Point charge inside thin insulated cone

[jozegorisek]

Homework Statement

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. Also, since this is a theoretical problem, the choice of parameters like cone dimensions is irrelevant.

Homework Equations

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:
$$(\mathbf{E_{out}}-\mathbf{E_{in}})\cdot \mathbf{\hat{n}} = \frac{\sigma}{\epsilon_0}$$
where $$\sigma$$ is the surface charge density.

The Attempt at a Solution

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. Or is there a way to solve this with variable separation anyhow?

Since there is a point charge inside the cone surface I understand that I am searching for the Green's 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.

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.

The formula for Green's function with Neumann boundary conditions (from Jackson) is:
$$\phi(x) = <\phi>_S + \frac{1}{4 \pi \epsilon_0} \int_V \rho(x') G_N(x,x')d^3x' + \frac{1}{4 \pi}\oint \frac{\partial \phi}{\partial n'} G_N da'$$
where $$\phi$$ is the potential, $$<\phi>_S$$ the average of the potential on the surface and $$G_N$$ is the Green's function for Neumann conditions.
1. How do I find the potential with boundary conditions (surface charge) which are the consequence of the very potential I'm looking for?
2. How to calculate the potential (Green's function)?

Thank you

 

[mark726]

for your question and for providing the relevant equations. Let me first clarify a few things about the problem at hand. The key concept in this problem is that of image charges. Since the cone is insulated and uncharged, it will remain uncharged regardless of the presence of the point charge inside it. This means that the charge distribution on the cone will be the same as if the point charge was not there, and the force between the point charge and the cone will also be the same.

Now, to answer your questions:

1. Finding the potential with the given boundary conditions can be done by using the method of images. This involves creating a "mirror" point charge outside the cone, such that the electric field at the cone's surface is the same as the one produced by the point charge inside the cone. This will ensure that the boundary conditions are satisfied. You can then use the method of separation of variables to solve for the potential in the region between the cone and the mirror charge.

2. The potential (Green's function) can be calculated using the method of separation of variables. This involves writing the potential as a sum of terms that satisfy Laplace's equation and the boundary conditions. The coefficients of these terms can be determined by using the boundary conditions and the method of images.

I hope this helps you in your solution. If you have any further questions, please don't hesitate to ask. Good luck!