Forces and Torques on Currents

The torque on a current loop is simply the twist a magnetic field puts on a loop of wire carrying current. (Torque comes from the Latin torquere, "to twist".) It's the reason a loudspeaker cone, a hard drive's read arm, an electric car's motor and an analog meter's needle all move. The formula $\tau = NIAB\sin\theta$ is just a product of plain pieces: $N$ (how many turns of wire), $I$ (the current), $A$ (the loop's area), $B$ (the field's strength) and $\sin\theta$ (how far the loop is from lining up with the field). It looks like a lot of letters, but each one just makes the twist bigger in the obvious way. Bear with it.

From charges to wires. Last lesson we found the push on one moving charge. A wire is full of them, so all we have to do is add them up. Take a short piece of wire of length $dl$ and cross-section $A$, with $n$ carriers per cubic meter, each of charge $q$ drifting at $\mathbf{v}_d$. Each carrier feels $q\mathbf{v}_d\times\mathbf{B}$, and there are $nA\,dl$ of them, so

From Current, Resistance and Ohm's Law, $I = nqAv_d$, and $\mathbf{v}_d$ points along the wire (against it for electrons, but then $q < 0$ too, so the product doesn't care). So $nqA\,dl\,\mathbf{v}_d = I\,d\mathbf{l}$, with $d\mathbf{l}$ pointing along the current, and

For a straight wire of length $L$ in a uniform field, the identical pieces add up to $\mathbf{F} = I\mathbf{L}\times\mathbf{B}$, of size $ILB\sin\theta$.

Curved wires. In a uniform field $\mathbf{F} = I\left(\int d\mathbf{l}\right)\times\mathbf{B}$, and $\int d\mathbf{l}$ is just the straight vector from the wire's start to its end. A curved wire feels the same force as a straight wire joining its ends, and a closed loop, whose ends meet, feels no net force at all.

But a loop does feel a torque. No net force doesn't mean nothing happens. Think of pushing up on one side of a steering wheel and down on the other: the pushes cancel, but the wheel turns. Take a rectangular loop with sides $a$ and $b$, carrying $I$ in a uniform field and free to turn about an axis parallel to the sides of length $b$ (and perpendicular to $\mathbf{B}$), with its normal at angle $\theta$ to $\mathbf{B}$. The forces on the two sides of length $a$ point along the axis and cancel. The two sides of length $b$ are perpendicular to $\mathbf{B}$ and feel equal and opposite forces $IbB$, perpendicular to the field. Their lines of action are offset by $a\sin\theta$, the part of the loop's width that lies along the field, so they form a couple:

With $N$ turns, multiply by $N$. In vector form, define the magnetic moment $\boldsymbol{\mu} = NI\mathbf{A}$, perpendicular to the loop by the right-hand rule (curl your fingers with the current; your thumb is $\boldsymbol{\mu}$). Then $\boldsymbol{\tau} = \boldsymbol{\mu}\times\mathbf{B}$, the same shape as torque $\mathbf{r}\times\mathbf{F}$ in mechanics. It works for any flat loop: chop the loop into tiny rectangles; the currents on shared edges cancel, as in the proof of Green's Theorem, and the tiny torques add up to $I$ times the total area.

Energy. Turning $\boldsymbol{\mu}$ away from $\mathbf{B}$ takes work against the torque: $U(\theta) = \int\mu B\sin\theta\,d\theta = -\mu B\cos\theta = -\boldsymbol{\mu}\cdot\mathbf{B}$. The loop is happiest aligned with the field, and flipping it all the way around costs $2\mu B$. A compass needle is a tiny magnetic moment settling into that minimum.

Motors. The torque $NIAB\sin\theta$ turns the coil toward alignment, then reverses once the coil overshoots, so a DC motor uses a commutator, a split ring that flips the current every half turn and keeps the torque pushing the same way. Real motors use many coils at different angles so the total torque barely ripples. The second picture further down shows this, and its averages are average values of $|\sin\theta|$.

A common mistake. It's tempting to measure $\theta$ from the loop's flat face, since that's the angle you can see. But $\theta$ is measured from the loop's normal, $\boldsymbol{\mu}$, and mixing them up swaps sine and cosine. So the torque is largest when the face is edge-on to the field, not when it faces the field. Two smaller slips: pointing $\mathbf{L}$ against the current, and forgetting the number of turns $N$.

Run a motor backwards, turning the coil by hand, and the same physics makes a voltage. That's the generator we'll meet in Faraday's Law, in the next unit.

Every electric motor, from a phone's vibration buzzer to a train's traction motor, turns $\boldsymbol{\tau} = \boldsymbol{\mu}\times\mathbf{B}$ into rotation, with commutators or electronics switching the current to keep the torque one way. Loudspeakers and hard-drive arms use $\mathbf{F} = I\mathbf{L}\times\mathbf{B}$ on a coil in a radial field, so the force follows the current exactly. MRI scanners work at the frequency set by $U = -\boldsymbol{\mu}\cdot\mathbf{B}$ for proton spins.