Classical Electrodynamics · Problem 3.5

Multipoles, Electrostatics of Macroscopic Media, Dielectrics — John David Jackson

Physics — E&M · boundary-value-problems-spherical-coordinates

Problem

A hollow sphere of inner radius

a

has the potential specified on its surface to be

\Phi = V(\theta, \phi)

. Prove the equivalence of the two forms of solution for the potential inside the sphere: (a)

\displaystyle\Phi(\mathbf{x}) = \frac{a(a^2 - r^2)}{4\pi} \int \frac{V(\theta', \phi')}{(r^2 + a^2 - 2ar\cos\gamma)^{3/2}}\, d\Omega'

where

\cos\gamma = \cos\theta\cos\theta' + \sin\theta\sin\theta'\cos(\phi - \phi')

. (b)

\displaystyle\Phi(\mathbf{x}) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} A_{lm} \left(\frac{r}{a}\right)^l Y_{lm}(\theta, \phi)

where

A_{lm} = \int d\Omega'\, Y^*_{lm}(\theta', \phi') V(\theta', \phi')