# 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 **r** ≠ **ζ**.

**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.