Magnetization is a property of some materials (e.g. magnets) that describes to what extent they are affected by magnetic fields, and also determines the magnetic field that the material itself creates. Magnetization is defined as the amount of magnetic moment per unit volume. The origin of the magnetic moments that create the magnetization can be either microscopic electric currents corresponding to the motion of electrons in atoms, or the spin of the electrons.

In some materials (e.g., ferromagnets) magnetization can exist even without an external magnetic field (spontaneous magnetization). In other types of materials, magnetization is induced only when an external magnetic field is present. Magnetization is not always homogeneous within a body, but rather a function of position.

Magnetization in Maxwell's equations

The behavior of magnetic fields ($\backslash mathbf\{B\}$, $\backslash mathbf\{H\}$), electric fields ($\backslash mathbf\{E\}$, $\backslash mathbf\{D\}$), charge density ($\backslash rho\backslash ,$), and current density ($\backslash mathbf\{J\}$) is described by Maxwell's equations. The role of the magnetization is described below.

Relations between B, H and M

The magnetization defines the auxiliary magnetic field $\backslash mathbf\{H\}$ as

$\backslash mathbf\{B\}\; =\; \backslash mu\_0(\backslash mathbf\{H\; +\; M\})$

which is convenient for various calculations.

A relation between $\backslash mathbf\{M\}$ and $\backslash mathbf\{H\}$ exists in many materials. In diamagnets and paramagnets, the relation is usually linear:

$\backslash mu\_0\backslash mathbf\{M\}\; =\; \backslash chi\_m\backslash mathbf\{H\}$

where $\backslash chi\_m\backslash ,$ is called the magnetic susceptibility.

In ferromagnets there is no one-to-one correspondence between $\backslash mathbf\{M\}$ and $\backslash mathbf\{H\}$ because of hysteresis.

Magnetization current

The magnetization $\backslash mathbf\{M\}$ makes a contribution to the current density $\backslash mathbf\{J\}$, known as the magnetization current or bound current:

$\backslash mathbf\{J\_m\}\; =\; \backslash nabla\backslash times\backslash mathbf\{M\}$

so that the total current density that enters Maxwell's equations is given by

$\backslash mathbf\{J\}\; =\; \backslash mathbf\{J\_f\}\; +\; \backslash nabla\backslash times\backslash mathbf\{M\}\; +\; \backslash frac\{\backslash partial\backslash mathbf\{P\}\}\{\backslash partial\; t\}$

where $\backslash mathbf\{J\_f\}$ is the electric current density of free charges, the second term is the contribution from the magnetization, and the last term is related to the electric polarization $\backslash mathbf\{P\}$.

Magnetostatics

In the absence of free electric currents and time-dependent effects, Maxwell's equations describing the magnetic quantities reduce to

$\backslash mathbf\{\backslash nabla\backslash cdot\; H\}\; =\; -\backslash mu\_0\; \backslash nabla\backslash cdot\backslash mathbf\{M\}\; \backslash qquad\; \backslash mathbf\{\backslash nabla\backslash times\; H\}\; =\; 0$

These equations can be easily solved in analogy with electrostatic problems where

$\backslash mathbf\{\backslash nabla\backslash cdot\; E\}\; =\; \backslash frac\{\backslash rho\}\backslash epsilon\_0\; \backslash qquad\; \backslash qquad\; \backslash qquad\; \backslash mathbf\{\backslash nabla\backslash times\; E\}\; =\; 0$

In this sense $-\backslash epsilon\_0\backslash mu\_0\backslash nabla\backslash cdot\backslash mathbf\{M\}$ plays the role of a "magnetic charge density" analogous to the electric charge density $\backslash rho\backslash ,$.

Types of magnetism

Diamagnetism

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 the Lenz's law.

Paramagnetism

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.

Ferromagnetism

Ferromagnetic materials present a magnetization much larger than other materials. Ferromagnetism arises from the strong coupling between the 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 Curie temperature (temperature of the phase change to paramagnetic behavior), coercive field (field strength needed do demagnetize the material) and remnant magnetization (magnetization at zero external field).

Electric and magnetic fields in matter

Magnetizace | Magnetisierung | 磁化向量