Classical Electrodynamics · Problem 3.18

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

Physics — E&M · mixed-boundary-conditions-cylindrical

Problem

The configuration of Problem 3.12 is modified by placing a conducting plane held at zero potential parallel to and a distance

L

away from the plane with the disc insert in it. For definiteness put the grounded plane at

z = 0

and the other plane with the center of the disc on the

z

axis at

z = L

. (a) Show that the potential between the planes can be written in cylindrical coordinates

(z, \rho, \phi)

\Phi(z, \rho) = V \int_0^{\infty} d\lambda\, J_1(\lambda) J_0(\lambda\rho/a) \frac{\sinh(\lambda z/a)}{\sinh(\lambda L/a)}

(b) Show that in the limit

a \rightarrow \infty

with

z

\rho

L

fixed the solution of part a reduces to the expected result. Viewing your result as the lowest order answer in an expansion in powers of

a^{-1}

, consider the question of corrections to the lowest order expression if

a

is large compared to

\rho

and

L

, but not infinite. Are there difficulties? Can you obtain an explicit estimate of the corrections? (c) Consider the limit of

L \rightarrow \infty

with

(L - z)

a

and

\rho

fixed and show that the results of Problem 3.12 are recovered. What about corrections for

L \gg a

, but not

L \rightarrow \infty