How an electric motor spins

A loop of wire in a magnetic field twists when current runs through it. That twist is a motor — once you solve the one problem that would make it stop after half a turn.

What a motor is

An electric motor is a coil of wire sitting in a magnetic field. Send current through the coil and the field pushes on the wire, turning the coil and the shaft it is fixed to. Electrical energy goes in; mechanical work comes out.

Why it matters

Motors are the most common use of magnetic force on a current-carrying wire, and the same arrangement, held back by a spring instead of allowed to spin, is the needle of an analog meter. But the naive version does not spin at all. Put current through a single loop and it swings toward one position and back again. Every motor that turns is a fix for that.

Where the twist comes from

A wire carrying a current in a magnetic field feels a force at right angles to both the current and the field. Take a rectangular loop on a vertical shaft, with the field running horizontally. On the top and bottom segments the force points along the shaft, so it cannot turn anything. On the two vertical sides the forces are equal and opposite — one pushes forward, the other back — so the loop as a whole does not move. But they act on opposite sides of the shaft, and a pair of opposite forces on opposite sides of a pivot is a torque: a twist.

How much twist depends on the loop's orientation. Call θ the angle between the field and a line perpendicular to the face of the loop. For a coil of N turns, each of area A, carrying current I in a field B, the torque is

τ = N I A B sin θ

Torque is largest when θ is 90°, with the face of the loop parallel to the field. It is zero when θ is 0°, with the face square to the field. A 100-turn square coil 10.0 cm on a side, carrying 15.0 A in a 2.00 T field, peaks at 30.0 N·m — enough to be useful in a motor.

Now the catch. The torque falls to zero at θ = 0°, and once the coil's momentum carries it past that point the torque reverses. The field now pulls it back. Left alone, the coil oscillates about θ = 0°. It is a compass needle, not a motor.

The fix is to reverse the current at the moment the coil passes through θ = 0°. Flip the current and the forces on the two sides swap, so the torque points the same way it did before. Automatic switches called brushes do this, reversing the current every half revolution. Each time the coil coasts through the zero-torque position, the current flips, the torque stays clockwise, and the coil keeps going.

In short

Current in a magnetic field feels a sideways force. On a loop, the forces on the two sides make a torque of N I A B sin θ, largest with the loop edge-on to the field and zero when it faces the field squarely. Past that point the torque turns around, so a plain loop only rocks back and forth. Reverse the current every half turn and the torque never turns around — and the loop becomes a motor.