Problem Assignment #1
A. W. Stetz
October 4, 2007
These problems are due Friday, Oct. 12.
1. Using the Dirac delta functions in the appropriate coordinates, express
the following charge distributions as three-dimensional charge densities
ρ(x).
(a) In spherical coordinates, a charge Quniformly distributed over a
spherical shell of radius R.
(b) In cylindrical coordinates, a charge λper unit length uniformly
distributed over a cylindrical surface of radius b.
(c) In cylindrical coordinates, a charge Qspread uniformly over a
flat circular disc of negligible thickness and radius R.
(d) The same as part (c), but using spherical coordinates.
2. The time-averaged potential of a neutral hydrogen atom is given by
φ=qe−αr
r1 + αr
2
where qis the magnitude of the electronic charge, and α−1=a0/2, a0
being the Bohr radius. Find the distribution of charge (both continu-
ous and discrete) that will give this potential and interpret your result
physically.
3. Given the definition,
D(x, α) = 1
√2πα2e−x2/2α2,
prove that in the limit α→0
D(x, α) = δ(x).
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