How Electric Motors Spin

Equivalent forms

\vec{\tau} = \vec{m}\times\vec{B}

m = N I A

The same τ = m×B that aligns a compass needle, harnessed and repeatedly re-aimed, powers civilization's machines.

Symbol	Name	SI unit	Dimension	Range
$\tau$	Torqueoutput Twisting moment on the coil	N·m	M·L²·T⁻²	0 – 100
$N$	Turns Number of coil turns	(dimensionless)	1	1 – 1000
$B$	Magnetic field Applied flux density	T	M·T⁻²·I⁻¹	0.01 – 2
$I$	Current Coil current	A	I	0.1 – 50
$A$	Coil area Area enclosed by one turn	m²	L²	0.0001 – 0.1
$\theta$	Angle Angle between coil plane and field	rad	1	0 – 3.1416

Unit systems

Symbol	SI
$\tau$	N·m
$N$	(dimensionless)
$B$	T
$I$	A
$A$	m²
$\theta$	rad

Where it holds

Valid for a planar coil in a uniform field. Real motors have angle-dependent fields, back-EMF that limits current at speed, and shaped pole pieces to flatten the torque curve — corrections to this idealized form.

Dimensional analysis

$N\cdot B\cdot I\cdot A = (T)$ (A $)(m^{2}) = (kg\cdot s^{-2}\cdot A^{-1})$ (A $)(m^{2}) = kg\cdot m^{2}\cdot s^{-2} = N\cdot m \checkmark$

Discovery

Michael Faraday · 1821

Faraday built the first device to turn electricity into continuous motion — a wire rotating around a magnet in mercury. The torque-on-a-loop principle grew into every motor on Earth.

Try this

From a phone's vibration buzzer to a Tesla, every electric motor turns by the same trick: a magnetic field shoving on a current. How much twist do you get?

A 100-turn square coil 5 cm on a side carries 2 A in a 0.4 T field. Find the maximum torque on the coil.

Research status: stable

Real-world applications

EV and industrial drive motors
Hard-drive voice coils
Galvanometers and analog meters
Loudspeaker and haptic actuators

Common misconceptions

Motors push on the magnetic poles — the force is on the moving charges (current) in the field
Torque is constant as it spins — it varies as $\sin \theta$ , which is why multi-coil rotors and commutation smooth it out

Experimental verification

Mount a coil on a torsion balance in a known field; measured torque tracks NBIA

\sin \theta

exactly, peaking when the coil plane is parallel to B and vanishing when aligned.

Derivation

Each side of the loop carrying current I in field B feels

F = BIL

.

The two sides of length L separated by width w form a couple of moment

\tau = F\cdot w\cdot \sin \theta = BIL\cdot w\cdot \sin \theta = BIA \sin \theta for

one turn

(A = Lw)

.

For N turns,

\tau

= NBIA

\sin \theta = mB \sin \theta

with

m = NIA

, i.e.

\tau = m\times B

.

Limiting cases

\theta \to 0

(coil aligned with field)⟶

\tau \to 0

When the magnetic moment lines up with B there is no twist — the dead spot a commutator must push past.

\theta \to 90^{\circ}

⟶

\tau \to \max

= NBIAMaximum leverage when the coil plane is parallel to the field.

What if…

What if the coil reaches the aligned position?

Torque drops to zero; a commutator reverses the current there so the push continues and the rotor keeps turning.

What if you double the turns and the current?

Torque quadruples — but more current also means more heating, the real design limit.

1

100-turn coil in 0.4 T

Given ·

N:: 100
B:: 0.4
I:: 2
A:: 0.0025
\theta:: 1.5708

Find · max torque

Steps

$A = (0.05\,\mathrm{m})^{2} = 2.5\times 10^{-3} m^{2}$
$\tau _\max$ = NBIA $(\sin 90^{\circ} = 1)$
$= 100\times 0.4\times 2\times 2.5\times 10^{-3} = 0.2 N\cdot m$

Result ·

0.2 N\cdot m

at the point of maximum leverage.

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