Page 278 - Electrical Properties of Materials

260 Magnetic materials

11.2 Macroscopic approach
By analogy with our treatment of dielectrics, I shall summarize here briefly the
main concepts of magnetism used in electromagnetic theory. As you know, the
presence of a magnetic material will enhance the magnetic flux density. Thus
the relationship
B = μ 0 H, (11.1)

M is the magnetic dipole moment valid in a vacuum, is modified to
per unit volume, or simply, mag-
netization. B = μ 0 (H+M) (11.2)

in a magnetic material. The magnetization is related to the magnetic field by
the relationship

χ m is the magnetic susceptibility. M = χ H. (11.3)
m
Substituting eqn (11.3) into eqn (11.2) we get

μ r is called the relative permeabil- B = μ 0 (1 + χ m ) H = μ 0 μ r H. (11.4)
ity.
11.3 Microscopic theory (phenomenological)

Our aim here is to express the macroscopic quantity, M, in terms of the
properties of the material at atomic level. Is there any mechanism at atomic
level that could cause magnetism? Reverting for the moment to the classical
picture, we can say yes. If we imagine the atoms as systems of electrons orbit-
ing around protons, they can certainly give rise to magnetism. We know this
from electromagnetic theory, which maintains that an electric current, I, going
∗ It is an unfortunate fact that the usual round in a plane will produce a magnetic moment, ∗
notation is μ both for the permeability
and for the magnetic moment. I hope μ m = IS, (11.5)
that, by using the subscripts 0 and r for
permeability and m for magnetic mo-
ment, the two things will not be con- where S is the area of the current loop. If the current is caused by a single
fused. electron rotating with an angular frequency ω 0 , then the current is eω 0 /2π,
and the magnetic moment becomes

eω 0 2
μ m = r , (11.6)
2
where r is the radius of the circle. Introducing now the angular momentum
2
= mr ω 0 , (11.7)
Remember that the charge of the we may rewrite eqn (11.6) in the form,
electron is negative; the magnetic
e
moment is thus in a direction op- μ m = . (11.8)
posite to the angular momentum. 2m
We now ask what happens when an applied magnetic field is present. Con-
sider a magnetic dipole that happens to be at an angle θ to the direction of the