第三课电磁场与电磁波英文版课件

3BoundaryValue

Problems3BoundaryValue

Problems1(1)Distributionproblems(2)Boundaryvalueproblems1.Thetypesofproblemsinelectrostaticfield2.Thecalculationmethodsofboundaryvalueproblems(1)Analyticalmethod(2)NumericalmethodBoundaryconditionsEquationsordq(1)Distributionproblems(2)B3.1TypesofBoundaryConditionsandUniquenessTheorem3.1TypesofBoundaryConditio

ConsideraregionVboundedbyasurfaceS.1.Dirichletboundaryconditionsspecifythepotentialfunctionontheboundary.wherefisacontinuousfunction.2.Neumannboundaryconditionsspecifythenormalderivativeofthepotentialfunctionontheboundary.wheregisacontinuousfunction.3.MixedboundaryconditionsThetypesofboundaryconditionsConsideraregionVbou

TheUniquenessTheoremTheuniquenesstheoremstatesthatthereisonlyone(unique)solutiontoPoisson’sorLaplace’sequationsatisfiedgivenboundaryconditions.

Poisson’sequationLaplace’sequationBoundaryconditionsTheUniquenessTheoremPois

Prove(proofbycontradiction)ConsideravolumeVboundedbysomesurfaceS.SupposethatwearegiventhechargedensitythroughoutVandthevalueofthescalarpotentialonsurfaceS.AssumethatthereexisttwosolutionsandofLaplace’sequationsubjecttothesameboundaryconditions.Then,Prove(proofbycontradict

and

LetthenApplyingGreen’sfirstidentity

Wehave

wherevolumeVboundedbytheenclosedsurfaceS.TheintegralisequaltozerosinceonthesurfaceS.Thus,TheintegralcanbezeroisifisaconstantWeknowthatonthesurfaceS,soweget第三课电磁场与电磁波英文版课件

Thatis,

ThroughoutVandonsurfaceS.OurinitialassumptionthatandaretwodifferentsolutionsofLaplace’sequations,satisfyingthesameboundaryconditions,turnsouttobeincorrect.Hence,thereisauniquesolutiontoLaplace’sequationsatisfiedgivenboundaryconditions.

Thatis,3.2DirectIntegrationNote:Thepotentialfieldisafunctionofonlyonevariable.3.2DirectIntegrationNote:TSolution

Sincethetwoconductorsofradiiaandbformequipotentialsurfaces,thepotentialmustbeafunctionofonly.Example3.2.1TheinnerconductorofradiusaofacoaxialcableisheldatapotentialofUwhiletheouterconductorofradiusbisgrounded.Determine(a)thepotentialdistributionbetweentheconductors,(b)thesurfacechargedensityontheinnerconductor,and(c)thecapacitanceperunitlength.SolutionExample3.2.1

Thus,Laplace’sequationreducesto

Integratingtwice,weobtain(1)whereandareconstantsofintegrationtobedeterminedfromtheboundaryconditions.

Substitutingandinto(1),wehave(2)

Substitutingandinto(1),wehave(3)Thus,Laplace’sequationrHence,thepotentialdistributionwithintheconductorsisTheelectricfieldintensityisThus,ThenormalcomponentofDatisequaltothesurfacechargedensityontheinnerconductor.Thus,Hence,thepotentialdistribut

ThechargeperunitlengthontheinnerconductorisFinally,weobtainthecapacitanceperunitlengthas

1Finally,weobtainthecapaproblemTheinnerconductorofradiusaofacoaxialcableisheldatapotentialofUwhiletheouterconductorofradiusbisgrounded.Thespacebetweentheconductorsisfilledwithtwoconcentriclayersofdielectric.Determine(a)thepotentialdistributionbetweentheconductorsand(b)blemExample3.2.2Asphericalcapacitorisformedbytwoconcentricsphericalshellsofradiiaandb,asshowninFigure3.2.2.TheregionbetweentheinnerandoutersphericalconductorsisfilledwithadielectricofpermittivityIfUisthepotentialdifferencebetweenthetwoconductors,determine(a)thepotentialdistributionbetweentheconductors,and(b)thecapacitanceofthesphericalcapacitor.Example3.2.2Solution

Sincethetwoconductorsofradiiaandbformequipotentialsurfaces,thepotentialmustbeafunctionofonly.Thus,Laplace’sequationreducesto(a<r<b)Integratingtwice,weobtainwhereC1andC2areconstantsofintegrationtobedeterminedfromtheboundaryconditions.(1)Solution(a<r<b)IntegratingSubstitutingboundaryconditionsandinto(1),weobtain

ThenormalcomponentofDatyieldsthesurfacechargedensityontheinnerconductor.Thus,Hence,thecapacitanceofthesystemisSubstitutingboundarycondi通解例体电荷均匀分布在一球形区域内，求电位及电场。解：采用球坐标系,分区域建立方程边界条件参考电位图1.4.2体电荷分布的球体0通解例体电荷均匀分布在一球形区域内，求电位及电场。解：电场强度（球坐标梯度公式）：得到图随r变化曲线0电场强度（球坐标梯度公式）：得到图随r变化曲线03.3SeparationofVariablesSolveLaplace’sequationTransformthethree-dimensionalpartialdifferentialequationintothreeone-dimensionalordinarydifferentialequations.3.3SeparationofVariablesSo

1.SeparationofVariablesforRectangularCoordinateSystemLaplace’sequationis

(1)Assumethatthepotentialcanbewrittenastheproductofone-dimensionalpotentials.(2)

whereisafunctionofxonly,isafunctionofyonlyandisafunctionofzonly.Substituting(2)into(1),andthendividingby

weobtain1.SeparationofVariables

(3)Sinceeachterminvolvesasinglevariable,theequationisrightifandonlyif

eachtermmustbeaconstant.Thus,welet

where,andarecalledtheseparationconstants.Note:Theformofgeneralsolutiontoeachequationdependsontheseparationconstant.where,andarTheformofsolutionofasfollows.1.Ifisarealnumber,thegeneralsolutionis

2.Ifisaimaginarynumberthegeneralsolutionis

3.Ifkxiszero,then

orwhereA1,A2,B1,B2,C1,C2,D1andD2arearbitraryconstants.hyperbolicsinehyperboliccosineTheformofsolutionof

Example3.3.1Aninfinitelylongrectangulartroughisformedbyfourconductingplanes,locateatandaandandbinair,asshowninFigure3.3.1.ThesurfaceatisatpotentialofU,theotherthreeareatzeropotential.Determinethepotentialdistributioninsidetherectangulartrough.

Two-dimensionalfieldExample3.3.1Two-dimension

Solution

Sincethetroughisinfinitelylonginthez-direction,thepotentialcanonlydependonxandy.Laplace’sequationreducesto(1)Let(2)Substituting(2)in(1)anddividingby(2),weget

LetThen,wehaveorSolutionLetThen,wehaveo

Solution

Sincethetroughisinfinitelylonginthez-direction,thepotentialcanonlydependonxandy.Laplace’sequationreducesto

Usingseparationofvariables,welet

Then,wehaveorSincetheboundaryconditionsareatandwecanassumeandTheformofgeneralsolutionofthepotentialiswhereA1,A2,B1andB2arearbitraryconstants.SolutionwhereA1,A2,B1TheconditionforallrequiresthenTheconditionforallrequiresthenTheconditionforallrequires

Thus,weassumethatthegeneralsolutionisTheconditionforallrequiresTheconditionLet

ThisisaFouriersineseries.Thecoefficientsaredeterminedby第三课电磁场与电磁波英文版课件WecangetThus,thegeneralsolutionofthepotentialis第三课电磁场与电磁波英文版课件第三课电磁场与电磁波英文版课件

Example3.3.2Twosemi-infiniteconductingplatesseparateddaregrounded,asshowninFigure.AconductingplatebetweentwoplatesisheldatapotentialU0.Findthepotentialbetweenthetwoconductingplates.SolutionThepotentialcanonlydependonxandy.Laplace’sequationreducesto(1)Example3.3.2Solution(1)

Let(2)Substituting(2)in(1)anddividingby(2),weget

LetTheformofgeneralsolutionofthepotentialisLet

Usingseparationofvariables,wecanwritetheformofthesolutionas

whereandseparationconstantkaredeterminedbytheboundaryconditions.SolutionThepotentialcanonlydependonxandy.Laplace’sequationreducesto(1)Solution(1)TheconditionforallrequiresTheconditionforallrequires

SincethepotentialisequaltozeroatandweobtainTheformofthegeneralsolutionreducesto

TheconditionforallrequiresThecondition

ThisisaFouriersineseries.Thecoefficientsaredeterminedby

Thus,thegeneralsolutionofthepotentialis

第三课电磁场与电磁波英文版课件

2.SeparationofVariablesforCylindricalCoordinateSystem(two-dimensionalfield)AssumethatthepotentialisthefunctionwithrespecttoandTheLaplace’sequationis

(2)(1)Let(4)Substituting(2)into(1),weobtainandthenmultiplyingbyweobtain(3)2.SeparationofVariablesInmanyproblems,thepotentialisafunctionwithrespecttowithaperiod.Thatis(7)Thus,kmustbeaninteger.Taking(6)becomes(8)Eachtermmustbeaconstant.Thus,welet(5a)

(5b)Solving(5b),wehave(6)Inmanyproblems,thepotentiaSubstitutingnfork,theequation(5a)

becomes

(9)

ThisisanEulerequationandthesolutionis(10)Thus,thegeneralsolutioncanbewrittenasn2Substitutingnfork,the

Example3.3.3Averylongdielectriccylinderofradiusaisalongzaxisinauniformelectricfieldwhichisinthedirectionofxaxis.Determinethepotentialandtheelectricfieldintensity.

Example3.3.3SolutionTheformofthesolutionoftheLaplace’sequationisTheboundaryconditionsareasfollows.(1)when(2)whenisfinite.(3)when(4)when(1)(2)Solution(1)(2)Fromboundarycondition(1),wehaveThus,

LetthenBoundarycondition(2)requiresThus,Usingcondition(3),weobtain(3)Fromboundarycondition(1),wThus,LetwehaveThen,thepotentialisFromcondition(4),wehave(4)(6)(5)(4)(6)(5)Solving(5)and(6),weobtainThus,thepotentialsareUsingSolving(5)and(6),weobtainWeobtainE2E0exWeobtainE2E0ex图均匀外电场中介质圆柱内外的电场图均匀外电场中介质圆柱内外的电场

3.SeparationofVariablesforSphericalCoordinateSystem(two-dimensionalfield)AssumethatthepotentialisthefunctionwithrespectivetorandTheLaplace’sequationreducesto(1)Let(2)Substituting(2)into(1),weobtain(3)r2r2f(r)3.SeparationofVariablesMultiplyingbyweobtain(4)

Let(5a)(5b)

Let(5b)becomes(6)

ThisisthegeneralizedLegendreequation.（勒让德方程）MultiplyingbyTake(7)

ThesolutionsareLegendrepolynomials

(8)TakeTheformertermsoftheare(9a)(9b)(9c)

(9d)(9e)

(9f)Theformertermsofthe第三课电磁场与电磁波英文版课件

Consider(5a).(10)

Thesolutionis(11)Thus,thepotentialis(12)

Constantsandaredeterminedbytheboundaryconditions.Consider(5a).

Example3.3.4Adielectricsphereofradiusaisinauniformelectricfieldwhichisinthedirectionofzaxis,asinshowninFigure3.3.5.Determinethepotentialandtheelectricfieldintensity.Example3.3.4

SolutionTheformofthesolutionoftheLaplace’sequationis

and

Useboundaryconditionstodetermineconstants.

（1）when

TheexpressionsofthefieldshaveonlythefirsttermThatis,（1）（2）（3）C1=-E0Solution（1）（2）（3）C1=-E(2)whenisfinitevalue.So(3)when(5)(4)whenweobtain

(7)Solving(5)and(7),weobtainn=1（4）（6）andThepotentialis(2)when

Thus,thepotentialsare

UsingweobtainThus,thepotentialsare图均匀场中放进了介质球的电场E0ez图均匀场中放进了介质球的电场E0ezAconductingsphereofradiusaisinauniformelectricfieldwhichisinthedirectionofzaxis,asinshowninFigure.Determinethepotentialandtheelectricfieldintensity.Aconductingsphereofradius3.4MethodofImages3.4MethodofImages

Example3.4.1Supposethatapointchargeqislocatedabovethesurfaceofaninfiniteconductingplaneandgrounded,asshowninFigure.Determine(a)theelectricpotentialandelectricfieldintensityabovetheplane,and(b)thetotalchargeinducedonthesurfaceoftheconductingplane.yzExample3.4.1yzAnelectricdipole：(1)Thepotentialatanypointonthebisectingplaneiszero,(2)Theelectricfieldintensityisnormaltotheplane.yzTheimaginarycharge–qissaidtobe

theimageoftherealcharge+qAnelectricdipole：yzTheimag

Solution(methodofimages)Theboundaryconditionis.Placeanimaginarycharge

–qat(0,0,-d)andtemporarilyignoretheexistenceoftheplane.ThepotentialatanypointPabovetheplaneis(zeropotentialatinfinity)TheelectricfieldintensityisSolution(methodofimages

whereand

ThenormalcomponentoftheDfieldmustbeequaltothesurfacechargedensityonthesurfaceoftheconductor.+d-dThus,thetotalchargeinducedonthesurfaceis+d-dThus,thetotalcharge

Basicprinciple(1)Imagechargesreplacethetotalchargeinduced.(2)Maintainthefieldequationandoriginalboundaryconditions.(3)Usetherealandimagechargestodeterminethefield.

Note：(1)Imagechargesarelocatedoutsidethefieldregion.(remainfieldequation)(2)Determinethenumber,magnitudeandpositionoftheimagecharges.BasicprincipleThenumberofimagechargesisThenumberofimagechargesis

Example3.4.2Apointchargeqislocatedatadistancedfromthecenterofagroundedconductingsphereofradiusa,asshowninFigure3.4.2.Computethesurfacechargedensityonthesurfaceoftheconductingsphere.Example3.4.2

Solution(methodofimages)TheboundaryconditionisPlaceanimaginarychargeat(0,0,d’)withinthesphere.Nowwetemporarilyignoretheexistenceoftheconductingsphere.ThepotentialoutsidethesphereisSolution(methodofimages

Fromtheboundaryconditions,weobtainSquaringbothsidesThisisanidentitywithrespectto.Thus,

Solvingtheseequations,weget

and第三课电磁场与电磁波英文版课件Chooseand.Thus,thepotentialoutsidethesphereis

SincethenormalcomponentoftheDfieldmustbeequaltothesurfacechargedensityonthesurfaceoftheconductor,wehaveChooseandThus,thetotalchargeinducedonthesurfaceisThus,thetotalchargeinduced图球外的电场分布图球外的电场分布（1）Apointchargeqisplacedatadistancedfromthecenterofaconductingsphereofradiusa.图不接地金属球的镜像（2）AconductingsphereofradiusahasachargeofQ.Apointchargeqisplacedatadistancedfromthecenterofthesphere.qdaConsider图点电荷位于不接地导体球附近的场图（1）Apointchargeqisplaced

Example3.4.5

Theplaneistheinterfacebetweentwodifferentdielectrics,asshowninFigure(a).Apointchargeqislocatedabovetheinterface.Determinethepotentialintworegions.

(a)(b)(c)Example3.4.5(b)(c)

SolutionTheboundaryconditionsareandattheplaneRegion1:Placeanimaginarychargeat(0,0,-d).Wetemporarilyassumethepermittivityofmedium2isalsoThepotentialatanypointPinregion1isSolutionRegion2:Wesuperposeanimaginarychargeonthechargeq.Wetemporarilyassumethepermittivityofmedium1isalsoThepotentialatanypointPinregion2is

Region2:WesuperposeanAnypointontheinterfacesatisfiesFromboundaryconditions,wehaveSolvingtheseequations,weobtain

AnypointontheinterfacesatThus,thepotentialatanypointinregion1isThepotentialatanypointinRegion2is

Thus,thepotentialatanypoi第三课电磁场与电磁波英文版课件

Example3.4.2Alongstraightlinechargeisparallelalongstraightgroundedwireofradiusa,asshowninFigure.Calculatethepotentialinspace.Example3.4.2第三课电磁场与电磁波英文版课件

Solution(methodofimages)Placeanimaginarylinechargeonthey=0planewithinthewire.isparallelto.Temporarilyremovethewire.ThepotentialatanypointPis

Sincethelongstraightwireofradiusaisgrounded,thepotentialsatpointsAandB

areequaltozero.Solution(methodofimages

Thus,wehaveThus,wehaveLetWeobtainandThus,thepotentialatanypointPis

Trialsolution试探解LetTrialsolution试探解

Thesurfacechargedensityonthesurfaceofthewireis

ThechargeinducedperunitlengthonthesurfaceisMethodofelectricaxisThesurfacechargedensity

Example3.4.4Twoparallelcylindricalconductorsofradiusaseparatedadistance2dformtransmissionline,asshowninFigure.Calculatethecapacitanceperunitlengthoftwocylindricalconductors.yxExample3.4.4yx

Solution

Usingmethodofelectricaxis,thepotentialatanypointoutsideoftheconductorsis

ThepotentialatApointis

ThepotentialatBpointis

Thepotentialdifferenceisyx(C=0zeroreferenceat

yaxis)

Solutionyx(C=0zerorefer

Thecapacitanceperunitlengthoftwocylindricalconductorsis

Whenweobtainyxyx第三课电磁场与电磁波英文版课件Twolongstraightlineswithequalandoppositebutuniformchargedistributionsareseparatedbyadistanceof2b.Findthepotentialinfreespace.(C=0,zeroreferenceat

yaxis)

LetTwolongstraightlineswitheAsetofeccentriccirclesThus,weobtainTheequationofequipotentiallinesiscenterRadiusWhentheradiusisacircleAsetofeccentriccirclesThus根据，得到Ex和Ey分量LinesofforcesofE根据，得到Ex和E例不同半径两平行长直导线（传输线）相距为d,确定电轴位置，单位长度的电容。图不同半径传输线的电轴位置解：例不同半径两平行长直导线（传输线）相距为d,确定电轴位ABAB

镜像法（电轴法）的理论基础是：镜像法（电轴法）的实质是：镜像法（电轴法）的关键是：镜像电荷（电轴）只能放在待求场域以外的区域。叠加时，要注意场的适用区域。用虚设的镜像电荷（电轴）替代未知电荷的分布，使计算场域为无限大均匀媒质；静电场惟一性定理；确定镜像电荷（电轴）的个数、大小及位置；应用镜像法（电轴法）解题时，注意：镜像法（电轴法）的理论基础是：镜像法（电轴法）的实质是3.5TheFinite-DifferenceMethod(FDM)3.5TheFinite-DifferenceMet

Numericalmethods:

finite-differencemethod(FDM),finiteelementmethod(FEM),methodofmoments(MOM).

Thefinite-differencemethod(FDM)

Numericalmethods:基本思想：将场域离散为许多网格，应用差分原理，将求解连续函数的微分方程问题转换为求解网格节点上的代数方程组的问题。Dividethesolutiondomainintofinitediscretepointsandreplacethepartialdifferentialequationwithasetofdifferenceequations.基本思想：将场域离散为许多网格，应用差分原理，将求Ifthepotentialvariationisindependentofz,Poisson’sequationsimplifiesto

(1)First,wedividetheregionintoafinitenumberofmeshes.Letusconsiderasquaremeshsurroundingapoint0.Ifthepotentialvariation

Letthepotentialatpoint0beequaltoandatthefoursurroundingpointsandThepotentialatpoint1,usingtheTaylorseriesexpansion,is

wherehisthemeshsize.andthepotentialatpoint3isLetthepotentialatpointThus,wehaveOmittinghighorderterms,weobtain

(2)Similarly,weget(3)Adding(2)and(3),weobtain(4)

Thus,wehaveSubstituting(1)into(4),wehave

Thus,thepotentialatpoint0isIncharge-freeregion,thepotentialatpoint0is

Thepotentialatapointmustbetheaverageofthepotentialatthefoursurroundingpoint.

Substituting(1)into(4),weExample

Arectangulartroughisformedbyfourconductingplanes.Therectangulartroughisinfinitelylonginthez-direction.Thesidesandbottomofthetroughareatzeropotential.Thelidisatapotentialof100V.Wedividetheregioninto16squares.Example第三课电磁场与电磁波英文版课件

Initerativemethod,thepotentialforthe(n+1)thiterationatnode(k

j,)is

Initerativemethod,thep

Successiveover-relaxation(SOR)iterativemethod

whereiscalledaccelerationfactor.Gauss-Seideliterativemethod高斯—赛德尔迭代法逐次超松弛迭代法迭代过程直到节点电位满足为止。收敛速度与电位初始值及网格剖分粗细有关；迭代次数与工程精度有关。Successiveover-relaxation边界节点赋已知电位值赋节点电位初始值累计迭代次数n=0n=n+1按超松弛法进行一次迭代，求打印NoYes程序框图下页上页返回边界节点赋已知电位值赋节点电位初始值累计迭代次数n=0上机作业要求：1.试用超松弛迭代法求解接地金属槽内电位的分布。给定边值：如图示；已知：计算：迭代次数N=?,分布。给定初值：误差范围：下页上页返回图接地金属槽的网格剖分上机作业要求：1.试用超松弛迭代法求解接地金属槽内电位的有限差分法有限元法边界元法矩量法积分方程法积分法分离变量法镜像法、电轴法微分方程法保角变换法计算法实验法解析法数值法实测法模拟法边值问题有限差分法有限元法边界元法矩量法积分方程法积分法分离变量法镜3BoundaryValue

Problems3BoundaryValue

Problems109(1)Distribu