Potential of Spherical Shell with Nonunifor Surface Charge

[azupol]

Homework Statement

A thin spherical shell of radius R carries a surface charge density of the form kcos ³ θ .
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

The Attempt at a Solution

I solved Laplace's equation using separation of variables, and got to where I would integrate to find the coefficients of the Fourier series, but that's where I'm stuck. I get that my coeffcient A_l = k/2εR^l-1∫cos³θ P_l(cos θ) sin θ dθ where P_l(cos θ) are the Legendre polynomials.

I don't know how to integrate this, but intuitively shouldn't all terms that aren't P₃ give me zero?

 

[Eats Dirt]

You need to write $$\cos^3\theta$$
In terms of legendre polynomials to do this integral easily. See wikipedia for the P polynomials http://en.wikipedia.org/wiki/Associated_Legendre_polynomials.

I'll try and get you started.
We start without cos^3theta term, and now we need to find the corresponding legendre polynomial.
$$P^{0}_{3}(\cos\theta)=\frac{1}{2}(5\cos^3\theta-3\cos\theta) \\ \frac{1}{5}(2P^{0}_3(\cos\theta)+3\cos\theta)=\cos^3\theta \\ now\, use\, the\, fact\, that\, P^{0}_{1}(\cos\theta)=\cos\theta\, and\, sub\, in \\ \frac{1}{5}(2P^{0}_3(\cos\theta)+3P^{0}_{1}(\cos\theta))=\cos^3\theta$$

now you can sub in and integrate!

 

[azupol]

Yes this is exactly what I was looking for thank you! When I integrate, I should get that all terms that are not P₁ and P₃ drop out due to the orthogonality of the Legendre Polynomials, yes?

 

[Eats Dirt]

Yes, the orthogonal P terms will cancel out!

 

[HalfLostAndSearching]

As a scientist, my response would be that the potential of a spherical shell with nonuniform surface charge has the potential to be a complex and challenging problem to solve. It requires a deep understanding of Laplace's equation and the use of advanced mathematical techniques such as separation of variables and integration. It is important to carefully consider the boundary conditions at the surface of the shell in order to accurately determine the electric field both inside and outside the sphere. Further research and analysis may be necessary to fully understand the behavior and implications of this potential solution.