Mathematical and Computational Methods for Engineers
E155B, Spring 2002
Problem Set #3
(Diffusion, Laplace, and Poisson equations in 1-, 2-, and 3-D)
Date: 4/17/2002 Due: 4/24/2002
Reading: Kreyszig 11.1, 11.5, 11.9
Problem 1 Consider an electrostatics problem in three dimensions governed by the Laplace equation:
The region is a rectangular box 0 < x < a, 0 < y < b, 0 < z < c. Assume that all sides of the box, except one at z = c, are at ground potential j = 0. The potential distribution at the surface z = c is given by j(x,y,c) = f(x,y). Such a configuration can be used, for instance, to analyze electric fields inside a shielded metal box containing an exposed electronic circuit board mounted to one of the sides. Use separation of variables to obtain an expression for the electric potential j(x,y,z). [Hint: use “double” Fourier series]
Problem 2 a) Consider the conduction of heat in a circular metal ring with insulated surface. If x is measured along the ring, the ring may be regarded as a rod of length L with two of its ends joined. Although the temperatures at the two ends are not known, it is known that:
These are known as periodic boundary conditions. Assuming that the temperature distribution at t=0 is given by , solve for
b) Show that in steady-state the temperature distribution is uniform along the ring and that the final temperature is equal to the average value of the initial distribution.
c) For L=2, , and use MATLAB to plot five reasonably spaced snapshots of in time. Confirm that the steady-state temperature is what you expect it to be.
Problem 3 Two-dimensional problems with cylindrical symmetry can be readily solved in polar coordinates. Laplace equation written in polar coordinates is given by:
Consider a semicircular plate of radius a as shown below. Let the bounding diameter be kept at 0oC and the circumference at a fixed temperature T0. Use separation of variables to determine the steady-state temperature distribution [Hint: for the radial part of the separated equation try as a solution]
b) For and use MATLAB to create a 2-D
black & white colormap and a contour plot of the
temperature distribution.
Problem 4 p.628 Exercise 6d)
Problem 5 p.609 Exercise 14
Problem 6 a) Consider a rectangular region , with specified boundary conditions: . Solution to the Poisson equation can be generally obtained by expanding as:
and by assumingto be of the form:
In this exercise, assume, for simplicity, that is a constant f0 and determine using the suggested procedure.
b) A long dielectric in the shape of a rectangular prism with m and m is uniformly charged with charge density C/m3. Assume that all sides of the dielectric are grounded. Ignoring any variation in potential in the length-wise direction, determine the potential distribution .
c) Use MATLAB to make a three-dimensional surface plot of the potential in part b).