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, expressthe following charge distributions as three-dimensional charge densitiesρ(x).

(a) In spherical coordinates, a charge Q uniformly distributed over aspherical shell of radius R.

(b) In cylindrical coordinates, a charge λ per unit length uniformlydistributed over a cylindrical surface of radius b.

(c) In cylindrical coordinates, a charge Q spread uniformly over aflat 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

r

(1 +

αr

2

)where q is 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 resultphysically.

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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