Classical Electrodynamics · Problem 3.3

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

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

Problem

A thin, flat, conducting, circular disc of radius

R

is located in the

x

y

plane with its center at the origin, and is maintained at a fixed potential

V

. With the information that the charge density on a disc at fixed potential is proportional to

(R^2 - \rho^2)^{-1/2}

, where

\rho

is the distance out from the center of the disc, (a) show that for

r > R

the potential is

\Phi(r,\,\theta,\,\phi) = \frac{2VR}{\pi} \frac{1}{r} \sum_{l=0}^{\infty} \frac{(-1)^l}{2l+1} \left(\frac{R}{r}\right)^{2l} P_{2l}(\cos\theta)

(b) find the potential for

r < R

. (c) What is the capacitance of the disc?