Lorrain&Corson111-113 charge in dielectric problem

3.7 CALCULATION OF ELECTRIC FELOS INVOLVING DIELECTRICS m1 Example |THEFREECHARGE DENSITY «,THE BOUND CHARGE DENSITY «, AND THE ELECTRIC DISPLACEMENT D AT ADIELECTRIC-CONDUCTOR BOUNDARY At the interface between adielectric and aconductor thete isa ‘bound surface charge density onthedielectric and afree surface charge density ontheconductor. For example, atthelower plate ofthe capacitor considered above, c= Pana -P, os since misthe outward normal tothe delectic surface and points : downward, while thelectric polarization Ppoints upward like E, Thus 1 =P= D- ok, G-65) =totp-2—4, 6-66) and aah G67) asin Eg. 3-60. Since &>1,the bound surface charge density ayand thefee surface charge density zyhave opposite signs. The bound chatge density isalso smaller inmagnitude than thefee charge density by the factor («.~1/er This telationship isalways true whenever a Cass Adielectric isin contact with aconductor, oFcarves afree surface charke density. The Fee charge density von aconductor incontact with any etectric isalways equal tothe electric displacement Djust inside the diciectrie. Ths follows from Gauss’ law, asinthecase ofthe parallel-plaie capacitor. Furthermore, ifaconducting surface is introduced into adieecric soastocoincide with anequipotemial surface, then thefree surface charge density ,ontheconductor at4particularpointisequalinmagnitude(otheelectricdisplacementDin thedieletric atthat point Example DIELECTRIC SPHERE WITH APOINT CHARGE AT ITS CENTER Let usconsideraClassAdielectricsphereofradiusRwithapoint charge Qembedded atthecenter, asin Figure 38. (Itisposible totap negative charges within dielectrics bybombarding them with high-energy electrons. This example isoflittle practical interest, tut itproves 10beanexcellent illustration ofthe behavior ofd- elects) uz I SO fae \: ee ee j jViee /jFao J SES: A Figure3-8,Dielectricspherewitha—_— pointchargeQatthecenter. Asain, wecan find Dfrom Gauss's law. Wedcaw animaginary sphere ofradius r<Rand use asaGaussian surface, Then =Binp=Sn, 08) =,2.pagan oe ealp ectopatitp tt2, 6-70 Attheoutersurfaceofthesphere, wna Patale oePn=P eR G7) F‘Thetotalamountofboundchargeontheoutersurfaceisthus o.-Sto. en ‘Since there isnovolume density offreecharge, there should be novolume density ofbound charge, aswesaw inthe last section. This iscorrect: LeaaOP=paOP)=0. (G-73) ‘There isalso abound charge onthesurface ofthecavity con- taining the free charge Qatthe center, This can beealeulated asfollows.Lettheradiusofthecavitybe&.Thenif—existhedensity ofboundchargeonthecavitysurface, thetotalboundchargeonthecavity is Ou=mended =—Prted==(%=4) deo,0-7 ent--() @ 75) ‘Thedielectric asawholetherefore remains neutral, asmustbeex-expected.Thenetchargeatthecenteristhen 3.7 CALCULATION OF ELECTRIC FIELDS INVOLVING DIELECTRICS 3 =! Qn=9-(*")o-2 6-76) and issmaller than thefree charge bythefactor ¢,.This accounts forthereduction oftheelectric field intensity within thedielectric bythefactor¢,,asinEq.3-69. Ttisinstcctive tocompare theelectric fed intensity outside the sphere and that inside. Outside, trom Gauss's lw, 2 1 p=Sin om t 2. 1 gE Ty (G-78) sthichstobecomparedfoBas.368and3-69forthefeldinsidethesphere. Atthesurface ofthespheretheelectricfieldintensityisdis- iSEnimuous themagnefotouethencebengnes {large asthemagnitude just inside thesurface. The diference isdue | tothebound charge, which produce anoppoting Seldwithin the ! dielectric. Example THEBARELECTRET Anelectet isthe electric equivalent of bar magnet. fnmost di- electrics thepolarization disappears immediately when theelectric field isremoved, butsome dielectrics retain their polatization fora ‘very long time Indeed, several polymers have extrapolated lifetimes (ofseveral thousand Yearsatroomtemperate ‘Onewayofchargingadektricistoplaceinastrongelectric field atsome high temperature. The bound charge density onthe surfaces then builds upslowly a8themolecules orient themselves. Free chargesarealsodeposiedonthesuaceswhensparkingoccars » Figure39.(a)Barleitpoliteuniformlyparalleltsaxis(0)TheBaldoftebareketcthesareesthtofaparofecularpnts crryingunform srace charge densities ofopposite polaris.