelation对于各向同性非铁磁性物质。M与H有简单的线性关系，即：Permeability磁导率相对磁导率回总目录574、Maxwell’sequationsinmediumItalsoincludingelectromagneticequationsinmedium.Asforisotropicmaterials:

DifferentialcoefficientformofOhm’slawWhereisconductivepermeability.回总目录58对于各向异性的非线性介质，则其关系为复杂的张量关系:其分量形式为

回总目录对于非线性情况下，D与E的高次量也有关系，即

此式在非线性电磁中非常重要。铁磁性物质的B与H也为非线性，与磁化过程有关。用磁化曲线和磁滞回线表示B与H的关系59§5电磁场边值关系（boundaryvaluerelationofelectromagneticfield)

BoundaryconditiondepicttherelationofE,B,D,H,chargeandcurrent.

边值关系就是描述界面两侧场量改变与界面上的电荷电流之间的关系式。由于在界面上的场量不连续。不能应用微分形式的麦克斯韦方程组。积分形式麦克斯韦方程可以应用任意不连续分布的电荷，电流激发的场。研究边值关系从积分形式麦克斯韦方程出发。1、积分形式的麦克斯韦方程

对于旋度表示的方程，进行面积分，并利用斯托克斯公式将左端化为曲线积分；对散度表示的方程在任一区域v上进行积分，利用高斯公式。可得积分形式的麦斯方程组。即：602、Discontinuityofnormalcomponentoffield法向量的跃变回总目录aNormalcomponentofelectricfield电场的法向分量Letusconstructthesmallpillbox-shapedsurfaceSthatintersectstheinterfaceandenclosesandarea∆Softheinterface,theheightofthepillboxbeingnegligiblysmallincomparisonwiththediameterofthebases.ThechargeenclosedbySis

取介质边界上取一面元为S扁平小柱体。高h为宏观小量但它包含足够多的分子层。应用于麦氏第三方程，即

61回总目录ThediscontinuityinthenormalcomponentofEisgivenbythesurfacedensityofexternalchargeontheinterface.

界面的电场强度法向分量跃变，其跃变与界面上的总电荷密度有关。Dealingwithpolariztionchargesameasit对于极化电荷在界面上类似处理：由公式

Ispolarizationchargesurfacedensity为极化电荷面密度

SumtwoequationandtakeviewoftheD,BNormalcomponentsofmagneticfield磁场的法向分量：ItshowthatthenormalcomponentsofBiscontinuity.说明B的法向分量总是连续的与边界上的电荷电流无关。

Withthesamemethods,wehave:类似于上述过程处理，可得623、切向分量的跃变thediscontinuityoftangentialcomponent.A磁场的切向分量thetangentialcomponentofmagneticfieldAboundaryconditiononH-fieldmaybeobtainbyapplyingthesecondmaxwell’sequation(extenionAmpere’slaw).Constructarectangleclosedpath,theborderlengthare△landh(sosmallcanbenegligible),respectively.Thecurrentthroughtherectangleisnegligibleunlessthereisatruesurfacecurrent.therefore取一小矩形长为△l。以界面为中心高为h高包含足够分子层

,且是宏观小量。应用磁场的第二麦氏方程有：63NowtheequationcometoBytakingthe

crossproductoftheequationwithn,theequationmaybewrittenas

Itshowsthatthetangentialcomponentofthemagneticintensityiscontinuousacrossaninterfaceunlessthereisatruesurfacecurrent.这说明磁场强度切向分量的跃变与界面上自由电流强度有关回总目录64ForthemagnetizationcurrentUsingthesamemethodwehavePlusthetwoequationwithaneyeto

将两式相加并考虑到Viz.Itshowsthatthetangentialcomponentofthemagneticinductionintensityisdiscontinuousacrossaninterface,andrelatetothetruesurfacecurrentandmagnetizationcurrent.说明，磁感应强度B切向分量的跃变与界面上的自由电流和磁化电流总和有关65B电场的切向分量Thetangentialcomponentoftheelectricfieldiscontinuousacrosstheinterface.

WiththesamemethoddealingthefirstMaxwell’equation,,andconsideringtheareacometozero.

类似地把第一麦氏方程应用到回路，并考虑到有界，而闭曲线所围面积趋于零，故其积分为零，可导出：Thetangentialcomponentoftheelectricfieldiscontinuousacrosstheinterface.此说明电场切向方向分量总是连续的。4、边值关系Boundarycondition：TheboundaryconditionofmediuminterfacecorrespondingtotheMaxwell’sequationsare

综上所述，与麦氏方程所对应的关于介质边界的边值关系如下：66切向tangentialcomponent法向normalcomponent67例

证明在导体界面上电流法向分量满足边值关系证，将积分形式的电荷守恒定律应用于扁平小柱体上，实际电流分布在柱体侧面上积分为零。导体面薄层电荷分布看成面电荷分布。体分布的电荷由于柱体积趋于零，对于右端积分为零。稳恒时稳恒电流的法向分量总是连续的。68解：因平板电容无穷大，由对称知，E垂直极板对于极板与介质1，由边值关系

对于极板与介质2，同理

例无穷大平板电容器内有两层介质极板上面电荷密度为求电场E和束缚电荷分布。

E2E169介质1与下板：

介质2与上板：对于两介质界面处，70§6电磁场的能量和能流Theenergyandenergyflowofelectromagneticfielda:theenergypropagateinspacewiththemotionoffield.Suchastheantennaradiatetheenergybyelectromagneticwave.场的能量随场的运动而在空间传播，如天线不停地通过电磁波把能量发射出去。Energydensityoffieldwistheenergyoffieldperunitvolume.场的能量密度W，单位体积内的场的能量

w=w(x,t)EnergyflowdensityoffieldSistheenergyflowingthroughthecrosssectionperunittimeandpercrosssectionareaalongthetransmittingdirection.场的能流密度S单位时间垂直流过单位的能量，其方向为传输方向，描述能量在场中的传播Theenergytransferbetweenthefieldandchargessystemduringtheinteractionhappen当场与电荷系统作用时，能量就在场和电荷系统间转移Theelectromagneticfieldhaveenergy,itisprovedthattheelectromagneticfieldinteractwiththeelectrifiedmaterial.电磁场这种特殊的物质同样具有能量。其能量能通过与带电物体作用表现出来。1、Lawofconservationofenergyofthefieldandchargessystem场和电荷系统的能量守恒定律的一般形式71b：场与电荷系统间的能量关系theenergyrelationshipoffieldandchargessystemConstructaclosedregionwithρ，J.fromtheconservationlawofenergyknowthattheenergyflowingintothevpertimeisequaltothesumoftheworkpoweroffieldtochargesandthefieldenergyincreasingrateinsidev.

取一闭区域V，截面为S，设V内电荷电流分布为ρ，J。由能量守恒定律知单位时间流入V内的能量等于场对V内电荷作功功率与V内电磁场能量增加率之和。Theworkpoweroffieldactwithcharges

场对电荷系统的作功功率为

TheinnerenergyofVincreasingrateis

V内场能量增加率

theenergyflowingintothevpertimeis则单位时间流入V内的能量变化率：72Usinggaussiantheorem

由高斯定理，得

（thedivergenceoftheenergyflowequaltothesumofworkpowerandtheenergychanging

rateIftheprobleminwholespace,thereisnoenergyflowinandout,therateofdecreasingenergyisus