The magnetization defines the auxiliary magnetic field as
which is convenient for various calculations.
where is called the volume magnetic susceptibility.
The magnetization makes a contribution to the current density , known as the magnetization current or bound current:
so that the total current density that enters Maxwell's equations is given by
where is the electric current density of free charges (also called the free current), the second term is the contribution from the magnetization, and the last term is related to the electric polarization .
In the absence of free electric currents and time-dependent effects, Maxwell's equations describing the magnetic quantities reduce to
These equations can be easily solved in analogy with electrostatic problems where
In this sense plays the role of a "magnetic charge density" analogous to the electric charge density .
Magnetization is volume density of magnetic moment. That is: if a certain volume has magnetization then volume element has magnetic moment of .
This is the most common magnetic behavior. The diamagnetic magnetization is proportional and opposing to the applied magnetic field. All materials present a diamagnetic response, although it may be shadowed by stronger magnetic behaviors. Diamagnetism can be explained by the normal response of the orbiting electrons considering Lenz's law. This is a weak form of magnetism that is non permanent and persists only while external field is applied. The magnitude of induced magnetic moment is very small and in a direction opposite to that of applied field. Therefore, relative permeability is less than 1 and magnetic susceptibility is negative. When placed between the poles of a strong electromagnet, diamagnetic materials are pushed out towards the region where the field is weaker.
Paramagnetic materials present a magnetization that is proportional to the applied field and reinforces it. This arises from the existence of magnetic dipoles in the material. Paramagnetism varies inversely with temperature and is characterized by the material's saturation magnetization. When placed between the poles of a strong electromagnet, paramagnetic materials are pulled towards the region where the field is stronger.
Ferromagnetic materials present a magnetization much larger than other materials. Ferromagnetism arises from the strong coupling between the neighboring magnetic dipoles in the material. Ferromagnetic materials can present spontaneous magnetization, and this gives rise to the hysteresis loops. Ferromagnetic materials can be characterized by their permeability, Curie temperature (temperature of the phase change to paramagnetic behavior), coercive field (field strength needed to demagnetize the material), and remnant magnetization (magnetization at zero external field).