[[Magnet]]
# Linear magnet

A **linear magnet** whose [[Magnetic dipole moment|magnetization]] is proportional to the magnetic field[^B], or equivalently
$$
\begin{align*}
\vab M = \chi_{m} \vab H
\end{align*}
$$
where $\vab H$ is the [[H-field]] and $\chi_{m}$ is the empirically determined **magnetic susceptibility**, and

- $\chi_{m} > 0$ for [[Paramagnetism]]
- $\chi_{m} < 0$ for [[Diamagnetism]]

The **permeäbility** of a linear magnet is then
$$
\begin{align*}
\mu = \mu_{0}(1+\chi_{m}) > 0
\end{align*}
$$
and the [[magnetic field]] is related to $\vab H$ by
$$
\begin{align*}
\vab H = \mu\vab B
\end{align*}
$$

  [^B]: Note that the electric field includes that caused by the magnet itself.

## Homogenous linear magnet

In case a volume $\Omega$ is occupied by a homogenous linear magnet,
i.e. one with constant $\chi_{m}$ we have
$$
\begin{align*}
\vab J_{b} = \chi_{m} \vab J_{f}
\end{align*}
$$
and so if there is no free current there is also no bound current,
meaning all bound current must be found on the surface $\partial\Omega$.

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