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.)