第一章麥克斯韋方程english含中文

Preface引言Wecanmastertheelectromagnetismsystemicallybylearningthiscourse.Learninghowtousethiselectromagnetismtheorytosettletheactualproblems,anditcandosomehelptosomestudiesinotherrelateddomain,suchastheoryandtechnologyofthewireless,thetechnologyoflaser,plasmaphysics,astrophysics.通過對它的學(xué)習(xí)可以指導(dǎo)系統(tǒng)地掌握電磁學(xué)理論。學(xué)會(huì)用此電磁學(xué)理論解決實(shí)際問題，同時(shí)有助于對其他有關(guān)領(lǐng)域的學(xué)科的學(xué)習(xí)，如無線電理論與技術(shù)，激光技術(shù)，等離子體理論，天體物理。Intheelectrodynamics,wewillstudythephenomenonaboutelectricityandmagnetisminthenaturefurther,togetherwithsomeexperimentallaws,motionallawsandthefundamentaltheoriesamongtheirmutualrelation,whichmakethetheoriesintegrate.

電動(dòng)力學(xué)進(jìn)一步深入研究自然界中電和磁現(xiàn)象及其許多實(shí)驗(yàn)定律，運(yùn)動(dòng)定律及其相互聯(lián)系的基本理論，使理論系統(tǒng)化。1方法：在學(xué)完每一章節(jié)后及時(shí)看書理解基本概念和規(guī)律、并輔以適當(dāng)?shù)木毩?xí)學(xué)習(xí)電動(dòng)力學(xué)的目的：掌握規(guī)律，理解電磁場的性質(zhì)（時(shí)空性）

獲得分析處理該類問題的能力

參考書：電動(dòng)力學(xué)，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(xué)習(xí)電動(dòng)力學(xué)的目的：掌握規(guī)律，理解電磁場的性質(zhì)（時(shí)空性）

獲得分析處理該類問題的能力Method:Reviewingthebasicconceptsandlawsaftertheclasses,withsomeproperexercises.方法：在學(xué)完每一章節(jié)后及時(shí)看書理解基本概念和規(guī)律、并輔以適當(dāng)?shù)木毩?xí)Referencebooks：Electrodynamics,Fields,Electromagnetismtheory,ElectromagnetismfieldsandtheElectromagnetismwave(inChinese),etc.F.D.JacksonClassicalElectrodynamicsNewYork1975J.A.KonyTheoryofElectromagneticWave，參考書：電動(dòng)力學(xué)，場論，電磁場理論，電磁場與電磁波等3AppendixVectorAnalysisandFields

附錄：矢量代數(shù)和場論1、Vectoralgebra矢量代數(shù)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(biāo)量場的梯度Thatis(4)、Integralform積分表達(dá)式BackDivergencetheorem散度公式Stokes’s

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

ez

arethethreeunitvectorsinrectangularcoordinates,respectively.

exeyez是直角坐標(biāo)系的三個(gè)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(gè)矢性微分算子；因此它在計(jì)算中具有矢性和微分的雙重性質(zhì)。

回總目錄2、Theoperationrulesofoperator

showthatwhen

worksonafunctionorvectorfunction,itcanbeexpressedas：2、算子▽的運(yùn)算規(guī)則表明，作用在一個(gè)數(shù)性函數(shù)或矢性函數(shù)上時(shí)，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(diǎn)：麥克斯韋方程組、邊界條件和羅侖茲力公式)chapterⅠGenerallawsofelectromagneticphenomenaDifficulty:Themediumcharactersandboundaryconditions(介質(zhì)特性和邊界條件)15Byanalyzingtheexperimentallaws,summarizethegenerallawsaboutelectromagnetism,setupMaxwellequationsandLorentz’s

formula.(分析實(shí)驗(yàn)規(guī)律，總結(jié)電磁普遍規(guī)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(zhì)的一種形式，由矢量函數(shù)E、B描述。由微分方程描述規(guī)律?；乜偰夸汿he(macroscopic)electricfieldistheforceperunitchargeonatestchargeembeddedinthedielectric,inthelimitwherethetestchargeissosmallthatitdoesnotitselfaffectthechargedistribution.（單位試驗(yàn)電荷所受的力，試驗(yàn)電荷足夠小，不影響原電場。）16§1ElectricchargeandtheelectricfieldBystudyinganexperimentallaw,

Gauss’law

,wecananalyzeGauss’law,divergenceandcurlinelectrostaticswhicharecalledthecharactersofelectrostaticfield.分析實(shí)驗(yàn)定律、高斯定律，及描述靜電場特征的電場散度和旋度?；乜偰夸?、Coulomb’slaw

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

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

issurfaceelement.回總目錄TocalculatethefluxofthechargeQ,youcanchoosesphericalsurfaceinwhichQlieoriginandradiusisrasgausssurface計(jì)算出該點(diǎn)電荷的電通量，可以以點(diǎn)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(diǎn)通量計(jì)算Page7）19Asforseveralchargesorthecontinuousdistributioncharge：多點(diǎn)分布電荷或連續(xù)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(shí)，上式變?yōu)镮tcanbeattainedbyGauss’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(diǎn)的散度只與其臨近區(qū)域的電荷密度有關(guān)，與遠(yuǎn)處的無關(guān)?；乜偰夸汫auss’slawlieslargelyinprovidingaveryeasywaytocalculateelectricfieldsinsufficientlysymmetricsituations.213、

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

divergence.計(jì)算電荷均勻的球內(nèi)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(shí)間垂直通過單位面積的電量

Thecurrentdistributionthroughawireisvarious,someuniformdistributingonthesectionoflead,whileotherdistributingunequally,forexampletheskineffectintheconditionofhighfrequency.

Weintroducecurrentdensity

Jtodescribethedistributionofcurrent.通過導(dǎo)線的電流分布是多樣化的。有的均勻分布在導(dǎo)線截面上，有的則不均勻，如高頻時(shí)的趨膚效應(yīng)。為表達(dá)電流在電流在導(dǎo)線上的分布，引入電流密度J，導(dǎo)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(yùn)動(dòng)電荷的電流，isJ=ρvAsafewparticleswithcharge：

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

回總目錄27FromGauss’slaw:Hence:Thisisequationofcontinuity,anditisthedifferentialformofchargeconservationlaw.這就是連續(xù)性方程，它是電荷守恒定律的微分形式?；乜偰夸汳ovetheminustotheleft,then:ThereforethetotalcurrentflowingintotheinterfaceequaltotherateofchargesincreasingwithinareaV.

（注意閉曲面的方向?yàn)橥夥ň€方向?yàn)檎颍?/p>

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(yuǎn)的場點(diǎn)x處的所產(chǎn)生的磁感應(yīng)強(qiáng)度Allcurrentlieonthelead,thenanelementofcurrent

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

ThisisAmpere’scircuitallaw.ThecirculationofBintheclosedcurveisμ0I.安培環(huán)路定律Pdlr31ForgeneralFromStockes’stheorem:4、

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

計(jì)算閉曲面的磁通量，以長直導(dǎo)線為對稱軸作一高斯面。

B回總目錄32Fromelectromagnetics,weknowthatthemagneticinductionlineproductbycurrentisclose.由電磁學(xué)知。電流激發(fā)的磁感應(yīng)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(diǎn)的微分，與源無關(guān)，故后二項(xiàng)為0，同時(shí)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(shí)被積分函數(shù)不為0，此時(shí)B為柱對稱，代入柱坐標(biāo)下的散度計(jì)算表達(dá)式，當(dāng)r≠0時(shí)，回總目錄被積函數(shù)35則上式可見，

和

由恒定電流下的畢奧－薩法定律導(dǎo)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(dǎo)線內(nèi)。求空間各點(diǎn)的磁感應(yīng)強(qiáng)度，并計(jì)算其旋度。

解

由對稱性知，B關(guān)于導(dǎo)線軸心軸對稱，以導(dǎo)線軸心為圓心作一垂直導(dǎo)線的圓。當(dāng)r>a時(shí)，圓心總電流為I，由安培環(huán)路定律得當(dāng)r<a時(shí)環(huán)路內(nèi)的電流回總目錄37FromAmpere’scircuitallaw,

38§3Maxwell’sEquations前兩節(jié)總結(jié)了恒定電磁場的基本規(guī)律（電磁場的與電荷,電流的關(guān)系，積分，微分，散度，旋度）本節(jié)研究變化的電磁場規(guī)律,建立描述電磁現(xiàn)象的普遍規(guī)律Maxwell’sEquations和Lorentzformula.1、電磁感應(yīng)定律(LawofElectromagneticInduction)Theinductionelectromotiveforce(emf)ofacircuitisproportionaltothedecreasingrateofmagneticfluxthroughthecircuit.閉合導(dǎo)體回路中的感生電動(dòng)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(fù)號表示阻止改變發(fā)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(fā)電磁場內(nèi)部運(yùn)動(dòng)規(guī)律。變化的電場和磁場也可以相互激發(fā)。這種相互激發(fā)在空間傳播形成電磁波，同時(shí)也體現(xiàn)了場能獨(dú)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(zhì)形態(tài)。434、洛侖茲力公式Lorentz’sformulaTheelectricfieldproducebychargescanexistandpropagateinitselfform,andcanactontheothercharges.

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

若電荷為連續(xù)分布，其密度為則電荷系統(tǒng)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(gè)帶電粒子受到的電磁力，即洛侖茲力的表達(dá)式反映電荷與場的作用關(guān)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(jìn)行，而實(shí)際生活中，電磁場幾乎都存在于介質(zhì)中。為此，完全理解，掌握和運(yùn)用電磁場理論于實(shí)際生活中，必須以此為基礎(chǔ)，結(jié)合介質(zhì)特性研究介質(zhì)中的電磁理論及邊值關(guān)系。45§4介質(zhì)的電磁性質(zhì)electromagneticpropertiesofmedium

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

Btheelectromagneticfieldcausesaforcetobeexertedoneachchargeparticle,thepositiveandnegativeparticleofeachmoleculearedisplacedfromtheequilibriumpositioninoppositedirection,itispolarize,ormagnetize.Thesedisplacementarelimitedbystrongrestoringforceswhicharesetupbythechangingchargeconfigurationinmolecule.Boundcharge,freecharge,magnetizationcurrent.當(dāng)其與電磁場作用時(shí)，就出現(xiàn)的電荷電流，它們也將產(chǎn)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(zhì)的極化Polarizationofdielectricmedium回總目錄ASortofMedium介質(zhì)分類Adielectricmediumiscomposedofatomormolecule.Itisclassifiedaspolarornonpolar.所有介質(zhì)都是由原子，分子構(gòu)成。

1.

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

有極分子正負(fù)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(zhì)極化Apolarizeddielectric,eventhoughitiselectricallyneutralontheaverage,producesanelectricfield,bothatexteriorpointandinsidethedielectricaswell.極化的電介質(zhì)即使總體表現(xiàn)為電中性，但在介質(zhì)內(nèi)和外都有電場。Thepolarizationofdielectricdependsonthetotalelectricfieldinthemedium,butapartoftheelectricfieldisproducedbythedielectricitself.Furthermore,thedistantelectricfieldofthedielectricmaymodifythefreechargedistributiononconductingbodies,andthisinturnwillchangetheelectricfieldinthedielectric.介質(zhì)極化依賴于介質(zhì)中的總電場，其中部分為介質(zhì)產(chǎn)生，介質(zhì)遠(yuǎn)處的場分布又改變電場分布，如此相互影響。48Ifthemediumispolarized,thenaseparationofpositiveandnegativechargehavebeeneffected,anditischaracterizedbytheelectricdipolemomentperunitvolume.

介質(zhì)極化，電場作用于介質(zhì)，使有極分子運(yùn)動(dòng)有序化，無極分子亦變?yōu)橛袠O性的并運(yùn)動(dòng)亦有序化的現(xiàn)象。

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

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

ρP對閉曲面積分，則在閉曲面內(nèi)的負(fù)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(zhì)內(nèi)），設(shè)偶極子的偶極矩Pi。在任意曲面上取一面元ds，設(shè)偶極子的正電荷在ds外，負(fù)電荷在ds內(nèi)，若單位體積分子數(shù)為N。則在ds外的正電荷50DBoundcharge介質(zhì)界面上的面束縛電荷Nowdiscussingtheboundchargeoftheinterfaceoftwomedium.Choosingathinlaminawhichcontaintheinterface.

現(xiàn)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(zhì)1進(jìn)入薄層的正電荷為

Theenteringnegativechargesfrommedium2tothelaminais

由介質(zhì)2進(jìn)入薄層的負(fù)電荷為

Thenetchargeinsidethelaminais薄層內(nèi)的凈余電荷為：回總目錄51E介質(zhì)中的電場，電位移矢量theelectricfieldinsideadielectric,thedisplacementvectorThepolarizationofdielectricdependsonthetotalelectricfieldinthemedium,butapartoftheelectricfieldisproducedbythedielectricitself.Furthermore,thedistantelectricfieldofthedielectricmaymodifythefreechargedistributiononconductingbodies,andthisinturnwillchangetheelectricfieldinthedielectric.Thefieldinsidethedielectricisproducedbythefreechargeandthepolarizationcharge.電場使介質(zhì)極化，極化電荷又產(chǎn)生場與原場疊加而成介質(zhì)中的場，因此介質(zhì)中的場是由兩個(gè)源產(chǎn)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(zhì)的介電常數(shù)（電容率）

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

D與E的關(guān)系隨介質(zhì)性質(zhì)而各異，一般各向同性線性介質(zhì)。極化強(qiáng)度D和E為線性關(guān)系。533、介質(zhì)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(zhì)內(nèi)的分子電流取向有序化，形成宏觀磁化電流密度JmB：磁化強(qiáng)度M與磁化電流密度的關(guān)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(nèi)，只有閉曲線穿過分子流線圈的分子流對電流有貢獻(xiàn)，其余則由分子的閉合性，而貢獻(xiàn)為0。也就是說只有閉曲線鏈環(huán)的分子電流有貢獻(xiàn)。54

C極化電流PolarizationcurrentThechangingelectricfieldcausesthepolarizationvectorchanging,andproducethepolarizationcurrent.

電場變化引起P變化，從而產(chǎn)生極化電流。與之和為介質(zhì)中的誘導(dǎo)電流abductioncurrent?；乜偰夸?5D介質(zhì)中的磁場問題，磁場強(qiáng)度themagneticfieldinmedium,themagneticintensityHTheabductioncurrentproducedbymagneticfieldfolduptheoriginmagneticfield.在介質(zhì)中磁場引起誘導(dǎo)電流，誘導(dǎo)電流又產(chǎn)生磁場為原場疊加，成介質(zhì)中點(diǎn)的場分布。即：Infact,wecanonlymeasuretheconventioncurrent,hereweintroduceanauxiliaryvector,themagneticintensity,H.

在實(shí)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(qiáng)度H這一輔助量。TheMaxwell’equationcometo回總目錄56Theauxiliaryvectorsimplifytheequation.

輔助量使介質(zhì)中的方程簡化。但H并不代表介質(zhì)內(nèi)場強(qiáng)。只有B才是一個(gè)基本物理量。因此要找到B和H的關(guān)系。TheconstitutiverelationofMandHofthenon-ferromagneticisotropyofmaterialislinearrelation對于各向同性非鐵磁性物質(zhì)。M與H有簡單的線性關(guān)系，即：Permeability磁導(dǎo)率相對磁導(dǎo)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(zhì)，則其關(guān)系為復(fù)雜的張量關(guān)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(guān)系，即

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

BoundaryconditiondepicttherelationofE,B,D,H,chargeandcurrent.

邊值關(guān)系就是描述界面兩側(cè)場量改變與界面上的電荷電流之間的關(guān)系式。由于在界面上的場量不連續(xù)。不能應(yīng)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(yīng)用任意不連續(xù)分布的電荷，電流激發(fā)的場。研究邊值關(guān)系從積分形式麥克斯韋方程出發(fā)。1、積分形式的麥克斯韋方程

對于旋度表示的方程，進(jìn)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(qū)域v上進(jìn)行積分，利用高斯公式?？傻梅e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(zhì)邊界上取一面元為S扁平小柱體。高h(yuǎn)為宏觀小量但它包含足夠多的分子層。應(yīng)用于麥?zhǔn)系谌匠?，?/p>

61回總目錄ThediscontinuityinthenormalcomponentofEisgivenbythesurfacedensityofexternalchargeontheinterface.

界面的電場強(qiáng)度法向分量躍變，其躍變與界面上的總電荷密度有關(guān)。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(xù)的與邊界上的電荷電流無關(guān)。

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(yīng)用磁場的第二麥?zhǔn)戏匠逃校?3NowtheequationcometoBytakingthe

crossproductoftheequationwithn,theequationmaybewrittenas

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

將兩式相加并考慮到Viz.Itshowsthatthetangentialcomponentofthemagneticinductionintensityisdiscontinuousacrossaninterface,andrelatetothetruesurfacecurrentandmagnetizationcurrent.說明，磁感應(yīng)強(qiáng)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(guān)65B電場的切向分量Thetangentialcomponentoftheelectricfieldiscontinuousacrosstheinterface.

WiththesamemethoddealingthefirstMaxwell’equation,,andconsideringtheareacometozero.

類似地把第一麥?zhǔn)戏匠虘?yīng)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(dǎo)出：Thetangentialcomponentoftheelectricfieldiscontinuousacrosstheinterface.此說明電場切向方向分量總是連續(xù)的。4、邊值關(guān)系Boundarycondition：TheboundaryconditionofmediuminterfacecorrespondingtotheMaxwell’sequationsare

綜上所述，與麥?zhǔn)戏匠趟鶎?yīng)的關(guān)于介質(zhì)邊界的邊值關(guān)系如下：66切向tangentialcomponent法向normalcomponent67例

證明在導(dǎo)體界面上電流法向分量滿足邊值關(guān)系證，將積分形式的電荷守恒定律應(yīng)用于扁平小柱體上，實(shí)際電流分布在柱體側(cè)面上積分為零。導(dǎo)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(wěn)恒時(shí)穩(wěn)恒電流的法向分量總是連續(xù)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(zhì)1，由邊值關(guān)系

對于極板與介質(zhì)2，同理

例無窮大平板電容器內(nèi)有兩層介質(zhì)極板上面電荷密度為求電場E和束縛電荷分布。

E2E169介質(zhì)1與下板：

介質(zhì)2與上板：對于兩介質(zhì)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(yùn)動(dòng)而在空間傳播，如天線不停地通過電磁波把能量發(fā)射出去。Energydensityoffieldwistheenergyoffieldperunitvolume.場的能量密度W，單位體積內(nèi)的場的能量

w=w(x,t)EnergyflowdensityoffieldSistheenergyflowingthroughthecrosssectionperunittimeandpercrosssectionareaalongthetransmittingdirection.場的能流密度S單位時(shí)間垂直流過單位的能量，其方向?yàn)閭鬏敺较?，描述能量在場中的傳播Theenergytransferbetweenthefieldandchargessystemduringtheinteractionhappen當(dāng)場與電荷系統(tǒng)作用時(shí)，能量就在場和電荷系統(tǒng)間轉(zhuǎn)移Theelectromagneticfieldhaveenergy,itisprovedthattheelectromagneticfieldinteractwiththeelectrifiedmaterial.電磁場這種特殊的物質(zhì)同樣具有能量。其能量能通過與帶電物體作用表現(xiàn)出來。1、Lawofconservationofenergyofthefieldandchargessystem場和電荷系統(tǒng)的能量守恒定律的一般形式71b：場與電荷系統(tǒng)間的能量關(guān)系theenergyrelationshipoffieldandchargessystemConstructaclosedregionwithρ，J.fromtheconservationlawofenergyknowthattheenergyflowingintothevpertimeisequaltothesumoftheworkpoweroffieldtochargesandthefieldenergyincreasingrateinsidev.

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

場對電荷系統(tǒng)的作功功率為

TheinnerenergyofVincreasingrateis

V內(nèi)場能量增加率

theenergyflowingintothevpertimeis則單位時(shí)間流入V內(nèi)的能量變化率：72Usinggaussiantheorem

由高斯定理，得

（thedivergenceoftheenergyflowequaltothesumofworkpowerandtheenergychanging

rateIftheprobleminwholespace,thereisnoenergyflowinandout,therateofdecreasingenergyisus