## 9.1

Magnetization Density

The sources of magnetic field in matter are the (more or less)
aligned magnetic dipoles of individual electrons or currents caused by
circulating electrons.

^{1} Magnetic monopoles, which would play a
role with respect to magnetic fields analogous to that of the charge
with respect to electric fields, may in fact exist, but are certainly
not of engineering significance. See * Science*, Research News,
"In search of magnetic monopoles," Vol. 216, p. 1086 (June 4,
1982). We now describe the effect on the magnetic field of a
distribution of magnetic dipoles representing the material.

In Sec. 8.3, we defined the magnitude of the magnetic moment *m* of a
circulating current loop of current *i* and area *a* as *m = ia*.
The moment vector, **m**, was defined as normal to the surface spanning
the contour of the loop and pointing in the direction determined by the
right-hand rule. In
Sec. 8.3, where the moment was in the *z* direction in spherical
coordinates, the loop was found to produce the magnetic field
intensity

This field is analogous to the electric field
associated with a dipole having the moment **p**. With **p** directed
along the *z* axis, the electric dipole field is given by taking the
gradient of (4.4.10).

Thus, the dipole fields are obtained from each other by making the
identifications

In Sec. 6.1, a spatial distribution of electric dipoles is
represented by the polarization density **P** = N**p**, where *N* is the
number density of dipoles. Similarly, here we define a
* magnetization density* as

where again *N* is the number of dipoles per unit volume. Note that
just as the analog of the dipole moment **p** is _{o} **m**, the
analog of the polarization density **P** is _{o} **M**.