FieldandWaveElectromagnetics電磁場(chǎng)與電磁波2012.3.611.ProductsofVectors2.OrthogonalCoordinateSystemsReviewCartesianCoordinatesPositionvector:ArbitraryVectorA:2Dotproduct:Crossproduct:Differentiallength:Differentialvolume:Differentialsurface:33.GradientofaScalarField4.DivergenceofaVectorField5.DivergenceTheorem46.CurlofaVectorField7.Stokes’sTheorem8.TwoNullIdentitiesifthenifthen59.Helmholtz’sTheoremHelmholtz’sTheorem:Avectorfield(vectorpointfunction)isdeterminedtowithinanadditiveconstantifbothitsdivergenceanditscurlarespecifiedeverywhere.6MaintopicStaticElectricFields1.FundamentalPostulatesofElectrostaticsinFreeSpace2.Coulomb’sLaw3.Gauss’sLawandApplications4.

ElectricPotential

71.FundamentalPostulatesofElectrostaticsinFreeSpace1.1.ElectricfieldintensityElectircfieldintensityisdefinedastheforce

perunitchargethataverysmallstationarytestchargeexperienceswhenitisplacedinaregionwhereanelectricfieldexists.Thatis,TheelectricfieldintensityEis,thenproportionaltoandinthedirectionoftheforceF.IfFismeasuredinnewtons(N)andchargeqincoulombs(C),thenEisinnewtonspercoulomb(N/C),

whichisthesameasvoltspermeter(V/m).AninverserelationofaboveEq.givestheforceFonastationarychargeqinanelectricfieldE:81.2.FundamentalPostulatesThetwofundamentalpostulatesofelectrostaticsinfreespacespecifythedivergenceandcurlofE.Theyare

isthevolumechargedensityoffreecharges(C/m3),and0

isthepermittivityoffreespace,auniversalconstant.Equationassertsthatstaticelectricfieldsareirrotational(conservative)andimpliesthatastaticelectricfieldisnotsolenoidalunless=0.Differentialform9DivergenceTheoremWhereQisthetotalchargecontainedinvolumeVboundedbysurfaceS.EquationisaformofGauss’slaw,whichstatesthatthetotaloutwardfluxoftheelectricfieldintensityoveranyclosedsurfaceinfreespaceisequaltothetotalchargeenclosedinthesurfacedividedby0.

Stokes’sTheoremwhichassertsthatthescalarlineintegralofthestaticelectricfieldintensityaroundanyclosedpathvanishes.Thescalarproductintegratedoveranypathisthevoltagealongthatpath.ThisEq.isanexpressionofKirchhoff’svoltagelawincircuittheorythatthealgebraicsumofvoltagedropsaroundanyclosedcircuitiszero.101.2.FundamentalPostulatesDifferentialformIntegralformPostulatesofelectrostaticsinfreespace112.Coulomb’sLaw2.1.Electricfieldduetoapointcharge12Example3-1p78場(chǎng)點(diǎn)P

(x,y,z)y源點(diǎn)Q(x’,y’,z’)zxO132.2Coulomb’sLawWhenapointchargeq2isplacedinthefieldofanotherpointcharge

q1attheorigin,aforceF12isexperiencedbyq2duetoelectricfieldintensityE12ofq1atq2.wehave2.3ElectricfieldduetoasystemofdiscretechargesSinceelectricfieldintensityisalinearfunctionof(proportionalto)aRq/R2,theprincipleofsuperpositionapplies,andthetotalEfieldatapointisthevectorsumofthefieldscausedbyalltheindividualcharges.WecanwritetheelectricintensityatafieldpointwhosepositionvectorisRas14Letusconsiderthesimplecaseofanelectricdipolethatconsistsofapairofequalandoppositecharges+qand–q,separatedbyasmalldistance,d,asshowninFig.Electricdipolemoment,p:15電偶極子的電場(chǎng)線和等位線16PV’2.4ElectricfieldduetoacontinuousdistributionofchargesTheelectricfieldcausedbyacontinuousdistributionofchargecanbeobtainedbyintegrating(superposing)thecontributionofan

elementofchargeoverthechargedistribution.dv’R(V/m)17Example3-4p85-87183.Gauss’sLawandApplicationsGauss’slawassertsthatthetotaloutwardfluxoftheelectricfieldintensityoveranyclosedsurfaceinfreespaceisequaltothetotalchargeenclosedinthesurfacedividedby0.Gauss’slawisparticularlyusefulindeterminingtheE-fieldofchargedistributionswithsomesymmetryconditions,suchthatthenormalcomponentoftheelectricfieldintensityisconstantoveranenclosedsurface.TheessenceofapplyingGauss’slawliesfirstintherecognitionofsymmetryconditionsandsecondinthesuitablechoiceofasurfaceoverwhichthenormalcomponentofEresultingfromagivenchargedistributionisaconstant.SuchasurfaceisreferredtoasaGaussiansurface.1920Example3-5p8821xzyr21rO例4求長度為L，線密度為的均勻線分布電荷的電場(chǎng)強(qiáng)度。

令圓柱坐標(biāo)系的z軸與線電荷的長度方位一致，且中點(diǎn)為坐標(biāo)原點(diǎn)。由于結(jié)構(gòu)旋轉(zhuǎn)對(duì)稱，場(chǎng)強(qiáng)與方位角

無關(guān)。因?yàn)殡妶?chǎng)強(qiáng)度的方向無法判斷，不能應(yīng)用高斯定律，必須直接求積。22

因場(chǎng)量與無關(guān)，為了方便起見，可令觀察點(diǎn)P

位于yz平面，即，那么xzyr21rO考慮到23求得當(dāng)長度L

時(shí)，1

0，2，則24Example3-6p8925Example3-7p9026274.ElectricPotentialWewanttomaketwomorepointsaboutEq.First,theinclusionofthenegativesignisnecessaryinordertoconformwiththeconventionthatingoingagainsttheEfieldtheelectricpotential

Vincreases.Second,whenwedefinedthegradientofascalarfield,thatthedirectionofVisnormaltothesurfacesofconstantV.hencethefieldlinesorstreamlinesareeverywhere

perpendiculartoequipotentiallinesandequipotentialsurfaces.28Electricpotentialdoeshavephysicalsignificance,anditisrelatedtotheworkdoneincarryingachargefromonepointtoanother.Aswedefinedtheelectricfieldintensityastheforceactingonaunittestcharge.Therefore,inmovingaunitchargefrompointP1

topointP2

inanelectricfield,workmustbedoneagainstthefield

andisequaltoAnalogoustotheconceptofpotentialenergyinmechanics,AboveequationrepresentsthedifferenceinelectricpotentialenergyofaunitchargebetweenpointP2andpointP1.

DenotingtheelectricpotentialenergyperunitchargebyV.theelectricpotential,wehave29Wehavedefinedapotentialdifference(electrostaticvoltage)betweenpointsP2andP1.Itmakesnomoresensetotalkabouttheabsolutepotentialofapointthanabouttheabsolutephaseofphasorortheabsolutealtitudeofageographicallocation;areferencezero-potentialpoint,areferencezero(usuallyatt=0),orareferencezeroaltitude(usuallyatsealevel)mustfirstbespecified.Inmost(butnotall)casesthezero-potentialpointistakenatinfinity.Whenthereferencezero-potentialpointisnotatinfinity,itshouldbespecificallystated.30ElectricPotentialduetoaChargeDistributionForasystemofndiscretepointchargesq1,q2,…,qnTheelectricpotentialdueto

onepointcharge31ForavolumechargedistributionForasurfacechargedistribution

Foralinechargedistribution

Asanexample,letusagainconsideranelectricdipoleconsistingofcharges+qand–qwithasmallseparationd.Calculatetheelectricfieldintensityproducedbytheelectricdipole.32TheelectricpotentialatPproducedbyanelectricdipolecanbewrittendowndirectly:Solution:33Makeatwo-dimensionalsketchoftheequipotentiallinesandtheelectricfieldlinesforanelectricdipole.34Example3-9P98-9935TheprecedingexampleillustratestheprocedurefordeterminingEbyfirstfindingVwhenGauss’slawcannotbeconvenientlyapplied.However,weemphasizethatifsymmetryconditionsexistsuchthataGaussiansurfacecanbeconstructedoverwhichE·dsisconstant,itisalwayseasiertodetermineEdirectly.ThepotentialV,if