Chapter 3.3, Problem 3.20P

Suppose the potential $V_0\left(\theta \right)$ at the surface of a sphere is specified, and there is no charge inside or outside the sphere. Show that the charge density onthe sphere is given by

   $\sigma \left(\theta \right)=\frac{{\in }_0}{2R}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(2l+1\right)}^2{C}_l{P}_l\left(\cos \theta \right)},$     (3.88)

where

   $C_l={\displaystyle {\int }_0^{\pi }{V}_0\left(\theta \right)P_l\left(\cos \theta \right)\sin \theta d\theta .}$     (3.89)