# 14.2.2 Electric Motor

The electric motor was invented soon after the discovery of $\displaystyle F=BIL$.

Below is the schematic of a very basic electric motor: a rectangular coil (length $\displaystyle L=PQ$ and width $\displaystyle W=QR$) of N turns, carrying current I, rotating in a uniform magnetic field B.

Firstly, we note that the currents I in PQ and RS are perpendicular to B. So sides PQ and RS both experience a magnetic force of

$\displaystyle {{F}_{b}}=NBIL$

(We can ignore the magnetic forces acting on the other two sides because they do not produce any moment about the rotational axis.)

Since the current in PQ and RS are running in opposite directions, the magnetic force Fb acting on them are also in opposite directions (using the FLHR, you can verify that Fb is upward for PQ, downward for RS). This pair of Fb forms a couple, producing a clockwise torque.

The torque of this couple t is dependent on the perpendicular distance between the two Fb, which in this case can be expressed as $\displaystyle W\cos \theta$. So

\displaystyle \displaystyle \begin{aligned}\tau &=NBIL\times W\cos \theta \\&=NABI\cos \theta \end{aligned}

where A is the area of the rectangular coil.

Do realize that the torque varies sinusoidally only because the couple’s perpendicular distance changes sinusoidally[1]. The magnetic force $\displaystyle {{F}_{b}}=NBIL$ is actually constant because, look carefully, I remain perpendicular to B even as the coil rotates.

Demonstration

Battery, Magnets and Wires (including spinning heart)

Concept Test

2823

[1] It is usually desirable to have a constant torque. Practical motors employ more sophisticated magnetic fields to achieve that.