# Multipole expansion on a sphere

1. Nov 7, 2009

### darkpsi

1. The problem statement, all variables and given/known data
A sphere of radius R, centered at the origin, carries charge density
ρ(r,θ) = (kR/r2)(R - 2r)sinθ,
where k is a constant, and r, θ are the usual spherical coordinates. Find the approximate potential for points on the z axis, far from the sphere.

2. Relevant equations
The multipole expansion of V and some manipulations on the charge density?

3. The attempt at a solution
So I thought I had solved the problem, but both the monopole and the dipole vanished and I believe the answer given was approximated by the dipole term. And when I decided to try the quadrupole, it got really messy. I ended up with (3πkR5/64εoz3. Is this right at all? I tried to change the charge density in terms of the vector, r', from the origin to the points on/in the sphere but I don't seem to be getting anywhere.
Here's what I did to find the dipole:

V(r,θ) = 1/4πεo * 1/r2 ∫∫∫ r'cosθ'kR(R-2r')sin2θ' dr'dθ'dφ'

Everything is fine until I integrate with respect to θ:

0πsin2θ'cosθ' dθ' = 1/4sin4θ' |0π = 0

The monopole did the same thing, but I expected that.

Last edited: Nov 7, 2009