# Homework Help: Non-Uniform Surface Charge Spherical Shell

1. Oct 14, 2014

### azupol

1. The problem statement, all variables and given/known data
A thin spherical shell of radius R carries a surface charge density of the form
$kcos3 \theta$
Find the electric field inside and outside the sphere and demonstrate explicitly that its
components satisfy the relevant boundary conditions at the surface.

2. Relevant equations
The solution to Laplace's Equation in spherical coordinates :

\Sigma Al rl Pl(cos \theta) (r≤R)

\Sigma Bl r-(l+1)Pl(cos \theta) (r≥R)

3. The attempt at a solution
I worked it through until

Al= k/2ε0Rl-1∫cos3 \theta Pl cos \theta sin \theta d \theta

Where do I go from here? Do I only need to consider the third Legendre polynomial?i.e.

$(5cos3\theta - 3 cos \theta)/2$
Are all the other coefficients zero?

EDIT: It seems TeX does not want to work for me, but I'm sure you get the idea

Last edited: Oct 14, 2014
2. Oct 20, 2014