`
`BOOKS IN PHYSICS
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`Editors: HENRY M. FOLEY AND
`MALVIN A. RUDERMAN
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`Concepts of Classical Optics
`John Strong
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`Thermophysics
`Allen L. King
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`X-Ray Diffraction in Crystals, Imperfect Crystals,
`and Amorphous Bodies
`A. Guinier
`
`Modern Quantum Theory
`Behram Kursunoglu
`
`Introduction to Electromagnetic Fields and Waves
`Dale R. Corson and Paul Lorrain
`
`INTRODUCTION TO
`
`ELECTROMAGNETIC
`FIELDS AND WAVES
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`
`
`
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`Dale R, Corson CORNELL UNIVERSITY
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`Paul Lorrain UNIVERSITY OF MONTREAL
`
`Onthe Interaction Between Atomic Nuclei
`and Electrons (A Golden Gate Edition)
`H. B. G. Casimir
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`SeOLON“X43d1SJONAIOSdONDYSAVE|e6eYg
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`_W.H. FREEMAN AND COMPANY.
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`Ss A N
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`F RAN OC
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`BAYER CROPSCIENCE LP EX. NO 1035
`Page 1
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`canmodifythechargedistributionwithinQ,,andinversely,givingtoacom-
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`betweenthecharges”hasnodefinitemeaning.Moreover,thepresenceofQ,
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`2.2.TheElectrostaticFieldIntensity
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`WethushaveCoulomb’slawforpointcharges:
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`lawasbeinganinteractionbetweenQ,andthefieldofQ,,andviceversa.We
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`WethinkoftheinteractionbetweenthepointchargesQ,andQ,inCoulomb's”
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`thechargescomplicatedinthatareextendedthesituationismorethe“distance
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`thetwocharges,(b)isproportionaltotheproductQ.Q,,and(c)isinversely
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`proportionaltothesquareofthedistancerseparatingthecharges.
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`themedium.WeshalldiscussconductorslateroninSection2-6,-Q.andQy,irrespectiveoftheotherforcesarisingdisplacementof.chargeswithinfromthe
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`chargesQ,andQ,ofthetypediscussedabove(a)actsalongthelinejoining
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`appliesindielectricsandconductorsifFayistakentobethedirectforcebetween
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`Thisistrueifthedimensionsofthechargesarenegligiblecomparedtor.If
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`andthendielectricsinChapter3.oe
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`Ithasbeenfoundexperimentallythattheforcebetweentwoelectrostatic
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`Equation2-2appliestoapairofpointchargessituatedinavacuum,Italso
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`definetheelectricfieldintensityEtobetheforceperunitchargeexertedona
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`leavinganelectrondeficiency(andthereforeanetpositivecharge)onone,and
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`silk,areprocessesinwhichelectronsaretransferredfromonebodytoanother,
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`Processeswhichseparatethetwotypesofcharge,aswhenglassisrubbedwith
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`anelectronexcess(andthereforeanetnegativecharge)ontheother.
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`wheretheforceFismeasuredinnewtons;thechargesQ,andQ,,incoulombs—
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`(themagnitudeofwhichwillbedefinedintermsofmagneticinteractionsin
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`formsoastosimplifyotherequationswhichareusedmuchmoreextensively
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`Chapter5);andthedistance7,inmeters.Thequantity47appearsinexplicit
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`2.1.Coulomb’sLaw
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`farad/meter.
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`thanCoulomb’slaw.Theconstant¢iscalledthepermittivityoffreespace:—é)=8.85x10-"
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`a8.
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`FuBate
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`plicatedvariationofforcewithdistance.
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`*Plimpton-andLawton,Phys,Rev$0,(1936),AYERCROPSCIENCELPEX.NG10461006
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`nuclei.Inunchargedprotonsarepresentinequalmatterelectronsandnumbers,
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`Fu=Are,rTy(2?)
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`andthatthebasicnegativechargesaretheelectronswhichsurroundtheatomic
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`Weshallassumethatthestudentisfamiliarwiththequalitativeaspectsof
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`Wenowknowthatthebasicpositivechargesaretheprotonsinatomicnuclei
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`on.IntheseunitsCoulomb’slawiswrittenas
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`repelother,twobodiescarryingunlikechargesattracteachother.eachwhereas
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`book,theunitsaredefinedfromotherrelationships,whichweshallstudylater
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`pendedoninsulatingthreadsonecanshowthattwobodiescarryinglikecharges
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`quantities.Intherationalizedm.k.s.system,whichweshallusethroughoutthis
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`withcat’sfuracquiresanegativecharge.Ifrodschargedinthiswayaresus-
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`¢.g.s.systemofunits,Kismadeunitybychoosingappropriateunitsforthese
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`rodrubbedwithsilkacquiresapositivecharge,whereasahardrubberrodrubbed
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`unitswhichareusedforthemeasurementofforce,charge,anddistance.Inthe
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`proportionalityKdependsonthe
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`electricchargesandtheirinteractions.Forexample,therearetwokindsof
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`electrostaticcharges,onecalledpositiveandanothercallednegative.Aglass
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`Themagaitudeoftheconstantofalongthelinejoiningthetwocharges.
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`anparinBeGrech
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`FieldsI
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`foundtobecorrecttobetterthanFigure2-1.ChargesQ,andQ,separated
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`squaredependenceoftheforceonQ,
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`somewhatindirectexperimentwhich
`Jomb’slawhasbeencarriedoutbya
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`distance.Theexponenthasbeen
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`CHAPTER2
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`tiveiftheyareofdifferentsign.
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`TheexactverificationofCou-
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`2-1.TheforceisrepulsiveifQ,andFQO,areofthesamesign,andisattrac-ab
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`measurestheexactnessoftheinverse
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`and’,isaunitvectorpointinginthedirectionfromQ,towardQ,,asinFigure
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`whereF,,istheforceexertedbyQ,onQs,Kisaconstantofproportionality,
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`[2.2]TheElectrostaticFieldIntensity29
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`Ifthepathisclosed,thetotalworkdoneis
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`dre
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`chargedisturbedbyitspresence.distributionsarenot
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`samevalueatthebeginningandattheendofthepath.Thenthelineintegral
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`doneagainstthefield.Hereagain,weassumethatQ’issmall,suchthatthe
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`not,evenifQ,islargecomparedtoQ,.
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`isgivenby
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`FuOn0,Aner?
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`alongthepathconsidered.Thenegativesignisrequiredheretoobtainthework
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`pathisgivenbythelineintegral
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`TheworkWrequiredtomoveitfromapointP,toapointP,alongagiven
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`Pr
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`ConsideratestpointchargeQ’whichcanbemovedaboutinanelectricfield.
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`2.3.TheElectrostaticPotential
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`position.
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`sumofalltheindividualfieldintensities.Thisiscalledtheprincipleofsuper-
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`oneproducesitsownfield,andtheresultantfieldintensityissimplythevector
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`inEq.2-7iszero,andthenetworkdoneinmovingapointchargeQ’aroundaclosedpathinthefieldofanother
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`fixedpointchargeQiszero.,
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`RedboneWhodb&edV,cooGblly
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`andwecanwritethat
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`VXE=0,(2-9)
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`E=—VvYP,(2-10)
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`since¥VXKVV=0.Wecanthusde-
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`whereVisascalarpointfunction,
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`w=-—|EoQ’dl(2-5)
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`negativesignisrequiredinEq.2-0
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`towardadecreaseininorderthattheelectricfieldintensityEcanpoint
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`convention.Itisimportanttonote
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`potential,accordingtotheusual
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`calledtheelectrostaticpotential.The
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`ofthefunctionV(x,y,z),whichis
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`—fEQ’-dl.(2-6)
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`thatwecanaddtoitanyconstant
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`thatVisnotuniquelydefinedin
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`scribethefieldcompletelybymeans
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`Thesumoftheincrementsof(1/r)overaclosedpathiszero,sincerhasthe
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`Nowthetermundertheintegralontherightsideissimplydr/r?or—d(1/r).
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`Letusevaluatethisintegral.Tosimplifymatters,weshallfirstconsiderthe
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`electricfieldproducedbyasinglepointchargeQ.Then
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`EQ’-al=22pa”ab),(2-7)
`electricfieldintensityE.Weshallgralistobecalculated.Thelight:ieinvestigate
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`wouldthendescribeonlypartoftheelementofthepathalongwhiehthelite.
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`vxEwouldnotbezeroandVViytheelectricfieldintensityauddbisaa:
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`dealingherewithelectrostatics.IfV,betweentwopointsésgivenby
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`{hereweremovingchargespresent,jJineintegralofE-dlfrom1to2,where
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`thismorecomplicated"¢Presentlinesofforee
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`WemustrememberthatweaieFigure2-2.Thepotentialdifference
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`caseinChapter6,
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`mightbedisturbedbytheintroductionofafinitetestchargeQ’,wecandefineEtobethe
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`limitingforceperunitchargeasthetestchargeQ’—>0:
`Iftheelectricfieldisproducedbymorethanonechargedistribution,each
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`E=lim4.2-
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`AnelectrostaticfieldisthereforeThen,conservative.fromStokes’salong@asitisalongb.theorem
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`thattheworkdoneingoingfromP,toP,alonga,andthenfromP;backtoPy
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`leadingfromP;toP2.Thenthesetwopathstogetherformaclosedcurvesuch
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`alongbiszero.ThentheworkdoneingoingfromP;toP,isthesame
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`(Section1.6),atallpointsinspace,
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`Theelectricfieldintensityismeasuredinvoltspermeter,Theelectricfielddue
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`tothepointchargeQ,isthesamewhetherthetestchargeQ,isinthefieldor
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`fE-al=0.28)
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` Eu=>.(2-3)
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`individualchargeofthedistributionareallzero.Thus,ingeneral,
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`WecannowshowthattheworkdoneinmovingatestchargefromapointPy
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`Inthecasewheretheelectricfieldisproducedbyachargedistributionwhich
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`toapointP,isindependentofthepath.Letaandbbeanytwodifferentpaths.
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`AccordingtoEq.2-10,
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`testchargeinthefield.ThustheelectricfieldintensityduetothepointchargeQ,
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`TftheelectricfieldisproducednotbyasinglepointchargeQbutbysome
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`30ELECTROSTATICFIELDS1[Chap.2]
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`{2.3}TheElectrostaticPotential31
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`fixedchargedistributioninspace,thelineintegralscorrespondingtoeach
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`BAYER CROPSCIENCE LP EX. NO 1035
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`Beforediscussingthecalculationofpotentialbat-apointinsideadistributed
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`pointatthecenterofthesphere,
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`protons,haveasmallbutextent,andwemustfiniteseewhetherornotthisfact
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`tributionarethesameastheywouldbeifthechargewereallconcentratedina
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`modifiesourresultsinanyway.WeshallseeinSection2.7.1thatthepotential
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`andtheelectricfieldintensityoutsideasphericallysymmetricalchargedis-
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`idealizedcaseofpointcharges.However,realcharges,thatis,electronsand
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`thefluxoftheelectricfieldintensityEthroughtheclosedsurfaceasfollows,
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`pointcharge@islocatedinsideaclosedsurfaceSatapointP.Wecancalcula
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`enclosedwithinthesurface.ConsiderthecaseshowninFigure2-3,inwhicha
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`Gauss’slawrelatesthefluxofEoveraclosedsurfacetothetotalcharge
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`ThefluxofBthroughtheelementofareadais
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`2.4.Gauss’sLaw
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`thefieldintensityEalsoconverges.ms
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`E‘daae
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`definethepotentialinagivenregionofspacetobezero.Itisusuallyconvenient
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`WhenthefieldisproducedbyasinglepointchargeQ,
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`charge.ThepotentialVisexpressedinjoulespercoulomb,orinvolts.isdefinedtobezerotothepointconsideredisVQ’.ThusVcanbewrittenasV=W/Q’andcanbedefinedtobetheworkperunit
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`whereristhedistancefromdrtoPand
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`dV=——P",(2-15):
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`TheworkWrequiredtobringachargeQ’fromapointatwhichthepotential
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`outsidedr
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`chargeisassumedtoextendtoinfinity.Anexampleofthiswillbediscussed
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`vidualcharges,wemayintegrateoverthecontinuousdistributionp(x,y,z).An
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`TheonlycaseinwhichwecannotsetVequaltozeroatinfinityisthatinwhich
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`thanAr.Thus,insteadofsummingthepotentialsofalargenumberofindi-
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`inSection2.7.3.
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`elementofchargepdrcontributesanelementofpotentialdVatapointP.
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`tochoosethepotentialatinfinilytobezero.ThenthepotentialVatthepointr
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`whereAQisthetotalchargewithinAr.ThevolumeArmaybeassumedlarge
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`isgivenby
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`V=[E-dl,(2-13)
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`enoughtoa.suchthatthefluctuationsinAQwithtime,orthosefromoneAr
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`remainsthoughsmaller”valideventhevolumeelementisnotallowedtobecome
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`neighboringone,arenegligible,yetsmallenoughsuchthatintegralcalculus
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`regr?Aree
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`v=[ot=2.(2-14)
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`r 4
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`Y=ine[Or2-16
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`chargeareconcentricspheres.WecanseefromEq.2-10thattheelectricfield
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`potentialsurfaces,accordingtoEq.2-10,ThevectorEiseverywheretangent
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`equipotentialsurface.Forexample,theequipotentialsurfacesaboutapoint
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`spacethatiscalleda/ineofforceandwhichiseverywherenormaltotheequi-
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`ItwillbeobservedthatthesignofthepotentialVisthesameasthatofQ.
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`intensityEiseverywhereequipotentialnormaltothesurfaces.
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`Ifwejoinend-to-endinfinitesimalvectorsrepresentingE,wegetacurvein
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`Ifwejoinallthepointsinspacewhichhavethesamepotential,weobtainan
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`considerthesituationwherethevolumeelementdrisasphericalshellof
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`tributedbythechargeelementpdratPisinfinite,sinceriszero.However,
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`atordecreasesmorerapidlythandoesrinthedenominator.Thepotentialve
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`atPadVof(p/4re\(4ar?/r)dr.Ifweconsideranothershellofsmallerradius.
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`thicknessdrandradiusrcenteredonP.Thechargeinthisshellcontributes
`weseethatitcontributesasmallerdVbecauseasrdecreases,drinthenumere
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`thereforeconverges,andtheintegralisfinite.Asimilarargumentshowsthat-
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`thecontinuouslydistributedcharge.AtfirstsightitappearsthatthedVcon-.
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`WemustnowaskwhetherornotthepotentialVisdefinedwhenPisinside
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`toalineofforce.
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`2.3.1.ThePotentialProducedbyaContinuousChargeDistribution.So
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`oftheelectrostaticpotentialatagivenpoint,wemustthereforearbitrarily
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`Then
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`12
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`ferencesbetweenthepotentialsattwodifferentpoints.Whenwewishtospeak
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`asinFigure2-2.NotethattheelectricfieldintensityEdeterminesonlydif-
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`21Ve-Vi=~{E-dl=[E-dl,(2-12)
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`electronsaresosmallcomparedtothedimensionsofordinaryapparatusthat
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`wemayconsiderasmallvolumeAranddefineanaveragechargedensityp,
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`discretechargesasthoughtheywerecontinuouslydistributed.Protonsand
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`charge,weshouldpointoutthatthereisoftengreatadvantageintreating
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`32ELECTROSTATICFIELDSI[Chap.2]
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`[2.4]Gauss’sLaw33
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`measuredincoulombs/meter’,as
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`AQ/ar,
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`farwehavelimitedourdiscussionoffieldintensitiesandpotentialstothe
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`ductorcanbecalculatedinseveralways,Considerthecaseinwhichtheconduc
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`4.3.5.ImageForces.Theforcewhichactsbetweenapointchargeandaeon
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`ThefleldofthechargeQpolarizesthedielectric,andanegativesurfaeeinduced
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`chargedensityao”isproducedonthesurface,Theresultantfeldatanypoint
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`infrontofasemi-infiniteblockofClassAdielectricmaterial,asinFigure4-13.
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`Thepresenceoftheplanethereforeincreasesthecapacitanceofthesphere.
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`fieldsinthepresenceodielectrics.Asanexample,considerthecaseofapointchargeQatadistaneeD-
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`Themethodofimagescanalsobeusedtodetermine
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`“Page5
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` 142ELECTROSTATICFIELDS11[Chap.4] Figure
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`4-11.Equipotentialsforachargedspherenearagroundedcon-
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`=dre+r+++),(4-51).
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`torisaninfiniteplane.Anannulusofchargeofwidthdsatdistancesfromthe
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`footoftheperpendicular,asinFigure4-12,exertsaforcedFonQinthe
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`[4.4]ImagesinDielectrics143
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`regiontotherightoftheplaneisequaltothatbetweenQanditsimage
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`ontheplane.lateddirectlyfromCoulomb’slaw<by
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`groundedconductingplanecanbecalcu-themagnitudeofthenetforceactingFromNewton’sthirdlaw,thisisalsotweenapointchargeQandaninfinite
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`onQmustbegivenbyEq.4-55inainturn,iscalculatedfromQandthein
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`simpleway.Sincethefieldinthe“8charge—Q.Theringelementshawn
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`Wecanseethattheforceactingusingtheinducedchargedensity0’.This,
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`...dsusedfortheintegration.The.forceis:
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`correctlygivenbyQanditsimage_9
`directionperpendiculartotheplane
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`Thiswillbeanattractiveforceifwe
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`usetheabsolutevalueoftheinduced
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`chargedensitycalculatedinEq.4-29.
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`or,expressingsandrintermsofthe
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`—Q,Coulomb’sJawleadstoEq.4-55
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`4.4.ImagesinDielectrics.Charge
`F=2cos?@sin6dé,
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`4nre(2DP:,..,Figureforceofattractionhe-.4-12.The
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`_QD2xsds_dF=DariAncor?cosé=(4-53)
`dB=GETScosy(4-52)
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`-2.(4-55)
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`NearaSemi-infiniteDielectric
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`BAYERCROPSCIENCELNO.1035
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`Thecapacitancebetweenthesphereandtheplaneisthen
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`ductingplane.
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`dreJo
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`2a{2
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`angle6,
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`(4-54)
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`Thus
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`asfollows:
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`4g€gh"
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`-@v=4rea
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`(4-50):
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`charges.
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`Q:makepotentialofthespherezero;sincethesameistrueofallthefollow-the
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`ingpairsofcharges,thepotentialofthesphereis
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`directly.Thisisalwaystrue.Theforcebetweenapointchargeandaconductor
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`isalwaysgivencorrectlybytheCoulombforcethepointchargeandtimagebetween
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`BAYER CROPSCIENCE LP EX. NO 1035
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`canbecalculated.
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`fieldisgivenby
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`P,=&K,—DE:
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`(4-56)
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`closedsurfacewithfacesoneithersideoftheboundary,wefindthatthenor-
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`beingnegativeifQispositive,andithasthesamemagnitudeoneitherside
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`oftheboundary.ThepartofthefieldarisingfromQisreadilycalculatedfrom
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`whichisthesame,bothinmagnitudeanddirection,oneithersideofthebound
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`ary.Altogetherthen,componentofthepolarizationjustthenormalinsideth
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`Itisorientedinthedirectionpointingtowardtheboundary,thesurfacecharges
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`treatingtheproblemasifthesecharges,togetherwithQ,wereinfreespace.
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`canthenbecomputedbyreplacingchargeandthedielectricwiththissurface
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`144ELECTROSTATICFIELDSIt[Chap.4]
`planesurfaceofalargeblockofdielec-o’=P,.
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`malcomponentofthefieldarisingfroma’is
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`Figure4-13.PointchargeQneartheand
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`ponentofEduetotheinducedsurface
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`chargedensity0’;Eq,isthenormalcom-
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`tric.ThequantityE),isthenormalcom-
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`ponentofEduetothechargeQ.
`=glx_12?aPasol=efKeDLaga+Diy~26,
`*3BDye
`Ei=2.—_>p__
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`GnAmeo(s?+D2?
`Eyn=ae(4-58).
`2D.(4-61)
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`justinsidethedielectricatadistance
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`chargeQ,andanotherarisingfrom
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`twoparts:onearisingfromthepoint
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`a’,ApplyingGauss’slawtoasuitable
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`izationproducedbythisresultant.
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`sfromthefootoftheperpendicular,
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`Thenormalcomponentofthepolar-
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`followingConsiderapointmanner.
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`asinFigure4-13.Atthispointthere
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`isaresultantelectrostaticfieldinten-
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`sityE;withanormalcomponent£,,;.
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`trostaticpotentialorfieldintensity
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`natesonthesurfacebeforetheelec-
`knownasafunctionofthecoordi-
`Thevalueofo’mustofcoursebe
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`NowE,,;isthesuperpositionof
`Tofindo’wecanproceedinthe
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`theimagechargetobeoutsidetheregioniichtheGeldisrequired,“TheBeirgetobeoutsidetheregioninWoehtigfellisrauineetTags.
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`whichreplacesQ.Thefirstofthesecanberuledoutbecausewealwayswant
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`whichwillgivethepropernormalcomponentofE:(a)thepointcharge|Qtogetherwithanimageinsidethedielectricthereare,atfirstglance,twosetsofpoint“chargeslocatedatadistanceDbehindtheboundary,asinFigure4-14a,==Forapoint
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`attheimagepositionbehindtheboundaryand(b)asingleimagecharge
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`thatE,,.willbeproperlygivenbythecharge+Qtogetherwithanimagecharge
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`charge.
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`acrosstheboundary?WecansavesometimeifwerecallthatancomponentsofEqs.4-58and4-59andalsomakethetangentialcomponentofEcontinuous
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`Weconsiderfirstapointoutsidethedielectricsurface.WeseefromEq.4-64
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`imagechargeisalwaysoutsidetheregioninwhichthefieldistobedetermined.
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`boundary.WeassuredthisconditionwhenweusedGauss’slawtocomputeo’,
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`ofthefieldarisingfromthesurfacechargeo’subtractsfromthatarisingfromQ.
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`downthenormalcomponentsoftheresultantfieldforapointjustinsidethe:
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`dielectric,£,;,andjustoutside,E,..Fortheinsidepointthenormalcomponent
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`Fortheoutsidepointtheyadd.Altogether,
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`chargeQ.Butthisisnotthesimplestwaytodealwiththisfield.Weshallfind:
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`integratingovertheo’distribution,andaddingthecontributionfronthepoint
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`point,eitherwithinthedielectricorinfreeSpace,byusingCoulomb’slaw,
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`[4.4]ImagesinDielectrics145
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`insteadasetofimagechargeswhichwillsatisfytheboundaryconditions.
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`Canwenowfindasetofimagechargeswhichwillgivethenormalfield
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`ItwillbeobservedthatthenormalcomponentofDiscontinuousacross.the
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`Tofindthesecharges,weconfineourattentiontotheboundaryandwrite
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`Atthisstagewecouldcalculatethepotentialandthefieldintensityatany
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`Ene=[:+(a~+i)|4re(s?+D2)?(4-64)
`~AKe+i)4re(s?+Dp?(4-65)
`~G4_QD~AK.+1/4re(s?+D2?(4-63)
`.=[1—-(Kect)\]_92
`=(25S)ome
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`2laDEheeeafh@)QK,--|Q,(448)
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`o-ESHows
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`dielectricisgivenby
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`Coulomb’slaw:
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`(K,
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`(4-60)
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`(4-59)
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`(4-57)
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`g'=4+4%:=Do
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`(K..-+1)
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`(4-67)
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`4-14b.
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`thedielectricwereplacedthedielectricbyanimagechargeQ’=anQgeneralmethodwhichwillsolvingequation,involvePoisson’s
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`Figure4-15
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`nearadielectric.Asusual
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`tialsforapointcharge
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`byarrows)andequipoten-
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`ty
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`notshown,“
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`nearthepointchargeare
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`row.The equipetentialy:
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`indicatedbythecurved-ar
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`thefigureabouttheaxis.
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`aregeneratedby.rotatiny
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`theequipotentialsurfae
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`*Seetions4,3to4.6.4maybeonsittedwithoutlosingcontinuity.
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`imagechargemustthereforebelocatedoutsideofthedielectric,asinFigure4.5.GeneralSolutionofLaplace'sEquation
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`regionpreviouslyoccupiedbythedielectric.
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`isremovedand[2/(K,+1)]QissubstitutedforQ,thefieldisunaffectedinthe
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`(K,+1)]Q,thefieldisunaffectedoutsidethedielectric.(b)Whenthedielectric
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`atthepositionofQgivesthefieldinthesecondmedium.
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`(a){b)
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`attheimagepositiontogetherwith@givesthefieldinthefirstmedium,and
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`solution.TheshapeofthefieldisshowninFigure4-15,
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`fieldarethefollowing:
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`PoegKe+KeQ
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`Ke—el
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`(4-69)
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`tions,weknowfromtheuniquenesstheoremthattheyprovidethecorrect
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`146ELECTROSTATICFIELDS11[Chap.4][4.5]GeneralSolutionofLaplace’sEquation147
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`singlechargeQ”=1GQatthepositionofQ,asinFigure4-14b.Since
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`||I
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`eee ee ee ee
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`thesecombinationsofchargesproducesatisfyfieldswhichtheboundarycondi-
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`confineourattentiontoproblemsinwhichthecharge
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`tofindthefieldinsidethedielectric,wereplacedbothQandthedielectricwitha.:::Tobeginwith,weshall
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`locatedatadistanceDbehindtheboundary,asinFigure4-14a.Then,inorderVV=—&(4-71)
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`Whatwehavedone,then,isthefollowing.Inordertofindthefieldoutside
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`electrostaticfieldsareusefulonlyinspecialcases.Weshalldiscusshereamore
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`Themethodswhichwehaveconsidereduntilnowforthecalculationof|
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`Figure(a)Whenthedielectricisreplacedbytheimagecharge—[CK,~1)/4-14.
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`.QTK,Q(470)
`Me2Kez
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`pointchargeQinthefirstmedium,thepointchargeswhichgivethecorrect
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`Ingeneral,fortwomediahavingdielectricconstantsKaandK.»,withthe
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`bytheprocessofvariableseparation.InCartesiancoordinaces,forexample,we
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`whereX(x),YC),andZ(z)arefunctionsonlyofthevariablesx,3ands,
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`(4-73).canusuallyfindasolutionoftheform:V=Xx)¥(y)2),
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`.BAYERCROPSCIENCELPEX.NO.1035°.,
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`solutionsofLaplace’sequationwhichwillsatisfyrequiredboundaryconditions”
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`4.5.1.SolutionsinRectangularCoordinates.Itisusuallypossibletofind
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`demonstratedreadilybysubstitutionintotheoriginalequation.
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`functions,wherethe4’sarearbitraryconstants,isalsoasolution.Thiscanbe
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`erties,ofwhichweshallusethefollowingone.IfthefunctionsVM,Vo,Vay<2
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`aresolutions,anylinearthencombinationA:Vi+A2V2+AsVs+«++ofthese”
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`areaninfinitenumberofthem.Thesefunctionshaveanumberofgeneralprop-
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`densitypisequaltozero,thusweshallLaplace’sequation,havetodealwith
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`vv=0.(4-72)
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`SolutionsofLaplace’sequationfunctions,andthereareknownasharmonic
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