PHY132H1/Unit 6

Unit 6 · Introduction to Physics II

Induction

The discovery that made electricity practical. A changing magnetic field creates an electric field, so motion can produce current. Essentially all the electricity in the world is generated by this one effect.

By the end you can

• Compute magnetic flux and its rate of change.
• Apply Faraday's law to find an induced voltage.
• Use Lenz's law to get the direction right.
• Explain generators, transformers and eddy-current braking.

Why it matters & where it comes from

Why it exists

Ørsted showed current produces magnetism. The obvious question was whether magnetism could produce current, and the answer took a decade: only if the field changes. Steady fields induce nothing.

In the real world

Every power station spins a magnet near coils. Transformers step voltage up for transmission and down for use. Induction cooktops, wireless charging and electric guitar pickups are the same physics.

Story & open questions

Faraday found induction in 1831. Asked by a politician what use it was, he is said to have replied that one day it would be taxed. Within fifty years it was electrifying cities.

Flux and Faraday's law

Definition

Magnetic flux through a loop of area $A$:

$\Phi = BA\cos\theta$, with $\theta$ between the field and the loop's normal, measured in webers.

Flux counts how much field passes through the loop. It changes if $B$ changes, if $A$ changes, or if the loop rotates.

Rule Faraday's law

$\mathcal{E} = -N\dfrac{\Delta\Phi}{\Delta t}$ for a coil of $N$ turns.

The induced voltage depends on the rate of change of flux, not on the flux itself. A strong steady field induces nothing at all; a weak field changing quickly induces a lot.

Example Three ways to change the flux

Change $B$: move a magnet toward the coil, or switch a nearby current on.

Change $A$: stretch or squash the loop.

Change $\theta$: rotate the loop. This is what a generator does, and it produces a sinusoidal voltage because $\cos\theta$ varies sinusoidally as the loop spins.

Lenz's law

Rule The induced effect opposes the change

The induced current flows in whichever direction produces a magnetic field opposing the change in flux that caused it. That is the meaning of the minus sign in Faraday's law.

Flux increasing → induced field opposes it, pointing the other way.

Flux decreasing → induced field maintains it, pointing the same way.

Example Lenz's law is energy conservation

Suppose the induced current instead helped the change.

Pushing a magnet toward a coil would then induce a field pulling it in faster, which would induce more current, which would pull harder.

The magnet would accelerate on its own and the current would grow without limit — energy from nothing.

So the sign must be opposition. You have to do work to push the magnet in, and that work is what becomes electrical energy. Lenz's law is not an extra rule; it is energy conservation expressed as a sign.

Example The magnet in the copper tube

Drop a magnet down a copper pipe and it falls far more slowly than through air.

The moving magnet changes the flux through each ring of the pipe, inducing circulating currents.

By Lenz's law those currents oppose the change — below the magnet they push up, above they pull back.

The magnet's lost gravitational energy becomes heat in the copper. Nothing touches it; it is slowed purely by induction.

Applications

Example The generator

A coil rotates in a magnetic field, so $\theta$ changes steadily and the flux varies as $\cos(\omega t)$.

Faraday's law then gives a voltage varying as $\sin(\omega t)$ — alternating current, and the reason mains electricity is sinusoidal.

A motor run backwards is a generator. The same device converts either way, which is why regenerative braking works in an electric car.

Example The transformer

Two coils on a shared iron core. Alternating current in the primary makes a changing flux, which induces a voltage in the secondary.

$\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p}$ — the voltage ratio is the turns ratio.

Power is roughly conserved, so raising the voltage lowers the current proportionally. That is exactly what Unit 4's $I^2R$ argument wanted for transmission.

Transformers only work on alternating current. With direct current the flux is constant and nothing is induced — which is the technical reason mains supply is AC.

Steady means nothing happens

A coil sitting in the strongest constant magnetic field available has no induced voltage whatever. Only change induces. This is the point students most often lose marks on.

Common pitfalls

• Thinking a large flux induces a large voltage. Only its rate of change matters.
• Getting the induced direction backwards. Lenz's law always opposes.
• Forgetting the number of turns $N$ multiplies the result.
• Expecting a transformer to work on direct current.
• Treating Lenz's law as an independent rule rather than as energy conservation.

Practice

Work each fully before opening the answer. Original exercises written for this guide in the style of the course's problem sets.

Practice 1 · Flux

Find the flux through a $0.02$ m$^2$ loop perpendicular to a $0.5$ T field.

Answer

$\Phi = (0.5)(0.02)(1) = 0.01$ Wb.

Practice 2 · No change

A coil sits motionless in a strong constant field. What is the induced voltage?

Answer

Zero. Nothing is changing, so there is no induction whatever the field strength.

Practice 3 · Faraday

A $200$-turn coil sees its flux change from $0.01$ to $0.03$ Wb in $0.1$ s. Find the induced voltage.

Answer

$\mathcal{E} = 200 \times 0.02/0.1 = 40$ V in magnitude.

Practice 4 · Direction

A magnet's north pole approaches a coil. Which way does the induced current flow?

Answer

In whichever direction makes the coil's near face a north pole, so it repels the approaching magnet and opposes the increase in flux.

Practice 5 · Transformer

A transformer has $100$ primary and $500$ secondary turns, with $120$ V in. Find the output.

Answer

$V_s = 120 \times 5 = 600$ V. The current correspondingly drops to a fifth.

Practice 6 · Energy

Where does a generator's electrical energy come from?

Solution

From whatever turns the coil. The induced current creates a magnetic force opposing the rotation, so work must be done against it — and the harder the electrical load, the harder the turbine is to turn. That resistance is Lenz's law.

Practice 7 · Why AC

Why is mains electricity alternating?

Answer

Because transformers need changing flux, and they are what make efficient long-distance transmission possible. Direct current cannot be stepped up and down this way.

Practice 8 · Eddy braking

How does a train's eddy-current brake work?

Answer

A magnet near the moving rail induces circulating currents in it. By Lenz's law those oppose the motion, and the kinetic energy becomes heat. No contact and so no wear, though it cannot hold the train stationary since it needs motion to work.

Expanded topic checklist

Course coverage. This unit is part of the complete PHY132H1 topic map. It should cover the following items before you move on:

• Flux. Its definition and the three ways it can change.
• Faraday's law. Rate of change, and the role of the turn count.
• Lenz's law. Direction of the induced effect, and why it is energy conservation.
• Applications. Generators, transformers, and eddy currents.

Textbook/source alignment

Induction completes the calendar's "magnetism" topic. Check the syllabus on Quercus for the assigned problems; Lenz's law direction questions are a standard exam item.

Core idea map

Changing magnetic flux induces a voltage, opposing the change that caused it. Combined with Unit 5's result that currents make magnetic fields, this gives a two-way coupling — and Unit 7 shows that coupling is what lets light exist.

Study routine. For every induction question ask two things in order: is the flux increasing or decreasing, and what would oppose that? Direction follows without any formula.

Deep practice extensions

Use these after the main practice set. They push on definitions and edge cases rather than routine calculation.

Extension 1 · What induces

What quantity must change for induction to occur?

Guide

The magnetic flux. A constant flux, however large, induces nothing.

Extension 2 · The minus sign

What does the minus sign in Faraday's law encode?

Guide

Lenz's law: the induced effect opposes the change, which is required by energy conservation.

Extension 3 · Three routes

Name three ways to change flux.

Guide

Change the field strength, change the loop area, or rotate the loop.

Extension 4 · Transformer limit

Why will a transformer not work on DC?

Guide

Direct current gives constant flux, so nothing is induced in the secondary.

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