III.Equipotential Sphere in a Uniform Electric Field →0 十十 E=Eoiz Eolir cose-iesine] limΦ(r,0)=-E。rcos0 [Φ=-Ez=-E。rcos0] Φ(r=R,0)=0 R3 cose This solution is composed of the superposition of a uniform electric field plus the field due to a point electric dipole at the center of the sphere: aple=Cos with p=4πeE.R3 4πe。r2 This dipole is due to the surface charge distribution on the sphere. aΦ cos0 =3e.E。c0s0 6.641,Electromagnetic Fields,Forces,and Motion Lecture 7 Prof.Markus Zahn Page 5 of 276.641, Electromagnetic Fields, Forces, and Motion Lecture 7 Prof. Markus Zahn Page 5 of 27 III. Equipotential Sphere in a Uniform Electric Field ( ) o oo r lim r, E r cos E z E r cos → ∞ Φ θ =− θ Φ=− =− θ ⎡ ⎤ ⎣ ⎦ Φ = θ= ( ) r R, 0 ( ) 3 o 2 R r, E r cos r ⎡ ⎤ Φ θ =− − θ ⎢ ⎥ ⎣ ⎦ This solution is composed of the superposition of a uniform electric field plus the field due to a point electric dipole at the center of the sphere: dipole 2 o p cos 4 r θ Φ = πε with 3 p 4 ER o o = πε This dipole is due to the surface charge distribution on the sphere. ( )( ) 3 s or o oo 3 rR rR 2R r R, E r R, E 1 cos r r = = ∂Φ ⎡ ⎤ σ = θ = = θ =− = + θ ⎢ ⎥ ∂ ⎣ ⎦ ε εε 3 E cos o o = ε θ