Surface charge density on conducting cone with point charge inside

[jozegorisek]

Homework Statement

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.

Homework Equations

The relevant equation is the Poisson equation
$$\nabla^2 \phi = \delta(\mathbf{r}-\mathbf{r_0})$$
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):
$$\phi|_{surface} = \phi_0$$
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:
$$\oint_S \rho dS = 0$$
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.

The Attempt at a Solution

I see no other solution than numerical computing of the poisson equation via the finite element method.
$$\Delta f(x,y) \approx \frac{f(x-h,y) + f(x+h,y) + f(x,y-h) + f(x,y+h) - 4f(x,y)}{h^2}$$

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

 

[mark726]

for your post! It seems like you have a good understanding of the problem and the relevant equations. As you mentioned, the Poisson equation is not separable in this case, so numerical methods are the best approach.

For the boundary conditions, you are correct that the potential on the cone's surface should be uniform, as it is a conducting surface. In addition, since the cone is insulated, the potential at infinity should be zero. This means that the boundary conditions for the potential should be:

1. On the surface of the cone: \phi|_{surface} = \phi_0
2. At infinity: \phi|_{\infty} = 0

As for obtaining the surface charge distribution from the potential, you can use the relation \rho = -\nabla^2\phi. This means that you can calculate the surface charge density at each point on the surface by taking the numerical derivative of the potential at that point. Then, you can integrate over the entire surface to find the total surface charge.

To calculate the force between the point charge and the cone, you can use the formula F = qE, where E is the electric field at the point charge's location. You can calculate the electric field using the potential and the relation E = -\nabla\phi.

I hope this helps and good luck with your calculations!