Find the potential in the interior of a sphere of unit radius when the potential on the surface f(θ)=cos^2(θ).
Should it be Like this:f(r,θ)=(r/R)cos^2(θ)?
Likely.
But I am writing this as a trial and error...
How to show this is the answer...i.e. what should be the correct procedure of approach?
OK,this was extraced from a mathematical physics problem sheet.
I think the correct procedure is to apply uniqueness theorem.We know when the potential at every point of the surface is given,and the potential in that region obeys Laplace's (Here, Poisson'sequation),the potential function is unique.
So,I think it would be the same inside the sphere.
The spherical one,
u_xx+u_yy+u_zz=0
You need to use change of variables,
sqrt{x^2+y^2+z^2}=rho^2
z=rho*sin(phi)
y=rho*sin(theta)*sin(phi)
x=rho*cos(theta)*sin(phi)
But I do not remember what is done after this. I gave back the math book to the library. And I remember reading this section a little bit. I think what happens next is that you need to use Legendre polynomials, here. I do not remember the exact details.
You can go here, look at the very end of the article. I think it gives the solution to Laplace equation in spherical coordinates in the end.