Page 267 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 249

Equation (21) merely says that if we go around a closed path and find dipole moments
going our way more often than not, there will be a corresponding current composed
of, for example, orbiting electrons crossing the interior surface.
This last expression has some resemblance to Amp`ere’s circuital law, and we
may now generalize the relationship between B and H so that it applies to media
other than free space. Our present discussion is based on the forces and torques on
differential current loops in a B field, and we therefore take B as our fundamental
quantity and seek an improved definition of H.We thus write Amp`ere’s circuital law
in terms of the total current, bound plus free,
B

· dL = I T (22)
µ 0
where
I T = I B + I
and I is the total free current enclosed by the closed path. Note that the free current
appears without subscript since it is the most important type of current and will be
the only current appearing in Maxwell’s equations.
Combining these last three equations, we obtain an expression for the free current
enclosed,
B

I = I T − I B = − M · dL (23)
µ 0
We may now define H in terms of B and M,
B
H = − M (24)
µ 0
and we see that B = µ 0 H in free space where the magnetization is zero. This rela-
tionship is usually written in a form that avoids fractions and minus signs:

B = µ 0 (H + M) (25)

We may now use our newly defined H field in (23),

H · dL (26)
I =
obtaining Amp`ere’s circuital law in terms of the free currents.
Using the several current densities, we have

I B = J B · dS
S

J T · dS
I T =
S

J · dS
I =
S