PDE questions

Modelling the Poisson equation

Adapted from Mark Newman’s book “Computational Physics, p. 412

In the tutorial we have used the relaxation method to solve Laplace’s equation for electrostatics:

\[\begin{equation} \nabla^2\phi = 0 \end{equation}\]

A more general form of this equation is Poisson’s equation:

\[\begin{equation} \nabla^2\phi = -\frac{\rho}{\epsilon_0}, \end{equation}\]

which governs the electric potential in the presence of charge density \(\rho\).

Assume, as in the tutorial example for Laplace’s equation, that there is a 1 metre square. However this time all four walls of the square at fixed at zero volts and that there are two 20cm x 20cm charged areas in the box. One charge has a density +1Cm\(^{-2}\), the other has a density -1Cm\(^{-2}\).

1. Modify the code for the Laplace’s equation example to solve this problem using the relaxation method.