Page 266 - Engineering Electromagnetics, 8th Edition

248 ENGINEERING ELECTROMAGNETICS

bound charges (orbital electrons, electron spin, and nuclear spin), and the field, which
has the dimensions of H, will be called the magnetization M. The current produced
by the bound charges is called a bound current or Amperian current.
Let us begin by defining the magnetization M in terms of the magnetic dipole
moment m. The bound current I b circulates about a path enclosing a differential area
2
dS, establishing a dipole moment (A · m ),
m = I b dS
If there are n magnetic dipoles per unit volume and we consider a volume ν, then
the total magnetic dipole moment is found by the vector sum
n ν

m total = m i (19)
i=1
Each of the m i may be different. Next, we define the magnetization M as the magnetic
dipole moment per unit volume,
n ν
1
M = lim m i
ν→0 ν
i=1
and see that its units must be the same as for H, amperes per meter.
Now let us consider the effect of some alignment of the magnetic dipoles as
the result of the application of a magnetic field. We shall investigate this alignment
along a closed path, a short portion of which is shown in Figure 8.9. The figure shows
several magnetic moments m that make an angle θ with the element of path dL; each
moment consists of a bound current I b circulating about an area dS.We are therefore
considering a small volume, dS cos θdL,or dS · dL, within which there are ndS · dL
magnetic dipoles. In changing from a random orientation to this partial alignment,
the bound current crossing the surface enclosed by the path (to our left as we travel in
the a L direction in Figure 8.9) has increased by I b for each of the ndS · dL dipoles.
Thus the differential change in the net bound current I B over the segment dL will be
dI B = nI b dS · dL = M · dL (20)
and within an entire closed contour,

I B = M · dL (21)

Figure 8.9 A section dL of a closed path along which magnetic dipoles have been
partially aligned by some external magnetic field. The alignment has caused the bound
current crossing the surface defined by the closed path to increase by nI b dS · dL A.