Home

PHY 415: Electromagnetic Theory I
Prof. S. Teitel stte@pas.rochester.edu ---- Fall 2017

## Problem Set 1

Due Tuesday, September 26, in lecture

• Problem 1 [10 points]

Using Dirac delta functions in the appropriate coordinate system, express the following charge distributions as three dimensional charge densities ρ(r). One way to check your answer is to integrate your ρ(r) over all space and see that you get the correct total charge.

a) In spherical coordinates, a charge Q uniformly distributed over an infinitesmally thin spherical shell of radius R.

b) In cylindrical coordinates, a charge λ per unit length uniformly distributed over an infinitely long cylindrical surface of radius b.

c) In cylindrical coordinates, a charge Q spread uniformly over a flat circular disk of negligible thickness and radius R, centered in the xy plane at z=0.

d) The same as (c), but using spherical coordinates.

• Problem 2 [10 points]

The time averaged electrostatic potential of a neutral hydrogen atom is given by

 φ(r) = q e-αrr (1+ αr2 )
where r=|r| is the radial distance, q is the magnitude of the electron charge, and α-1=a0/2 with a0 the Bohr radius. Find the distribution of charge (both continuous and discrete) which will give rise to this potential, and interpret your result physically.

• Problem 3 [10 points]

Consider a plane polarized electromagnetic wave in the vacuum described by the vector and scalar potentials,

A(r, t) = Aoei(k·r-ωt),      φ(r, t) = φoei(k·r-ωt)

where the orientation of the vector Ao is arbitrary.

a) Using Maxwell's equations, find the relationship that must hold between Ao and φo.

b) Using the principal of gauge invariance, show that one can transform to a new but physically equivalent vector potential which is transversely polarized, i.e. Ao·k=0. Explicitly find the gauge transformation function χ that does this.

c) What is the scalar potential φ(r, t) in the gauge of part (b)?

• Problem 4 [10 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 4π ∫ d3r′ ∇′ ⋅ f(r′) |r-r′| ]

and

 f⊥(r) = ∇x [ 1 4π ∫ d3r′ ∇′ x f(r′) |r-r′| ]

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

 f(r) = 1(2π)3 +∞∫−∞ d3k eik ⋅ r f(k)

Using the results above, find expressions for the Fourier transforms of the curlfree (longitudinal) and divergenceless (transverse) parts of f(r). Your results should tell you why these parts have the names that they do!

It may help to know that the Fourier transform of 1/|r-r'| is (4π/k2)e-ikr', and that the Fourier transform of the Dirac delta function δ(r-r') is e-ikr'.