= Current loop
{title2=$\mathbf m=I\mathbf A$}
A <current loop> is a closed path carrying <electric current>. For a thin wire following a closed <curve> $C$, the <volume integral> defining its <magnetic dipole moment> becomes a <line integral>:
$$
\mathbf m=\frac I2\oint_C\mathbf r\times d\mathbf r=I\mathbf A,
\qquad
\mathbf A=\frac12\oint_C\mathbf r\times d\mathbf r.
$$
The <vector area> $\mathbf A$ is $A\widehat{\mathbf n}$ for a planar <current loop>. For a nonplanar <current loop>, use the <vector area>, rather than the scalar <area> of an arbitrarily chosen spanning surface. Reversing the <electric current> reverses $\mathbf m$ without changing the geometry. <Classical Electromagnetism by Richard Fitzpatrick/Localized Current Distribution>{ref}.
\Image[/current-loop-moment-right-hand-rule.png]
{title=Current direction and the right-hand rule for a planar magnetic dipole moment}
{description=Viewed face-on, counterclockwise <conventional current> gives a <magnetic dipole moment> out of the page, shown by a dot; reversing the <electric current> gives a <magnetic dipole moment> into the page, shown by a cross. The curled fingers follow the <electric current> and the thumb follows the unit <normal vector>.}
For $N$ closely spaced identical turns carrying the same <electric current>, the <magnetic dipole moments> add: $\mathbf m=NIA\widehat{\mathbf n}$. Each turn contributes its own <vector area>; unequal areas or differently oriented turns require a <vector> sum. As a calculated example, twelve circular turns of <radius> $R=3.0\,\mathrm{cm}$ carrying $I=0.40\,\mathrm A$ give $m=NI\pi R^2=1.36\times10^{-2}\,\mathrm{A\,m^2}$. This is an idealized example, not an experimental measurement.
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