= Magnetic dipole moment
{title2=$\mathbf m$}
{wiki}
= Magnetic moment
{synonym}
A <magnetic dipole moment> is a <vector> that summarizes the strength and orientation of a localized magnetic source. For a planar <current loop>, it is the <electric current> times the <vector area>: $\mathbf m=IA\widehat{\mathbf n}$. The <right-hand rule> fixes the unit <normal vector> $\widehat{\mathbf n}$: curl the fingers of the right hand along the <conventional current>, and the thumb points along $\mathbf m$. This single <vector> predicts both the distant <magnetic dipole field> and the turning effect of an imposed <magnetic field>.<University Physics Volume 2 by Samuel J. Ling and others/11.5 Force and Torque on a Current Loop>[§11.5, equations 11.20–11.22: loop moment, winding count and torque]{ref}
The <SI> unit is <ampere> times square metre, $\mathrm{A\,m^2}$, equivalently <joule> per <tesla (unit)>. A <magnetic dipole moment> is neither a <magnetic field> nor a <magnetic monopole>: a closed <current loop> can have a nonzero <magnetic dipole moment> while its net outward <magnetic flux> through any closed surface remains zero, as required by <Gauss's law for magnetism>.
For a localized, steady <electric current density> $\mathbf J(\mathbf r)$, the more general definition is
$$
\mathbf m=\frac12\int_{\mathbb R^3}\mathbf r\times\mathbf J(\mathbf r)\,d^3r.
$$
Here the <cross product> is integrated over the complete source, including return paths. Steadiness requires $\nabla\cdot\mathbf J=0$, and localization requires sufficiently rapid decay for the boundary terms to vanish. Under these assumptions the <localized steady-current moment identities> give $\int\mathbf J\,d^3r=0$: shifting the origin by $\mathbf a$ changes the proposed <magnetic dipole moment> by $-\tfrac12\mathbf a\times\int\mathbf J\,d^3r=0$. An isolated open wire segment does not satisfy these assumptions. <Classical Electromagnetism by Richard Fitzpatrick/Localized Current Distribution>{ref}.
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