Self-Induction, Mutual Induction and Eddy Currents

1A spark from a 1.5 V cell

Switch off a large electric motor, a car's ignition coil or even a relay, and a spark may jump across the switch. A few volts in the circuit, yet a spark needs hundreds of volts or more. The coil itself is the source: when its current collapses, its own magnetic flux collapses, and by Faraday's law it induces a large emf in itself. This lesson looks at induction within a single coil (self-induction), between two coils (mutual induction), and in solid lumps of metal (eddy currents), and at the uses of each, from transformers to induction cookers.

2Self-inductance

A current $II$ in a coil produces a magnetic field and so a flux linkage $N\mathrm{\Phi }N\backslash Phi$ through the coil itself, proportional to $II$. The constant of proportionality is the coil's self-inductance $LL$:

By Lenz's law this back emf opposes the change in current: it resists a rise when the circuit is switched on and tries to keep the current flowing when it is switched off. An inductor is the electrical equivalent of inertia.

Inductance of a long solenoid

A solenoid of $NN$ turns, length $ll$ and cross-section $AA$ has an internal field $B=\frac{{\mu }_{0}NI}{l}B\; =\; \backslash frac\{\backslash mu\_0\; NI\}\{l\}$ (S5), so

An iron core multiplies $LL$ many times, which is why large inductors are wound on iron.

3Demonstration 4.1: the neon lamp that lights at switch-off

Apparatus: a 1.5 V cell, a switch, a large coil on a closed iron core (for example 0.50 H with a resistance of $5.0\mathrm{\Omega }5.0\backslash ,\backslash Omega$), a small neon lamp, leads (Figure 4.1).

Figure 4.1: A neon lamp needs roughly 70 V to 90 V to light, far more than the cell provides.

1. Close the switch. The lamp stays dark: the p.d. across it is only about 1.5 V. A steady current of $\frac{1.5}{5.0}=0.30\mathrm{A}\backslash frac\{1.5\}\{5.0\}\; =\; 0.30\backslash ,\backslash mathrm\{A\}$ flows through the coil.
2. Open the switch. The lamp flashes. The current in the coil collapses in about a millisecond, inducing an emf of about $L\frac{\mathrm{\Delta }I}{\mathrm{\Delta }t}=0.50\times \frac{0.30}{0.001}=150\mathrm{V}L\backslash frac\{\backslash Delta\; I\}\{\backslash Delta\; t\}\; =\; 0.50\; \backslash times\; \backslash frac\{0.30\}\{0.001\}\; =\; 150\backslash ,\backslash mathrm\{V\}$, which drives a brief current through the lamp.

4Growth of current and the time constant

When a circuit of emf $EE$, resistance $RR$ and inductance $LL$ is switched on, the back emf holds the current back at first, so it rises smoothly rather than jumping:

Figure 4.2: After one time constant $\tau =\frac{L}{R}\backslash tau\; =\; \backslash frac\{L\}\{R\}$ the current has reached $1-e^{-1}\approx 63\mathrm{\%}1\; -\; e^\{-1\}\; \backslash approx\; 63\backslash \%$ of its final value.

For the coil of Demonstration 4.1, $\tau =\frac{0.50}{5.0}=0.10\mathrm{s}\backslash tau\; =\; \backslash frac\{0.50\}\{5.0\}\; =\; 0.10\backslash ,\backslash mathrm\{s\}$. You will meet inductors in AC circuits in the next unit.

5Energy stored in a magnetic field

While the current rises, the source must do work against the back emf. In a time $dtdt$ the work done is $dW=\epsilon Idt=\left(L\frac{dI}{dt}\right)Idt=LIdIdW\; =\; \backslash varepsilon\; I\backslash ,dt\; =\; \backslash left(L\backslash frac\{dI\}\{dt\}\backslash right)I\backslash ,dt\; =\; LI\backslash ,dI$. Building the current up from 0 to $II$:

This energy is stored in the magnetic field. For a solenoid, substitute $L=\frac{{\mu }_0{N}^{2}A}{l}L\; =\; \backslash frac\{\backslash mu\_0\; N^2\; A\}\{l\}$ and $I=\frac{Bl}{{\mu }_{0}N}I\; =\; \backslash frac\{Bl\}\{\backslash mu\_0\; N\}$: $W=\frac{B^2}{2{\mu }_0}AlW\; =\; \backslash frac\{B^2\}\{2\backslash mu\_0\}Al$. Dividing by the volume $AlAl$ gives the energy density of any magnetic field:

This stored energy is what lit the neon lamp.

6Mutual inductance

When two coils are close together, a changing current $I_{1}I\_1$ in the first changes the flux linking the second and induces an emf in it:

The mutual inductance $MM$ depends on the numbers of turns, the shapes of the coils, how close they are and whether they share an iron core; it is largest when all the flux of one coil passes through the other. A transformer (lesson 3) is two coils with a large $MM$ on one core. The pad that charges a phone without a cable uses the same effect across a small air gap.

7Eddy currents

A changing flux induces emfs not only in wires but in any piece of metal. In a solid block the induced currents swirl round inside the metal in closed loops, called eddy currents. By Lenz's law they oppose the change that causes them, and because the resistance of a solid block is small they can be large, heating the metal.

8Experiment 4.2: the eddy-current pendulum

Apparatus: a strong U-shaped magnet (or two magnets on an iron yoke), a clamp stand, two pendulums each carrying a copper or aluminium plate, one solid and one with slots cut into it (Figure 4.3).

Figure 4.3: Slots break up the paths of the eddy currents.

1. Set the magnet so the plate passes through the gap between the poles. Release the solid plate from the same angle each time and count the swings before it stops. Repeat with the magnet removed.
2. Repeat with the slotted plate.
3. The solid plate stops within a swing or two: as it enters and leaves the field, eddy currents in it produce forces that oppose its motion (magnetic damping). The slotted plate swings on, because the slots cut the loops in which eddy currents can flow.

9Useful and unwanted eddy currents