Paramagnetic materials increase the magnetic field in a solenoid (with constant current forced through it) when they occupy the space inside the coil. However, since the susceptibility is on the order of .00001 for most paramagnets, the effect is not very large. In particular if we want a stronger magnetic field it is much easier to increase the current by .001% than to fill the coil with a paramagnetic material.
There are, however, other materials whose magnetic dipole moments align much more readily with an applied field. In this case the magnetic field caused by the polarization is not small. In fact this part of the magnetic field can be the dominant effect. At low applied fields, the susceptibility of materials like iron, nickel and cobalt is on the order of 1000 or more. Since
and
we can write
and when
, the net magnetic field can be huge even when the applied magnetic field (
) is small. For these materials, we think of the small applied field as our method of controlling the large field due to polarization. The contribution of the applied field to the net field can be negligible.
When the magnetic field due to polarization is larger than the applied field, a remarkable phenomenon can occur. Imagine that we introduce a piece of iron (unpolarized) into the center of a solenoid with a field
(comparable to the earth's magnetic field). As the iron moves into the high field region, it begins to polarize. 
grows, approaching 1 Tesla, and causes more of the dipole moments to align. This increases 
, which in turn causes more polarization, etc.
In an ideal case, the polarization and its magnetic field would become infinite, as the sample polarizes itself (triggered by the small applied field). In practice, several factors might limit the degree of polarization:
1) Thermal fluctuations tend to randomize the directions of the dipole moments, in spite of the torques pushing them to align.
2) Some dipole moments are hindered in their rotations by their microscopic environments, and thus do not align completely.
3) If there are a finite number of microscopic dipoles, then the maximum polarization occurs when all of then are aligned. If matter is composed of atoms, and each atom has a certain dipole moment, this would explain the observed upper limit on polarization. Thus magnetic experiments can be used to verify some aspects of the atomic theory of matter.
Although the polarization does not become infinite, it does become so large that
is strong enough to keep the material polarized even when the applied field is turned off. The magnetic object holds itself in a polarized state, and produces a magnetic field, both inside and out. If the object is shaped like a bar, and if it is polarized parallel to the length of the bar, then it produces a good approximation of a dipole magnetic field.
Materials which display permanent magnetization like this are called ferromagnets, after the most common example, iron. Until the advent of superconducting magnets, iron core solenoids were the most practical method of producing magnetic fields above 1 Tesla. For some applications such as magnetic latches and loudspeakers, permanent magnets are still the most effective sources of magnetic field.
To study ferromagnetism, we want to find the relation between
and 
, just as for paramagnetism. To do this, we build a solenoid and measure 
as a function of current through the solenoid. From these data we calculate 
as a function of current.
Then we fill the solenoid with ferromagnetic material and measure the magnetic field inside as a function of current. With these data, we can calculate
as a function of current.
|
| Magnetization curve for a ferromagnetic sample which was initially unmagnetized. |
| Increasing caused an increase in , up to (almost) the saturation asymptote. |
Finally we can plot
versus , as in the figure. The experiment represented by this plot began with a piece of completely unpolarized material in the coil, with zero current.
Then the current was turned up, the fields were measured and the magnetic dipole moment per unit volume (
) was calculated. The fact that stops rising with increasing is significant.
(It suggests a finite number of magnetic dipoles, all of which are now lined up with the magnetic field. An increase in applied field produces no new alignment. This in turn suggests a molecular model for the structure of matter. For now, we turn away from microscopic models for magnetism.)
This phenomenon is called "saturation." Since the slope of the
versus curve is not constant, the polarizability must not be constant: The simple picture represented by 
clearly fails. It only works for very small fields in ferromagnets. (It works much better in paramagnets.)
Hysteresis
|
| Changing changes as shown. The sample saturates in both |
| the + and the - direction, as the system executes a "hysteresis loop." |
The figure shows a sequence of changes in the
field. Starting at zero, is raised to + saturation, and then reduced to zero. rises to saturation, but does not fall to zero when H returns to zero. The sample is now "permanently magnetized."
When
changes direction (to -), does not immediately change directions. However, as the magnitude of increases, does decrease, become zero, and then finally change directions to agree with .
This continues as
increases in the - direction until the material reaches saturation again (in the - direction). Reducing the magnitude of to zero now leaves the sample permanently magnetized in the - direction. Increasing in the + direction will carry the sample back to + saturation, but not through zero magnetization. The path described by a complete cycle is called a "hysteresis loop." It represents the fact that the state of a magnetic object depends on its previous treatment - its history.
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