Homework #3

Due Friday of week 3 at 5pm

1. From Mahajan text book: Problems 3.36, 3.37, 3.38. The “enjoyable additional problem” suggested in 3.36 is not required.

2. Perform the integration step of the Green's function solution for the stretched string example given in class. I suggest performing the integration numerically (on a computer). Plot both your numerical result and the exact solution. The form of the exact solution should be calculated analytically (pen & paper, show your working). It is possible to compute the Green's function integral analytically, or you can solve the inhomogeneous ODE with some other analytical approach.

To create a piecewise function in PyLab, here is some example code to play with (make sure you understand how the code is working):

# CODE FOR A SIMPLE PIECEWISE FUNCTION
from __future__ import division
from pylab import *

L = 1 #meters
step = 0.01*L #meters
x = arange(0,L,step)
G = zeros(x.size)
i = G.size/2
G[0:i]=x[0:i]
G[i:G.size]=2*x[i] - x[i:G.size]
show(plot(x,G))

3. Show that ∇²G = δ(r - ζ) is satisfied by G = 1/(4π|r - ζ|) when rζ.

4. Consider the following problem: ∇²φ = ρ/ε in the region above a metal surface. The metal surface is in the x-y plane. At the metal surface φ(r)=0.

Question: What is the Green's function for this problem?

Hint: If you are not familiar with “the method of images”, please read the appropriate section in the EM textbooks by Griffiths or Jackson.