Electromagnetic Induction (Faraday's Law)

Electromagnetism & Relativity

Course 2 · Chapter 5 · Electromagnetic Induction (Faraday's Law)

Chapter 4 closed with a promise: that its own motor mechanism — a current-carrying wire pushed by a magnetic field — runs in reverse. This chapter delivers on it, with Michael Faraday returning from Chapter 2's field concept to make one more real, foundational discovery.

Faraday's 1831 Discovery

On 29 August 1831, Faraday wrapped two separate coils of wire around opposite sides of an iron ring and connected one coil to a battery. He observed something genuinely subtle: a current appeared in the second, unconnected coil only at the exact moments the battery circuit was connected or disconnected — not while a steady current was already flowing. The key real insight was that a changing magnetic field induces a current; a constant one, however strong, does not.

Faraday's Law

Faraday's law quantifies this exactly: the induced electromotive force (EMF, effectively an induced voltage) in a circuit equals the negative rate of change of magnetic flux through it:

ε = −N(dΦ/dt)

Here N is the number of turns in a coil, and magnetic flux Φ is:

Φ = BA

(for a field B perpendicular to an area A). Flux measures how much magnetic field genuinely passes through a given loop — and it's a change in this quantity, however it happens (a changing field, a changing area, or a changing angle), that induces an EMF.

Lenz's Law: The Meaning of the Minus Sign

The negative sign in Faraday's law is not a mathematical afterthought — it encodes a real physical principle, known as Lenz's law: an induced current always flows in the direction that opposes the very change that produced it. If a magnet's approach is increasing the flux through a coil, the coil's own induced current creates a magnetic field opposing that increase; if the magnet withdraws, the induced current instead opposes the decrease. This is a direct, real consequence of Chapter 3's own conservation of energy (inherited from Course 1's own Chapter 3): if induction worked the other way, reinforcing rather than opposing the change, it would create energy from nothing, with no external input required.

Worked Example: EMF from a Changing Field

A 100-turn coil has an area of 0.05 m². A perpendicular magnetic field through it rises steadily from 0.2 T to 0.8 T over 0.5 s. What EMF is induced?

ΔΦ = ΔB × A = (0.8 − 0.2) × 0.05 = 0.03 Wb
ε = N(ΔΦ/Δt) = 100 × (0.03/0.5)
ε = 6 V

(The magnitude is 6 V; Lenz's law separately determines the induced current's own direction — opposing the rising field.)

The Faraday Disk: The First Real Generator

In that same year, 1831, Faraday built the first working homopolar generator — a conducting copper disk rotating between the poles of a magnet, with electrical contacts at the disk's centre and at its rim. As the disk spun through the magnetic field, it genuinely generated a small, continuous electric current directly from mechanical motion — the real historical beginning of every modern dynamo and generator, though Faraday's own original disk was far too inefficient to serve as any kind of practical power source.

🔗 A Motor Run Backward Chapter 4 described an electric motor: current through a wire, sitting in a magnetic field, produces a mechanical force (F = BIL) that spins a coil. The Faraday disk runs the identical physical setup in reverse: mechanical motion through a magnetic field produces a current instead. Motors and generators are, at the level of the real underlying physics, the same device used in opposite directions — a genuine, elegant symmetry running straight through both this chapter and the one before it.

Joseph Henry's Real, Delayed-Credit Story

Working independently in the United States, Joseph Henry discovered both self-inductance and mutual inductance around the very same period as Faraday — some real historians credit Henry's own observations as potentially predating Faraday's, made while Henry was building electromagnets at Albany Academy. But Henry did not publish his results promptly, and Faraday's own 1831 publication came first in print. As a direct, real consequence, Faraday became — and remains — the officially recognised discoverer of electromagnetic induction, even though Henry may genuinely have observed the effect independently around the same time, or possibly even earlier.

💡 A Recurring Real Pattern in This Two-Course Project This joins a genuine, recurring theme spanning both this course and Classical Mechanics & Thermodynamics: Cavendish's unpublished discovery of Coulomb's law (this course, Ch.1), Ohm's initially dismissed 1827 publication (Ch.3), and now Henry's own real loss of credit specifically for not publishing in time. Priority in the history of science, again and again, has genuinely depended on publication as much as on the underlying discovery itself — which is exactly why the SI unit of inductance is still named the henry (H), a real, lasting acknowledgement of Henry's own genuine, independent contribution, even without full credit for priority.

Motors and Generators: A Direct Symmetry

Property	Motor (Chapter 4)	Generator (This Chapter)
Input	Electrical current	Mechanical motion
Output	Mechanical force/motion	Electrical current
Governing formula	F = BIL	ε = −N(dΦ/dt)
Real historical example	Any modern electric motor	The Faraday disk, 1831

Hands-On Exercises

Exercise 1

A 50-turn coil has an area of 0.02 m^2. The magnetic field through it, initially 0.6 T, drops to 0.1 T over 0.25 s. Calculate the magnitude of the induced EMF.

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Exercise 2

A bar magnet's north pole is pushed toward a coil of wire, increasing the magnetic flux through the coil. Using Lenz's law, explain (in terms of opposing the change) what direction the induced current must flow, and why - without needing to specify a literal clockwise/counterclockwise answer.

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Exercise 3

Explain, in your own words, why Joseph Henry lost priority credit for discovering electromagnetic induction despite possibly observing the effect independently around the same time as Faraday, or even earlier - and name at least one other real example from this two-course project where the same kind of publishing-related credit issue occurred.

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Quick Reference

Faraday discovered induction on 29 August 1831: only a CHANGING current/field induces a current, not a steady one
Faraday's law: ε = −N(dΦ/dt), where Φ = BA
Lenz's law: induced current always opposes the change that produced it (a real consequence of energy conservation)
The Faraday disk (1831) was the first real generator — a motor's own mechanism, run in reverse
Joseph Henry independently discovered inductance around the same time but lost priority credit by not publishing first; the henry (H) unit still honours him

Next chapter: Maxwell's Equations — where Faraday's own intuitive field concept (Chapter 2), together with everything else this course has covered, gets unified into one complete mathematical theory of electricity, magnetism, and light.