1The idea in one paragraph

Move a wire across a magnetic field and the free electrons inside it, moving with the wire, feel a magnetic force that pushes them along it: an emf appears, and if the wire is part of a circuit a current flows. This is electromagnetic induction. The general rule is Faraday's law: an emf is induced whenever the magnetic flux through a circuit changes, and the faster the change, the larger the emf. The flux can change because the field changes, because the circuit moves in or out of the field, or because the circuit turns. Lenz's law gives the direction: the induced current always opposes the change that caused it, because if it helped, you would get energy for nothing. A coil turning at a steady rate in a uniform field produces an emf that rises and falls as a sine wave, which is how almost all of the world's electricity is generated.

2Magnetic flux

Magnetic flux Φ measures how much magnetic field passes through an area. For a flat area A in a uniform field B:

Φ = BA cos θ

where θ is the angle between the field and the normal to the area (the line at right angles to its surface). The unit is the weber, Wb, and 1 Wb = 1 T m². Figure 1 shows why the cosine is there. With the coil facing the field (θ = 0) every field line passes through it, and Φ = BA is as large as it can be. Tilt the coil and fewer lines pass through. Edge-on (θ = 90°), none do and Φ = 0.

The commonest error in the subtopic is taking θ as the angle between the field and the plane of the coil. Always find the normal first.

A coil of N turns has the same flux through each turn, so the total is the flux linkage, NΦ, also in webers (sometimes written Wb-turns).

Worked example 1. A rectangular coil of 50 turns, 4.0 cm by 5.0 cm, sits in a uniform field of 0.30 T with its normal at 60° to the field. Find the flux through one turn and the flux linkage.

3Faraday's law

Faraday's law of induction: the emf induced in a circuit is proportional to the rate of change of the flux linkage through it.

ε = −N ΔΦ ÷ Δt

The minus sign is Lenz's law (section 5); for the size of the emf, use N ΔΦ/Δt. The important word is change. A strong field that stays the same induces nothing, however strong it is; a weak field that changes quickly can induce a large emf.

Φ = BA cos θ has three factors, so there are three ways to change it, and the guide asks you to know an example of each. Figure 2 shows them.

• A time-varying magnetic field. Change B. In Figure 2(b) the current in one coil is varied, so its field changes, and the changing field threads a second coil nearby that is not connected to it. An alternating current in the first coil gives a continuously changing field, which is how a transformer works.
• Relative motion between a conductor and a field. Change the area of field inside the circuit. Push a magnet into a coil (Figure 2a), move a coil into or out of a field, or slide a rod along rails (section 4). Only the relative motion matters: moving the magnet towards the coil and the coil towards the magnet give the same emf.
• A coil rotating in a field. Change θ. As the coil in Figure 2(c) turns, the flux through it rises and falls continuously. This is a generator (section 7).

Reading emf from a graph. Because ε = −N ΔΦ/Δt, the emf at any moment is minus the gradient of the graph of flux linkage NΦ against time. Figure 3 shows a coil in a field that is switched on steadily, held, then switched off twice as fast.

Worked example 2. The coil of worked example 1 is turned to face the field (θ = 0). The field rises steadily from 0 to 0.30 T in 0.10 s, stays constant for 0.15 s, then falls to zero in 0.050 s. Find the emf in each stage.

The same change of flux produced twice the emf when it happened in half the time. The sign flips because the flux rose in one stage and fell in the other: the induced current flows one way while the field grows and the other way while it collapses. The area under the ε–t graph over each stage is NΔΦ, the same size for both.

4A straight conductor moving across a field: ε = BvL

The simplest case to calculate is a straight rod of length L moving at speed v at right angles to a uniform field B, with the rod itself at right angles to the field as well. It can be derived in two ways, and both are worth knowing.

From the force on the charges. Every free electron in the rod moves with it at speed v, so each feels a magnetic force evB along the rod (D.3). The electrons pile up at one end until the electric field this creates pushes back equally. The charges then feel no net force when eE = evB, so E = vB, and the pd between the ends is E × L:

From Faraday's law. In a time Δt the rod moves vΔt and sweeps out an area LvΔt. The flux through that area is ΔΦ = B × LvΔt, so

ε = BvL for a straight conductor moving at right angles to a uniform field.

Worked example 3. An aircraft with a wingspan of 35 m flies horizontally at 240 m s⁻¹ where the vertical component of the Earth's field is 5.0 × 10⁻⁵ T. Find the emf between its wingtips.

Only the vertical component counts, because the wings move horizontally and only the field at right angles to the motion is cut. The emf is real but harmless, and it cannot drive a current round a circuit on the plane, because every wire moving with the plane has the same emf induced in it.

The rod on rails. Figure 4 turns the rod into a circuit. It slides along two parallel rails joined by a resistor R, in a field into the page. The moving rod is the source of emf, and a current flows round the loop.

Now the current in the rod is itself in the field, so the rod feels a force F = BIL (D.3). Using the right-hand rule, that force points backwards, against the motion. To keep the rod moving at constant speed, you must pull it forwards with an equal force.

Worked example 4. In Figure 4, L = 0.20 m, B = 0.50 T and R = 0.60 Ω; the rails and rod have negligible resistance. The rod is pulled at a steady 3.0 m s⁻¹. Find the emf, the current, the force needed and the power, and compare the power with the rate at which energy is transferred in the resistor.

The two powers are equal. Every joule transferred to the resistor as internal energy was supplied by the person pulling the rod. That is not a coincidence, and it is the subject of the next section.

5Lenz's law, and why it has to be true

Lenz's law: the direction of the induced emf is such that the current it drives opposes the change in flux that produced it.

In the rod example, the flux through the circuit grows as the rod moves right. The induced current flows anticlockwise round the loop, which by the right-hand grip rule makes its own field out of the page inside the loop, against the growing flux into the page. The force on the rod opposes its motion. Every case works the same way: the induced current always fights the change.

Figure 5 shows the classic demonstration. Push the N pole of a magnet towards a coil, and the induced current makes the near face of the coil an N pole, which repels the magnet. Pull the magnet away, and the current reverses, the near face becomes an S pole, and it attracts the magnet back. Either way, you have to do work to move the magnet.

Why it follows from energy conservation. The guide expects this argument. Suppose the induced current helped the change instead: the coil in Figure 5(a) made an S pole and pulled the magnet in. The magnet would speed up, which would increase the rate of change of flux, which would increase the current, which would pull harder. Kinetic energy and electrical energy would both grow from nothing. That breaks the conservation of energy, so the induced current must oppose the change. Put the other way round, the work you do against the opposing force is exactly the electrical energy that appears in the circuit, as worked example 4 showed.

Finding the direction in a question. Use whichever route suits the question.

• For a coil and a magnet: decide which pole the coil face must present to oppose the motion, then use the grip rule to find the current (anticlockwise, seen from outside, for an N face).
• For a moving rod: use the right-hand rule on the positive charges in the rod (thumb along v, fingers along B, palm gives the push along the rod), or find the current that makes the force BIL oppose the motion. The two always agree.

6Coils moving into and out of fields, and a magnet on a spring

The guide limits quantitative work to straight conductors and to rectangular coils moving in and out of fields or rotating in them. A coil moving into a field is a rod on rails in disguise: only the leading side is cutting field lines while the coil enters, so ε = NBvL for that side.

Worked example 5. A square coil of 20 turns and side 0.10 m moves at a steady 0.25 m s⁻¹ through a region of uniform field 0.40 T that is 0.30 m wide, as in Figure 6. Describe and calculate the emf, taking t = 0 as the moment the leading side enters the field.

While the coil is wholly inside, both sides cut field lines, but their emfs are equal and opposite round the loop. Faraday's law says the same thing more simply: the flux is not changing. The emf while leaving has the opposite sign to the emf while entering, because the flux is falling instead of rising, and by Lenz's law the force on the coil opposes its motion both on the way in and on the way out.

A magnet oscillating on a spring above a coil. This is the guide's other named example. As the magnet moves down, the flux through the coil increases; as it moves up, the flux decreases. So the emf alternates, as in Figure 7. At each turning point the magnet is momentarily still, the flux is momentarily not changing, and the emf is zero. The emf is larger when the magnet moves faster.

Connect the coil to a resistor and the oscillation dies away faster than it would with the coil unconnected. The induced current opposes the magnet's motion both ways (Lenz's law), and energy is transferred from the oscillation to internal energy in the circuit. This is electromagnetic damping, used in some brakes and in damping sensitive meters.

7A coil rotating in a uniform field: the generator

A rectangular coil of N turns and area A turning at a steady angular velocity ω in a uniform field B is a generator. Let θ = ωt be the angle between the field and the normal to the coil. Then the flux linkage is

It rises and falls as a cosine, from +NBA through zero to −NBA and back, once per revolution: the "negative" flux means the field is threading the coil from the other side.

Why the emf is a sine wave. Use ε = BvL on the two long sides of the coil, each of length L, a distance b/2 from the axis, so that A = Lb. Each side moves at speed v = ω(b/2). Its velocity is along the coil's normal, at angle θ to the field, so only the component v sin θ cuts across the field:

Figure 8 puts the two graphs together. The emf is largest when the flux is zero, with the plane of the coil along the field: the sides are then moving straight across the field lines and the flux is changing fastest. The emf is zero when the flux is largest, with the coil facing the field: the sides are then moving along the field lines, and the flux is momentarily not changing. The emf graph is the flux graph a quarter of a cycle later, which is exactly what "minus the gradient" does to a cosine.

Changing the frequency of rotation. Spin the coil faster and two things change at once. The flux changes faster, so the peak emf rises in proportion to ω; and each cycle takes less time, so the period T = 1/f falls. Doubling the frequency doubles the peak emf and halves the period, as Figure 9 shows.

Worked example 6. A generator coil of 200 turns, 0.10 m by 0.12 m, turns at 25 revolutions per second in a uniform field of 0.050 T. Find the peak emf and the period, and then both again at 50 revolutions per second.

Increasing a generator's output. This is the guide's second guiding question. The peak emf is NBAω, so there are four levers: more turns, a stronger field (stronger magnets, or an iron core to concentrate the field), a larger coil area, and a faster rotation. Each one raises the rate of change of flux linkage. A larger emf drives a larger current through the same load, and the power delivered rises with both. The energy still comes from whatever turns the coil. By Lenz's law, the more current the generator delivers, the harder it is to turn, and the turbine must supply more work each second.

Induction and industry. Michael Faraday announced his discovery of electromagnetic induction in 1831. Before it, electricity came from chemical cells and was expensive and weak. Induction made it possible to turn mechanical work, from steam engines, water and later turbines, into electrical energy on a large scale, and to change alternating voltages with transformers so that energy could be sent long distances with little loss. By the end of the nineteenth century, generators were supplying electricity to whole cities.

8Self-induction

A coil carrying a current sits in its own magnetic field. If that current changes, the coil's own flux linkage changes, and by Faraday's law an emf is induced in the coil itself. This is self-induction. By Lenz's law, the self-induced emf opposes the change in current that produced it, so it is often called a back emf.

Two consequences are worth knowing, qualitatively only. When a circuit containing a large coil is switched on, the current rises gradually rather than instantly, because the back emf opposes the rise. When the circuit is switched off, the current tries to collapse almost instantly, the rate of change of flux is enormous, and the self-induced emf can be large enough to make a spark jump across the opening switch. The guide does not require inductance or calculations with it.