The electric motor

A current loop in a magnetic field experiences a torque that tends to make it rotate: this is the heart of the electric motor, the device that converts electrical energy into mechanical energy.

History — From Faraday's wire to trams

The idea of the electric motor was born in 1821: Michael Faraday built the first rudimentary prototype, in which a current-carrying wire rotated around a magnet immersed in mercury. The first practical motor was built in 1834 by Moritz von Jacobi, a Prussian engineer, and used to propel a boat on the Neva river in Saint Petersburg. Today electric motors move everything, from high-speed trains to hard disks to the fan inside a PC.

Elementary electric motor: loop between two permanent magnets, vertical rotation axis, brush commutator fed by the generator.

Force analysis. Consider a rectangular loop with sides $\ell$ (parallel to the axis) and $h$ (perpendicular to the axis), carrying current $i$ in a uniform field $\vec{B}$. Each side carries current in a field, so it feels $\vec{F} = i\vec{L}\times\vec{B}$:

• The two sides parallel to the axis (length $\ell$) receive forces $F_1 = i\,\ell\,B$ in opposite directions: they form a couple that makes the loop rotate.
• The two sides perpendicular to the axis (length $h$) receive forces along the axis, which balance out and produce no rotation.

Torque as a function of the angle. When the plane of the loop forms an angle with $\vec{B}$, each long side acts with a moment arm $(h/2)\sin\beta$, where $\beta$ is the angle between $\vec{\mu}$ and $\vec{B}$. The total torque is:

$|\vec{M}| = 2\cdot F_1\cdot\frac{h}{2}\sin\beta = i\,\ell\,B\,h\,\sin\beta = i\,A\,B\,\sin\beta$

where $A = \ell h$ is the area of the loop. Using $|\vec{\mu}| = iA$:

Law — Torque on a loop in a magnetic field

$\ev{|\vec{M}| = |\vec{\mu}|\,|\vec{B}|\,\sin\beta = i\,A\,B\,\sin\beta} \qquad \vec{M} = \vec{\mu}\times\vec{B}$ The torque is maximum for $\beta = 90^\circ$ (loop parallel to $\vec{B}$) and zero for $\beta = 0^\circ$ (loop perpendicular to $\vec{B}$, i.e. $\vec{\mu}$ aligned with the field).

Collegamenti

Argomenti: Magnetismo Concetti: Dipolo magnetico · Forza di Laplace · Momento di una forza

Esercizi collegati: Worked exercise — rectangular loop in a uniform field · Torque on a square loop · Which field gives maximum torque