Magnetization

 ( Electric And Magnetic Fields In Matter ) 

In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Accordingly, physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume.

It is represented by a pseudovector M. Magnetization can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics.

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 that result from those interactions.

The origin of the magnetic moments responsible for magnetization can be either microscopic resulting from the motion of in , or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field.

Paramagnetic materials have a weak induced magnetization in a magnetic field, which disappears when the magnetic field is removed. Ferromagnetic and ferrimagnetic materials have strong magnetization in a magnetic field, and can be magnetized to have magnetization in the absence of an external field, becoming a permanent magnet. Magnetization is not necessarily uniform within a material, but may vary between different points.

Where $\backslash mathrm\{d\}\backslash mathbf\{m\}$ is the elementary magnetic moment and $\backslash mathrm\{d\}V$ is the volume element; in other words, the M-field is the distribution of magnetic moments in the region or manifold concerned. This is better illustrated through the following relation: $$\backslash mathbf\; m=\backslash iiint\; \backslash mathbf\; M\backslash ,\backslash mathrm\{d\}V$$ where m is an ordinary magnetic moment and the triple integral denotes integration over a volume. This makes the M-field completely analogous to the electric polarization field, or P-field, used to determine the electric dipole moment p generated by a similar region or manifold with such a polarization: $$\backslash mathbf\; P\; =\; \{\backslash mathrm\{d\}\backslash mathbf\; p\; \backslash over\; \backslash mathrm\{d\}V\},\; \backslash quad\; \backslash mathbf\; p\; =\; \backslash iiint\; \backslash mathbf\; P\backslash ,\backslash mathrm\{d\}V\; ,$$ where $\backslash mathrm\{d\}\backslash mathbf\{p\}$ is the elementary electric dipole moment.

Those definitions of P and M as a "moments per unit volume" are widely adopted, though in some cases they can lead to ambiguities and paradoxes.

The M-field is measured in per meter (A/m) in SI units.

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

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

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

$\backslash mathbf\{M\}\; =\; \backslash chi\backslash mathbf\{H\},\; \backslash ,\; \backslash mathbf\{B\}\; =\; \backslash mu\backslash mathbf\{H\}=\backslash mu\_0\; (1+\backslash chi)\backslash mathbf\{H\},$

$-\backslash mathbf\{M\}\; \backslash cdot\; \backslash mathbf\{B\}=-\backslash chi\backslash mathbf\{H\}\; \backslash cdot\; \backslash mathbf\{B\}=-\backslash frac\{\backslash chi\}\{1+\backslash chi\}\backslash frac\{\backslash mathbf\{B\}^2\}\{\backslash mu\_0\},$

In diamagnets () and paramagnets ($\backslash chi\; >0$), usually $|\backslash chi|\backslash ll\; 1$, and therefore $\backslash mathbf\{M\}\; \backslash approx\; \backslash chi\backslash frac\{\backslash mathbf\{B\}\}\{\backslash mu\_0\}$.

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

$\backslash mathbf\{B\}=\backslash mu\_0\backslash mathbf\{H\}\; +\; \backslash mathbf\{I\}$ (SI units).

This is by direct analogy to the electric polarization, $\backslash mathbf\{D\}=\backslash varepsilon\_0\backslash mathbf\{E\}\; +\; \backslash mathbf\{P\}$. The magnetic polarization thus differs from the magnetization by a factor of :

$\backslash mathbf\{I\}=\backslash mu\_0\backslash mathbf\{M\}$ (SI units).

Whereas magnetization is given with the unit ampere/meter, the magnetic polarization is given with the unit tesla.

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

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