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1、Chapter4SelectedTopicsforCircuitsandSystems4-1Poisson’sandLaplace’sEquationsvρ2Poission’sequation:∇⋅εE=ρ⇒∇V=−ε2Laplace’sequation:Ifnochargeexists,ρ=0,∇V=0Eg.Thetwoplatesofaparallel-platecapacitorareseparatedbyadistancedandmaintainedatpotentials0andV0.Assumenegligiblefringingeffectattheedges,determi
2、ne(a)thepotentialatanypointbetweentheplates,(b)thesurfacechargedensitiesontheplates.[清大電研]2dVV(Sol.)=0⇒V=cy+c,V()()y=0=,0Vy=d=V⇒V=0y2120dydvdVV0vρE=−yˆ=−yˆ,En=aˆn⋅E=dydεεV0Atthelowerplate:aˆn=yˆ,ρsl=−.dεV0Attheupperplate:aˆ=−yˆ,ρ=nuldEg.Theupperandlowerconductingplatesofalargeparallel-platecapacito
3、rareseparatedbyadistancedandmaintainedatpotentialsV0and0,respectively.Adielectricslabofdielectricconstantεranduniformthickness0.8displacedoverthelowerplate.Assumingnegligiblefringingeffect,determine(a)thepotentialandelectricfielddistributioninthedielectricslab,(b)thepotentialandelectricfielddistrib
4、utionintheairspacebetweenthedielectricslabandtheupperplate.[台大電研]v(Sol.)SetVd()y=c1y+c2,Ed=−ˆcy1,Dd=−yˆε0εrc1vV()y=cy+c,E=−ˆcy,D=−yˆεca34a3a03Vd()0=,0Va()d=V0,Vd()8.0d=Va(8.0d),Dd(8.0d)=Da(8.0d)VεV(1−ε)V0r0r0⇒c=,c=,0c=d,c=1()23()48.0+2.0εd8.0+2.0ε1+.025εrrr5yVv5V5εy−4(ε−1)dv5εV00rrr0()aV=,E=−yˆ()bV
5、=V,E=−yˆd()d()a()0a()4+εd4+εd4+εd4+εdrrrrEg.Showthatuniquenessofelectrostaticsolutions.2ρ2ρ(Proof)LetV1andV2satisfy∇V=−and∇V=−.DefineVd=V1-V2,12εε2▽Vd=01.Ontheconductingboundaries,Vd=0⇒V1=V2v2.Letf=Vd,A=▽Vdvv222∇⋅()Vd∇Vd=∇⋅()Af=f∇⋅A+A⋅∇f=Vd∇Vd+∇Vd⇒∫∫()Vd∇Vd⋅aˆnds=∫∫∫∇Vddvsv′2R→∞,V=V−V∝1,∇V∝1,ds∝R2⇒
6、∫∫()V∇V⋅aˆds→,0∴∫∫∫∇Vdv=0⇒d12d2ddndRRsvVd=0⇒V1=V2ImageTheoremP(x,y,z)inthey>0regionisQ⎛11⎞V()x,y,z=⎜⎜−⎟⎟,whereR+andR-are4πε0⎝R+R−⎠thedistancesfromQand–QtothepointP,respectively.Eg.ApointchargeQexistsatadistancedabovealargegroundedconductingplane.Determine(a)thesurfacechargedensityρs,(b)thetotalchar
7、geinducedontheconductingplane.[交大光電所]vQQd(Sol.)E/y=0=−yˆ2⋅2sinθ=−yˆ22324πε0R2πε0()d+rvQd∞()aρ=yˆ⋅εE=−,()bρ2πrdr=−Qs/y=0∫s()223202πd+rEg.Twodielectricmediawithdielectricconstantsε1andε2areseparatedbyaplanebo
