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 forces that result from those interactions. 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. 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.
Definition The magnetization field or M-field can be defined according to the following equation:
M = d m d V {\displaystyle \mathbf {M} ={\frac {\mathrm {d} \mathbf {m} }{\mathrm {d} V}}}
Where d m {\displaystyle \mathrm {d} \mathbf {m} } is the elementary magnetic moment and d V {\displaystyle \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:
m = ∭ M d V {\displaystyle \mathbf {m} =\iiint \mathbf {M} \,\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:
P = d p d V , p = ∭ P d V , {\displaystyle \mathbf {P} ={\mathrm {d} \mathbf {p} \over \mathrm {d} V},\quad \mathbf {p} =\iiint \mathbf {P} \,\mathrm {d} V,}
where d p {\displaystyle \mathrm {d} \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 amperes per meter (A/m) in SI units.
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
The magnetization defines the auxiliary magnetic field H as
B = μ 0 ( H + M ) {\displaystyle \mathbf {B} =\mu _{0}(\mathbf {H+M} )} (SI)
B = H + 4 π M {\displaystyle \mathbf {B} =\mathbf {H} +4\pi \mathbf {M} } (Gaussian system) which is convenient for various calculations. The vacuum permeability μ0 is, approximately, 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:
M = χ H , B = μ H = μ 0 ( 1 + χ ) H , {\displaystyle \mathbf {M} =\chi \mathbf {H} ,\,\mathbf {B} =\mu \mathbf {H} =\mu _{0}(1+\chi )\mathbf {H} ,}
where χ is called the volume magnetic susceptibility, and μ is called the magnetic permeability of the material. The magnetic potential energy per unit volume (i.e. magnetic energy density) of the paramagnet (or diamagnet) in the magnetic field is:
− M ⋅ B = − χ H ⋅ B = − χ 1 + χ B 2 μ 0 , {\displaystyle -\mathbf {M} \cdot \mathbf {B} =-\chi \mathbf {H} \cdot \mathbf {B} =-{\frac {\chi }{1+\chi }}{\frac {\mathbf {B} ^{2}}{\mu _{0}}},}
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