Non-Uniform Surface Charge Spherical Shell

[azupol]

Homework Statement

A thin spherical shell of radius R carries a surface charge density of the form
<br /> <br /> kcos³ \theta<br />
Find the electric field inside and outside the sphere and demonstrate explicitly that its
components satisfy the relevant boundary conditions at the surface.

Homework Equations

The solution to Laplace's Equation in spherical coordinates :\Sigma A_l r^l P_l(cos \theta) (r≤R)\Sigma B_l r^-(l+1)P_l(cos \theta) (r≥R)

The Attempt at a Solution

I worked it through untilA_l= k/2ε₀R^l-1∫cos³ \theta P_l cos \theta sin \theta d \thetaWhere 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

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[azupol]

https://www.physicsforums.com/threa...nonunifor-surface-charge.776855/#post-4887044