Preface引言Wecanmastertheelectromagnetismsystemicallybylearningthiscourse.Learninghowtousethiselectromagnetismtheorytosettletheactualproblems,anditcandosomehelptosomestudiesinotherrelateddomain,suchastheoryandtechnologyofthewireless,thetechnologyoflaser,plasmaphysics,astrophysics.通過對它的學習可以指導系統地掌握電磁學理論。學會用此電磁學理論解決實際問題，同時有助于對其他有關領域的學科的學習，如無線電理論與技術，激光技術，等離子體理論，天體物理。Intheelectrodynamics,wewillstudythephenomenonaboutelectricityandmagnetisminthenaturefurther,togetherwithsomeexperimentallaws,motionallawsandthefundamentaltheoriesamongtheirmutualrelation,whichmakethetheoriesintegrate.

電動力學進一步深入研究自然界中電和磁現象及其許多實驗定律，運動定律及其相互聯系的基本理論，使理論系統化。1方法：在學完每一章節后及時看書理解基本概念和規律、并輔以適當的練習學習電動力學的目的：掌握規律，理解電磁場的性質（時空性）

獲得分析處理該類問題的能力

參考書：電動力學，場論，電磁場理論，電磁場與電磁波等F.D.JacksonClassicalElectrodynamicNewyork1975J.A.KonyTheoryofElectromagneticWave，2Theaim:Tomasterthelawsandtheory,understandthecharacteroftheelectromagnetism(itspropertyofspaceandtime)Toobtaintheabilityofanalyzingandsettlingthosekindofproblems學習電動力學的目的：掌握規律，理解電磁場的性質（時空性）

獲得分析處理該類問題的能力Method:Reviewingthebasicconceptsandlawsaftertheclasses,withsomeproperexercises.方法：在學完每一章節后及時看書理解基本概念和規律、并輔以適當的練習Referencebooks：Electrodynamics,Fields,Electromagnetismtheory,ElectromagnetismfieldsandtheElectromagnetismwave(inChinese),etc.F.D.JacksonClassicalElectrodynamicsNewYork1975J.A.KonyTheoryofElectromagneticWave，參考書：電動力學，場論，電磁場理論，電磁場與電磁波等3AppendixVectorAnalysisandFields

附錄：矢量代數和場論1、Vectoralgebra矢量代數Thetriplevectormixtureproductofthreevectors三矢量混合積Theproductisunchangedbyanexchangeofdotandcrossorderbyacyclicpermutionofthethreevector.

Thetriplevectorproductofthreevectors三矢量矢積42、Divergence，CurlandGradient散度、旋度、梯度(1)、Divergenceofvector散度(2)、Curlofvector旋度when回總目錄5(3)、Gradientofscalarquantity標量場的梯度Thatis(4)、Integralform積分表達式BackDivergencetheorem散度公式Stokes’s

theorem斯托克斯公式6(5)、Theexpressionaboutdivergence，curlandandgradientinrectangularcoordinates直角坐標系中散度、旋度和梯度的表達式exey

ez

arethethreeunitvectorsinrectangularcoordinates,respectively.

exeyez是直角坐標系的三個單位矢量回總目錄76、Del(

)operator算符回總目錄Usingthisoperator,wecanave

:利用算符：8Notice：

1、Del(

)isadifferentialoperatorinthatitisusedonlyinfrontofafunction,whichitdifferentiates;itisavectorinthatitobeythelawsofvectoralgebra.soithasthetwocharactersofvectoranddifferentialduringthecalculation.注意：1、算子▽的定義，表明▽是一個矢性微分算子；因此它在計算中具有矢性和微分的雙重性質。

回總目錄2、Theoperationrulesofoperator

showthatwhen

worksonafunctionorvectorfunction,itcanbeexpressedas：2、算子▽的運算規則表明，作用在一個數性函數或矢性函數上時，其方式有如下三種：9Usefullformulae:有用公式回總目錄Formulaeaboutoperator▽:10Formulaeaboutoperator▽:▽算符公式11公式證明示例12Laplace’soperatorandequation

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

Lorentz

formulae(重點：麥克斯韋方程組、邊界條件和羅侖茲力公式)chapterⅠGenerallawsofelectromagneticphenomenaDifficulty:Themediumcharactersandboundaryconditions(介質特性和邊界條件)15Byanalyzingtheexperimentallaws,summarizethegenerallawsaboutelectromagnetism,setupMaxwellequationsandLorentz’s

formula.(分析實驗規律，總結電磁普遍規律，建立麥克斯韋理論和羅侖茲力公式)

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

of

electromagneticfieldstate,whileusingitsdifferentialequationstodescribeitsrules.物質的一種形式，由矢量函數E、B描述。由微分方程描述規律。回總目錄The(macroscopic)electricfieldistheforceperunitchargeonatestchargeembeddedinthedielectric,inthelimitwherethetestchargeissosmallthatitdoesnotitselfaffectthechargedistribution.（單位試驗電荷所受的力，試驗電荷足夠小，不影響原電場。）16§1ElectricchargeandtheelectricfieldBystudyinganexperimentallaw,

Gauss’law

,wecananalyzeGauss’law,divergenceandcurlinelectrostaticswhicharecalledthecharactersofelectrostaticfield.分析實驗定律、高斯定律，及描述靜電場特征的電場散度和旋度。回總目錄1、Coulomb’slaw

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

Fromthesuperpositionprinciple,wecanknowthefieldofasystemofnpointcharges：由疊加原理，點電荷組的場：回總目錄Whenchargeisacontinuouschargedistribution,wecanhave：連續分布P(x,y,z)andItistheintegralformofelectricfieldintheelectrostaticsfield.Thesourceofcontinuouschargedistributioncanbevolume,surface,orlineardistribution,andwecanchangetheexpressionofdQaccordingly.Theintegralformreflectselectrostaticsfield’seffectsbroughtupbychargemacroscopically,whileitsdifferentialformshowstherelationshipbetweenachargeandotherfieldnearbydirectly.連續分布有線分布、面分布、體分布，積分反映宏觀，微分反映場點臨域的微觀。182、Gauss’lawandthedivergenceintheelectricfieldTheabilityofchargesproducingafieldcanbereflectbytheflux,whilethefluxofthefieldthroughaclosedsurfacewiththetotalchargeQenclosedbythesurfaceis通量反映電荷產生場的能力，電場對閉曲面積分。Thesurfaceintegralofthenormalcomponentofthiselectricfieldoveraclosedsurfacethatenclosetheoriginandconsequently,thechargeQ.ds

issurfaceelement.回總目錄TocalculatethefluxofthechargeQ,youcanchoosesphericalsurfaceinwhichQlieoriginandradiusisrasgausssurface計算出該點電荷的電通量，可以以點電荷為球心選一半徑為r的球面為高斯面（書上給出了一般閉曲面的點通量計算Page7）19Asforseveralchargesorthecontinuousdistributioncharge：多點分布電荷或連續電荷：Therightofequationincludeallchargesenclosedbytheclosedsurface.右端必須為閉曲面所全包含的電荷Whenthevolumeoftheclosedsurfacecometozero，thepreviousequationcanbywrittenas：所選閉曲面所圍體積元趨于零時，上式變為ItcanbeattainedbyGauss’law：直接由高斯定理回總目錄20ThedifferentialequationofGauss’lawshowsthedivergenceofacertainfieldspointinthenearareaonlyrelatesthechargedensityonthispoint,havingnomatterwithotherchargeinthedistance.Chargeonlyproducethefieldofitsnearregion,thefieldofotherregionistransferredbyitself.Thedifferentialequationofgauss’slawisuniversalrightforchargeproducingfield.高斯定理的微分形式反映確定場點的散度只與其臨近區域的電荷密度有關，與遠處的無關。回總目錄Gauss’slawlieslargelyinprovidingaveryeasywaytocalculateelectricfieldsinsufficientlysymmetricsituations.213、

CurlinElectrostaticField（靜電場的旋度）.Thecirculationproducedbypointchargeis（點電荷的環量）電場為有源場。起于點電荷，止于負電荷或無窮遠，場在自由空間連續分布。靜電場為無旋電場。Curloffieldreflectthecharacterofcircumfluence.(旋度反映電場的環流特性)回總目錄E22sample.ChargeQisdistributinguniformlywithinaspherewhichradiusisa.Calculatetheelectricfieldofeverypoint,andits

divergence.計算電荷均勻的球內外的場Answer：Thisproblemissphericalsymmetry,wemayconstructagaussiansurfacethatisasphericalsurface,sphericalcenterisorigin,radiusisr.1、whenr≥a，chargesenclosedinGaussiansurfaceareQ,wecanhavethefollowingequationfromGauss’law:Divergenceformulainsphericalcoordinate：回總目錄ra23Forthesphericalcase:

2、

whenr<a,chargesenclosedinGausssurface:FromGauss’slaw:

回總目錄24Whileitsdivergence：also,Ehasthesamedirectionasradius回總目錄25§2Electriccurrentandmagneticfield電流和磁場1、Chargeconversationlaw電荷守恒定律thecurrentdensity

Jisthequantityofchargesthroughperunitcrosssectionareaandpertimealongthedirectionofcurrent電流密度J為沿電流方向上單位時間垂直通過單位面積的電量

Thecurrentdistributionthroughawireisvarious,someuniformdistributingonthesectionoflead,whileotherdistributingunequally,forexampletheskineffectintheconditionofhighfrequency.

Weintroducecurrentdensity

Jtodescribethedistributionofcurrent.通過導線的電流分布是多樣化的。有的均勻分布在導線截面上，有的則不均勻，如高頻時的趨膚效應。為表達電流在電流在導線上的分布，引入電流密度J，導線上的任一截面的電流元ThecurrentthroughthesurfaceS,anarbitrarilyshapedsurfaceareaofmacroscopicsize,isgivenbytheintegral:

A

Currentdensity電流密度26Thecurrentofmovingcharges,supposethechargedensityρ

with

thesame

velocity

v

運動電荷的電流，isJ=ρvAsafewparticleswithcharge：

BEquationofcontinuity――ChargeconservationlawChargeconservationlaw:Thechargecanneitherbecreatednordestroyed;thechargechanginginacertainclosedsurfaceequalstothesumofchargesflowingoutandinthearea.Itcanbedescribedbycontinuitylaw:電荷既不能被創造也不能被破壞，閉曲面內的電荷變化等于流入流出該區域的總電荷。WhichdenotethatthetotalcurrentflowingoutfromtheinterfaceequaltotherateofchargesdecreasingwithinregionV.流出電荷等于區域內的減少。

回總目錄27FromGauss’slaw:Hence:Thisisequationofcontinuity,anditisthedifferentialformofchargeconservationlaw.這就是連續性方程，它是電荷守恒定律的微分形式。回總目錄Movetheminustotheleft,then:ThereforethetotalcurrentflowingintotheinterfaceequaltotherateofchargesincreasingwithinareaV.

（注意閉曲面的方向為外法線方向為正向）

28Discuss:

1whenVisfullspace,thennocurrentoutorin,thetotalchargeremainthesame.全空間無流入流出，so,Forsteadycurrent:

▽·J＝0whileitsdistributionisacloselinewithoutsource.292、

ThelawofBiotandSavart

Thelawdescribesthemagneticeffectsofcurrent

Itdenotethatthemagneticinductionatfieldpointxwasproducedbythesourcecurrent,thepositionvectorrdirectsfromthesourcepoint(currentelement)tofieldpoint.

表示源電流在離源r遠的場點x處的所產生的磁感應強度Allcurrentlieonthelead,thenanelementofcurrent

回總目錄HencetheforceofcurrentelementIdlatthispointis303、CirculationandcurlofB磁場的環量和旋度Forconvenience,themagneticinductionofainfinitecurrentlinearleadis考慮無限長直導線。Fromthesymmetry,themagneticfieldofP:回總目錄ThecirculationofBis（∵B與dl同向）

ThisisAmpere’scircuitallaw.ThecirculationofBintheclosedcurveisμ0I.安培環路定律Pdlr31ForgeneralFromStockes’stheorem:4、

Thedivergenceandmagneticfluxofmagneticfield磁場的磁通量和散度Tostudyacertaindivergence,wefirstlycalculatethemagneticfluxoftheclosedsurface.choosingthegaussiansurfaceiscylinderwhichsymmetricalaxisisthelead

計算閉曲面的磁通量，以長直導線為對稱軸作一高斯面。

B回總目錄32Fromelectromagnetics,weknowthatthemagneticinductionlineproductbycurrentisclose.由電磁學知。電流激發的磁感應線為閉合線。故B為無源場。則B對任意閉曲面的總通量為05、證明磁場的旋度和散度公式∵▽是對場點的微分，與源無關，故后二項為0，同時積分為對源積分。33Itisvectorpotential。Fromthevectoranalysis,thedivergenceofcurliszero回總目錄b.（∵▽和積分分別對場和源）34對于此積分只有上時被積分函數不為0，此時B為柱對稱，代入柱坐標下的散度計算表達式，當r≠0時，回總目錄被積函數35則上式可見，

和

由恒定電流下的畢奧－薩法定律導出，但前者是在任何磁場都成立的。后者僅在恒定下成立。

36例，電流I

在均勻分布于半徑為a的無限長直導線內。求空間各點的磁感應強度，并計算其旋度。

解

由對稱性知，B關于導線軸心軸對稱，以導線軸心為圓心作一垂直導線的圓。當r>a時，圓心總電流為I，由安培環路定律得當r<a時環路內的電流回總目錄37FromAmpere’scircuitallaw,

38§3Maxwell’sEquations前兩節總結了恒定電磁場的基本規律（電磁場的與電荷,電流的關系，積分，微分，散度，旋度）本節研究變化的電磁場規律,建立描述電磁現象的普遍規律Maxwell’sEquations和Lorentzformula.1、電磁感應定律(LawofElectromagneticInduction)Theinductionelectromotiveforce(emf)ofacircuitisproportionaltothedecreasingrateofmagneticfluxthroughthecircuit.閉合導體回路中的感生電動勢與通過以該回路為邊界的任一曲面的磁通量減少率成正比。39ThenegativesigninFaraday’slawindicatethatthedirectionoftheinducedemfissuchastoopposethechangethatproduceit.ItisLenz’slaw:incaseofachangeinamagneticsystem,thatthinghappenswhichtendstoopposethechange.

(負號表示阻止改變發送，即楞次定律)Usingthecirculationofelectricfielddenotetheelectromotiveforce.

Ifthecircuitisarigidstationarycircuit,thetimederivativecanbetakeninsidetheintegral,whereitbecomesapartialtimederivative,then回總目錄402、位移電流(displacementcurrent)

回總目錄Fromtheequationofcontinuity,wehave：ItisthedifferentialformofFaraday’slaw,anditisanindependentexperimentallaw–itcannotbederivedfromotherexperimentallaw.Theinductionfieldisacurlfield.ItrepresentsoneofMaxwell’smajorcontributiontoelectromagnetictheory.Therehasacontradiction.41回總目錄MaxwellbringDisplacementcurrentJDforwardtosolvethiscontradiction.Itis

Fromthechargeconversationlawandthedifferentialformofthegauss’slaw,wehave：

Fromthiswecanseethatdisplacementcurrentisthechangingrateofelectricfield.423、

麥克斯韋方程組Maxwell’sequations

Maxwellgivestheuniversallawoftheelectomagneticfieldbymaxwell’sequations回總目錄Eachoftheseequationsrepresentsageneralizationofcertainexperimentobservation:thefirstisthedifferentialformofFaraday’slawofelectromagneticinduction;thesecondisanextentionofAmpere’slaw;thethirdisgauss’slaw,whichinturnderivesfromthecoulomb’slaw;thelastisthefactthatsinglemagneticpoleshaveneverbeenobserved.它反映了一般情況下電荷電流激發電磁場內部運動規律。變化的電場和磁場也可以相互激發。這種相互激發在空間傳播形成電磁波，同時也體現了場能獨立于電荷之外而存在（如空間的無限電波）可見場亦是一種物質形態。434、洛侖茲力公式Lorentz’sformulaTheelectricfieldproducebychargescanexistandpropagateinitselfform,andcanactontheothercharges.

電場由電荷產生，并能以其自身的形態存在、傳播，它亦能作用于電荷、電流。TheelectromagneticforceofaunitvolumeofcontinuousdistributionchargesisLorentz’formulaforforcedensity.

若電荷為連續分布，其密度為則電荷系統單位體積所受的力密度f為洛侖茲力密度公式

若v為電荷為e的粒子速度Theelectromagneticforcetothesinglechargeoftheelectromagneticfieldis則可得單個帶電粒子受到的電磁力，即洛侖茲力的表達式反映電荷與場的作用關系回總目錄theelectrostaticforce:Themagneticforceofsteadycurrentelement:44前面這些討論都是基本于真空和無界情況下進行，而實際生活中，電磁場幾乎都存在于介質中。為此，完全理解，掌握和運用電磁場理論于實際生活中，必須以此為基礎，結合介質特性研究介質中的電磁理論及邊值關系。45§4介質的電磁性質electromagneticpropertiesofmedium

1、介質及其于電磁場的相互作用(Interactofdielectricmediumandelectromagnetic)ADielectricmediumarecomposedofmolecules,havequantitiesmovingchargedparticle,buttheoveralleffectfromthemacroscopicpointofviewisthatthereisnonetcharge.介質就是在空間一定區域中聚集的大量的運動著的帶電粒子，其宏觀特性就是電中性如半導體材料等物體，某些液體。

Btheelectromagneticfieldcausesaforcetobeexertedoneachchargeparticle,thepositiveandnegativeparticleofeachmoleculearedisplacedfromtheequilibriumpositioninoppositedirection,itispolarize,ormagnetize.Thesedisplacementarelimitedbystrongrestoringforceswhicharesetupbythechangingchargeconfigurationinmolecule.Boundcharge,freecharge,magnetizationcurrent.當其與電磁場作用時，就出現的電荷電流，它們也將產生新的場分布，從而影響原場的分布。此那種電荷電流分別稱為束縛電荷，磁化電流。場作用主要使有極分子趨向有序化。無極分子被極化并有序取向。462、

介質的極化Polarizationofdielectricmedium回總目錄ASortofMedium介質分類Adielectricmediumiscomposedofatomormolecule.Itisclassifiedaspolarornonpolar.所有介質都是由原子，分子構成。

1.

nonpolarmoleculehavethesame‘centersofthegravity’ofthepositiveandnegativechargedistribution.無極分子正負電荷中心重合。2.Apolarmoleculehavedifferent‘centerofthegravity’ofthepositiveandnegative,ithasapermanentdipolemomentelectricdipolewithanelectricdipolemoment.

有極分子正負電荷中心不重合，可看成偶極子，有一定的的電偶極距。47BPolarizingofdielectric介質極化Apolarizeddielectric,eventhoughitiselectricallyneutralontheaverage,producesanelectricfield,bothatexteriorpointandinsidethedielectricaswell.極化的電介質即使總體表現為電中性，但在介質內和外都有電場。Thepolarizationofdielectricdependsonthetotalelectricfieldinthemedium,butapartoftheelectricfieldisproducedbythedielectricitself.Furthermore,thedistantelectricfieldofthedielectricmaymodifythefreechargedistributiononconductingbodies,andthisinturnwillchangetheelectricfieldinthedielectric.介質極化依賴于介質中的總電場，其中部分為介質產生，介質遠處的場分布又改變電場分布，如此相互影響。48Ifthemediumispolarized,thenaseparationofpositiveandnegativechargehavebeeneffected,anditischaracterizedbytheelectricdipolemomentperunitvolume.

介質極化，電場作用于介質，使有極分子運動有序化，無極分子亦變為有極性的并運動亦有序化的現象。

wedefineapolarizationvectorintermofthenumberofdipolemomentsperunitvolume.用矢量p描述介質的宏觀電偶極距的分布，即電極化強度是矢量Isverysmallvolumeinclusionquantitydipolefromthemacroscopicviewpoint

為包含一定量的電偶極子的物理小體積。49C計算束縛電荷密度與電極化強度間的關系Relationofboundchargedensityandtheelectricpolarizationvector.回總目錄Integraltheclosedsurface,wehavethetotalnegativechargequantitiesinsidetheclosedsurfacewhichisequaltothepositivechargeoutoftheclosedsurface.Thepolarizationchargedensity,δP,

ρP對閉曲面積分，則在閉曲面內的負電荷等于曲面外的正電荷。即：由高斯公式Choosingaclosedsurfaceinsidethemediumwithsurfaceelementds,Ifthepositivechargeofthedipoleisoutoftheds,theunitvolumemoleculeisn,thequantitiespositivechargeintheoutsidedsare:取一閉曲面S（在介質內），設偶極子的偶極矩Pi。在任意曲面上取一面元ds，設偶極子的正電荷在ds外，負電荷在ds內，若單位體積分子數為N。則在ds外的正電荷50DBoundcharge介質界面上的面束縛電荷Nowdiscussingtheboundchargeoftheinterfaceoftwomedium.Choosingathinlaminawhichcontaintheinterface.

現討論面界面間的束縛電荷，在界面取一薄層，包含兩界面。則Theenteringpositivechargesfrommedium1tothelaminais

由介質1進入薄層的正電荷為

Theenteringnegativechargesfrommedium2tothelaminais

由介質2進入薄層的負電荷為

Thenetchargeinsidethelaminais薄層內的凈余電荷為：回總目錄51E介質中的電場，電位移矢量theelectricfieldinsideadielectric,thedisplacementvectorThepolarizationofdielectricdependsonthetotalelectricfieldinthemedium,butapartoftheelectricfieldisproducedbythedielectricitself.Furthermore,thedistantelectricfieldofthedielectricmaymodifythefreechargedistributiononconductingbodies,andthisinturnwillchangetheelectricfieldinthedielectric.Thefieldinsidethedielectricisproducedbythefreechargeandthepolarizationcharge.電場使介質極化，極化電荷又產生場與原場疊加而成介質中的場，因此介質中的場是由兩個源產生，即：自由電荷，束縛電荷。

即有:

Nowtherelationofthesourceandfieldis:

即:Infact，wecannotmeasuretheboundcharge,nowsupposeadisplacementvector

為引入電位移矢量,電位移矢量D是輔助場量:52回總目錄

isthepermittivityofthematerial.為介質的介電常數（電容率）

為相對介電常數（相對電容率）TherelationofDandEisdifferentfromothermedium.TheDandEoflinearisotropydielectricobeythelinearlaws

D與E的關系隨介質性質而各異，一般各向同性線性介質。極化強度D和E為線性關系。533、介質磁化mediummagnetization回總目錄Current：aconventioncurrentandatomcurrentbothkindsofcurrentmayproducemagneticmoment,m=ia.A：磁化(magnetization)

theatomcurrentchangetosamedirectionundertheexteriormagneticfield,itbecomeamacroscopiccurrent,magnetizationcurrentIm,themagnetizationcurrentdensityJm.在磁場作用下，介質內的分子電流取向有序化，形成宏觀磁化電流密度JmB：磁化強度M與磁化電流密度的關系relationofmagnetizationMandthemagnetizationcurrentdensity

ThemagnetizationcurrentofasurfaceisThemagnetizationcurrentisthesumoftheallatomcurrent.Thecurrentsinthevariousloopstendtocanceleachotherout,andthereisnotneteffectivecurrentintheinteriorofthematerial,theneteffectivecurrentistheclosedlinelinkingatomcurrent.另一方面，磁化電流由磁化分子電流總和組成，在曲面內，只有閉曲線穿過分子流線圈的分子流對電流有貢獻，其余則由分子的閉合性，而貢獻為0。也就是說只有閉曲線鏈環的分子電流有貢獻。54

C極化電流PolarizationcurrentThechangingelectricfieldcausesthepolarizationvectorchanging,andproducethepolarizationcurrent.

電場變化引起P變化，從而產生極化電流。與之和為介質中的誘導電流abductioncurrent。回總目錄55D介質中的磁場問題，磁場強度themagneticfieldinmedium,themagneticintensityHTheabductioncurrentproducedbymagneticfieldfolduptheoriginmagneticfield.在介質中磁場引起誘導電流，誘導電流又產生磁場為原場疊加，成介質中點的場分布。即：Infact,wecanonlymeasuretheconventioncurrent,hereweintroduceanauxiliaryvector,themagneticintensity,H.

在實際工作中只能控制自由電流，磁化電流則不然。為此引入磁場強度H這一輔助量。TheMaxwell’equationcometo回總目錄56Theauxiliaryvectorsimplifytheequation.

輔助量使介質中的方程簡化。但H并不代表介質內場強。只有B才是一個基本物理量。因此要找到B和H的關系。TheconstitutiverelationofMandHofthenon-ferromagneticisotropyofmaterialislinearrelation對于各向同性非鐵磁性物質。M與H有簡單的線性關系，即：Permeability磁導率相對磁導率回總目錄574、Maxwell’sequationsinmediumItalsoincludingelectromagneticequationsinmedium.Asforisotropicmaterials:

DifferentialcoefficientformofOhm’slawWhereisconductivepermeability.回總目錄58對于各向異性的非線性介質，則其關系為復雜的張量關系:其分量形式為

回總目錄對于非線性情況下，D與E的高次量也有關系，即

此式在非線性電磁中非常重要。鐵磁性物質的B與H也為非線性，與磁化過程有關。用磁化曲線和磁滯回線表示B與H的關系59§5電磁場邊值關系（boundaryvaluerelationofelectromagneticfield)

BoundaryconditiondepicttherelationofE,B,D,H,chargeandcurrent.

邊值關系就是描述界面兩側場量改變與界面上的電荷電流之間的關系式。由于在界面上的場量不連續。不能應用微分形式的麥克斯韋方程組。積分形式麥克斯韋方程可以應用任意不連續分布的電荷，電流激發的場。研究邊值關系從積分形式麥克斯韋方程出發。1、積分形式的麥克斯韋方程

對于旋度表示的方程，進行面積分，并利用斯托克斯公式將左端化為曲線積分；對散度表示的方程在任一區域v上進行積分，利用高斯公式。可得積分形式的麥斯方程組。即：602、Discontinuityofnormalcomponentoffield法向量的躍變回總目錄aNormalcomponentofelectricfield電場的法向分量Letusconstructthesmallpillbox-shapedsurfaceSthatintersectstheinterfaceandenclosesandarea?Softheinterface,theheightofthepillboxbeingnegligiblysmallincomparisonwiththediameterofthebases.ThechargeenclosedbySis

取介質邊界上取一面元為S扁平小柱體。高h為宏觀小量但它包含足夠多的分子層。應用于麥氏第三方程，即

61回總目錄ThediscontinuityinthenormalcomponentofEisgivenbythesurfacedensityofexternalchargeontheinterface.

界面的電場強度法向分量躍變，其躍變與界面上的總電荷密度有關。Dealingwithpolariztionchargesameasit對于極化電荷在界面上類似處理：由公式

Ispolarizationchargesurfacedensity為極化電荷面密度

SumtwoequationandtakeviewoftheD,BNormalcomponentsofmagneticfield磁場的法向分量：ItshowthatthenormalcomponentsofBiscontinuity.說明B的法向分量總是連續的與邊界上的電荷電流無關。

Withthesamemethods,wehave:類似于上述過程處理，可得623、切向分量的躍變thediscontinuityoftangentialcomponent.A磁場的切向分量thetangentialcomponentofmagneticfieldAboundaryconditiononH-fieldmaybeobtainbyapplyingthesecondmaxwell’sequation(extenionAmpere’slaw).Constructarectangleclosedpath,theborderlengthare△landh(sosmallcanbenegligible),respectively.Thecurrentthroughtherectangleisnegligibleunlessthereisatruesurfacecurrent.therefore取一小矩形長為△l。以界面為中心高為h高包含足夠分子層

,且是宏觀小量。應用磁場的第二麥氏方程有：63NowtheequationcometoBytakingthe

crossproductoftheequationwithn,theequationmaybewrittenas

Itshowsthatthetangentialcomponentofthemagneticintensityiscontinuousacrossaninterfaceunlessthereisatruesurfacecurrent.這說明磁場強度切向分量的躍變與界面上自由電流強度有關回總目錄64ForthemagnetizationcurrentUsingthesamemethodwehavePlusthetwoequationwithaneyeto

將兩式相加并考慮到Viz.Itshowsthatthetangentialcomponentofthemagneticinductionintensityisdiscontinuousacrossaninterface,andrelatetothetruesurfacecurrentandmagnetizationcurrent.說明，磁感應強度B切向分量的躍變與界面上的自由電流和磁化電流總和有關65B電場的切向分量Thetangentialcomponentoftheelectricfieldiscontinuousacrosstheinterface.

WiththesamemethoddealingthefirstMaxwell’equation,,andconsideringtheareacometozero.

類似地把第一麥氏方程應用到回路，并考慮到有界，而閉曲線所圍面積趨于零，故其積分為零，可導出：Thetangentialcomponentoftheelectricfieldiscontinuousacrosstheinterface.此說明電場切向方向分量總是連續的。4、邊值關系Boundarycondition：TheboundaryconditionofmediuminterfacecorrespondingtotheMaxwell’sequationsare

綜上所述，與麥氏方程所對應的關于介質邊界的邊值關系如下：66切向tangentialcomponent法向normalcomponent67例

證明在導體界面上電流法向分量滿足邊值關系證，將積分形式的電荷守恒定律應用于扁平小柱體上，實際電流分布在柱體側面上積分為零。導體面薄層電荷分布看成面電荷分布。體分布的電荷由于柱體積趨于零，對于右端積分為零。穩恒時穩恒電流的法向分量總是連續的。68解：因平板電容無窮大，由對稱知，E垂直極板對于極板與介質1，由邊值關系

對于極板與介質2，同理

例無窮大平板電容器內有兩層介質極板上面電荷密度為求電場E和束縛電荷分布。

E2E169介質1與下板：

介質2與上板：對于兩介質界面處，70§6電磁場的能量和能流Theenergyandenergyflowofelectromagneticfielda:theenergypropagateinspacewiththemotionoffield.Suchastheantennaradiatetheenergybyelectromagneticwave.場的能量隨場的運動而在空間傳播，如天線不停地通過電磁波把能量發射出去。Energydensityoffieldwistheenergyoffieldperunitvolume.場的能量密度W，單位體積內的場的能量

w=w(x,t)EnergyflowdensityoffieldSistheenergyflowingthroughthecrosssectionperunittimeandpercrosssectionareaalongthetransmittingdirection.場的能流密度S單位時間垂直流過單位的能量，其方向為傳輸方向，描述能量在場中的傳播Theenergytransferbetweenthefieldandchargessystemduringtheinteractionhappen當場與電荷系統作用時，能量就在場和電荷系統間轉移Theelectromagneticfieldhaveenergy,itisprovedthattheelectromagneticfieldinteractwiththeelectrifiedmaterial.電磁場這種特殊的物質同樣具有能量。其能量能通過與帶電物體作用表現出來。1、Lawofconservationofenergyofthefieldandchargessystem場和電荷系統的能量守恒定律的一般形式71b：場與電荷系統間的能量關系theenergyrelationshipoffieldandchargessystemConstructaclosedregionwithρ，J.fromtheconservationlawofenergyknowthattheenergyflowingintothevpertimeisequaltothesumoftheworkpoweroffieldtochargesandthefieldenergyincreasingrateinsidev.

取一閉區域V，截面為S，設V內電荷電流分布為ρ，J。由能量守恒定律知單位時間流入V內的能量等于場對V內電荷作功功率與V內電磁場能量增加率之和。Theworkpoweroffieldactwithcharges

場對電荷系統的作功功率為

TheinnerenergyofVincreasingrateis

V內場能量增加率

theenergyflowingintothevpertimeis則單位時間流入V內的能量變化率：72Usinggaussiantheorem

由高斯定理，得

（thedivergenceoftheenergyflowequaltothesumofworkpowerandtheenergychanging

rateIftheprobleminwholespace,thereisnoenergyflowinandout,therateofdecreasingenergyisus