22.8 Torque on a Current Loop: Motors and Meters

Torque on a Current Loop: Motors and Meters

Electric motors convert electrical energy into mechanical work by exploiting the torque that a magnetic field exerts on a current-carrying loop. This same principle also drives the operation of analog meters (galvanometers). Understanding how torque arises on a current loop connects the force laws you've already learned to real, rotating devices.

Conversion of Energy in Motors

A current-carrying loop placed in an external magnetic field experiences forces that create a torque, spinning the loop and converting electrical energy into rotational kinetic energy. That's the core idea behind every DC motor.

The force on each current-carrying segment of the loop comes from the magnetic force law:

$\vec{F} = I\vec{L} \times \vec{B}$

• $I$ is the current in the wire
• $\vec{L}$ is a length vector pointing in the direction of current flow
• $\vec{B}$ is the external magnetic field

Because the cross product produces a force perpendicular to both $\vec{L}$ and $\vec{B}$, opposite sides of a rectangular loop get pushed in opposite directions. Those two forces don't cancel; instead, they form a couple that produces a net torque, spinning the loop.

To find the direction of the force on each segment, use the right-hand rule: point your fingers in the direction of the current, curl them toward $\vec{B}$, and your thumb points in the direction of $\vec{F}$.

Torque Calculation for Current Loops

Rather than computing forces on each side of the loop separately, you can use the compact torque formula:

$\vec{\tau} = \vec{\mu} \times \vec{B}$

Here $\vec{\mu}$ is the magnetic dipole moment of the loop:

$\vec{\mu} = NIA\hat{n}$

• $N$ = number of turns (loops of wire)
• $I$ = current
• $A$ = area of one loop
• $\hat{n}$ = unit vector perpendicular to the plane of the loop (direction found by right-hand rule: curl fingers with current, thumb gives $\hat{n}$)

The magnitude of the torque is:

$\tau = \mu B \sin\theta = NIAB\sin\theta$

where $\theta$ is the angle between $\vec{\mu}$ and $\vec{B}$.

Key cases to remember:

• Maximum torque ($\tau = NIAB$) when $\theta = 90°$, meaning the plane of the loop is parallel to the field.
• Zero torque when $\theta = 0°$ or $180°$, meaning $\vec{\mu}$ is aligned (or anti-aligned) with $\vec{B}$.

Quick example: A 50-turn rectangular coil (0.10 m × 0.20 m) carries 2.0 A in a 0.30 T field. At $\theta = 90°$:

$\tau = NIAB = (50)(2.0)(0.10 \times 0.20)(0.30) = 0.60 \text{ N·m}$

Function of Brushes and Commutators

Without some clever engineering, the loop would just swing to $\theta = 0°$ (where torque is zero) and stop, or oscillate back and forth. DC motors solve this with brushes and a commutator.

Here's how they work, step by step:

1. The commutator is a split ring (two or more insulated copper segments) mounted on the shaft and rotating with the armature (the coil assembly).
2. Each segment connects to one end of the armature winding.
3. Brushes are stationary contacts, typically made of carbon or graphite, that press against the commutator and connect to the external power supply.
4. As the armature rotates past the zero-torque position, the commutator segments swap which brush they touch. This reverses the current direction in the loop at exactly the right moment.
5. Because the current reverses every half-turn, the torque always pushes the loop in the same rotational direction, producing continuous spinning instead of oscillation.

The brushes also serve as the electrical bridge between the stationary external circuit (the stator side) and the spinning armature.

Electromagnetic Principles in Motors

As the loop spins in the magnetic field, the changing magnetic flux through it induces a voltage according to Faraday's law. This induced voltage is called back EMF, and it opposes the current driving the motor (by Lenz's law).

Back EMF has a practical consequence: as a motor speeds up, the back EMF increases, which reduces the net current through the motor. This is why a motor draws a large current when it first starts (low back EMF) but much less current at full speed. If the motor is stalled or under heavy load, the back EMF stays low and the current stays high, which can overheat the windings.