Question

The potential at the surface of a sphere (radius R) is given by Vo=Kcos3D, Where D is theta,K is a constant. Find the potential inside and outside the sphere, as well as the surface charge density s(D) on the sphere.(Assume there is no charge inside or outside the sphere).

Answer 1

The potential inside and outside must have a form

(1)

By continuity of the potential,

The normal component of the electric field is discontinuous by / ,   

Thus,

We can find A_l ,

Since is constant, A0 = R/₀ and is 0 for all other . Then the potential is given by

(2)

The potential at the surface of a sphere (radiusR ) is given by

(3)

First, notice that . Potential is given by Eq. 1. One can find A_l by comparing with .

The charge density