# Numeric method to calculate the charge distribution on a conducting surface?

If I have an arbitrary (closed?) conducting surface and a nearby charge density, is there a simple numeric way of computing the induced charge distribution on the surface?

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A metallic surface or a dielectric or what? –  BebopButUnsteady Oct 31 '11 at 23:57
It said in the title it's a conductor - I edited that into the question. –  David Z Nov 1 '11 at 0:43
If you're interested in more information about numerical solutions, you could ask on scicomp.stackexchange.com –  David Ketcheson Jan 13 '12 at 20:04

## 1 Answer

There is no simple way. The "standard" way is to solve Poisson equation with proper boundary conditions (constant $\varphi$ at the surface). Out of potential distribution it is easy to extract charge distribution.

For simple shapes (infinite plane, sphere, etc) it is possible to solve the problem analytically. For arbitrary shape there is no simple solution.

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However, the numerical solution of Poisson's equation is to my knowledge a very tractable computational problem, since you can just simulate the heat equation. –  BebopButUnsteady Nov 1 '11 at 14:29
For those who don't know, the famous way to solve analytically for simple shapes only is the image charge method. –  Steve B Nov 6 '11 at 16:43