, is defined so
that it is the magnetic dipole moment per unit volume of the material. The
relation between polarization, applied magnetic field, and resultant magnetic
field, is similar to the analogous properties for dielectrics:Go directly to derivation of magnetic effects.
The response of many common solids to an applied magnetic field is such that the magnetic field inside the sample is greater than the applied electric. It is as if the solid has produced its own current distribution - become polarized - so that it produces a magnetic field which adds, in support of, the applied field. Materials that do this are called paramagnetic or ferromagnetic materials. Typically, the resultant field points in the same direction as the applied field.
Magnetic polarization,
, is defined so
that it is the magnetic dipole moment per unit volume of the material. The
relation between polarization, applied magnetic field, and resultant magnetic
field, is similar to the analogous properties for dielectrics:
where 
is
called the magnetic polarizability of the material, and
is the usual
mks magnetic constant.
A new field,
, is defined so
that
( has been
called the "magnetic intensity.") Then
where 
is
called the magnetic susceptibility.
For anisotropic materials we allow the possibility that a magnetic field in one direction might produce a polarization in another (as is sometimes observed experimentally). We can write in vector components,
where we use the Einstein Summation Convention, that a repeated index (in this case the repeated j index) is summed. Thus typically the ith component of polarizability is calculated as the sum of three contributions; one from each component of the applied field.
It remains true that
While this leads to the same kind of eigen value discussion as the electric field case, magnetic fields have more interesting common properties: Saturation, and Ferromagnetism . While the polarization field in a dielectric has a plausible maximum effect (net field equal to zero, which occurs when the dielectric response is extremely high, as in a metal), there is no such obvious limit for magnetic materials.
Since in the magnetic polarization, the resulting field is larger, one might expect the polarization to increase; then the net field to increase, etc. In practice, an applied field has a well-defined, non-infinite, resulting field. The polarization increases as the applied field increases, up to a limit. When the polarization ceases to increase, it is said to be "saturated." (A cause for this saturation is found in the atomic theory of matter, using quantum mechanics.)
For paramagnetic materials, polarization returns to zero when the applied field is removed. However, for some materials (most famously, iron) the polarization remains greater than zero. This phenomenon called "ferromagnetism," and is independent of saturation. Ferromagnetic materials that have been put in such a polarized state are said to be "permanent" magnets, although it is possible to return the polarization to zero. Navy ships are routinely "depolarized" to make them less detectable. Ferromagnets also exhibit saturation. For iron, the polarization saturates at about 20,000 times the earth's magnetic field.
OTHER Properties
The elastic, electric, and magnetic properties have been discussed here as though they were independent. In practice, they are not. For example, applying a magnetic field to a material causes a small change in the dimensions of the sample. This phenomenon is called "magnetostriction." There are also "electrostrictive" effects associated with electric fields. It is also true that, for example, a mechanical stress can produce electric polarization. Materials that do this are generally called "piezoelectric," although some refinement of that term is often called for.
The elastic constants and the magnetic and electric polarizability constants are all temperature sensitive. Typically solid materials become less stiff and their polarizabilities often decrease at higher temperatures. Gases at constant volume become more difficult to compress.
The simplest example of this coupling is in paramagnetic materials. It is called Curie's Law, and may be expressed in an equation:
where 
is
the absolute temperature, originally derived from ideal gas experiments, and
is called the
Curie constant.
Finally there is the familiar thermal expansion, in which an increase in temperature causes an increase in the dimensions of a sample. The thermal expansion for many materials is a simple affair, reminiscent of the elastic, magnetic, and electric properties of materials. For a sample of length L,
where 
is
now the thermal expansion coefficient.
Return to index.