Magnetization
= Magnetization
{title2=$\mathbf M$}
{wiki}
<Magnetization> is <magnetic dipole moment> per unit <volume>, described by a <vector field> $\mathbf M(\mathbf r)$. Average the microscopic <magnetic dipole moments> in a small region of <volume> $\Delta V$ around the <point> $\mathbf r$:
$$
\mathbf M(\mathbf r)\simeq\frac{1}{\Delta V}\sum_{i\in\Delta V}\mathbf m_i,
\qquad
\mathbf m_{\rm total}=\int_V\mathbf M\,dV.
$$
The averaging region should contain many microscopic constituents while being small compared with the distance over which the macroscopic <magnetization> varies. With <number density> $n$ of identical constituents, this becomes $\mathbf M=n\langle\mathbf m\rangle$. <Magnetic dipole moment> has <SI> units $\mathrm{A\,m^2}$, so <magnetization> has units $\mathrm{A/m}$: <amperes> per metre. The average is a <vector> sum, not a sum of magnitudes.<Classical Electromagnetism: An intermediate level course by Richard Fitzpatrick/Magnetization>[Classical Electromagnetism: An intermediate level course by Richard Fitzpatrick; Magnetization, equations (856)–(857) and the surrounding averaging definition]{ref}
Alignment can change <magnetization> without changing the magnitude of any individual <magnetic dipole moment>. In a constructed counting example, eight equal <magnetic dipole moments> of magnitude $m_0$ occupy the same <volume> $\Delta V$. Six along $+z$ and two along $-z$ give $\mathbf M=4m_0\widehat{\mathbf z}/\Delta V$; turning the two opposing <magnetic dipole moments> to $+z$ doubles <magnetization> to $8m_0\widehat{\mathbf z}/\Delta V$. Both arrangements have the same sum of magnitudes, $8m_0$. Changing the individual <magnetic dipole moments> provides another possible response; their alignment and their magnitudes are distinct ingredients. Microscopic <spin> contributes <magnetic dipole moment> without being a literal rotating charged ball.
\Image[/magnetization-dipole-vector-averaging.png]
{title=<Magnetization> depends on the <vector> average of <magnetic dipole moments>}
{description=Original eight-<magnetic dipole moment> counting model, with equal arrow lengths and equal averaging <volumes>. Reorienting two arrows doubles the net <magnetic dipole moment> without enlarging any individual <magnetic dipole moment>. The small count illustrates the sum; a macroscopic averaging region contains many more constituents.}
Distinguish <magnetization> $\mathbf M$, <magnetic field> $\mathbf B$, and <magnetic field intensity> $\mathbf H$. In <SI>,
$$
\mathbf B=\mu_0(\mathbf H+\mathbf M),\qquad
[\mathbf M]=[\mathbf H]=\mathrm{A/m},\qquad [\mathbf B]=\mathrm T.
$$
Here $\mu_0$ is the <vacuum permeability>, and $\mathrm T$ denotes the <tesla (unit)>. <Magnetization> describes matter's <magnetic dipole moment> density; <magnetic field> enters the <Lorentz force>; <magnetic field intensity> separates the free <electric current> contribution in the macroscopic equations. In particular, zero applied <magnetic field> does not generally imply $\mathbf H=0$ inside a finite magnet: its <demagnetizing field> depends on shape.<Classical Electromagnetism: An intermediate level course by Richard Fitzpatrick/Magnetization>[Classical Electromagnetism: An intermediate level course by Richard Fitzpatrick; Magnetization, equations (860)–(864): current decomposition and definition of H]{ref}
For stationary <electric polarization>, <magnetization> produces <bound current> in the <volume> and on its bounding <surface>:
$$
\mathbf J_{\rm b}=\nabla\times\mathbf M,
\qquad
\mathbf K_{\rm b}=\mathbf M\times\mathbf n.
$$
The <curl> gives a <current density> in $\mathrm{A/m^2}$; $\mathbf K_{\rm b}$ is a <surface current density> in $\mathrm{A/m}$. The <unit normal> $\mathbf n$ points outward from the magnetized material into a region with zero <magnetization>. Across an interface between two magnetized media, the net bound <current sheet> is
$$
\mathbf K_{\rm interface}=(\mathbf M_1-\mathbf M_2)\times\mathbf n,
$$
where the <unit normal> points from medium 1 to medium 2. Both faces contribute: medium 2 has the opposite outward <normal vector>. This fixes the <cross product> sign. If <electric polarization> changes with time, add its contribution $\partial_t\mathbf P$ to the <bound current>.<Classical Electromagnetism by Richard Fitzpatrick/Permanent Ferromagnets>[Classical Electromagnetism by Richard Fitzpatrick; Permanent Ferromagnets, equation (716): volume and outward-surface magnetization currents]{ref}
A short proof outline explains why both contributions are needed. Write $\mathbf M_{\rm ext}=\mathbf M\,\chi_V$, where $\chi_V$ is the <indicator function> of the material's <volume>. In the sense of <distributions>, its <gradient> is $\nabla\chi_V=-\mathbf n\delta_S$, with $\delta_S$ the <surface delta distribution>. The <product rule> for the <curl> gives
$$
\nabla\times(\chi_V\mathbf M)
=\chi_V\nabla\times\mathbf M+(\mathbf M\times\mathbf n)\delta_S.
$$
Thus a constant interior <magnetization> can have zero interior <curl> and still produce a nonzero boundary <electric current>. Equivalently, neighboring ideal microscopic <current loops> cancel along shared edges while uncompensated edges remain on the outside. This <bound current> representation predicts the same macroscopic <magnetic field> as the <magnetization> distribution; it does not require a wire or free charge transport around the magnet. Its usefulness is that the familiar <Biot-Savart law> can now calculate the <magnetic field> from a specified <magnetization>.
A time-dependent average <magnetization> need not vary smoothly under a smooth drive. The <Barkhausen effect> resolves collective jumps through their induced <electromotive force>; <Barkhausen pulse area> explains how that derivative readout relates to the underlying change.