Problem Assignment #1 - Oregon State University
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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 Q uniformly distributed over a spherical shell of radius R. (b) In cylindrical coordinates, a charge λ per unit length uniformly distributed over a cylindrical surface of radius b. (c) In cylindrical coordinates, a charge Q spread 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 = q e -αr r 1+ αr 2 where q is the magnitude of the electronic charge, and α -1 = a 0 /2, a 0 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 √ 2πα 2 e -x 2 /2α 2 , prove that in the limit α → 0 D(x, α)= δ(x). 1
Transcript of Problem Assignment #1 - Oregon State University
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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