# Multipole expansion of point charge placed inside a cube

1. Sep 29, 2012

### amccaffrey

1. The problem statement, all variables and given/known data

A cube of side a is fi lled with a uniform charge density distribution of total charge Q. A point
charge +Q is placed at the center of the cube.

Show all odd electrostatic multipole moments vanish. (i.e., 2^l poles with odd l). Show that
among the even moments, those with l = 0; 2, vanish.

2. Relevant equations

Classical Electrodynamics, Jackson 4.3

http://tinypic.com/view.php?pic=wh4z5&s=6

3. The attempt at a solution

I can't seem to figure out how to go about this. There's two main things that I'm stuck with:

How to deal with the point charge within the cube

How to solve the multipole moments in cartesian (Jackson)

I believe I would have to split the integrals into two parts where one goes from 0 to the cubes surface and the other goes from the cubes surface to infinity but it doesn't seem to be working out.

I've been trying to find an example of solving the multipole moments in Cartesian and finding the moments of a volume with a charge in the middle but I've had no luck.

Any help with this is greatly appreciated.