Phys690: Homework Assignment 6


==> 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.)