Calculate the force on a current-carrying conductor in a magnetic field and explain its role in the operation of a motor.

What this dot point is asking

You need to calculate the force on a current-carrying conductor in a magnetic field and explain how this force drives an electric motor.

The motor effect

A current is just charges in motion, so the same magnetic force that acts on moving charges acts on a current-carrying wire. Summing the force over all the charges in a length $L$ of wire gives:

Direction of the force

Use the right-hand rule (treating conventional current as the direction of motion of positive charge): point fingers along the current $I$, curl toward the field $B$; the thumb/palm gives the force. Equivalently the flat-hand rule: fingers point along $B$, thumb along $I$, palm pushes in the force direction. The force is perpendicular to both the current and the field.

How a DC motor works

A simple motor is a current-carrying coil free to rotate in a magnetic field:

1. Current flows through the coil. The two sides of the coil carry current in opposite directions.
2. By the motor effect, the two sides feel forces in opposite directions (one up, one down), producing a turning effect (torque) on the coil.
3. The coil rotates. As it passes the vertical, the forces would now oppose the rotation.
4. A split-ring commutator reverses the current direction in the coil every half-turn, so the torque keeps driving the coil the same way and rotation continues.

The turning effect is largest when the coil is parallel to the field (forces have the longest lever arm) and zero when the coil is perpendicular to the field - but the coil's momentum carries it through this point.