Physics · Electricity and magnetism · Grade 10

Self-induction

Why does a spark appear when a switch is opened — the current in the coil cannot stop at once?

The situation

If you break a circuit containing a large coil with a switch, a bluish spark flashes at the switch contacts — even though the battery is only a few volts. A car's ignition coil makes tens of thousands of volts from 12 V, and a strong magnet drops through a copper tube as slowly as if it were falling through honey. There is one reason for all of it: the current in a coil cannot change suddenly, and if the magnetic flux through a conductor changes, a current appears that opposes the change. How strong is this “electrical inertia”, and where is the energy stored?

Topic

Self-induction, energy of a magnetic field. Spark when a switch is opened: the current in a coil does not stop at once

Key ideas

Self-induced EMF
Inductance L
Energy of the field

Formulas

ℰ_{s} = - L \frac{d i}{d t}, Φ = L I

Self-induction EMF: a coil resists changes in the current. The unit of inductance is the henry (H).

i (t) = \frac{ℰ}{R} (1 - e^{- t / τ}), τ = \frac{L}{R}

When a circuit is closed the current does not rise instantly but over a time constant $τ$ ; when it is broken, $i = I_{0} e^{- t / τ}$ .

L = μ_{0} μ n^{2} V = \frac{μ_{0} μ N^{2} S}{l}

Inductance of a long coil (solenoid); $n = N / l$ , $V = S l$ .

W = \frac{L I^{2}}{2}, w = \frac{B^{2}}{2 μ_{0} μ} = \frac{B H}{2}

Energy of the magnetic field and its energy density (J/m³).

ℰ_{2} = - M \frac{d I_{1}}{d t}, M = \frac{μ_{0} μ N_{1} N_{2} S}{l}; F = k v

Mutual induction and the braking force of eddy (Foucault) currents — proportional to speed.

Theory

Self-induction: the “inertia” of current

If the current through a coil changes, its own magnetic flux $Φ = L I$ also changes and an EMF appears in the coil: $ℰ_{s} = - L d i / d t$ . By Lenz's law it opposes the change in current: when the circuit is closed it slows the rise of the current, and when it is opened it tries to keep the current going. In “Switch on” mode the lamp with the coil lights later than the lamp with the rheostat: $i = \frac{ℰ}{R} (1 - e^{- t / τ})$ , $τ = L / R$ . At $t = τ$ the current reaches 63 % of its steady value. These additional currents, which appear when a circuit is closed and opened, are called extra currents.

On breaking the circuit — a strong EMF

When the switch is opened the current falls very quickly, $| d i / d t |$ is large — and the EMF can exceed the battery voltage hundreds of times. In “Switch off” mode a neon lamp flashes from a 4.5 V battery, although it strikes at about 80 V. Without the neon the energy goes into a spark at the switch contacts (air breaks down at a few kilovolts). This phenomenon is used in a car's ignition coil; in electronics a protective diode is placed across the coil. The energy released comes from the $W = L I_{0}^{2} / 2$ stored in the coil — compare it with the energy the neon received, shown on the screen.

Inductance

Inductance $L = Φ / I$ is a quantity that depends on the shape and size of the conductor and on the medium; it does not depend on the current. For a long coil $L = μ_{0} μ n^{2} V$ : it is proportional to the square of the number of turns, and an iron or ferrite core increases it a thousandfold (we assume that the core closes the magnetic circuit; for a short coil the formula is approximate). In an alternating-current circuit a coil presents an inductive reactance $X_{L} = ω L$ : the current lags the voltage by a quarter of a period, and the total resistance (impedance) is $Z = \sqrt{R^{2} + {(ω L)}^{2}}$ — phasor diagrams are in the “Alternating current and RLC” experiment.

Energy and density of the magnetic field

To raise the current from zero to $I$ the source does work against the self-induction EMF: $W = \int L i d i = L I^{2} / 2$ . This energy is stored in the field inside the coil, with $w = B^{2} / (2 μ_{0} μ)$ per unit volume. In a long coil $B = μ_{0} μ n I$ and $w V = L I^{2} / 2$ — both are shown in “Inductance” mode. For the same $B$ the energy density inside iron is $μ$ times less than in air — most of the energy is concentrated not in the iron but in air gaps and empty space. The magnetic properties of matter — in the “Magnetic properties of matter” experiment.

Mutual induction — the idea of the transformer

If the current in one coil changes, its flux also passes through the second coil and an EMF arises in it: $ℰ_{2} = - M d I_{1} / d t$ . For two coils on a common core $M = μ_{0} μ N_{1} N_{2} S / l = \sqrt{L_{1} L_{2}}$ . With alternating current $U_{2} / U_{1} = N_{2} / N_{1}$ — this is the basis of the transformer (details — in the “Transformer” experiment).

Eddy currents, the quasi-static approximation and the skin effect

If a magnet moves near a bulk conductor, ring-shaped eddy (Foucault) currents arise inside the conductor. They oppose the magnet's motion: in a copper tube the ring current below the magnet pushes it, and the one above it pulls. The force is proportional to the speed, $F = k v$ , so the magnet soon falls at a constant speed $v = m g / k$ ; for a thin-walled tube $k = \frac{45 μ_{0}^{2} p_{m}^{2} σ w}{1024 a^{4}}$ ( $p_{m}$ — the magnetic moment of the magnet, $σ$ — the conductivity of copper, $w$ — the wall thickness, $a$ — the radius of the tube). If the tube is cut lengthwise, the ring currents circulating around it are broken and the braking weakens, but does not vanish: the currents turn at the edges of the slit and close as local eddies, so even in a real slit tube the magnet slows noticeably. The model calculates only the ring currents — in “Slit copper” the magnet falls almost freely. This phenomenon is used in the magnetic brakes of trains and fairground rides and in induction furnaces, and transformer cores are built of thin laminations to reduce eddy currents. Treating the current as the same along the whole circuit rests on the quasi-static condition: the size of the circuit is much smaller than the wavelength ( $c / f$ ). At high frequency the current is squeezed towards the surface of the conductor (the skin effect): $δ = \sqrt{2 ρ / (ω μ_{0} μ)}$ , and for copper at 50 Hz $δ \approx 9$ mm.

Did you know?

In 1832 the American physicist Joseph Henry, noticing a large spark when the circuit of a powerful electromagnet was broken, discovered self-induction. The unit of inductance — the henry (H) — is named after him.
For a current of frequency 50 Hz the skin depth in copper is about 9 mm: at this depth the current density is $e \approx 2.7$ times less than at the surface. That is why the inner part of very thick conductors carries current poorly.

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.

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