Classical Electrodynamics · Problem 3.21

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

Physics — E&M · capacitance-variational-methods

Problem

(a) By using the Green function of Problem 3.17b in the limit

L \rightarrow \infty

, show that the capacitance of a flat, thin, circular, conducting disc of radius

R

located parallel to, and a distance

d

above, a grounded conducting plane is given by

\frac{4\pi\epsilon_0}{C} = \int_0^{\infty} dk\,(1 - e^{-2kd}) \frac{\left[\int_0^R \rho J_0(k\rho)\sigma(\rho)\, d\rho\right]^2}{\left[\int_0^R \rho\sigma(\rho)\, d\rho\right]^2}

where

\sigma(\rho)

is the charge density on the disc. (b) Use the expression in part a as a variational or stationary principle for

C^{-1}

with the approximation

\sigma(\rho) =

constant. Show explicitly that you obtain the correct limiting value for

C^{-1}

d \ll R

. Determine an approximate value of

C^{-1}

for an isolated disc (

d \gg R

) and evaluate the ratio of it to the exact result,

4\pi\epsilon_0/C = (\pi/2)R^{-1}

. (c) As a better trial form for

\sigma(\rho)

consider a linear combination of a constant and

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

, the latter being the correct form for an isolated disc. For part b the following integrals may be of use:

\int_0^{\infty} dt\left[\frac{J_1(t)}{t}\right]^2 = \frac{4}{3\pi}, \qquad \int_0^{\infty} \frac{dt\, J_1^2(t)}{t} = \frac{1}{2}