= Magnetic dipole torque
{title2=$\boldsymbol\tau=\mathbf m\times\mathbf B_{\mathrm{ext}}$}
An externally imposed <magnetic field> that is nearly uniform over a rigid <current loop> produces the <torque>
$$
\boldsymbol\tau=\mathbf m\times\mathbf B_{\mathrm{ext}},
\qquad
U=-\mathbf m\cdot\mathbf B_{\mathrm{ext}}=-mB\cos\theta.
$$
The <angle> $\theta$ is measured between the <magnetic dipole moment> and the external <magnetic field>, not between the plane of the <current loop> and the <magnetic field>. The <magnetic dipole potential energy> uses a <dot product>, whereas the <torque> uses a <cross product>. The <torque> is largest when the <magnetic dipole moment> is perpendicular to the <magnetic field>; parallel alignment minimizes the <potential energy>.<University Physics Volume 2 by Samuel J. Ling and others/11.5 Force and Torque on a Current Loop>[§11.5, equations 11.22–11.23: vector torque and orientation energy]{ref}
For the sign convention $\mathbf B_{\mathrm{ext}}=B\widehat{\mathbf x}$ and $\mathbf m=m(\cos\theta\widehat{\mathbf x}+\sin\theta\widehat{\mathbf y})$, positive rotation increases $\theta$ and
$$
\tau_z=-mB\sin\theta=-\frac{dU}{d\theta}.
$$
Thus at $0<\theta<\pi$ the <torque> turns the <magnetic dipole moment> toward the external <magnetic field>. Both parallel and antiparallel alignment have zero <torque>, but they are respectively a <stable equilibrium> and an <unstable equilibrium>. With no <damping>, a freely released <current loop> can undergo <oscillation> around the <stable equilibrium>; settling requires <damping>.
\Image[/magnetic-dipole-angular-energy-torque.png]
{title=Angular potential energy and signed torque of a magnetic dipole in an imposed field}
{description=The ideal model has $U/(mB)=-\cos\theta$ and $\tau_z/(mB)=-\sin\theta$. The diagram fixes the external <magnetic field> along positive $x$, so the <torque> is clockwise for the illustrated positive <angle>. These curves are calculated, not measured data.}
In the twelve-turn example, take $B=0.020\,\mathrm T$ and $\theta=60\degree$. The predicted <torque> is $\tau_z=-2.35\times10^{-4}\,\mathrm{N\,m}$ and the <magnetic dipole potential energy> is $U=-1.36\times10^{-4}\,\mathrm J$. The unit check is $(\mathrm{A\,m^2})\mathrm T=\mathrm{N\,m}=\mathrm J$; <torque> and <energy> have the same <physical dimensions> but different physical meanings.
A short derivation starts with the <magnetic force on a current-carrying wire>, $d\mathbf F=I\,d\mathbf r\times\mathbf B_{\mathrm{ext}}$. In an exactly uniform <magnetic field>, the net <force> is $I(\oint_Cd\mathbf r)\times\mathbf B_{\mathrm{ext}}=0$, but its <torque> need not vanish: opposite sides produce a turning <couple>. Integrating $\mathbf r\times d\mathbf F$ gives $\mathbf m\times\mathbf B_{\mathrm{ext}}$. Integrating $dU=-\tau_z\,d\theta$ at fixed $m$ and $B$ gives the orientation <potential energy>. This is the mechanical orientation <potential energy> for a maintained <electric current>, with the external source held fixed; the <energy> accounting for field and power supply is a separate problem. A varying external <magnetic field> requires corrections across a finite <current loop>, and can also give a net <force>.
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