Classical Electrodynamics · Problem 3.26

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

Physics — E&M · neumann-green-function-spherical

Problem

Consider the Green function appropriate for Neumann boundary conditions for the volume

V

between the concentric spherical surfaces defined by

r = a

and

r = b

a < b

. To be able to use (1.46) for the potential, impose the simple constraint (1.45). Use an expansion in spherical harmonics of the form,

G(\mathbf{x}, \mathbf{x}') = \sum_{l=0}^{\infty} g_l(r, r') P_l(\cos\gamma)

where

g_l(r, r') = r_<^l/r_>^{l+1} + f_l(r, r')

. (a) Show that for

l > 0

, the radial Green function has the symmetric form

g_l(r, r') = \frac{r_<^l}{r_>^{l+1}} + \frac{1}{(b^{2l+1} - a^{2l+1})}\left[\frac{l+1}{l}(rr')^l + \frac{l}{l+1}\frac{(ab)^{2l+1}}{(rr')^{l+1}} + a^{2l+1}\left(\frac{r^l}{r'^{l+1}} + \frac{r'^l}{r^{l+1}}\right)\right]

(b) Show that for

l = 0

g_0(r, r') = \frac{1}{r_>} - \left(\frac{a^2}{a^2 + b^2}\right)\frac{1}{r'} + f(r)

where

f(r)

is arbitrary. Show explicitly in (1.46) that answers for the potential

\Phi(\mathbf{x})

are independent of

f(r)

. [The arbitrariness in the Neumann Green function can be removed by symmetrizing

g_0

r

and

r'

with a suitable choice of

f(r)