Surface charge density on conducting cone with point charge inside

jozegorisek

1. The problem statement, all variables and given/known data
Calculate the surface charge density on a thin insulated and uncharged cone, which has a point charge inside of it on the cone axis. Furthermore, calculate the force between the point charge and the cone.


2. Relevant equations
The relevant equation is the Poisson equation
[tex]\nabla^2 \phi = \delta(\mathbf{r}-\mathbf{r_0})[/tex]
I'm not so sure about the appropriate boundary conditions although three things are certain:
1. The potential on the cone's surface is uniform (since the surface is conducting):
[tex]\phi|_{surface} = \phi_0[/tex]
2. The tangential component of the electrical field on the surface is zero
3. Since the cone was uncharged in the beginning, the total charge on the surface must remain zero:
[tex]\oint_S \rho dS = 0[/tex]
Since the Poisson equation is not separable in a way that the coordinate surfaces would coincide with the surfaces of the cone and since the method of images works only with plane or spherical surfaces, I presume this is a numerical problem.

3. The attempt at a solution
I see no other solution than numerical computing of the poisson equation via the finite element method.
[tex]\Delta f(x,y) \approx \frac{f(x-h,y) + f(x+h,y) + f(x,y-h) + f(x,y+h) - 4f(x,y)}{h^2}[/tex]

I can imagine computing the potential inside the cone this way but I have two problems:
1. What are the correct boundary conditions for the potential?
2. How should I obtain the surface charge distribution from the numerical data of the potential?

Thank you