Induction

Currents make magnetic fields, so the obvious experiment is to put a coil near a strong magnet and look for a current, and through the 1820s several capable people did exactly that and found nothing at all.

They found nothing because there was nothing to find. A steady magnetic field, however strong, drives no current in a stationary circuit. What Michael Faraday established on 29 August 1831, and Joseph Henry had found independently at Albany the year before but published later, is that the effect appears only while something is changing. This lesson assumes the magnetic force law and the field of a solenoid from the two previous lessons, and it is where the subject stops being a collection of static results.

Faraday's ring

Faraday's apparatus was a soft iron ring about 15 cm across with two separate coils wound on opposite sides of it, touching nowhere. One coil went to a battery, the other to a galvanometer some distance away. No current from the first coil could reach the second: the only connection was the iron.

Closing the battery circuit produced a momentary kick of the galvanometer needle, which then fell back to zero and stayed there while the current in the first coil ran steadily. Opening the circuit produced another kick, in the opposite direction. The effect was in the transitions, not the state.

Faraday spent the following months isolating what mattered, and the sequence in his diary is a model of experimental method. He replaced the battery coil with a permanent magnet thrust into a coil: same kick, and only while it moved. He moved the coil towards a stationary magnet instead: same kick again. He turned a copper disc between the poles of a magnet and drew a steady current from its rim to its axis, which was the first dynamo. By the end he had reduced everything to one quantity.

Flux, and Faraday's law

The quantity is magnetic flux, defined exactly as electric flux was:

Φ_{B} = B A \cos θ

for a uniform field through a flat area, and $\int B \cos θ d A$ in general, in units of tesla square metres, called webers. It counts the field lines threading the circuit.

Faraday's law is that the EMF induced in a circuit is the rate of change of the flux through it:

E = - N \frac{d Φ_{B}}{d t}

with $N$ the number of turns, since each turn is threaded separately. Only the rate matters. A large flux held constant produces nothing; a small flux changed quickly produces a lot.

That single statement covers every one of Faraday's arrangements, and the three ways of changing the flux are worth separating because they feel unrelated in the laboratory. $B$ can change, as when the current in a neighbouring coil is switched. $A$ can change, as when a circuit is stretched or a rod slides along rails. And $θ$ can change, as when a coil rotates, which is a generator. The flux does not care which.

Example. A coil of 200 turns encloses 25 cm² and sits in a field perpendicular to its plane. The field rises steadily from 0.10 T to 0.60 T over 0.20 s. What EMF appears?

The area and angle are fixed, so the flux change per turn is $Δ Φ = A Δ B = (25 \times 10^{- 4}) (0.50) = 1.25 \times 10^{- 3}$ Wb. The rate is $1.25 \times 10^{- 3} / 0.20 = 6.25 \times 10^{- 3}$ Wb/s, and with 200 turns the EMF is $200 \times 6.25 \times 10^{- 3} = 1.25$ V.

Now you. A coil of 150 turns encloses 40 cm² perpendicular to a field that falls steadily from 0.80 T to 0.20 T in 0.15 s. What EMF appears?

Answer

$Δ Φ = (40 \times 10^{- 4}) (0.60) = 2.4 \times 10^{- 3}$ Wb per turn, over 0.15 s giving $1.6 \times 10^{- 2}$ Wb/s, and with 150 turns, $E = 2.4$ V.

Where the EMF comes from, when the circuit moves

Faraday's law can be quoted, and it is more satisfying to derive the moving case from what is already known.

Take a conducting rod of length $L$ sliding at speed $v$ along two rails, all inside a uniform field $B$ perpendicular to the plane of the circuit. Every free charge in the rod is being carried along at $v$ , so each feels a magnetic force $q v B$ directed along the rod. The force per unit charge is $v B$ , and pushing charge from one end of a length $L$ to the other at that rate means an energy per unit charge of

E = B L v

which is an EMF by the definition of the sixth lesson, produced with no battery and no chemistry.

Check it against Faraday's law. The circuit's area grows at $L d x / d t = L v$ , so the flux grows at $B L v$ , and the two agree exactly. The rod is a source of EMF driving current round the circuit, and the magnetic force on the carriers is doing the driving.

There is a subtlety here that is worth not glossing over, because it is the entrance to relativity. The magnetic force does no work, as the seventh lesson insisted, so it cannot be the ultimate source of the energy that the circuit dissipates. It is not: the energy comes from whatever is pushing the rod. Once current flows, the current-carrying rod in the field feels a retarding force $B I L$ , and the agent pushing it must work against that. The magnetic force acts as an intermediary, redirecting the work of the pusher into the circuit, and the books balance exactly.

Lenz's law

The minus sign in Faraday's law is a separate physical statement, articulated by Heinrich Lenz in 1834: the induced current flows in the direction that opposes the change producing it. Push a magnet's north pole towards a loop and the loop's induced current makes a north pole facing the magnet, pushing back. Pull it away and the loop makes a south pole, pulling after it.

The justification is energy conservation, and the argument by contradiction is short. Suppose the current flowed the other way, aiding the change. Then pushing the magnet in would produce a force pulling it in further, accelerating it, producing more current, producing more force. Energy would appear from nothing, with the current in the loop and the kinetic energy of the magnet both increasing without any work being done. Lenz's law is therefore not an extra empirical fact but the only sign consistent with the first law of thermodynamics.

The practical consequences are large. Every generator resists being turned, and resists more when more current is drawn, which is why a power station burns more fuel when demand rises: the connection between the grid and the boiler is Lenz's law. Every motor generates a back EMF opposing the supply that drives it, which is why a motor draws a huge current at the instant of starting, when it is not yet turning and there is no back EMF, and a modest one at speed.

Example. A rod of length 0.25 m slides at 4.0 m/s along rails in a 0.50 T field, with the circuit closed by a 2.0 Ω resistance. Find the EMF, the current, the force needed to keep the rod moving, and check the power balance.

$E = B L v = (0.50) (0.25) (4.0) = 0.50$ V. The current is $I = E / R = 0.25$ A. The rod now carries current in a field, so it feels a retarding force $F = B I L = (0.50) (0.25) (0.25) = 3.13 \times 10^{- 2}$ N, and by Lenz's law it opposes the motion, so the same force must be applied to keep the speed constant. The mechanical power is $F v = (3.13 \times 10^{- 2}) (4.0) = 0.125$ W, and the electrical power dissipated is $E^{2} / R = 0.25 / 2.0 = 0.125$ W. They agree, as they must.

Now you. A rod of length 0.40 m slides at 2.5 m/s along rails in a 0.80 T field, with a 4.0 Ω resistance closing the circuit. Find the EMF, the current, the retarding force and the power.

Answer

$E = (0.80) (0.40) (2.5) = 0.80$ V, so $I = 0.80 / 4.0 = 0.20$ A. The force is $B I L = (0.80) (0.20) (0.40) = 6.4 \times 10^{- 2}$ N, and the mechanical power is $F v = 0.16$ W, matching $E^{2} / R = 0.64 / 4.0 = 0.16$ W.

The generator

Rotate a coil of $N$ turns and area $A$ at angular speed $ω$ in a uniform field $B$ . The angle between the field and the coil's normal is $θ = ω t$ , so the flux is $Φ = B A \cos ω t$ and

E = - N B A \frac{d}{d t} \cos ω t = N B A ω \sin ω t

an alternating EMF of peak value $N B A ω$ . This is the origin of alternating current, and of the fact that mains electricity is a sine wave: it is what a rotating machine naturally produces, and rotation is what a turbine naturally provides.

Example. A generator coil of 500 turns and area 100 cm² rotates at 50 revolutions per second in a 0.20 T field. What is the peak EMF, and what is the root mean square value?

$ω = 2 π f = 2 π (50) = 314.2$ rad/s. The peak EMF is $N B A ω = (500) (0.20) (0.0100) (314.2) = 314$ V. The rms value of a sine wave is the peak divided by $\sqrt{2}$ , so 222 V, which is roughly a European mains supply from a coil the size of a hand.

Now you. A coil of 250 turns and area 80 cm² rotates at 60 revolutions per second in a 0.35 T field. What is the peak EMF?

Answer

$ω = 2 π (60) = 377.0$ rad/s, so the peak EMF is $(250) (0.35) (80 \times 10^{- 4}) (377.0) = 264$ V.

Induction is not always wanted. A solid conductor moving in a non-uniform field, or sitting in a changing one, has induced currents circulating within its own bulk. These eddy currents obey Lenz's law and so oppose the motion, which is the whole of magnetic braking: drop a strong magnet down a copper pipe and it takes seconds to fall a metre, because the induced currents in the pipe wall produce a field that repels it from below and attracts it from above. The pipe is neither magnetic nor in contact with the magnet. In transformer cores the same currents are pure loss, and are suppressed by building the core from thin insulated laminations that break the circulating paths.

The mechanism problem

Two of Faraday's experiments give the same EMF and appear to have nothing in common.

Move the loop, hold the magnet still, and the explanation is the one derived above: charges in the moving conductor feel $q v \times B$ and are pushed round the circuit. No electric field is involved anywhere.

Move the magnet, hold the loop still, and that explanation is unavailable: the charges in the loop are not moving, so $q v \times B$ is zero. Something must still push them, and the only candidate is an electric field. So a changing magnetic field creates an electric field, in empty space, whether or not any conductor is there to reveal it.

Only relative motion matters, and the measured EMF is identical in the two cases, yet classical theory offers two entirely different mechanisms for it. Einstein opened his 1905 paper on special relativity with precisely this observation, noting that the asymmetry in the explanation has no counterpart in the phenomena, and taking that as evidence that the notion of absolute rest is empty. The two mechanisms are the same thing seen from two frames: what one observer calls a magnetic field, another moving relative to them calls a partly electric one. Electricity and magnetism are not two fields that interact but one field seen from different states of motion.

The death of potential

The second half of that argument has an immediate consequence, and it wrecks something built earlier in this course.

Written for the induced electric field, Faraday's law says

\oint E \cos φ d l = - \frac{d Φ_{B}}{d t}

The circulation of the electric field round a closed path is not zero. Take a loop of wire encircling a solenoid whose current is being ramped up, and carry a charge once round the loop: it returns to where it started having gained energy.

Every appearance of electric potential in this course assumed the opposite. The fourth lesson defined $V$ by the work done between two points, which is only a function of position if the work round any closed path is zero. Under a changing magnetic field it is not, so there is no potential function for an induced electric field, and asking for the voltage at a point in such a region is asking a question with no answer. Two voltmeters connected to the same two points on a loop encircling a changing flux, with their leads routed on opposite sides, read different values, and both are correct.

Kirchhoff's loop rule survives only by being rewritten: the sum of potential drops round a loop equals the induced EMF rather than zero, which is exactly how circuit analysis handles inductors. Within a circuit whose changing flux is confined to identifiable components, the old bookkeeping works. Outside such a circuit, potential is simply not available.

What comes next

Faraday's law says a circuit responds to a change in the flux through it, and says nothing about where that flux came from. In particular it does not exclude the circuit's own field.

A coil carrying a changing current therefore induces an EMF in itself, opposing its own change, and by Lenz's law that opposition acts to keep the current where it was. A circuit acquires something very like inertia. Working out how much, where the energy goes, and what happens when such a circuit is connected to a capacitor is the next lesson, and the answer to the last part is an oscillation whose frequency is built from $ε_{0}$ and $μ_{0}$ alone.