February 7, 2001
PHY 712 – Problem Set # 9
Read Chapter 3 of Jackson.
1. Convince yourself that Eqs. 3.62 and 3.70 are correct by expanding the expressions to
second order. That is, verify the following:
Pl (r̂ · r̂0 ) =
and
l
4π X
Y ∗ (r̂)Ylm (r̂0 )
2l + 1 m=−l lm
∞ X
l
l
X
1
4π r<
∗
0
=
l+1 Ylm (r̂)Ylm (r̂ ).
|r − r0 | l=0 m=−l 2l + 1 r>
(1)
(2)
2. Consider a charge distribution of the form:
ρ(r, θ, φ) = ρ0 r2 e−αr cos2 (θ),
(3)
where ρ0 and α are constants.
(a) Express ρ(r, θ, φ) as a sum of radial functions time spherical harmonic functions
in the form:
X
ρ(r, θ, φ) =
ρlm (r) Ylm (θ, φ).
(4)
lm
(b) Using the results of the previous problem, find the corresponding electrostatic
potential which vanishes at ∞.