Potential of NonUniformly Charged Spherical Shell
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.
So, I can work this all the way to finding the coefficients of the infinite series (i.e. I've solved Laplace's equation and need to find the coefficients to satisfy the boundary conditions.) I know the BCs (potential is discontinuous at the surface, electric field isn't) so my question is, when I'm integrating against the Legendre polynomials, will my charge density be orthogonal to all of them except P3(cos theta)= (5cos^3 θ 3cosθ)/2 or do I have to integrate for all them?
