Thread: potential problem

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?

Originally Posted by kolahalb

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?

I do not know if this is a physics questions or a math question. When you say potention what do you want me to do?

I am assuing you want to solve the Dirichlet problem to the Laplace Equation where the boundry is a circle? Is that it?

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.

OK,I was wrong.I have to solve Laplace equation first for spherical polar co-ordinates.Then I am to feed the boundary condition.

Originally Posted by kolahalb

OK,I was wrong.I have to solve Laplace equation first for spherical polar co-ordinates.Then I am to feed the boundary condition.

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.

x=r*sin theta cos phi
y=r*sin theta sin phi
z=r*cos theta