# Phys690: Homework Assignment 6

4/11/07
==> * *** Due on 4/25, Wed in class** <==

** (1). Numerical solution of Poisson's equation.**

We consider the case of solving Laplace's equation in the presence of
an interior charge density. The
static potential in this case is given by Poisson's equation. We will
consider a square of linear dimension L=25 whose boundary is fixed at
a potential of V=10.

**Finite-differencing.**
First assume the charge distribution is restricted to a 5x5 square in
the middle of the large square. Assume each interior cell in the 5x5
has a uniform charge density rho such that the total charge is 1. We
want to compute the potential inside the large square. Work out the
corresponding (Jacobi or Gauss-Seidel) scheme. Write a program to
compute the potential distribution. Use a finite difference grid that
coincides with the cells, i.e., of dimension 25x25. Qualitatively
justify your results. Now use a finer mesh. Compare the result with
that of the coarse mesh. Where do the two results agree the best and
the worst? Why?

**Point charge.**
What if the unit charge is distributed in the center only in a 1x1
instead of 5x5 square? Use Gauss-Seidel to compute the potential
distribution. Compare your result to that of a point unit charge
enclosed by a circle of radius 15 which is fixed at V=10. First plot
the two potentials along the x-axis, then along the diagonal
direction. Justify/explain your results. (Hint: To obtain the point
charge potential, recall that the system we are dealing with is
confined in a 2-dimensional space. That is, in real life (3-d) it
corresponds to a infinite, straight wire rather than a point. Its
potential satisfies the 2-dimensional Poisson's equation whose
solution is in the form of `ln' rather than `1/r' --- to see how these
solutions work, apply the Laplacian in 2-d and 3-d respectively, and
see how the Cartesian components cancel.)