Introduction to Electrodynamics · Problem 3.43

Potentials — David J. Griffiths

Physics · separation-of-variables-spherical

Problem

A conducting sphere of radius

a

, at potential

V_0

, is surrounded by a thin concentric spherical shell of radius

b

, over which someone has glued a surface charge

\sigma(\theta) = k\cos\theta,

where

k

is a constant and

\theta

is the usual spherical coordinate. (a) Find the potential in each region: (i)

r > b

, and (ii)

a < r < b

. (b) Find the induced surface charge

\sigma_i(\theta)

on the conductor. (c) What is the total charge of this system? Check that your answer is consistent with the behavior of

V

at large

r

\left[Answer: V(r,\theta) = \begin{cases} aV_0/r + (b^3 - a^3)k\cos\theta/3r^2\epsilon_0, & r \geq b \\ aV_0/r + (r^3 - a^3)k\cos\theta/3r^2\epsilon_0, & r \leq b \end{cases}\right]