工程电磁场(国外教材)

EngineeringElectromagneticsW.H.HaytJr.andJ.A.BuckChapter5:ConductorsandDielectricsCurrentandCurrentDensityCurrentisafluxquantityandisdefinedas:Currentdensity,J,measuredinAmps/m2,yieldscurrentinAmpswhenitisintegratedoveracross-sectionalarea.TheassumptionwouldbethatthedirectionofJisnormaltothesurface,andsowewouldwrite:CurrentDensityasaVectorFieldnInreality,thedirectionofcurrentflowmaynotbenormaltothesurfaceinquestion,sowetreatcurrentdensityasavector,andwritetheincrementalfluxthroughthesmallsurfaceintheusualway:whereS=ndaThen,thecurrentthroughalargesurfaceisfoundthroughthefluxintegral:RelationofCurrenttoChargeVelocityConsiderachargeQ,occupyingvolumev,movinginthepositivexdirectionatvelocityvxIntermsofthevolumechargedensity,wemaywrite:Supposethatintimet,thechargemovesthroughadistancex=L=vxtThenorThemotionofthechargerepresentsacurrentgivenby:RelationofCurrentDensitytoChargeVelocityWenowhaveThecurrentdensityisthen:Sothatingeneral:ContinuityofCurrentQi(t)SupposethatchargeQiisescapingfromavolumethroughclosedsurfaceS,toformcurrentdensityJ.Thenthetotalcurrentis:wheretheminussignisneededtoproducepositiveoutwardflux,whiletheinteriorchargeisdecreasingwithtime.Wenowapplythedivergencetheorem:sothatorTheintegrandsofthelastexpressionmustbeequal,leadingtotheEquationofContinuityEnergyBandStructureinThreeMaterialTypesa)Conductorsexhibitnoenergygapbetweenvalenceandconductionbandssoelectronsmovefreelyb)Insulatorsshowlargeenergygaps,requiringlargeamountsofenergytoliftelectronsintotheconductionbandWhenthisoccurs,thedielectricbreaksdown.c)Semiconductorshavearelativelysmallenergygap,somodestamountsofenergy(appliedthroughheat,light,oranelectricfield)mayliftelectronsfromvalencetoconductionbands.ElectronFlowinConductorsFreeelectronsmoveundertheinfluenceofanelectricfield.TheappliedforceonanelectronofchargeQ=-ewillbeWhenforced,theelectronacceleratestoanequilibriumvelocity,knownasthedriftvelocity:whereeistheelectronmobility,expressedinunitsofm2/V-s.Thedriftvelocityisusedtofindthecurrentdensitythrough:fromwhichweidentifytheconductivityforthecaseofelectronflow:Theexpression:isOhm’sLawinpointformS/mInasemiconductor,wehaveholecurrentaswell,andResistanceConsiderthecylindricalconductorshownhere,withvoltageVappliedacrosstheends.Currentflowsdownthelength,andisassumedtobeuniformlydistributedoverthecross-section,S.First,wecanwritethevoltageandcurrentinthecylinderintermsoffieldquantities:UsingOhm’sLaw:Wefindtheresistanceofthecylinder:abGeneralExpressionforResistanceabElectrostaticPropertiesofConductors1.Chargecanexistonlyonthesurfaceasasurfacechargedensity,s--notintheinterior.2.Electricfieldcannotexistintheinterior,norcanitpossessatangentialcomponentatthesurface(aswillbeshownnextslide).3.Itfollowsfromcondition2thatthesurfaceofaconductorisanequipotential.s+++++++++++++++++++++EsolidconductorElectricfieldatthesurfacepointsinthenormaldirectionE=0insideConsideraconductor,onwhichexcesschargehasbeenplacedTangentialElectricFieldBoundaryConditionconductordielectricnOvertherectangularintegrationpath,weuseTofind:orThesebecomenegligibleashapproacheszero.ThereforeMoreformally:BoundaryConditionfortheNormalComponentofDndielectricconductorsGauss’Lawisappliedtothecylindricalsurfaceshownbelow:Thisreducesto:ashapproacheszeroThereforeMoreformally:SummaryAtthesurface:TangentialEiszeroNormalDisequaltothesurfacechargedensityMethodofImagesTheTheoremofUniquenessstatesthatifwearegivenaconfigurationofchargesandboundaryconditions,therewillexistonlyonepotentialandelectricfieldsolution.Intheelectricdipole,thesurfacealongtheplaneofsymmetryisanequipotentialwithV=0.Thesameistrueifagroundedconductingplaneislocatedthere.Sotheboundaryconditionsandchargesareidenticalintheupperhalfspacesofbothconfigurations(notinthelowerhalf).Ineffect,thepositivepointchargeimagesacrosstheconductingplane,allowingtheconductortobereplacedbytheimage.Thefieldandpotentialdistributionintheupperhalfspaceisnowfoundmuchmoreeasily!FormsofImageChargesEachchargeinagivenconfigurationwillhaveitsownimageExampleoftheImageMethodInthiscase,wearetofindthesurfacechargedensityontheconductingplaneatthepoint(2,5,0).A30-nClinechargeliesparalleltotheyaxisatz=3.Thefirststepistoreplacetheconductingplanebyalinechargeof-30nCatz=-3.Example(continued)Wenowaddthetwofieldstoget:Referringtothefigure,wefind:ThenExample(concluded)WenowhavetheelectricfieldatpointP:NownDTofindthechargedensity,usen=azThereforewhereElectricDipoleandDipoleMomentQdp=QdaxIndielectric,chargesareheldinposition(bound),andideallytherearenofreechargesthatcanmoveandformacurrent.Atomsandmoleculesmaybepolar(havingseparatedpositiveandnegativecharges),ormaybepolarizedbytheapplicationofanelectricfield.Considersuchapolarizedatomormolecule,whichpossessesadipolemoment,p,definedasthechargemagnitudepresent,Q,timesthepositiveandnegativechargeseparation,d.Dipolemomentisavectorthatpointsfromthenegativetothepositivecharge.ModelofaDielectricAdielectriccanbemodeledasanensembleofboundchargesinfreespace,associatedwiththeatomsandmoleculesthatmakeupthematerial.Someofthesemay