Point charge inside insulated thin cone

[jozegorisek]

My problem is the following:There is a point charge inside a thin uncharged and insulated metal cone. Calculate the charge distribution on the cone and the force between the point charge and the cone.
I presume "thin cone" means only the infinitely narrow
surface of a cone.The relevant equation is the Poisson equation$$\nabla^2 U = \delta(\mathbf{r}-\mathbf{r_0})$$
Since the cone surface is insulated and not grounded, I presume that the boundary conditions on the cone surface are of the Neumann type, that is
$$\frac{\partial U}{\partial \mathbf{\hat{n}}} = \mathbf{E} = 0$$

I apologize for possibly trivial questions, the thing is quite some time has passed since I last studied electrostatics therefore I am very rusty in this subject. So I just want to clear a few things up.
I have studied Jackson's Classical electrodynamics a bit, to try find a way towards a solution, but I am a bit lost at this point.

Had the problem been something without the point charge it would be no problem. It would mean solving the Laplace equation in cylindrical coordinates and the result would probably be something like a sum of sine/cosine*bessel function terms.

Since there is a point charge inside the cone surface I understand that I am searching for the Green function with Neumann boundary conditions for the cone surface. This is something I have never dealt with before and I am grateful for any direction/advice you can give me. I only want to know if I am thinking in the right direction and how I should approach this.

Thank you

 

[Liquidxlax]

a little hard to solve without knowing anything about the dimensions of the cone like the r to h ratio

 

[jozegorisek]

It's a theoretical problem so the choice of $$\frac{r}{h}$$ is arbitrary. I'm not looking for a complete solution of the problem, just for some insight how to approach the problem. How do you calculate the Green function for this kind of boundary geometry and conditions?