# I Induced charge density -- non-zero potential case

1. Apr 30, 2017

### mertcan

Hi, Let's think 2 arbitrary shape conductors with non zero charged. If these 2 conductors are closed, there will be induced charge density over surfaces of these conductors. I have not seen such an example, instead there are lots of problems which involve zero(grounded) potential case and method of images theorem is applied. So, I am asking how can we mathematically find induced charge density over conductors' surfaces including non zero potential case(not grounded)?
Thanks....

Last edited by a moderator: Apr 30, 2017
2. Apr 30, 2017

### Henryk

Finding charge density distribution involves solving Poisson's equation $\nabla ^2 U = - \frac {\rho}{\epsilon \epsilon_0}$.
This equation can be solved analytically only in some simple cases when the geometry is rather regular (e.g. planar, cylindrical, spherical, etc.) and these you find in textbooks. In other cases, you can get a power (or other) series expansion. But if you want to solve the Poisson's equation for a completely arbitrary shapes (conductors, dielectrics) you have to use Finite Element Analysis or similar methods.

3. Apr 30, 2017

### mertcan

Then, please let me ask the following question: I am really looking for that kind of examples, could you tell me how I can find some examples and solutions related to calculating charge density with finite element analysis ?