February 5, 2000
PHY 712 { Problem Set # 8
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:
X
l
( )Y ( 0 )
0 ) = 4
Ylm
lm
2l + 1 m= l
Pl (^r ^r
and
jr
1
r0j
1 X
X
=
l
^r
^r
(1)
4 r<l
( )Y ( 0 ):
Ylm
lm
l
+1
l=0 m= l 2l + 1 r>
^r
^r
(2)
2. Consider a charge distribution of the form:
(r; ; ) = 0 r2 e
r cos2 ( );
(3)
where 0 and are constants.
(a) Express (r; ; ) as a sum of radial functions time spherical harmonic functions
in the form:
(r; ; ) = lm (r) Ylm (; ):
(4)
X
lm
(b) Using the results of the previous problem, nd the corresponding electrostatic
potential which vanishes at 1.