Physics · Electricity and magnetism · Grade 8

Electric motor

What makes a current-carrying wire turn, and why doesn't a motor speed up without limit?

The situation

A blender, a hairdryer, an electric scooter, a toy car — all of them contain an electric motor. The demonstration motor in a school laboratory is built in the same way: a wire coil between two permanent magnets, a copper commutator on its axle and two carbon brushes touching it. Connect a power supply and the coil starts to spin, and its speed settles at a steady value. What turns the wire, and why does the motor not speed up without limit?

Topic

Magnetic field, Ampère force. How an electric motor turns

Key ideas

Ampère force
Left-hand rule
Back EMF

Formulas

F = B I l \sin α, A = I Δ Φ

The Ampère force (its direction is given by the left-hand rule) and the work it does when a current-carrying conductor is moved: $Δ Φ$ is the magnetic flux that the conductor “cuts across”.

M = p_{m} B | \sin θ |, p_{m} = N I S

The torque that turns the coil; $p_{m}$ is the magnetic moment of the current. $θ$ is the angle between the normal to the coil and $\vec{B}$ ; the modulus sign is the work of the commutator.

ℰ = N B S ω | \sin θ |

Back emf: it is induced in the rotating coil and acts against the supply voltage.

I = \frac{U - ℰ}{R}

The current in the coil. In a stalled motor $ℰ = 0$ and the current is greatest: $I = U / R$ .

ω_{0} \approx \frac{4 U}{π N B S}

The steady speed with no load and no friction: the $ω$ at which the average torque over one revolution is zero.

Theory

The Ampère force and the left-hand rule

A force $F = B I l$ acts on a current-carrying wire perpendicular to a magnetic field. Left-hand rule: hold your left hand so that the field lines (from N to S) enter the palm and your four fingers point along the current; the thumb, opened out at 90°, shows the direction of the force. On the two sides of the coil the current flows in opposite directions, so in the scene the red arrows point one up and one down — together they form a couple that turns the coil. Two parallel currents also interact with this force: $F / l = μ_{0} I_{1} I_{2} / (2 π d)$ — currents in the same direction attract, currents in opposite directions repel; until 2019 the ampere was defined by exactly this force (more in the “Magnetic field” experiment).

What is the commutator for?

If the direction of the current did not change, after half a turn the coil would feel a reverse torque and would oscillate to a stop. The commutator is two half-rings mounted on the axle. At the moment when the plane of the coil is perpendicular to the field (the dead point, $θ = 0$ or $180 °$ ) the brushes pass from one half-ring to the other and the current in the coil reverses. As a result the torque always acts the same way: $M = N B I S | \sin θ |$ . If the motor stops exactly at the dead point, the torque is zero and it cannot start by itself — it has to be given a push by hand. At slow speed (0.02×) watch how the yellow current arrows reverse at exactly this moment.

Back emf and steady speed

A rotating coil is a generator at the same time: an emf $ℰ = N B S ω | \sin θ |$ is induced in it, acting against the supply (Lenz's law). As the speed rises the current $I = (U - ℰ) / R$ falls, and so does the torque. The motor turns steadily at the speed where the torque balances the load and friction on average. If you increase the load, the speed drops, while the current and the power drawn rise. In a stalled motor $ℰ = 0$ : all the power $U^{2} / R$ heats the coil — which is why a jammed motor can burn out.

Magnetic moment and the work of the Ampère force

A current-carrying loop in a field behaves like a compass needle. Its magnetic moment $p_{m} = N I S$ is a vector perpendicular to the plane of the loop (right-hand screw rule); the torque $\vec{M} = {\vec{p}}_{m} \times \vec{B}$ tends to turn it along $\vec{B}$ , and the energy is $W = - {\vec{p}}_{m} \cdot \vec{B}$ . This holds not only for a rectangle: for any flat loop $p_{m} = I S$ , and in general the moment of a line current is ${\vec{p}}_{m} = \frac{1}{2} \oint \vec{r} \times I d \vec{l}$ . When a current-carrying conductor is displaced by $Δ x$ , the Ampère force does work $A = B I l Δ x = I Δ Φ$ . When the coil turns through half a revolution, the flux through it changes from $N B S$ to $- N B S$ , while the commutator reverses the current — so the work adds up with the same sign every half-turn. This work is taken from the supply: $A = \int ℰ I d t$ (see $A$ and $N Φ$ in the “Measurements” table).

The galvanometer, measuring instruments and the electric car

A moving-coil instrument is a “relative” of the motor: the coil sits in a radial field between the pole pieces and an iron core, so the torque $M = N B I S$ does not depend on the angle. Instead of a commutator, a spiral spring provides the opposing torque $k θ$ : $θ = N B S I / k$ — the deflection is proportional to the current, and the scale is even. Eddy currents in the aluminium frame damp the oscillations of the pointer. If a small shunt is connected in parallel with the galvanometer, it becomes an ammeter; if a large multiplier resistor is connected in series, it becomes a voltmeter (the “Circuits and Kirchhoff's rules” experiment); in engineering such instruments show current, voltage and (via sensors) temperature or pressure. An electric car has its wheels turned by electric motors supplied from the battery through an inverter. When braking, the motor works as a generator: the back emf exceeds the battery voltage, the current flows in reverse and the battery is recharged (regenerative braking).

Model and simplifications

The field between the poles is taken to be uniform, and the inductance of the coil is neglected (for a small air-cored coil $L / R \approx 0.1$ ms — much less than the time of half a turn). The coil: $l = 5$ cm, width 4 cm, $S = 20$ cm². Friction is a constant torque of 0.5 mN·m plus air resistance $b ω$ ( $b = 2 \cdot 10^{- 6}$ N·m·s); the rotor's moment of inertia is $J = 5 \cdot 10^{- 5}$ kg·m². The equation of motion $J \dot{ω} = M - M_{f r} - M_{l o a d}$ is solved by the Runge–Kutta (RK4) method with a 20 µs step. The speed and current of a single-coil motor fluctuate every half-turn; in real motors many coils and a multi-segment commutator smooth out these fluctuations. The galvanometer model: coil 2.2 × 1.6 cm, moment of inertia $3.3 \cdot 10^{- 8}$ kg·m², the field is fully radial within the deflection range (±50°), damping torque $\propto B^{2} ω$ ; in its mode time runs at 1× speed.

Did you know?

In 1820 Hans Christian Ørsted discovered that a wire carrying a current deflects a compass needle. In the same year André-Marie Ampère showed by experiment that current-carrying conductors attract or repel each other, and studied the law of this force.
In 1834 Moritz Jacobi built in Königsberg a rotary electric motor that could be used in practice. In 1838 in St Petersburg a boat fitted with his motor and powered by batteries carried passengers on the River Neva.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Electrification, Coulomb's law — Grade 7
Electric circuit, Ohm's law — Grade 7
Conductor, dielectric, lightning — Grade 7
Power sources, EMF, internal resistance — Grade 7
Work and power of an electric current — Grade 8
Aurora and the Lorentz force — Grade 8
Electric field and potential — Grade 8
Capacitor and capacitance — Grade 8
Circuits and Kirchhoff's rules — Grade 8
Electric current in metals — Grade 8