Page 279 - Electrical Properties of Materials

Microscopic theory (phenomenological) 261
B
magnetic field (Fig. 11.1). The magnetic field produces a torque μ × B on
m
the magnetic dipole. Noting that the rate of change of the angular momentum
is equal to the torque we may obtain, with the aid of eqn (11.8), the equation
dμ/dt =(e/2m)(μ × B). (11.9a)

In a constant magnetic field the solution of this equation gives precession μ m
around the magnetic field at a frequency of

ω L = eB/2m, (11.9b) θ
which is called the Larmor frequency. If the magnetic dipole precesses, some
electric charge must go round. So we could use eqn (11.6) to calculate the
magnetic moment due to the precessing charge. Replace ω 0 by ω L ; we get Fig. 11.1
A magnetic dipole precessing around
2 2
Br e a static magnetic field.
(μ m ) ind = , (11.10)
4 m
where r is now the radius of the precessing orbit. The sign of this induced
magnetic moment can be deduced by remembering Lenz’s law. It must oppose
the magnetic field responsible for its existence.
We are now in a position to obtain M from the preceding microscopic con-
siderations. If there are N a atoms per unit volume and each atom contains Z
electrons, the total induced magnetic dipole moment is
M = N a Z(μ m ) ind . (11.11)

Hence, the magnetic susceptibility is
2 2
M N a Ze r μ 0
χ m = =– . (11.12)
H 4m
–5
Rarely exceeding 10 , χ m given by the above equation is a small number,
but the remarkable thing is that it is negative. This is in marked contrast
with the analogous case of electric dipoles, which invariably give a positive
contribution. The reason for this is that the electric dipoles line up, whereas ∗ The electric susceptibility can also be
∗
the magnetic dipoles precess in a field. Magnetic dipoles can line up as well. negative, but that is caused by free
charges and not by electric dipoles.
The angle of precession will stay constant in the absence of losses but not
otherwise. In the presence of some loss mechanism the angle of precession
gradually becomes smaller and smaller as the magnetic dipoles lose energy; in
other words, the magnetic dipoles do line up. They will not align completely
because they occasionally receive some energy from thermal vibrations which
frustrates their attempt to line up. This is exactly the same argument we used
for dielectrics, and we can therefore use the same mathematical solution. Re-
placing the electric energy in eqn (10.13) by the magnetic energy, we get the
average magnetic moment in the form,
μ m μ 0
μ m = μ m L(a), a = H. (11.13)
kT
Denoting by N m the number of magnetic dipoles per unit volume, we get for
the total magnetic moment,
M = N m μ m . (11.14)