Page 271 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 253

is determined by allowing the surface to cut a small cylindrical gaussian surface.
Applying Gauss’s law for the magnetic field from Section 7.5,

B · dS = 0
S
we find that
B N1 S − B N2 S = 0
or

B N2 = B N1 (32)
Thus
µ 1
H N2 = H N1 (33)
µ 2
The normal component of B is continuous, but the normal component of H is discon-
tinuous by the ratio µ 1 /µ 2 .
The relationship between the normal components of M,of course, is fixed once
the relationship between the normal components of H is known. For linear magnetic
materials, the result is written simply as

µ 1 χ m2 µ 1
M N2 = χ m2 H N1 = M N1 (34)
µ 2 χ m1 µ 2
Next, Amp`ere’s circuital law

H · dL = I
is applied about a small closed path in a plane normal to the boundary surface, as
shown to the right in Figure 8.10. Taking a clockwise trip around the path, we find
that
H t1 L − H t2 L = K L

where we assume that the boundary may carry a surface current K whose component
normal to the plane of the closed path is K. Thus

H t1 − H t2 = K (35)

The directions are specified more exactly by using the cross product to identify the
tangential components,
(H 1 − H 2 ) × a N12 = K
where a N12 is the unit normal at the boundary directed from region 1 to region 2. An
equivalent formulation in terms of the vector tangential components may be more
convenient for H:
H t1 − H t2 = a N12 × K