PHY132H1/Unit 7

Unit 7 · Introduction to Physics II

Electromagnetic Waves & Light

The synthesis. A changing magnetic field makes an electric field, and a changing electric field makes a magnetic one — so the two can sustain each other and travel with nothing to travel through. That travelling disturbance is light.

By the end you can

• Describe how the two fields sustain one another.
• Explain why $c$ follows from measured electric and magnetic constants.
• Place the parts of the electromagnetic spectrum.
• Relate energy, intensity and polarisation.

Why it matters & where it comes from

Why it exists

Every previous unit dealt with fields tied to their sources. Here the fields let go: a wave carries energy away and keeps going with no charges present at all. That is how sunlight crosses empty space.

In the real world

Radio, microwaves, wifi, X-rays, and vision itself. All the same phenomenon at different wavelengths, all travelling at the same speed.

Story & open questions

Maxwell computed the wave speed in 1862 from constants measured in laboratory experiments on charges and currents. It came out equal to the measured speed of light, and he concluded light must be electromagnetic — one of the great unifications in physics, done on paper.

How the wave sustains itself

Rule Each field regenerates the other

Unit 6: a changing magnetic field induces an electric field.

Maxwell's addition: a changing electric field induces a magnetic field.

So an oscillating electric field creates an oscillating magnetic field, which recreates the electric field a little further on, and the pair propagates.

No medium is needed. This was the hardest part to accept — the nineteenth century searched hard for a substance called the ether for light to wave in, and never found it.

Definition

In an electromagnetic wave in vacuum:

$E$ and $B$ are perpendicular to each other and both perpendicular to the direction of travel — the wave is transverse.

They oscillate in phase: both peak together.

Their magnitudes are related by $E = cB$.

Rule The speed comes out of the constants

$c = \dfrac{1}{\sqrt{\varepsilon_0\mu_0}}$

Substituting the values from Units 2 and 5 gives about $3.00 \times 10^8$ m/s.

Nothing about light went into the calculation. Two constants from electrostatics and magnetostatics produce the speed of light, which is the whole argument that light is electromagnetic.

In a medium light slows down

$v = c/n$, where $n$ is the refractive index. The frequency is unchanged — it is set at the source — so the wavelength shortens. Unit 8 builds refraction entirely on this.

The spectrum

Rule One phenomenon, many wavelengths

All obey $c = f\lambda$. In order of increasing frequency:

Radio — metres to kilometres. Broadcasting.

Microwave — centimetres. Radar, wifi, heating water molecules.

Infrared — micrometres. Heat, remote controls.

Visible — about $400$ to $700$ nm. The narrow band our eyes evolved for, because it is what the Sun emits most of.

Ultraviolet — enough energy to damage molecules, hence sunburn.

X-ray and gamma — penetrating, and ionising. Imaging and radiotherapy.

Example Working with the relation

An FM station broadcasts at $100$ MHz. Find the wavelength.

$\lambda = c/f = (3 \times 10^8)/(10^8) = 3$ m.

Antenna lengths are chosen as fractions of the wavelength, which is why an FM aerial is around a metre and a mobile phone antenna is a few centimetres.

Energy, intensity, polarisation

Rule Intensity

Intensity is power per unit area, in W/m$^2$, and it is proportional to $E^2$.

From a point source it falls off as $1/r^2$, since the same power spreads over a sphere of area $4\pi r^2$.

Doubling the distance quarters the brightness — the same geometry as Coulomb's law, for the same reason.

Definition

Polarisation is the direction the electric field oscillates in. Sunlight is unpolarised — a random mixture of all directions. A polarising filter transmits only one.

Unpolarised light through one filter loses half its intensity, whatever the filter's orientation.

Already-polarised light through a filter at angle $\theta$ transmits $I = I_0\cos^2\theta$ — Malus's law.

Two crossed filters at $90$ degrees transmit nothing.

Example Three filters, and a surprise

Two crossed filters block all light. Insert a third at $45$ degrees between them.

Light now gets through — adding a filter increases transmission.

The middle filter rotates the polarisation to $45$ degrees, which is no longer perpendicular to the third. Each stage passes $\cos^2 45 = \tfrac12$, so an eighth of the original unpolarised intensity emerges.

The lesson: a polariser does not merely filter, it reorients. This is the standard demonstration and it reliably surprises.

Example Polarised sunglasses

Light reflecting off water or road surfaces is partly horizontally polarised. Sunglasses with a vertical axis block most of it, cutting glare while passing ordinary light — which is why tilting your head sideways makes them stop working.

Common pitfalls

• Thinking light needs a medium. It does not.
• Expecting frequency to change when light enters glass. The wavelength changes.
• Forgetting that $E$ and $B$ are perpendicular to the travel direction and to each other.
• Applying Malus's law to unpolarised light. The first filter always passes half.
• Assuming more filters always mean less light.

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 · Wavelength

Find the wavelength of a $2.4$ GHz wifi signal.

Answer

$\lambda = (3 \times 10^8)/(2.4 \times 10^9) = 0.125$ m.

Practice 2 · No medium

How does sunlight cross space?

Answer

The electric and magnetic fields sustain each other, so no medium is required. The search for the ether failed because there is none.

Practice 3 · Into glass

Light with $n = 1.5$ enters glass. What changes?

Answer

Speed drops to $2 \times 10^8$ m/s and the wavelength shortens by the same factor. Frequency is unchanged.

Practice 4 · Intensity

You move three times further from a lamp. What happens to the brightness?

Answer

It drops to a ninth, by the inverse-square law.

Practice 5 · One filter

Unpolarised light of intensity $I_0$ passes one polariser. What emerges?

Answer

$I_0/2$, regardless of the filter's orientation.

Practice 6 · Malus

Polarised light meets a filter at $30$ degrees. What fraction gets through?

Answer

$\cos^2 30 = 0.75$, so three quarters.

Practice 7 · Three filters

Compute the output for unpolarised light through filters at $0$, $45$ and $90$ degrees.

Solution

First: $I_0/2$. Second: $\times\cos^2 45 = I_0/4$. Third: $\times\cos^2 45 = I_0/8$. Removing the middle filter would give zero — adding a filter increased the transmission.

Practice 8 · Maxwell's argument

Why did computing $1/\sqrt{\varepsilon_0\mu_0}$ prove anything?

Answer

Because both constants came from experiments on charges and currents with no reference to light, yet the combination gave the measured speed of light exactly. That coincidence is too strong to be one.

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:

• Self-sustaining fields. How each induces the other and why no medium is needed.
• The speed of light. Derived from $\varepsilon_0$ and $\mu_0$.
• The spectrum. One phenomenon across many wavelengths.
• Intensity and polarisation. Inverse square, Malus's law, and the three-filter result.

Textbook/source alignment

The calendar names "light". This unit bridges the electromagnetism half of the course and the optics half; check the syllabus on Quercus for the assigned problems.

Core idea map

Units 1 to 6 built two fields that turned out to be coupled. Here the coupling lets them travel on their own, at a speed fixed by the constants of the earlier units. Units 8 and 9 then study what that wave does.

Study routine. Keep straight which quantity survives a change of medium. Frequency does; speed and wavelength do not. Almost every optics error traces back to that.

Deep practice extensions

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

Extension 1 · Medium

What does an electromagnetic wave travel through?

Guide

Nothing is required. The fields sustain each other, so it propagates in vacuum.

Extension 2 · The constants

What is the significance of $c = 1/\sqrt{\varepsilon_0\mu_0}$?

Guide

It derives the speed of light from purely electrical and magnetic measurements, identifying light as an electromagnetic wave.

Extension 3 · Entering a medium

Which wave property is unchanged?

Guide

Frequency. It is fixed at the source, so speed and wavelength change together.

Extension 4 · Polarisers

Why can a third filter increase transmission?

Guide

Because a polariser rotates the polarisation as well as filtering it, so the light is no longer perpendicular to the final filter.

← Unit 06Induction Unit 08 →Geometric optics