Page 269 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 251

The magnetization is M = χ m H,or39, 000 A/m. The alternate ways of relating B
and H are, first,

B = µ 0 (H + M)
or
−7
0.05 = 4π × 10 (796 + 39, 000)
showing that Amperian currents produce 49 times the magnetic field intensity that
the free charges do; and second,

B = µ r µ 0 H
or
0.05 = 50 × 4π × 10 −7 × 796

where we use a relative permeability of 50 and let this quantity account completely
for the notion of the bound charges. We shall emphasize the latter interpretation in
the chapters that follow.

The first two laws that we investigated for magnetic fields were the Biot-Savart
law and Amp`ere’s circuital law. Both were restricted to free space in their application.
We may now extend their use to any homogeneous, linear, isotropic magnetic material
that may be described in terms of a relative permeability µ r .
Just as we found for anisotropic dielectric materials, the permeability of an
anisotropic magnetic material must be given as a 3 × 3 matrix, and B and H are
both 3 × 1 matrices. We have

B x = µ xx H x + µ xy H y + µ xz H z
B y = µ yx H x + µ yy H y + µ yz H z
B z = µ zx H x + µ zy H y + µ zz H z

For anisotropic materials, then, B = µH is a matrix equation; however, B =
µ 0 (H + M) remains valid, although B, H, and M are no longer parallel in general.
The most common anisotropic magnetic material is a single ferromagnetic crystal,
although thin magnetic films also exhibit anisotropy. Most applications of ferromag-
netic materials, however, involve polycrystalline arrays that are much easier to make.
Our definitions of susceptibility and permeability also depend on the assumption
of linearity. Unfortunately, this is true only in the less interesting paramagnetic and
diamagnetic materials for which the relative permeability rarely differs from unity
by more than one part in a thousand. Some typical values of the susceptibility for
−5
−5
diamagnetic materials are hydrogen, −2 × 10 ; copper, −0.9 × 10 ; germanium,
−5
−5
−5
−0.8 × 10 ; silicon, −0.3 × 10 ; and graphite,−12 × 10 .Several representative
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−5
paramagnetic susceptibilities are oxygen, 2×10 ; tungsten, 6.8×10 ; ferric oxide
−3
−6
(Fe 2 O 3 ), 1.4 × 10 ; and yttrium oxide (Y 2 O 3 ), 0.53 × 10 .Ifwe simply take the
ratio of B to µ 0 H as the relative permeability of a ferromagnetic material, typical