Page 268 - Engineering Electromagnetics, 8th Edition

250 ENGINEERING ELECTROMAGNETICS

With the help of Stokes’ theorem, we may therefore transform (21), (26), and (22)
into the equivalent curl relationships:

∇× M = J B
B
∇× = J T
µ 0

∇× H = J (27)

We will emphasize only (26) and (27), the two expressions involving the free
charge, in the work that follows.
The relationship between B, H, and M expressed by (25) may be simplified for
linear isotropic media where a magnetic susceptibility χ m can be defined:

M = χ m H (28)

Thus we have

B = µ 0 (H + χ m H)
= µ 0 µ r H
where

µ r = 1 + χ m (29)
is defined as the relative permeability µ r .Wenext define the permeability µ:
(30)
µ = µ 0 µ r
and this enables us to write the simple relationship between B and H,
B = µH (31)

EXAMPLE 8.5
Given a ferrite material that we shall specify to be operating in a linear mode with
B = 0.05 T, let us assume µ r = 50, and calculate values for χ m , M, and H.
Solution. Because µ r = 1 + χ m ,wehave

χ m = µ r − 1 = 49
Also,
B = µ r µ 0 H

and
0.05
H = = 796 A/m
50 × 4π × 10 −7