Electrostatics: Surface charge density-numerical solution

juvan

Hi, I'm very new here, 10min old, but the problem with my knowledge, or better, lack there of, at this time is very hard, so I need help. I am trying to numerically calculate a certain electrostatic problem (attached an image "prob.jpg" to clarify). I have a disk at potential V0, and with this equation

[itex]V(\rho,\varphi)=\frac{1}{4\pi\epsilon_0}\int\int \frac{\sigma(\rho',\varphi')\rho'd\rho'd\varphi'}{ \sqrt{\rho^2+\rho'^2 - 2\rho\rho'cos(\varphi-\varphi')}} (1)[/itex]
I'm trying to then numerically calculate for σ (attached 2 images "disk.png","charge_on_disk.png" to clarify), this integral equation. So I basically get an equation A*σ=b (A-matrix, σ,b-vectors). As for [itex] A_{ij} [/itex] where [itex]i \neq j[/itex], i think there should be no problem with equation (1), but what about [itex]A_{ii}[/itex], what is.. or how should I set up the equation for these terms, since if I used the same equation (1), I would get [itex]\frac{something}{0}[/itex]. I have no idea where to even start the thought process, to go about setting up the equation.
For the sake of being brief, I will stop, and ask that if I have made anything unclear just say so and I will try to explain further(better).

Thank you.

EDIT: I tried setting all the terms to 1 [itex]A_{ii}=1[/itex] and the resulting graph seems correct, but that's just guess work, I have no basis for setting it to a constant 1.

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