Feb 23, 2002

Notes for Lecture Notes for Lecture #15

Vector potentials in magnetostatics

The vector potential corresponding to a current density distribution J(r) is given by

A(r) = m0
4 p
ó
õ
d3r¢ J(r¢)
|r - r¢|
.
(1)
This expression is useful if the current density J(r) is confined within a finite region of space. Consider the following example corresponding to a rotating charged sphere of radius a, with r0 denoting the uniform charge density within the sphere and w denoting the angular rotation of the sphere:
J(r¢) = ì
í
î
r0 w× r¢
for r¢ £ a
0
otherwise
(2)

In order to evaluate the vector potential (1) for this problem, we can make use of the expansion:

1
|r - r¢|
=
å
lm 
4 p
2 l +1
r < l
r > l+1
Ylm( ^
r
 
)Ylm*( ^
r¢
 
).
(3)
Noting that
r¢ = r¢   æ
 ú
Ö

4p
3
 
æ
ç
ç
ç
è
Y1-1( ^
r¢
 
)
^
x
 
+ ^
y
 

Ö2
+ Y11( ^
r¢
 
)
^
x
 
- ^
y
 

Ö2
+ Y10( ^
r¢
 
) ^
z
 
ö
÷
÷
÷
ø
,
(4)
we see that the angular integral result takes the simple form:
ó
õ
dW¢
å
m 
Ylm( ^
r
 
)Ylm*( ^
r¢
 
)   r¢ = r¢
r
r   dl1.
(5)
Therefore the vector potential for this system is:
A(r) = m0 r0 w×r
3 r
ó
õ
a

0 
dr¢   r¢3 r <
r > 2
,
(6)
which can be evaluated as:
A(r) = ì
ï
ï
ï
ï
ï
í
ï
ï
ï
ï
ï
î
m0 r0 w× r
3
æ
ç
è
a2
2
- 3 r2
10
ö
÷
ø
for r £ a
m0 r0 w× r
3 r3
a5
5
for r ³ a
.
(7)

As another example, consider the current associated with an electron in a spherical atom. In this case, we assume that the current density is due to an electron in a bound atomic state with quantum numbers |n l ml ñ, as described by a wavefunction ynlml(r), where the azimuthal quantum number ml is associated with a factor of the form ei ml f. For such a wavefunction the quantum mechanical current density operator can be evaluated:

J(r) = -e (h/2p)
2 me i
(ynlml*Ñynlml-ynlml Ñynlml* ).
(8)
Since the only complex part of this wavefunction is associated with the azimuthal quantum number, this can be written:
J(r) = -e (h/2p)
2 me i rsinq
æ
ç
è
ynlml*
f
ynlml-ynlml
f
ynlml* ö
÷
ø
^
f
 
=
-e (h/2p) ml ^
f
 

me r sinq
|ynlml|2.
(9)

where me denotes the electron mass and e denotes the magnitude of the electron charge.

For example, consider the |nlm = 211ñ state of a H atom:

J(r¢) = -e(h/2p)
64 m pa5
e-r¢/a    ^
z
 
×r¢,
(10)
where a here denotes the Bohr radius. Using arguments similar to those above, we find that
A(r) =
-e(h/2p) m0 ^
z
 
×r

192 mpa5 r
ó
õ
¥

0 
dr¢   r¢3   e-r¢/a    r <
r > 2
.
(11)
This expression can be integrated to give:
A(r) =
-e (h/2p) m0 ^
z
 
×r

8 mpr3
é
ê
ë
1 - e-r/a æ
ç
è
1 + r
a
+ r2
2 a2
+ r3
8 a3
ö
÷
ø
ù
ú
û
.
(12)


File translated from TEX by TTH, version 2.20.
On 23 Feb 2002, 14:03.