Charge Density in Particle-in-Cell Simulation

[jj442434]

I am creating a particle in cell simulation that models an electron plasma in a cylindrical container. Part of this process is assigning charge density to grid points based on the position of each particle. My question is this: for the examples of these simulations that I have seen, the charge density is given by the amount of charge on a grid point divided by the volume of a unit cell, or: $$\rho=\frac{q}{d\tau}$$

which for cylindrical coordinates is: $$\rho=\frac{q}{r\bigtriangleup_r \bigtriangleup_z \bigtriangleup_{\theta}}$$, where the deltas represent the grid spacing that we have specified. For r and z, these are well defined.

however, we are taking the system to be symmetric about $\theta$, meaning that this simulation is basically in two dimensions, r and z, and that $\bigtriangleup_{\theta}=0$. How would I resolve this, because it looks to me like this would blow up?

 

[Vailanor]

The resolution to this issue lies in the fact that, due to the symmetry of the system, we can effectively treat it as a two-dimensional system, with regard to the charge density. This means that we don't need to include \theta in our calculation of the charge density, and therefore \bigtriangleup_{\theta} can be set to 1. This allows us to calculate the charge density as: \rho = \frac{q}{r \bigtriangleup_r \bigtriangleup_z}.