Introduction to Electrodynamics · Problem 3.20

Potentials — David J. Griffiths

Physics · separation-of-variables-spherical

Problem

Suppose the potential

V_0(\theta)

at the surface of a sphere is specified, and there is no charge inside or outside the sphere. Show that the charge density on the sphere is given by

\sigma(\theta) = \frac{\epsilon_0}{2R}\sum_{l=0}^{\infty}(2l+1)^2 C_l P_l(\cos\theta),

where

C_l = \int_0^{\pi} V_0(\theta) P_l(\cos\theta)\sin\theta\, d\theta.