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1、Chapter1ElectromagneticFieldTheory1-1ElectricFieldsandElectricDipolesv→rQ1Gauss’slawofE:∫∫E⋅dS==∫∫∫ρdv'sεεdivergence→1→ρ1theorm−9⎯⎯→⎯⎯∫∫∫∇⋅Edv'=∫∫∫ρdv'⇒∇⋅E=andε0=×10(F/m)inthevεv'ε36πfreespace.→→rr→→→aqˆRR'q(R−R)'R−R'ForqatR',fieldpointatR⇒E=→→=→→andaˆRR
2、'=rr.4πε
3、R−R
4、'24πε
5、R−R
6、'3R−R'→→→→1nq(R−R')RkEduetoasystemofdiscretecharges:E=∑→→4πεk=1
7、R−R'
8、3k→→1∧ρ1ρRVolumesourceρ⇒E=adv'=dv'∫∫∫vv''R2∫∫∫→4πεR4πε3
9、R
10、→1∧ρ→1∧ρslSurfacesourceρs⇒E=adS'Linesourceρl⇒E=adl'∫∫s'R2∫l'R24πεR4πεRvr→∧qqqqRRv1212Eg.ShowthatCoulomb’
11、slawF=aR=,whereaˆ=,R=R.23R4πεR4πεRRvr→→→q12→∧qqR11(Proof)∵F=qEandE⋅dS==4πrE,E=aR=2∫∫23ε4πεR4πεR→∧qq12∴F=aR24πεREg.Determinetheelectricfieldintensityofaninfinitelylonglinechargeofauniformdensityρlinair.(Sol.)→→L2π∫∫E⋅dS=∫∫Erdφdz=2πrLEs00ρL→∧ρll2πrLE=,E=ar
12、ε2πεr00Eg.Determinetheelectricfieldintensityofaninfiniteplanarchargewithauniformsurfacechargedensityρs.→→ρAs(Sol.)∫E⋅dS=2ES=2EA,2EA=sε0⎧∧ρs⎪=z,z>0→⎪2ε⇒E=0⎨∧ρ⎪=−zs,z<0⎪2ε⎩0Eg.Alinechargeofuniformdensityρlinfreespaceformsasemicircleofradiusb.Determinethema
13、gnitudeanddirectionoftheelectricfieldintensityatthecenterofthesemicircle.[高考]→ρ(bdφ)→∧∧ρπ∧ρlll(Sol.)dE=−⋅sinφ,E=yE=−ysinφdφ=−yy2y∫04πε0b4πε0b2πε0bEg.Determinetheelectricfieldcausedbysphericalcloudofelectronswithavolumechargedensityρ=-ρ0for0≤R≤b(bothρ0and
14、barepositive)andρ=0forR>b.[交大電子物理所](Sol.)(a)R≥b4π→Q∧ρb330Q=−ρb,E=aˆ=−a0R2R234πεR3εR00(b)0≤R≤b→∧→∧→→2E=aE,ds=ads,E⋅dS=EdS=E4πRRR∫∫ssii4π→∧ρR30Q=ρdv=−ρdv=−ρR,E=−a∫∫∫vv0∫∫∫0R33ε0Eg.AtotalchargeQisputonathinsphericalshellofradiusb.Determinetheelectricalfield
15、intensityatanarbitrarypointinsidetheshell.[台大電研](Sol.)Qρ⎛dSdS⎞ρ=,dE=s⎜1−2⎟s4πb24πε⎜r2r2⎟0⎝12⎠dS1dS2ρs⎛dΩdΩ⎞dΩ=cosα=cosα,dE=⎜−⎟=022r1r24πε0⎝cosαcosα⎠Electricdipole:Apairofequalbutoppositechargeswithseparation.→→→→d→d→dd2−3−2/32−2/3
16、R−
17、=[(R−)⋅(R−)]=[R−R⋅d+
18、]2224→→→→−3R⋅d−2/3−33R⋅d≅R1[−]≅R1[+]22R2R→→→→d−3−33R⋅d
19、R+
20、≅R1[−]222R⎧⎫⎪→→⎪→d→d⎪R−R+⎪→→→→⎡⎤⎡⎤→q⎪22⎪qR⋅d→→1R⋅P→→⇒E=⎨3−3⎬≅3⎢32R−d⎥=3⎢32R−P⎥4πε⎪→→⎪4πεR⎢R⎥4πεR⎢R⎥→d→d⎣⎦⎣⎦⎪R−R+⎪⎪22⎪⎩⎭⎛→∧∧∧→→⎞⎛⎞⎜p=zp=p⎜aRcosθ−aθsinθ⎟R⋅p=pcosθ⎟⎝⎝⎠
