Torque on a Current Loop Placed in a Magnetic Field

Consider a rectangular loop $PQRS$ (sides $a$, $b$) carrying current $I$, in a uniform field $\vec B$, tilted so its normal $\hat n$ makes angle $\theta$ with $\vec B$. Working out the force on each side using $F=BIl\sin\theta$ from §3.10.5: the forces on $QR$ and $SP$ (each of length $b$) turn out equal, opposite, and collinear, so they cancel completely. The forces on $PQ$ and $RS$ (each of length $a$) are $F_{PQ}=F_{RS}=IaB$, equal and opposite but not collinear -- separated by a perpendicular distance $b\sin\theta$ -- so they form a couple. Taking moments,

In vector form, $\vec\tau = (I\vec A)\times\vec B = \vec p_m\times\vec B$ (using $p_m=IA$ from §3.8.5)

-- the torque always tends to rotate the loop so its normal aligns with $\vec B$. For $N$ turns,

$\boxed{\tau = NIAB\sin\theta}$ …