Calculating the charge distribution on the surface of an assymetric conductor

[Chitran]

How do I calculate the charge distribution on the surface of any asymmetric closed conducting surface? Is it possible for me to calculate the surface charge density 'σ' as a function of '$\bar{r}$' the position vector in a spherical co-ordinate system in space, provided I know that the conductor has been qiven a net charge 'Q' and the equation of the conductor in space is ((x/a)^2)+((y/b)^2)+((z/c)^2)=1...

 

[NoPoke]

For any conducting surface you will have to resort to an approximation.

In brief: use a numerical method to establish the Electric field for the system. Use that E field result to establish surface charge distribution.

 

[Chitran]

But how do I calculate $\bar{E}$ if I don't know my σ? It's like the chicken egg problem except that it's not what comes first that matters, but I need one to know the other. All I know unfortunately is the shape of the conductor and the total charge Q which according to the uniqueness theorum has a unique way of settling on the surface in the abscence of any external electric field...

 

[NoPoke]

you don't need to know anything about the charge. Field lines are normal to a conducting surface and themselves normal to equipotentials. This means that you can set up a grid with random starting values for potential and by recalculating each value in turn end up closer to the correct solution.

a quick google gave this http://www.physics.hku.hk/~phys3231/pdf/P1%20-%20Static%20Electric%20Field%20-%20Laplaces%20equation%20in%202D.pdf The 3d process is very similar.

Also http://www.ece.msstate.edu/~donohoe/ece3323analytical_numerical_techniques.pdf you need the last few pages.

The procedure looks hard but really isn't: Though it is a long time since I last did one!

 

[gabbagabbahey]

Just solve Laplace's equation for the potential. You know that the potential on the surface is a constant (which you may as well set to zero) and that the potential at large distances should look like that of a point charge, so you have your 2 boundary conditions. From the potential, you can determine the charge density.