工程电磁场第三章解读课件

3.1ElectricFluxDensity2.ElectricFlux:3.ElectricFluxDensity(coulombs/squaremeter):direction(thedirectionofthefluxlinesatthatpoint)andmagnitude(thenumberoffluxlinescrossingasurfacenormaltothelinesdividedbytheS.area).4.Shownintherightfigure1.Faraday’sexperiment:alargerpositivechargeontheinnersphereinducedacorrespondinglargernegativechargeontheoutersphere.3.1ElectricFluxDensity2.El13.1ElectricFluxDensity5.IftheinnerspherebecomesapointchargeofQ,westillhave6.Comparedwith,wehave7.Forageneralvolumechargedistribution:8.Fordielectrics,therelationshipbetweenandwillbemorecomplicated3.1ElectricFluxDensity5.If23.1ElectricFluxDensity9.Example3.1:findDintheregionaboutauniformlinechargeof8nC/mlyingalongthezaxisinfreespace.10.Exercise:D3.1,D3.23.1ElectricFluxDensity9.Ex33.2Gauss’sLaw1.Gauss’slaw:Theelectricfluxpassingthroughanyclosedsurfaceisequaltothetotalchargeenclosedbythatsurface.---thegeneralizationofFaraday’sexperiment2.Acloudofpointcharges(totalchargeQ)areshowninthefollowingFigure.ThereissomevalueDSateverypointonthesurface.

Howtodescribeanincrementalelementofarea?

ThefluxcrossingisThetotalfluxpassingthroughtheclosedsurfaceis3.Considerthenatureofanincrementalofthesurface:3.2Gauss’sLaw1.Gauss’slaw:43.2Gauss’sLaw4.Toagaussiansurface,themathematicalformulationofGauss’slaw5.Thelastformisusuallyused6.Forexample:placingapointchargeQattheoriginofasphericalcoordinatesystemandchooseasphereofradiusaasthegaussianS.7.Exercise:D3.3(P61)3.2Gauss’sLaw4.Toagaussian53.3ApplicationofGauss’sLaw:somesymmetricalcharge...1.DeterminingifthechargedistributionisknownChooseaclosedsurfaceinwhichiseverywhereeithernormalortangentialtotheclosedsurfaceOnthatportionoftheclosedsurfaceforwhichisnotzero,=constant2.Example:apointchargeQattheoriginofasphericalcoordinatesystem,andtheresultsagreewiththoseofChap.2.Chooseaclosedsurfacecenteredattheorigin:asphereofradiusr

3.3ApplicationofGauss’sLaw63.3ApplicationofGauss’sLaw3.Asecondexample:theuniformlinechargedistributionlyingalongthezaxisandextendingfrom-∞to+∞OnlytheradialcomponentofDispresentSowecanchooseacylindricalsurfacetowhichiseverywherenormal4.Ifsymmetrydoesnotexist,wecannotuseGauss’sLawtoobtainasolution.Theradialcomponentisafunctionofρonly3.3ApplicationofGauss’sLaw73.3ApplicationofGauss’sLaw5.Acoaxialcable,theinnerofradiusaandtheouterradiusb,achargedistributionofontheoutersurfaceoftheinnerconductorChoosearightcircularcylindricaloflengthLandradiusand,sowehave3.3ApplicationofGauss’sLaw83.3ApplicationofGauss’sLaw6.Example3.2:Selecta50-cmlengthofcoaxialcablehavinganinnerradiusof1mm,andanouterradiusof4mm.Thespacebetweenconductorsisassumedtobefilledwithair.Thetotalchargeontheinnerconductoris30nC.Wewishtoknowthechargedensityoneachconductor,andEandDvectorfields.7.Exercise:D3.5(P66)3.3ApplicationofGauss’sLaw93.4ApplicationofGauss’sLaw:DifferentialVolumeElement1.ApplyGauss’sLawtotheproblemwithoutanysymmetry.Chooseaverysmallclosedsurface:DisalmostconstantandthesmallchangeinDcanberepresentedbythefirsttwotermsofTaylor’s-seriesexpansionforD.2.TheaimistoreceivesomeinformationaboutthewayDvariesintheregionofoursmallsurface.3.Consideranypoint,showninthefollowingFigure.Choosethesmallrectangularboxastheclosedsurface3.4ApplicationofGauss’sLaw103.4ApplicationofGauss’sLaw:DifferentialVolumeElement3.Consideranypoint,showninthefollowingFigure.Choosethesmallrectangularboxastheclosedsurface3.4ApplicationofGauss’sLaw113.4ApplicationofGauss’sLaw:DifferentialVolumeElementBythesameprocess,wecanobtain,Theseresultsmaybecollectedtoyield4.ApplyGauss’slawtotheclosedsurfacesurroundingthevolumeelementandhaveanapproximateresult3.4ApplicationofGauss’sLaw123.4ApplicationofGauss’sLaw:DifferentialVolumeElement5.Example3.3:findanapproximatevalueforthetotalchargeenclosedinanincrementalvolumeof10-9m3locatedattheorigin,if6.Exercise:D3.6(P70)3.4ApplicationofGauss’sLaw133.5Divergence1.ThisequationcanbewrittenasOrasalimit:2.Thelasttermisthevolumechargedensity,hence3.ThemethodscouldhavebeenusedonanyvectorA3.5Divergence1.Thisequation143.5Divergence4.ThedivergenceofthevectorfluxdensityAis:theoutflowoffluxfromasmallclosedsurfaceperunitvolumeasthevolumeshrinkstozero.5.Apositivedivergenceindicatesasource;anegativedivergenceindicatesasink.6.Incartesiancoordinatesystem:7.Note:Divergenceisperformedonavector;buttheresultisascalar.3.5Divergence4.Thedivergenc153.5Divergence8.Example3.4:FinddivDattheoriginif9.Exercise:D3.7(P73)3.5Divergence8.Example3.4:163.6Maxwell’sFirstEquation(Electrostatics)1.Thefirstisthedefinitionofdivergence;Thesecondistheresultofapplyingthedefinitiontoadifferentialvolumeelement.2.ThisisthefirstofMaxwell’sequationsastheyapplytoelectrostaticsandsteadymagneticfields.Itstatesthattheelectricfluxperunitvolumeleavingavanishinglysmallvolumeunitisexactlyequaltothevolumechargedensitythere.3.Maxwell’sfirstequation

isdescribedasthedifferential-equationformofGauss’slawandGauss’slawisrecognizedastheintegralformofMaxwell’sfirstequation.3.6Maxwell’sFirstEquation(173.6Maxwell’sFirstEquation(Electrostatics)4.ConsiderthedivergenceofDintheregionaboutapointchargeQlocatedattheorigin.5.Exercise:D3.8(P74)3.6Maxwell’sFirstEquation(183.7Thevectoroperator▽

andthedivergencetheorem1.Definethedeloperator▽asavectoroperator,2.Theuse