Simulating Electrostatic potential

[philm001]

Hello everyone,

I am currently learning how to use a simulation C++ library (for those wondering, it is the deal.II library) by simulating a "simple" problem where I have a charged parallel plate in free space and I am solving for the electrostatic potential around the plates.

For those that do not know, the deal.II library solves PDEs via the weak form. Once I have it in the weak form, I approximate the integral by quadrature.

Now, I am new to finite element solvers such as deal.II so I am not too sure how I can properly setup my problem. I did make an attempt and I would like to check with the community on how to setup the physics. I am pretty sure that my coding is correct. However, I need to be sure that my equations that I am using are correct hence me asking this question to the physics forum and not the deal.II forum.

So, as it stands, I am currently solving for this equation:

-∇²*V=ρ / ε

where V is the electrostatic potential and ρ is the charge density and ε is the electrical permittivity. (since I am working in the 2-D space, ρ has the units C/m^2)

With this method, I am able to determine the weak form and reduce it to quadrature. However, my only issue is that I am not sure how I should interpret ρ. Would ρ be the charge density on the plate? Or should I calculate ρ for every point around the plates. I know that V and E are both related and that E decreases as the distance from one point to another decreases hence the V decreases. So, I am wondering, how is the distance taken into account? Is it in the V term or the ρ a function of distance (r in this case).

Or, is this a terrible way of thinking about it and I can solve it with a better method in the simulator?

 

[Orodruin]

Rho is the volume charge density in the volume you are solving the PDE in. If you have a charge surface density on a plate at the boundary, this will translate into the boundary conditions. If you have a charged plate inside your volume you need to approximate the surface density with some finite but thin volume density to solve it numerically.

-∇2*V=ρ / ε

The * should not be here. In fact, it just makes it more unclear what you actually mean (I know what you mean because I know Gauss's law, but keep these things in mind).