MAGNETIZATION AND SPIN MAGNETIC MOMENTS

1
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
The coordinates of the line element dl are (a,
p/2, f).
dl
x
2
From equation 7 in the previous lecture we will
write A as
Note that J is tangential to the circle.
We will assume that r gtgt a, so then the distance
r-r can be approximated by
3
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.
4
The contribution to the vector potential dA is
then for a distributed magnetization
dm
r-r
dA
r
r
5
Since we can write
Equation (5) can be recast using product rule 7
from reference 2.
The second integral in eqn (6) can be converted
into a surface integral ( see ref. 2).
6
The conversion of the second volume integral to a
surface integral is done like this ( again see
ref. 2)
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 from the magnetic moments of the
fermions in the nucleus
7
The expression for A combining both these sources
is
We derived eqn (8) based on the assumption that
the size of the dipole, altltr, 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,
8

• REFERENCES
• Classical Electrodynamics, 2nd Edition, John
  David Jackson, John Wiley and Sons, 1975
• 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