Torque on a Current Loop in a Uniform Magnetic Field

Setup: a rectangular loop in a uniform field. Consider a single rectangular current loop

$PQRS$, of sides $a$ and $b$ (area $A = ab$), carrying current $I$, free to rotate about an axis

through its centre, placed in a uniform magnetic field $\vec{B}$. Let $\theta$ be the angle between

the field $\vec{B}$ and the NORMAL $\hat{n}$ to the plane of the loop (the normal direction is itself

fixed, by the right-hand rule, by the sense in which the current circulates around the loop).

Force on each side. By the result of Section 4.10, each of the four straight sides of the loop

experiences a magnetic force $F=BIL\sin\phi$, where $\phi$ is the angle that particular side makes

with $\vec{B}$. Consider the pair of sides of length $a$ that are oriented PERPENDICULAR to

$\vec{B}$ in projection (i.e. lying along the rotation axis): each experiences a force of magnitude

$F=BIa$, and, because current flows in OPPOSITE directions along these two opposite sides of the

loop, the two forces are equal in magnitude but point in OPPOSITE directions -- an equal, opposite,

non-collinear pair of forces is precisely a couple. (The other pair of sides, of length $b$,

also experience forces, but these act along the SAME line -- through the centre of the loop -- and

so contribute zero net torque about the central axis; they need not be tracked further for the

torque calculation, though they are not zero individually.)

Deriving the torque. A couple's torque is the product of one of the equal forces and the

PERPENDICULAR distance between the two lines along which the forces act (the couple arm). Here the

two forces $F=BIa$ act along the two sides of length $a$, separated by the OTHER side's length $b$;

but the perpendicular distance between them, projected onto the direction perpendicular to $\vec{B}$,

is $b\sin\theta$ rather than the full $b$ (since it is the LOOP'S ROTATION, by angle $\theta$ away

from the field-aligned position, that determines how far apart the two force-lines are when measured

perpendicular to the forces themselves). The torque is therefore

using $A=ab$. For a coil of $N$ turns rather than a single loop, each turn contributes the same

torque, so the total is $N$ times as large:

Magnetic moment and the compact vector form. Defining the loop's (or coil's) magnetic dipole moment as the vector $\vec{m} = NI\vec{A}$ (magnitude $m=NIA$, direction along the normal

$\hat{n}$, fixed by the right-hand rule applied to the current's sense of circulation), the torque

result above can be written compactly as the vector cross product

exactly analogous in form to the torque on an electric dipole in a uniform electric field,

$\vec{\tau} = \vec{p}\times\vec{E}$, from this unit's first sub-topic.

Extreme cases and potential energy. When $\theta=90^\circ$ (the loop's plane parallel to

$\vec{B}$, i.e. its normal perpendicular to the field), $\sin\theta=1$ and the torque is at its

MAXIMUM, $\tau_{\max}=NIAB$ -- the orientation exploited directly by the moving-coil galvanometer of …