Problem Set 1

Due Monday, September 23, in lecture

Problem 1 [10 points]
Suppose the electric and magnetic fields, E and B, are given by scalar and vector potentials, and A, that are not in the Lorentz gauge. Show that one can always find a gauge transformation that takes one to new potentials which are in the Lorentz gauge. Specifically, if is the scalar function of the gauge transformation, find an equation to solve for in terms of the orignial and A, such that the new ´ and A´ are in the Lorentz gauge.
Problem 2 [20 points]
In lecture we discussed the decomposition of any vector function f(r) into its curlfree and divergenceless parts, f_||(r) and f(r), as given by Helmholtz's theorem:
f(r) = f_||(r) + f (r)
where

f_||(r) = — [ 1
4
d³r´ ´ · f(r´)
| r-r´|
]

and

f(r) = [ 1
4
d³r´ ´ f(r´)
| r-r´|
]

Consider f(k), the Fourier transform of f(r):

f(r) = 1
(2)³ +

- d³k e^{ik · r} f(k)

Using the results above, find expressions for the Fourier transforms of the curlfree (longitudinal) and divergenceless (transverse) parts of f(r).
You will need to know that the Fourier transform of 1/|r| is 4/k², and that the Fourier transform of the Dirac delta function (r) is e^{ik · r}

Last update: Tuesday, August 21, 2007 at 1:47:10 PM.