MAGNETIZATION AND SPIN MAGNETIC MOMENTS
Among macroscopic objects we find those which have a permanent magnetic
field, even if there are no obvious macroscopic currents. On the atomic and
nuclear level we find particles, such as the electron, which themselves act as
permanent magnets. These particles are said to have a magnetic dipole moment,
or simply a magnetic moment for short. We would like to be able to express
magnetic fields, B, in term of the vector potential, A. We will work out a specific
case, but it turns out that it can easily be generalized. Consider the vector
potential created by a circular current loop of radius a. There is a constant current
I in the loop. The loop has a cross sectional area S so that I = JS, where J is the
magnitude of the current density
P
z
r
Calculate the field A at point p.
The coordinates of P are (r,q,g)
q
I
y
r’
f
dl
x
The coordinates of the line element dl
are (a, p/2, f).
From equation 7 in the previous lecture we will write A as
 
  1 3 J (r ' )
A(r )   d r '   (1)
c
| r  r '|

p
p
3
ˆ
d r '  Sdl , J  J (cos(f  ) x  sin( f  ) yˆ )
2
2
Note that J is tangential to the circle.

r  r (sin q cos g xˆ  sin q sin g yˆ  cos q zˆ )

r '  a( cos f xˆ  sin f yˆ )
We will assume that r >> a, so then the distance |r-r’| can be approximated by
 
a
| r  r ' | r (1  sin q (cos g cos f  sin g sin f ))
r
2p
  Ia 2
A( r )  2 sin q  df ( xˆ sin f  yˆ cos f )(cos g cos f  sin g sin f )
cr
0
In the last step we used the approximat ion,
1
 1 x
1 x
for small x.
  Ia 2p
Ia 2p
A(r ) 
sin q ( xˆ sin g  yˆ cos g ) 
sin q ( xˆr sin g  yˆr cos g )
2
3
cr
cr
  Ia 2p
A(r ) 
( yxˆ  xyˆ ), let
3
cr
 Ia 2p
m
zˆ, (2)
c
 
  m
r
so then A(r )  3 , (3) for a  r.
r
m is called the magnetic dipole moment. Although we worked this out for a
specific current distribution eqn (3) gives the correct leading order term for A for
any current distribution. For macroscopic media then magnetic moment might
be due to a domain. In the atom, the intrinsic spins of the electrons or other
fermions will contribute to the vector potential A through eqn (3).
We can define a magnetization M to be the magnetic dipole density.

 dm

M
, where dV is a volume element and dm
dV
is the magnetic dipole moment of the element.
The contribution to the vector potential dA is then for a distributed magnetization
dm
  
 dm
 (r  r ' )
dA 
  3
| r  r '|
|r-r’|
dA
r’
r

A
V
  
  
dm  (r  r ' )
M  (r  r ' )
  3   dV
  3 , (4)
| r  r '|
| r  r '|
V
Since we can write
 
r r'
1


'
 
 
| r  r ' |3
| r  r '|
 

1
A(r )   dV M  '   ,
| r  r '|
V
(5)
Equation (5) can be recast using product rule 7 from reference 2.

 
  fV  f (  V )  V  f , where f is a scalar function and

V is a vector function.
 
 
 
'M (r ' )
M (r ' )
A(r )   dV     dV '    , (6)
| r  r '|
| r  r '|
V
V
The second integral in eqn (6) can be converted into a surface integral ( see
ref. 2).

 
 
 
'M (r ' )
M (r ' )  dS
A(r )   dV    
  , (7)
| r  r '|
| r  r '|
V
S
The conversion of the second volume integral to a surface integral is done
like this ( again see ref. 2)
Evaluate

dV


v
.


Choose any vector c.
V
Then from the divergence theorem we have

 
 
 dV   ( v  c)   ( v  c)  dS. We can rearrange the cross products such as
V
S


 
dV
(
c

(


v
)
v

(


c
)

 


c

(d
S

v
).
But
if
c
is a constant

V
S


 
c  (  dV   v   v  dS)  0.
V
S
The surface integral in eqn (7) becomes important if there is a discontinuity in the
magnetization M. This happens in textbook examples. A physical magnetization
will change continuously, so the first volume integral of eqn (7) is the only
important term for our discussion of atomic nuclei. In the nucleus we have
charged particles in motion, contributing to a conduction current Jc (r’) that we
included before, and now also a term coming
the magnetic moments of the
 from

fermions in the nucleus
c'  M(r ')
The expression for A combining both these sources is
 
 
  1
J (r ' )  c'M (r ' )
A(r )   dV c
, (8)
 
cV
| r  r '|
We derived eqn (8) based on the assumption that the size of the dipole, a<<r, is
small. For the elementary fermions, such as electrons and quarks, this should be
valid because they are assumed to be point particles. The spins and magnetic
moments of these fermions are understood in terms of relativistic quantum
mechanics. Classical pictures can only be taken as a qualitative guide in
developing a mental picture of spin or magnetic moments of the charged
fermions. The current density which gives rise to the vector potential is thus,
 
 
 
J (r ' )  J c (r ' )  c'M (r ' ), (9)
REFERENCES
1) “Classical Electrodynamics”, 2nd Edition, John David Jackson, John
Wiley and Sons, 1975
2) “Introduction to Electrodynamics”, 2nd edition, David J. Griffiths,
Prentice-Hall, 1989 ( This is an excellent text book. )
3) “Electrodynamics”, Fulvio Melia, University of Chicago Press, 2001
4) “Relativistic Quantum Mechanics and Field Theory”, Franz Gross, John
Wiley and Sons, 1993