Classical Electrodynamics · Problem 4.3

Magnetostatics, Faraday's Law, Quasi-Static Fields — John David Jackson

Physics — E&M · cartesian-vs-spherical-multipole-moments

Problem

The

l

th term in the multipole expansion (4.1) of the potential is specified by the

(2l+1)

multipole moments

q_{lm}

. On the other hand, the Cartesian multipole moments,

Q^{(l)}_{\alpha\beta\gamma} = \int \rho(\mathbf{x}) x^\alpha y^\beta z^\gamma \, d^3x

with

\alpha

\beta

\gamma

nonnegative integers subject to the constraint

\alpha + \beta + \gamma = l

, are

(l+1)(l+2)/2

in number. Thus, for

l > 1

there are more Cartesian multipole moments than seem necessary to describe the term in the potential whose radial dependence is

r^{-(l+1)}

. Show that while the

q_{lm}

transform under rotations as irreducible spherical tensors of rank

l

, the Cartesian multipole moments correspond to reducible spherical tensors of ranks

l,\, l-2,\, l-4,\, \ldots,\, l_{\min}

, where

l_{\min} = 0

1

for

l

even or odd, respectively. Check that the number of different tensorial components adds up to the total number of Cartesian tensors. Why are only the

q_{lm}

needed in the expansion (4.1)?