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Point charge inside thin insulated cone

  1. Nov 17, 2011 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations
    The relevant equation is the Poisson equation[tex]\nabla^2 U = \delta(\mathbf{r}-\mathbf{r_0})[/tex]
    Since the cone surface is insulated and not grounded, I presume that the boundary conditions on the cone surface are:
    [tex](\mathbf{E_{out}}-\mathbf{E_{in}})\cdot \mathbf{\hat{n}} = \frac{\sigma}{\epsilon_0}[/tex]
    where [tex]\sigma[/tex] is the surface charge density.

    3. 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:
    [tex]\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' [/tex]
    where [tex]\phi[/tex] is the potential, [tex]<\phi>_S[/tex] the average of the potential on the surface and [tex] G_N[/tex] 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
     
  2. jcsd
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