[[Electrodynamics MOC]]
# Magnetic dipole moment

The **magnetic dipole moment** $\vab m$ is defined as a [[pseudovector]] relating the aligning [[torque]] on an object from an externally applied [[magnetic field]] such that
$$
\begin{align*}
\vab\tau = \vab m \times \vab B
\end{align*}
$$
and measures the strength of a [[Magnet]], determining the dipole term in the [[Multipole expansion of the magnetostatic potential]].
For a closed loop carrying current $I$ around an [[Surface orientation|Orientated surface]] $\Sigma$, the magnetic moment is given by
$$
\begin{align*}
\vab m = I \oint_{\Sigma} d\vab a
\end{align*}
$$
It is useful to introduce the concept of **magnetization**,
so that the magnetic dipole moment due to a volume $\Omega$ is given by
$$
\begin{align*}
\vab m = \iiint_{\Omega} \vab M(\vab r') \,d\tau'
\end{align*}
$$
Analysis of the [[Magnetic potential of a magnetized material]] motivates the surface and volume **bound current densities**
$$
\begin{align*}
\vab K_{b}(\vab r') &= \vab M(\vab r') \times \unitv n &
\vab J_{b}(\vab r') = \vab{\nabla}\times\vab M(\vab r')
\end{align*}
$$

## See also

See [[Perfect magnetic dipole]].

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