Magnetization

This article is about magnetization as it appears in Maxwell's equations of classical electrodynamics. For a microscopic description of how magnetic materials react to a magnetic field, see magnetism. For mathematical description of fields surrounding magnets and currents, see magnetic field.

In physics, magnetization is a measure of a material's response to an applied magnetif field. Magnetization is defined as the quantity of magnetic moment per unit volume. Physicists and engineers use magnetization to precisely describe the way that a material responds to a magnetic field. Magnetization describes the way that a material's properties change the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the corresponding material response to an electric field in electrostatics.

Magnetization can be defined according to the following equation:

$\mathbf {M} ={\frac {N}{V}}\mathbf {m} =n\mathbf {m}$

Here, M represents magnetization; m is the vector that defines the magnetic moment; V represents volume; and N is the number of magnetic moments in the sample. The quantity N/V is usually written as n, the number density of magnetic moments. The M-field is measured in amperes per meter (A/m) in SI units.

The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always homogeneous within a body, but rather a function of position.

Magnetization in Maxwell's equations

The behavior of magnetic fields (B, H), electric fields (E, D), charge density (ρ), and current density (J) is described by Maxwell's equations. The role of the magnetization is described below.

Relations between B, H, and M

Main article: Magnetic field

The magnetization defines the auxiliary magnetic field H as

$\mathbf {B} =\mu _{0}\mathbf {(H+M)}$ (SI units)

$\mathbf {B} =\mathbf {(H+4\pi M)}$ (Gaussian units)

which is convenient for various calculations. The vacuum permeability μ₀ is, by definition, 4π×10⁻⁷ V·s/(A·m).

A relation between M and H exists in many materials. In diamagnets and paramagnets, the relation is usually linear:

$\mathbf {M} =\chi _{m}\mathbf {H}$

where χ_m is called the volume magnetic susceptibility.

In ferromagnets there is no one-to-one correspondence between M and H because of hysteresis.

Magnetization current

The magnetization M makes a contribution to the current density J, known as the magnetization current or bound current:

$\mathbf {J_{m}} =\nabla \times \mathbf {M}$

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

$\mathbf {J} =\mathbf {J_{f}} +\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}$

where J_f 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 P.

Magnetostatics

Main article: Magnetostatics

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

${\begin{aligned}\mathbf {\nabla \cdot H} &=-\nabla \cdot \mathbf {M} \\\mathbf {\nabla \times H} &=0\end{aligned}}$

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

${\begin{aligned}\mathbf {\nabla \cdot E} &={\frac {\rho }{\epsilon }}_{0}\\\mathbf {\nabla \times E} &=0\end{aligned}}$

In this sense $-\nabla \cdot \mathbf {M}$ plays the role of a "magnetic charge density" analogous to the electric charge density $\rho$ (see also demagnetizing field).

Magnetization is volume density of magnetic moment. That is: if a certain volume has magnetization $\mathbf {M}$ then the volume element $dV$ has a magnetic moment of $d\mathbf {m} =\mathbf {M} \,dV$

Magnetization dynamics

Main article: Magnetization dynamics

The time-dependent behavior of magnetization becomes important when considering nanoscale and nanosecond timescale magnetization. Rather than simply aligning with an applied field, the individual magnetic moments in a material begin to precess around the applied field and come into alignment through relaxation as energy is transferred into the lattice.

See also

• Permeability (electromagnetism)
• Magnetic susceptibility
• Earth's magnetic field
• Geomagnetic reversal
• Geomagnetic excursion

The dictionary definition of magnetization at Wiktionary