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Electric motors

Electricity and magnetism · level 7, lesson 3 of 6

A question to start

The motor effect pushes a wire sideways. How do we turn that sideways push into something that spins round and round?

The idea

Put a rectangular coil in a magnetic field. Current flows one way along one side and back the other way along the other side. So the two sides are pushed in opposite directions, and the coil turns.

The turning effect, the torque, is greatest when the coil's plane lies along the field:

\[\tau = NBIA,\]

for \(N\) turns of area \(A\). As the coil turns, the torque falls, and after half a turn it would push back. So the current must be reversed every half turn:

  • In a simple DC motor, a split-ring commutator with brushes does this.
  • In a brushless motor, electronics switch the currents in fixed coils, and the magnet spins instead. These last longer, as there are no brushes to wear out.

The power a motor delivers is \(P = \tau\omega\), where \(\omega\) is its spin rate in radians per second.

NS force upforce downthe coil turns

Worked examples

1. Torque. 100 turns, \(B = 0.5\) T, \(I = 2\) A, \(A = 4\) cm² \(= 4\times10^{-4}\) m²: \(\tau = 100 \times 0.5 \times 2 \times 4\times10^{-4} = 0.04\) N m.

2. Power. At 3,000 turns a minute, \(\omega = 3000 \times 2\pi/60 = 314\) rad/s: \(P = 0.04 \times 314 = 12.6\) W.

3. More torque. Double the current and the torque doubles: 0.08 N m.

Try it

  1. Find the torque on a 50-turn coil of area 0.01 m² carrying 3 A in a 0.2 T field.
  2. Why must the current in the coil reverse every half turn?
  3. A motor gives 0.5 N m at 251 rad/s. What is its power?
  4. Why do brushless motors last longer?
Answers
  1. \(50 \times 0.2 \times 3 \times 0.01 = 0.3\) N m.
  2. Otherwise, after half a turn, the forces would turn the coil back the other way.
  3. \(0.5 \times 251 = 126\) W.
  4. They have no brushes rubbing on a commutator to wear out.

Common mistake

"The torque is always \(NBIA\)." That is the largest value, when the coil lies along the field. When the coil faces the field squarely, the forces pull straight outwards and the torque is zero.

Where it leads