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dikmikkel
#1
Sep16-11, 09:27 AM
P: 169
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^2-3x^2-3y^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 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

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.
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