Expressing a potential inside a spherical shell as

[dikmikkel]

Homework Statement

The potential inside a spherical shell is given by:
V_{-}(x,y,z)= \frac{V_0}{R^2}(6z^2-3x^2-3y^2)
P_n(\cos(\theta )) where \theta is the polar angle.

The potential on the surface carries a surface charge density \sigma. Besides this, ther's no other charges and no outher field. The potential is rotational symmetric around the z-axis inside and outside, and goes to 0 far away from the sphere.

b) express the potential inside the spherical shell using a LegendrePolynomial

Homework Equations

In spherical coordinates i have:
V(r,\theta ) = \sum\limits_{l=0}^{\infty}(A_lr^lP_l(\cos(\theta)) = V_0(\theta)

The Attempt at a Solution

This is how far i made it. Now i suppose i could multiply it with P_{l'}(\cos( \theta ))\sin(\theta) and integrate, but i can't 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.

 

[vela]

You're making the problem more complicated than you need to. First, express the given V in terms of spherical coordinates and write all the trig functions in terms of cos θ. Then compare what you get to the Legendre polynomials.

 

[dikmikkel]

Ha ha, yeah now i get it :) It's because normally we want to find the potential. But now i get it as the Legendre from 0 to 2, which makes sense by the order of the cartesian polynomial. So i am getting
\frac{3V_0r^2}{R^2}(3\cos(\theta)^2-1)