SPH4U · Unit 4 test review

The Wave Nature of Light: Test Review

Everything for the Unit 4 test on one page: the key facts from all 5 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Interference and Diffraction of Waves

Superposition: where waves overlap, the resulting displacement is the sum of the individual displacements.
For two in-phase sources, a point is on an antinodal line if $∣ P S_{1} - P S_{2} ∣ = mλ$ and on a nodal line if $∣ P S_{1} - P S_{2} ∣ = (n - \frac{1}{2}) λ$ .
For distant points: $sin θ_{n} = (n - \frac{1}{2}) \frac{λ}{d} \approx \frac{x _{n}}{L}$ .
Shorter wavelength or larger source separation gives more nodal lines, closer together.
Diffraction is significant when $λ / w$ is roughly 1 or larger, and negligible when the opening is many wavelengths wide.
The wave equation still links the quantities: $v = f λ$ .

Huygens' Principle and the Particle-vs-Wave Debate

Huygens' principle: every point on a wavefront is a source of wavelets moving at the wave speed; the new wavefront is the line tangent to all the wavelets.
Snell's law in wave form: $\frac{sin θ _{1}}{sin θ _{2}} = \frac{v _{1}}{v _{2}} = \frac{λ _{1}}{λ _{2}} = \frac{n _{2}}{n _{1}} .$
Crossing a boundary changes a wave's speed and wavelength but not its frequency; in a medium of index $n$ , $v = c / n$ and $λ = λ_{v a c uu m} / n$ .
Newton's particle model predicted light is faster in water; the wave model predicted slower. Foucault (1850) found it slower.
Key evidence for waves: Young's interference (1801 to 1803), the Poisson–Arago spot (1818 to 1819), Foucault's speed in water (1850), Maxwell's electromagnetic waves (1860s).

Young's Double-Slit Experiment and Gratings

Double slit or grating, bright fringes: $d sin θ_{m} = mλ$ ; dark fringes: $d sin θ_{n} = (n - \frac{1}{2}) λ$ .
Small-angle fringe spacing: $Δ y = \frac{λ L}{d}$ , so $λ = \frac{d Δ y}{L}$ .
Grating spacing: $d = \frac{1}{lines per unit length}$ ; the highest visible order is the largest whole number $m$ with $m \leq d / λ$ .
Single slit, dark fringes: $w sin θ_{n} = nλ$ ; the central maximum is about $2 λ L / w$ wide.
Young's interference fringes showed that light is a wave and gave a way to measure its wavelength directly.

Thin Films and Polarization

Wavelength inside a film: $λ_{f i l m} = λ / n_{f i l m}$ ; the frequency does not change.
Reflection off a higher-index medium flips the wave by half a wavelength; reflection off a lower-index medium does not.
One phase change: bright when $2 t = (m + \frac{1}{2}) λ_{f i l m}$ . Zero or two: bright when $2 t = m λ_{f i l m}$ . The dark conditions swap.
The thinnest non-reflecting coating (two phase changes) has $t = \frac{λ}{4 n}$ .
Unpolarized light through one filter keeps half its intensity; crossed filters block it; Malus's law $I = I_{1} cos^{2} θ$ gives the in-between cases.
Only transverse waves can be polarized, so polarization shows light is transverse.

Light as an Electromagnetic Wave

Electromagnetic waves are produced by accelerating charges, such as an oscillating electric dipole in an antenna.
In a vacuum all electromagnetic waves travel at $c = 3.00 \times 1 0^{8} m/s$ and obey $c = f λ$ .
Longest to shortest wavelength: radio, microwave, infrared, visible, ultraviolet, X-ray, gamma ray.
Dispersion: the refractive index depends on wavelength, so violet bends more than red in glass.
Resolution limit: $θ_{min} \approx 1.22 λ / D$ in radians. Bigger openings and shorter wavelengths see finer detail.

Where marks get lost

Using $mλ$ for a nodal line. Nodal lines need an odd number of half-wavelengths, $(n - \frac{1}{2}) λ$ ; whole wavelengths give antinodal lines. (Interference and Diffraction of Waves)
Counting the central line as a nodal line. With in-phase sources the bisector is always antinodal, and the first nodal line is $n = 1$ on either side of it. (Interference and Diffraction of Waves)
Forgetting to double when asked for the total number of nodal lines. The inequality counts the lines on one side only. (Interference and Diffraction of Waves)
Thinking a bigger opening causes more diffraction. It is the reverse: the narrower the opening compared with the wavelength, the more the wave spreads. (Interference and Diffraction of Waves)
Using $sin θ \approx x / L$ for points close to the sources. The approximation needs $L$ to be much larger than $d$ . (Interference and Diffraction of Waves)
Thinking the frequency changes when light enters a new medium. The frequency is set by the source; only the speed and the wavelength change. (Huygens' Principle and the Particle-vs-Wave Debate)
Writing Snell's law upside down in speed form. The larger angle goes with the larger speed: $sin θ_{1} / sin θ_{2} = v_{1} / v_{2}$ , but $= n_{2} / n_{1}$ for indices. (Huygens' Principle and the Particle-vs-Wave Debate)
Saying Newton's model failed to explain refraction. It explained the bending well; it failed because it predicted the wrong speed in water. (Huygens' Principle and the Particle-vs-Wave Debate)
Treating Huygens' wavelets as separate waves you could see. They are a construction: only their common tangent, the new wavefront, is observed. (Huygens' Principle and the Particle-vs-Wave Debate)
Assuming the wave model is the final word. Light also behaves as particles (photons), so both models capture part of the truth. (Huygens' Principle and the Particle-vs-Wave Debate)
Mixing units in $Δ y = λ L / d$ : nanometres for $λ$ , millimetres for $d$ and metres for $L$ . Convert everything to metres first. (Young's Double-Slit Experiment and Gratings)
Using $Δ y = λ L / d$ for a diffraction grating. Grating angles are large, so use $d sin θ = mλ$ and then trigonometry if a screen distance is needed. (Young's Double-Slit Experiment and Gratings)
Using the line density as $d$ . A grating with 600 lines/mm has $d = 1/600 mm$ , not $600$ . (Young's Double-Slit Experiment and Gratings)
Applying the double-slit bright-fringe rule to a single slit. For one slit, $w sin θ = nλ$ gives dark fringes. (Young's Double-Slit Experiment and Gratings)
Thinking a wider slit separation spreads the fringes out. $Δ y$ is inversely proportional to $d$ , so closer slits give wider spacing. (Young's Double-Slit Experiment and Gratings)
Skipping the phase-change count. The same film thickness can be bright or dark depending on whether one or two reflections are flipped, so always count first. (Thin Films and Polarization)
Using the wavelength in air for the film. The path difference is inside the film, so divide by the film's refractive index. (Thin Films and Polarization)
Forgetting that the path difference is $2 t$ , not $t$ . The second ray goes down through the film and back up. (Thin Films and Polarization)
Thinking every reflection flips the wave. Only reflection off a higher-index (slower) medium does. (Thin Films and Polarization)
Treating the first polarizer as a $cos^{2} θ$ problem. Unpolarized light has no single angle; one filter always passes half. (Thin Films and Polarization)
Thinking different electromagnetic waves travel at different speeds in a vacuum. They all travel at $c$ ; only frequency and wavelength differ. (Light as an Electromagnetic Wave)
Forgetting to convert MHz or GHz to hertz before using $c = f λ$ . $1 MHz = 1 0^{6} Hz$ and $1 GHz = 1 0^{9} Hz$ . (Light as an Electromagnetic Wave)
Saying red light bends more than violet in a prism. Violet has the higher refractive index in glass, so it bends more. (Light as an Electromagnetic Wave)
Using degrees in $θ_{min} = 1.22 λ / D$ . The formula gives radians. (Light as an Electromagnetic Wave)
Believing a bigger lens gives more diffraction blur. A larger opening gives a smaller minimum angle, so finer detail. (Light as an Electromagnetic Wave)

Practice test

25 questions from across the unit. Each one shows the lesson it comes from.

Interference and Diffraction of Waves
1
Two in-phase sources produce waves with a wavelength of $1.2 cm$ . What is the path difference, in centimetres, for a point on the third nodal line?
cm
Interference and Diffraction of Waves
2
Two in-phase sources are $4.0 cm$ apart and produce waves of wavelength $1.0 cm$ . At what angle from the perpendicular bisector is the second nodal line? Give your answer in degrees to one decimal place.
°
Interference and Diffraction of Waves
3
Straight water waves of wavelength $2.0 cm$ pass through a gap. Which gap width produces the most diffraction?
Interference and Diffraction of Waves
4
Why can you hear someone talking in the next room through an open door, but not see them?
Interference and Diffraction of Waves
5
Order these steps for finding the wavelength from a photograph of a two-source ripple-tank pattern.
1. 1Solve for the wavelength and check it against v = f times wavelength
2. 2Set the path difference equal to (n - 1/2) times the wavelength
3. 3Subtract to find the path difference
4. 4Identify which nodal line the chosen point lies on, counting outward from the bisector
5. 5Measure the distances from the point to each source
Huygens' Principle and the Particle-vs-Wave Debate
6
Diamond has a refractive index of $2.42$ . What is the speed of light in diamond, in units of $1 0^{8} m/s$ ? Give three significant figures.
× 10⁸ m/s
Huygens' Principle and the Particle-vs-Wave Debate
7
Water waves travelling at $30 cm/s$ meet the boundary of a shallow region at an angle of incidence of $5 0^{\circ}$ . In the shallow region they travel at $20 cm/s$ . What is the angle of refraction, in degrees? Give three significant figures.
°
Huygens' Principle and the Particle-vs-Wave Debate
8
What did Arago observe when he shone light past a small round disc?
Huygens' Principle and the Particle-vs-Wave Debate
9
Which principle says that every point on a wavefront acts as a source of new wavelets? Give its name.
Huygens' Principle and the Particle-vs-Wave Debate
10
Match each observation to the conclusion it supports.
Dark fringes between two lit slits
Light is slower in water than in air
Bright spot in a round shadow
Electromagnetic waves travel at the measured speed of light
Young's Double-Slit Experiment and Gratings
11
Light of wavelength $480 nm$ passes through two slits $0.30 mm$ apart onto a screen $2.0 m$ away. What is the fringe spacing in millimetres?
mm
Young's Double-Slit Experiment and Gratings
12
For the same grating ( $5000$ lines/cm) and light ( $600 nm$ ), what is the highest order maximum that can appear?
Young's Double-Slit Experiment and Gratings
13
Laser light of wavelength $633 nm$ passes through a single slit onto a screen $1.5 m$ away. The first dark fringe is $8.0 mm$ from the centre. How wide is the slit, in millimetres? Give three significant figures.
mm
Young's Double-Slit Experiment and Gratings
14
White light passes through a diffraction grating. Order these colours in the first-order spectrum from smallest angle to largest angle.
1. 1Red
2. 2Yellow
3. 3Green
4. 4Violet
5. 5Blue
Young's Double-Slit Experiment and Gratings
15
Which observation from the double-slit experiment most directly contradicts a simple particle model of light?
Thin Films and Polarization
16
Light travelling in air reflects off the surface of a pane of glass. What happens to the reflected wave?
Thin Films and Polarization
17
A coating with $n = 1.25$ is applied to glass with $n = 1.52$ . What is the minimum coating thickness, in nanometres, that cancels reflected light of wavelength $500 nm$ ?
nm
Thin Films and Polarization
18
Two polarizing filters are crossed so that no light gets through. A third filter is slid between them with its axis at $4 5^{\circ}$ . What happens?
Thin Films and Polarization
19
An air wedge lit with $632.8 nm$ light has a dark fringe at the contact edge. How thick, in nanometres, is the air gap at the 8th dark fringe beyond the contact edge?
nm
Thin Films and Polarization
20
Order the steps for solving a thin-film interference problem.
1. 1Solve 2t = (condition) for the thickness or wavelength
2. 2Find the wavelength inside the film using lambda over n
3. 3Choose the bright or dark condition that matches that count
4. 4Identify the refractive index on each side of both reflecting surfaces
5. 5Count how many of the two reflections are flipped
Light as an Electromagnetic Wave
21
A Wi-Fi router transmits at $2.4 GHz$ . What is the wavelength of its signal, in centimetres?
cm
Light as an Electromagnetic Wave
22
Order these parts of the electromagnetic spectrum from longest wavelength to shortest.
1. 1Ultraviolet
2. 2Gamma rays
3. 3X-rays
4. 4Radio waves
5. 5Microwaves
6. 6Infrared
7. 7Visible light
Light as an Electromagnetic Wave
23
Why can a Blu-ray disc hold much more data than a CD of the same size?
Light as an Electromagnetic Wave
24
What is the name for the separation of white light into its colours because the refractive index depends on wavelength?
Light as an Electromagnetic Wave
25
Light enters glass with a refractive index of $1.50$ . What is its speed in the glass, in units of $1 0^{8} m/s$ ?
× 10⁸ m/s

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…