Electromagnetic induction generators and motors

When a magnetic field passes through a closed loop, the magnetic flux $\Phi = B A \cos\theta$ describes how many field lines pass through the loop. Faraday discovered that when this flux changes, an induced emf appears in the loop: $\mathcal{E} = -N \frac{\Delta\Phi}{\Delta t}$. The minus sign expresses Lenz's law: the induced current is directed so that its magnetic field opposes the change in flux. In a generator, mechanical rotational energy is converted into electrical energy: a coil rotates in a magnetic field, the flux changes periodically, and an alternating emf is produced. In a motor, the process is reversed: current in conductors in a magnetic field experiences the Ampère force, creating a torque, and electrical energy is converted into mechanical energy. Thus, a generator and a motor are two aspects of the same electromechanical energy-conversion process: a motor is essentially a generator driven in reverse, and its rotating coil also exhibits electromagnetic induction, which produces a back-emf that opposes the supply current.

A coil has $N = 100$ turns. The magnetic flux through each turn increases uniformly from $0.02$ Wb to $0.08$ Wb in $\Delta t = 0.40$ s. Find the average induced emf.

A resistor is connected to a coil. A magnet's north pole is moved toward the coil. Determine the direction of the induced current, which pole appears at the near end of the coil, and the direction of the force on the magnet. Justify with Lenz's law.

In a generator, a rectangular coil of area $A = 0.020$ m$^2$ has $N = 50$ turns and rotates at frequency $f = 60$ Hz in a uniform magnetic field $B = 0.10$ T. 1) Find the maximum emf. 2) At what rotation frequency would the maximum emf be $12$ V? Take $\omega = 2\pi f$.

Safe self-study investigation with a simulation.

1. Open an electromagnetic induction simulation (for example, PhET «Faraday's Electromagnetic Lab» or a similar one) in the «Pickup Coil» mode.
2. Slowly move a magnet into and out of the coil; record the direction of the needle deflection.
3. Repeat at a higher speed; compare the size of the deflection.
4. Change the number of turns in the coil and check how the current changes.
5. Switch to the «Generator» mode: rotate the source and observe the alternating current; then in the «Motor» mode, supply current and observe rotation.
   Do not connect real devices to the mains; for a real experiment use only a low-voltage source (for example, a battery) and a low-power LED.

Show the link between changing magnetic flux, induced current, and Lenz's law, and compare a generator and a motor as mutually inverse devices.

Diagram: a coil with a galvanometer, a magnet, and arrows showing its motion; curved arrows show the direction of the induced current and the coil's magnetic field opposing the motion. Next to it, two panels: a generator (rotation of a coil in field $B$ → alternating current) and a motor (current in a coil in field $B$ → rotation), with labels for fluxes and forces.

A cyclist rides at $18\ \text{km/h} = 5\ \text{m/s}$. The dynamo roller of radius $r = 1.5\ \text{cm}$ is pressed against the tyre of radius $R = 0.34\ \text{m}$, so the roller spins at $f = v/(2\pi r) \approx 53\ \text{Hz}$. A permanent magnet rotates inside, and the magnetic flux through the coil reverses sign $2f \approx 106$ times per second. The generator produces an alternating voltage and lights a $6\ \text{V}$, $3\ \text{W}$ lamp (current $I = 0.5\ \text{A}$).

This is a generator: rotational energy becomes electrical energy. Faraday's law $\varepsilon = -N\dfrac{\Delta\Phi}{\Delta t}$ applies; for steady rotation $\Phi(t) = BA\cos(\omega t)$ with $\omega = 2\pi f$, so the emf amplitude $\varepsilon_0 = NBA\omega$ grows with rotation rate, field $B$, coil area $A$ and number of turns $N$.

A $1500\ \text{kg}$ electric car brakes from $20\ \text{m/s}$ ($72\ \text{km/h}$); its kinetic energy is $\tfrac12 mv^2 = 3.0\times10^5\ \text{J}$. The motor is switched to generator mode: the wheels turn the rotor and current flows into a $400\ \text{V}$ battery. Returning about $60\%$ of the energy puts $1.8\times10^5\ \text{J}$ into the battery, that is a charge $Q = E/U = 450\ \text{C} \approx 0.13\ \text{A·h}$.

This shows machine reversibility: the same motor becomes a generator when the rotor is turned from outside. Lenz's law fixes the direction of the induced current, and the resulting braking torque $M = BINA$ (for $N$ turns, current $I$, coil area $A$) opposes the rotation, while energy conservation links the stored charge to the loss of $\tfrac12 mv^2$.

An induction hob transfers $2000\ \text{W}$ into the base of a cast-iron pan of diameter $20\ \text{cm}$. The coil under the glass is fed with alternating current at $25\ \text{kHz}$, so the magnetic flux through the steel base reverses $50\,000$ times per second. Eddy currents in the metal heat the base to $200\ ^\circ\text{C}$ and above, the glass stays near $60\ ^\circ\text{C}$, and the transferred fraction of power reaches $90\%$.

Faraday's law is applied to a solid conductor: $\varepsilon = -\dfrac{\Delta\Phi}{\Delta t}$ drives eddy currents, and Joule heating $P = I^2R$ warms the base. Lenz's law explains why the eddy currents oppose the change of flux, while the high frequency and large field $B$ make $\Delta\Phi/\Delta t$ large. Here induction works without any wire with terminals.