Feb 22, 2001
Notes for Lecture
Notes for Lecture #16
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:
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
|
|
|
æ
ç
ç
ç
è
|
Y1-1( |
^
r¢
|
) |
Ö2
|
+ Y11( |
^
r¢
|
) |
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
|
|
| |
| |
| |
|
. |
| (7) |
As another example, consider the current associated with an
electron in the |nlm = 211ñ state of a H atom:
|
J(r¢) = |
-e(h/2p)
64 m pa5
|
e-r¢/a |
^
z
|
×r¢, |
| (8) |
where a here denotes the Bohr radius. Using arguments similar to
those above, we find that
|
A(r) = |
192 mpa5 r
|
|
ó
õ
|
¥
0
|
dr¢ r¢3 e-r¢/a |
r <
r > 2
|
. |
| (9) |
This expression can be integrated to give:
|
A(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
|
. |
| (10) |
File translated from TEX by TTH, version 2.20.
On 22 Feb 2001, 11:14.