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1. The problem statement, all variables and given/known data

The potential at the surface of a sphere of a radius R is given by [itex] V_{0} = k\cos 3\theta [/itex]. Find the potential inside and outside the sphere.

2. Relevant equations

Solution to Laplace's equation in spherical coords is given by

[tex] V(r,\theta)=\sum_{l=1}^{\infty}\left(A_{l}r^l + \frac{B_{l}}{r^{l+1}}\right)P_{l}(\cos\theta) [/tex]

[tex] \cos 3\theta = 4\cos^3\theta-3\cos\theta [/tex]

3. The attempt at a solution

The only problem really is finding the coefficient A. B is related to A like

[tex] B_{l}=-A_{l}R^{2l+1}[/tex]

I used the expansion of the cosine and found a linear combination of the Legendre Polynomials such that

[tex] V_{0} = \frac{8}{5}kP_{3}(\cos\theta)-\frac{3}{5}kP_{1}(\cos\theta) [/tex]

From here can i just say that

[tex] A_{3} = \frac{8k}{5R^3} [/tex] and

[tex] A_{1} = \frac{-3k}{5R} [/tex]

or do i have to solve for A using

[tex] A_{l} = \frac{2l+1}{2R^l} \int_{0}^{\pi} V_{0}(\theta) P_{l}(\cos\theta)\sin\theta d\theta [/tex]

Thanks for your help!

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# Homework Help: Potential of a sphere

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