Classical Electrodynamics · Problem 3.6

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

Physics — E&M · multipole-expansion-spherical-harmonics

Problem

Two point charges

q

and

-q

are located on the

z

axis at

z = +a

and

z = -a

, respectively. (a) Find the electrostatic potential as an expansion in spherical harmonics and powers of

r

for both

r > a

and

r < a

. (b) Keeping the product

qa \equiv p/2

constant, take the limit of

a \rightarrow 0

and find the potential for

r \neq 0

. This is by definition a dipole along the

z

axis and its potential. (c) Suppose now that the dipole of part b is surrounded by a grounded spherical shell of radius

b

concentric with the origin. By linear superposition find the potential everywhere inside the shell.