# 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 classical electromagnetism, magnetization[1] or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. 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 varies between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes 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 measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers define magnetization as the quantity of magnetic moment per unit volume. It is represented by a vector M.

## Definition

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.[2]

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

which is convenient for various calculations. The vacuum permeability μ0 is, by definition, ×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 χm is called the volume magnetic susceptibility.

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

### Magnetization current

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

$\mathbf{J_m} = \nabla\times\mathbf{M}$

and for the bound surface current:

$\mathbf{K_m} = \mathbf{M}\times\mathbf{\hat n}$

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 Jf 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{align} \mathbf{\nabla\cdot H} &= -\nabla\cdot\mathbf{M}\\ \mathbf{\nabla\times H} &= 0 \end{align}

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

These equations can be solved in analogy with electrostatic problems where

\begin{align} \mathbf{\nabla\cdot E} &= \frac{\rho}\epsilon_0\\ \mathbf{\nabla\times E} &= 0 \end{align}

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

It is important to note that there is no such thing as a "magnetic charge," but that issue was still debated through the whole 19th century. Other concepts, that went along with it, such as the auxiliary field H, also have no real physical meaning in their own right. However, they are convenient mathematical tools, and are therefore still used today for applications such as modeling the magnetic field of the Earth.

## 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.

## Demagnetization

Demagnetization is the reduction or elimination of magnetization.[3] One way to do this is to heat the object above its Curie temperature, where thermal fluctuations have enough energy to overcome exchange interactions, the source of ferromagnetic order, and destroy that order. Another way is to pull it out of an electric coil with alternating current running through it, giving rise to fields that oppose the magnetization.[4]

One application of demagnetization is to eliminate unwanted magnetic fields. For example, magnetic fields can interfere with electronic devices such as cell phones or computers, and with machining by making cuttings cling to their parent.[4]