Electromagnetic Induction

Lessons

Senior 6 · Physics

Electromagnetic Induction

4 lessons

Lesson 1 of 4

Faraday's Experiments, Magnetic Flux and Faraday's Law

25 min

Electricity from magnetism

In 1820 Oersted showed that an electric current makes a magnetic field; you used that in S5 to explain electromagnets and motors. Michael Faraday asked the reverse question: can a magnet make a current? On 29 August 1831 he found the answer, and it is the reason we have electricity in our homes. Almost every kilowatt-hour on Rwanda's national grid, whether it comes from a hydropower station, a methane-gas plant on Lake Kivu or a diesel generator, is produced by the effect he discovered: electromagnetic induction.

Faraday's induction ring: two separate coils of insulated wire wound on one soft-iron ring. Switching a current on or off in one coil gave a brief current in the other. Photo from a 1922 magazine, public domain, via Wikimedia Commons

Experiment 1.1: Faraday's magnet and coil

Apparatus: a coil of several hundred turns of insulated copper wire, a centre-zero galvanometer, connecting leads, two bar magnets (one stronger), a coil with fewer turns (Figure 1.1).

Figure 1.1: The galvanometer shows a current only while the magnet moves.

Action	Galvanometer
push the N pole into the coil	deflects, say to the right
hold the magnet still inside the coil	reads zero
pull the N pole out	deflects to the left
push the S pole in	deflects to the left
move the magnet faster	larger deflection
use a stronger magnet, or a coil with more turns	larger deflection
hold the magnet still and move the coil	deflects, as when the magnet moves

Conclusions: an emf is induced only while the magnetic field through the coil is changing; its direction depends on the direction of the change; and its size grows with the speed of the change, the strength of the field and the number of turns. Only relative motion matters.

Warning

Safety. Strong magnets can snap together and pinch fingers: slide them apart, do not pull. Ceramic magnets are brittle and can chip if dropped. Keep magnets away from phones, bank cards and the galvanometer itself. The galvanometer is delicate: never connect it directly across a cell.

Magnetic flux and flux linkage

Faraday pictured the "amount of field" passing through a coil. We measure it by the magnetic flux. For a uniform field $B$ crossing a flat area $A$ whose normal makes an angle $\theta$ with the field:

Figure 1.2: Only the component of $B$ along the normal, $B\cos\theta$ , passes through the loop.

\Phi = BA\cos\theta

The flux is greatest ( $B A$ ) when the field is perpendicular to the plane of the loop ( $\theta = 0$ ) and zero when the field lies in the plane ( $\theta = 90^\circ$ ). A coil of $N$ turns has a flux linkage $N\Phi$ .

Definition

The magnetic flux $\Phi$ through an area $A$ in a uniform field $B$ is $\Phi = BA\cos\theta$ , where $\theta$ is the angle between $B$ and the normal to the area. Its unit is the weber (Wb): $1\,\mathrm{Wb} = 1\,\mathrm{T\,m^2}$ . The flux linkage of a coil of $N$ turns is $N\Phi$ (Wb-turns).

Worked example 1: flux through a tilted coil

A square coil of 200 turns and side 5.0 cm lies in a uniform field of 0.40 T. The field makes an angle of 30° with the PLANE of the coil. Find the flux through each turn and the flux linkage.

The angle between the field and the normal is $90^\circ - 30^\circ = 60^\circ$ , and $A = (0.050)^2 = 2.5 \times 10^{-3}\,\mathrm{m^2}$ .

\Phi = BA\cos 60^\circ = 0.40 \times 2.5 \times 10^{-3} \times 0.5 = 5.0 \times 10^{-4}\,\mathrm{Wb}

N\Phi = 200 \times 5.0 \times 10^{-4} = 0.10\,\mathrm{Wb}

Faraday's law of electromagnetic induction

All of Faraday's observations are summed up in one law: the induced emf equals the rate of change of flux linkage.

\varepsilon = -N\frac{d\Phi}{dt} \qquad \text{or, on average,} \qquad \varepsilon = -N\frac{\Delta\Phi}{\Delta t}

The minus sign expresses Lenz's law (lesson 2): the emf acts to oppose the change that causes it. Since $\Phi = BA\cos\theta$ , the flux, and so an emf, can be changed in three ways: change $B$ (move a magnet, or switch a current on in a nearby coil), change $A$ (stretch the loop, or slide part of it), or change $\theta$ (rotate the coil, as in a generator).

Definition

Faraday's law: the emf induced in a circuit is directly proportional to the rate of change of the magnetic flux linkage through it: $\varepsilon = -N\frac{d\Phi}{dt}$ . An emf is induced whenever the flux linking the circuit changes; a current flows if the circuit is closed.

Worked example 2: switching off a field

The field through the coil of Worked example 1 is now made perpendicular to its plane and then reduced steadily from 0.40 T to zero in 0.050 s. Find the average induced emf.

|\varepsilon| = N\frac{\Delta\Phi}{\Delta t} = 200 \times \frac{0.40 \times 2.5 \times 10^{-3} - 0}{0.050} = 4.0\,\mathrm{V}

Worked example 3: a field that grows with time

A 50-turn coil of area $0.010\,\mathrm{m^2}$ and resistance $2.0\,\Omega$ lies perpendicular to a field $B = (0.20 + 0.30t)\,\mathrm{T}$ , where $t$ is in seconds. Find the induced emf and current.

|\varepsilon| = NA\frac{dB}{dt} = 50 \times 0.010 \times 0.30 = 0.15\,\mathrm{V}

I = \frac{\varepsilon}{R} = \frac{0.15}{2.0} = 0.075\,\mathrm{A}

The steady part of the field, 0.20 T, induces nothing: only the CHANGE matters.

Worked example 4: turning a coil

A 100-turn coil of area $4.0 \times 10^{-3}\,\mathrm{m^2}$ lies with its plane perpendicular to a 0.050 T field. It is turned through 90° in 0.10 s, so that its plane becomes parallel to the field. Find the average emf.

The flux through each turn falls from $B A$ to zero:

|\varepsilon| = N\frac{BA - 0}{\Delta t} = \frac{100 \times 0.050 \times 4.0 \times 10^{-3}}{0.10} = 0.20\,\mathrm{V}

Try it in a simulation

Open Faraday's Electromagnetic Lab and choose the Pickup Coil screen. Move the bar magnet through the coil and watch the light bulb or voltmeter. Check each row of the table in Experiment 1.1: moving faster, adding loops, flipping the magnet, and holding it still.

Simulation by PhET Interactive Simulations, University of Colorado Boulder, licensed under CC BY 4.0 (https://phet.colorado.edu)

Warning

Common mistakes. (1) Thinking a strong STEADY field induces an emf: only a changing flux does. (2) Using the angle between $B$ and the PLANE in $\Phi = BA\cos\theta$ : $\theta$ is measured from the normal. (3) Forgetting to multiply by $N$ . (4) Leaving areas in cm²: $1\,\mathrm{cm^2} = 10^{-4}\,\mathrm{m^2}$ .

Tip

No change, no emf. Ask first: "What is changing: $B$ , $A$ or $\theta$ ?" Then $\varepsilon = N\times$ (change in flux) $\div$ (time).

What is electromagnetic induction? (iKen Edu). This animation describes Faraday's observations and the size and direction of the induced emf; connect each step to the table in Experiment 1.1.

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Practice

Define the weber and express it in base SI units.
Find the flux through a circular loop of radius 5.0 cm in a uniform field of 0.15 T perpendicular to its plane.
A coil of 300 turns and area $2.0 \times 10^{-3}\,\mathrm{m^2}$ lies perpendicular to a field that rises steadily from 0 to 0.50 T in 0.20 s. Find the induced emf.
A 20-turn coil of area $0.050\,\mathrm{m^2}$ and resistance $4.0\,\Omega$ lies perpendicular to a field $B = (0.10 + 0.40t)\,\mathrm{T}$ . Find the induced current.
A 150-turn coil of area $3.0 \times 10^{-3}\,\mathrm{m^2}$ is turned in 0.050 s from being parallel to a 0.20 T field to being perpendicular to it. Find the average emf.
State three different ways of inducing an emf in a coil that is not connected to any battery.
Uwase holds a magnet very still inside a coil and says, "This magnet is so strong that the galvanometer must show a big current." Evaluate her statement.

Show answersHide answers

The flux through $1\,\mathrm{m^2}$ perpendicular to a field of 1 T: $1\,\mathrm{Wb} = 1\,\mathrm{T\,m^2} = 1\,\mathrm{kg\,m^2\,s^{-2}\,A^{-1}}$ .
$\Phi = B\pi r^2 = 0.15 \times \pi \times 0.050^2 = 1.2 \times 10^{-3}\,\mathrm{Wb}$
$|\varepsilon| = \frac{300 \times 0.50 \times 2.0 \times 10^{-3}}{0.20} = 1.5\,\mathrm{V}$
$|\varepsilon| = 20 \times 0.050 \times 0.40 = 0.40\,\mathrm{V}$ ; $I = \frac{0.40}{4.0} = 0.10\,\mathrm{A}$
The flux rises from 0 to $B A$ : $|\varepsilon| = \frac{150 \times 0.20 \times 3.0 \times 10^{-3}}{0.050} = 1.8\,\mathrm{V}$
Move a magnet towards or away from it (change $B$ ); switch a current on or off in a nearby coil (change $B$ ); rotate the coil in a field (change $\theta$ ); change the area of the coil inside a field (change $A$ ).
Wrong. An emf is induced only while the flux is CHANGING. A magnet held still gives a steady flux, however strong, so the galvanometer reads zero. She must move the magnet (or the coil).

Key points

An emf is induced whenever the magnetic flux linking a circuit changes.
$\Phi = BA\cos\theta$ (Wb); flux linkage $N\Phi$ .
Faraday's law: $\varepsilon = -N\frac{d\Phi}{dt}$ ; change $B$ , $A$ or $\theta$ .

Sources and credits

23.4: Induced Emf and Magnetic Flux and 23.5: Faraday's Law of Induction: Lenz's Law, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
13.2: Faraday's Law, from University Physics Volume 2 by Samuel J. Ling, Jeff Sanny and William Moebs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Simulation: Faraday's Electromagnetic Lab 1.0.5, PhET Interactive Simulations, University of Colorado Boulder, licensed under CC BY 4.0 (linked, not copied).
Photo: Faraday ring transformer.jpg, from a 1922 technical magazine, Wikimedia Commons, public domain.
Rwanda's power stations: Power Plant list, Rwanda Energy Group (REG) (facts quoted).
Video: What is Electromagnetic Induction? | Faraday's Laws and Lenz Law, iKen Edu, embedded from YouTube.

The experiment, the two diagrams, the worked examples and the practice exercises are written by Ganzaa.

1Electricity from magnetism

2Experiment 1.1: Faraday's magnet and coil

3Magnetic flux and flux linkage

4Faraday's law of electromagnetic induction

5Try it in a simulation

Electricity from magnetism

Experiment 1.1: Faraday's magnet and coil

Magnetic flux and flux linkage

Faraday's law of electromagnetic induction

Try it in a simulation