1. The problem statement, all variables and given/known data
The potential inside a spherical shell is given by:
[itex]V_{}(x,y,z)= \frac{V_0}{R^2}(6z^23x^23y^2)[/itex]
[itex]P_n(\cos(\theta ))[/itex] where [itex] \theta [/itex] is the polar angle.
The potential on the surface carries a surface charge density [itex]\sigma[/itex]. Besides this, ther's no other charges and no outher field. The potential is rotational symmetric around the zaxis inside and outside, and goes to 0 far away from the sphere.
b) express the potential inside the spherical shell using a LegendrePolynomial
2. Relevant equations
In spherical coordinates i have:
[itex]V(r,\theta ) = \sum\limits_{l=0}^{\infty}(A_lr^lP_l(\cos(\theta)) = V_0(\theta)[/itex]
3. The attempt at a solution
This is how far i made it. Now i suppose i could multiply it with [itex]P_{l'}(\cos( \theta ))\sin(\theta)[/itex] and integrate, but i cant figure out how to simplify it and extract the solution.
I'm aware that the functions' are orthogonal, but the integral of a sum is something i've never done before.
