How can I calculate the charge density of a conductor in an electric field? I have tried many ways but I always seem to not understand how to do it. How can I do this mathematically?
EDIT: The conductor is isolated from other conductors. The external electric field is produced from insulators.(the insulators charge distribution doesnt vary no matter what.) The electric field that's acting on the conductor is given and the potential and charge distribution both have to be found. The surface is parametric. The conductor is hollow and has charges present in it.( again the charges are stationary and do not move). THE TOTAL CHARGE OF THE CONDUCTOR IS ZERO.
Best Answer
The question is interesting as one is usually tasked with finding the potential or the field for a given charge distribution.
In its full generality this is a complicated problem not done analytically except in some specialized cases where one can use symmetry arguments to determine some feature of the fields. If there is symmetry, Gauss's law or the method of images might work, although these approaches usually start with known charge distributions and obtain the ensuing field or potential subject to the appropriate boundary conditions.
When solving for the potential the simplest general numerical method is often to use Poisson's equation $\nabla^2 V=-\rho_f/\epsilon$, where $\rho_f$ is the local density of free charge. I do not know if this powerful method can be inverted easily to find the densities given the potential (and hence the field). It is easily applied with known charge densities so this could be an obvious starting point.
If you try to invert Poisson's equation, it will have to be interatively and you will benefit from the condition that, to a good approximation $V$ is constant inside a conductor, or alternatively there is no free charge or $\vec E$ field inside the conductor.