E. The Wave Nature of Light · The Wave Nature of LightLight as a Wave: Refraction, Index of Refraction and Dispersion
E2.1 E2.2 E2.3 E3.1 E3.2 E3.3 A1.12 A1.13About 70 minutes
Learning goals- I can describe a wave using wavefronts and rays, and explain refraction as a change in wave speed.
- I can use v = fλ, n = c/v and λₙ = λ/n to find the speed and wavelength of light in a medium.
- I can use Snell's law in its wave form: sin θ₁/sin θ₂ = v₁/v₂ = λ₁/λ₂ = n₂/n₁.
- I can explain dispersion and use it to explain why a prism separates white light into colours.
Words to know
wavefront
A line (or surface) joining neighbouring points of a wave that are in the same phase, such as one crest.
ray
An arrow showing the direction a wave travels. A ray is always perpendicular to the wavefronts.
phase
Where a point is in its cycle of motion (crest, trough or in between). Two points are in phase if they are at the same stage of the cycle at the same time.
refraction
The change in direction of a wave as it crosses a boundary at an angle into a medium where its speed is different.
index of refraction (n)
The ratio n = c/v of the speed of light in a vacuum to its speed in a medium. It has no units.
dispersion
The separation of light into its colours because the index of refraction of a medium depends on wavelength.
monochromatic
Light of a single wavelength (one colour), such as the light from a laser.
Why does the straw look broken?
Put a straw in a glass of water and it seems to snap at the surface. In Grade 10 (Light and Geometric Optics) you explained this with rays and Snell's law. You drew rays bending, but you never asked why light bends. This unit answers that question, and many more, by treating light as a wave.
The wave model will explain the rainbow colours on a soap bubble, the tiny tracks on a Blu-ray disc, the anti-glare coating on eyeglasses and the way polarized sunglasses cut glare off a snowy road. We start with the wave idea you already know best: a wave changes direction when it changes speed.
Remember from Grade 11
In Waves and Sound (SPH3U) you used the universal wave equation v = fλ and learned that when a wave crosses into a new medium, its frequency stays the same (it is set by the source), while its speed and wavelength change. In Grade 10 you used n = c/v and n₁ sin θ₁ = n₂ sin θ₂, with c = 3.00 × 10⁸ m/s.
Wavefronts and rays in two dimensions
In a ripple tank (a shallow glass-bottomed tank of water lit from above), a vibrating bar makes straight waves and a dipper makes circular waves. Seen from above, each bright line is a crest. A line joining points in the same phase (for example, one crest) is called a wavefront. The distance between neighbouring wavefronts is one wavelength λ.
A ray shows which way the wave travels. Rays are always perpendicular to the wavefronts. Straight wavefronts have parallel rays; circular wavefronts have rays pointing outward from the source like spokes.
Water waves travel more slowly in shallow water. If you put a glass plate in part of the tank to make it shallow, the waves slow down when they reach it. The dipper still taps at the same rate, so f does not change. Since v = fλ, a smaller v means a smaller λ: the wavefronts crowd together.
If the waves hit the boundary at an angle, one end of each wavefront slows down first while the other end keeps going fast. The wavefront swings around and the wave changes direction. That is refraction.
Snell's law, wave version
The angle between the incident ray and the normal equals the angle between the wavefront and the boundary, so you can measure either. Geometry of the wavefronts in the diagram gives:
sin θ₁ / sin θ₂ = v₁ / v₂ = λ₁ / λ₂
This works for every kind of wave: water, sound and light. For light, the speed in a medium is v = c/n, so v₁/v₂ = n₂/n₁ and the ratio becomes the Snell's law you know:
sin θ₁ / sin θ₂ = v₁ / v₂ = λ₁ / λ₂ = n₂ / n₁, or n₁ sin θ₁ = n₂ sin θ₂
Two results you will use all unit:
- Speed in a medium: v = c/n.
- Wavelength in a medium: λₙ = λ/n, where λ is the wavelength in a vacuum (air is close enough: n_air = 1.0003 ≈ 1.00). The frequency does not change.
Worked exampleExample 1: Refraction in a ripple tank
Straight water waves with wavelength 2.0 cm travel at 24 cm/s in deep water. They reach a shallow region with an angle of incidence of 40° and refract at 28°. Find the frequency, the speed in the shallow water and the wavelength in the shallow water.
- Frequency (same in both regions): f = v₁/λ₁ = 24 ÷ 2.0 = 12 Hz.
- Speed: v₂ = v₁ × sin θ₂/sin θ₁ = 24 × sin 28°/sin 40° = 24 × 0.4695/0.6428 = 17.5 cm/s.
- Wavelength: λ₂ = λ₁ × sin θ₂/sin θ₁ = 2.0 × 0.7304 = 1.46 cm.
- Check: f λ₂ = 12 × 1.46 = 17.5 cm/s. ✓
f = 12 Hz in both regions; v₂ ≈ 17.5 cm/s; λ₂ ≈ 1.46 cm (1.5 cm to two significant figures).
Worked exampleExample 2: A laser beam in water
A helium-neon laser produces red light with a wavelength of 633 nm in air. The beam enters water (n = 1.33). Find the speed, wavelength and frequency of the light in the water.
- Speed: v = c/n = (3.00 × 10⁸ m/s) ÷ 1.33 = 2.26 × 10⁸ m/s.
- Wavelength: λₙ = λ/n = 633 nm ÷ 1.33 = 476 nm.
- Frequency (unchanged): f = c/λ = (3.00 × 10⁸) ÷ (633 × 10⁻⁹ m) = 4.74 × 10¹⁴ Hz.
- Check: v/λₙ = 2.26 × 10⁸ ÷ 476 × 10⁻⁹ = 4.74 × 10¹⁴ Hz. ✓
v = 2.26 × 10⁸ m/s, λ = 476 nm, f = 4.74 × 10¹⁴ Hz. A swimmer underwater still sees the beam as red: colour depends on frequency, and 476 nm is only the wavelength inside the water.
Dispersion: why a prism makes a rainbow
For glass, water and most transparent materials, the index of refraction is slightly different for each wavelength. In ordinary glass, n is a little larger for violet light than for red light. So violet light slows down more, bends more, and separates from red. This spreading of white light into colours is called dispersion.
A triangular prism makes the separation easy to see because the light bends twice in the same direction, on the way in and on the way out. Rainbows (which you studied in Grade 10) are dispersion inside raindrops. Mirages are refraction too: air just above a hot road is less dense, so it has a slightly smaller index of refraction, and light from the sky curves upward into your eyes, looking like a puddle.
Dispersion is the first of several ways this unit will separate white light into colours. Later you will see colours made by interference in thin films, by diffraction gratings, and by polarization in stressed plastic.
Worked exampleExample 3: How far apart do red and violet travel?
White light strikes a block of glass at an angle of incidence of 50.0°. For this glass, n = 1.514 for red light and n = 1.528 for violet light. Find the angle of refraction of each colour and the angle between them.
- Red: sin θ₂ = (1.00 × sin 50.0°)/1.514 = 0.7660/1.514 = 0.5060, so θ₂ = 30.40°.
- Violet: sin θ₂ = 0.7660/1.528 = 0.5013, so θ₂ = 30.09°.
- Difference: 30.40° − 30.09° = 0.31°.
Red refracts at 30.40°, violet at 30.09°; they separate by about 0.31°. Violet, with the larger n, bends more toward the normal.
Just for funWhy did the light beam refuse to leave the swimming pool? It was afraid of being bent out of shape again on the way out.
Hands-on activityMeasure the index of refraction of water
You need: a tall clear glass or jar, water, a coin, a ruler, a pencil and a strip of masking tape.
- Put the coin in the bottom of the empty glass. Stick the tape up the outside of the glass.
- Fill the glass with water to a depth of about 10 cm. Measure the real depth d from the water surface to the coin.
- Look straight down at the coin from above. Slide the pencil tip up or down the outside of the glass until it seems level with the coin's image. Mark that height on the tape. The distance from the water surface to the mark is the apparent depth d′.
- For viewing straight down, n ≈ d/d′. Repeat three times and average.
- Bonus (dispersion): on a sunny day, set a full glass of water on a windowsill so sunlight passes through it onto a white sheet of paper on the floor. Look for a small spectrum at the edge of the bright patch. Never look at the Sun directly.
What to look for: d/d′ close to 1.33 (for example, real depth 10.0 cm and apparent depth about 7.5 cm). Your answer will be less exact than a lab value because judging the image position is hard; write down how you could reduce that uncertainty (more trials, a deeper container, a thinner marker). In the spectrum, red is on the side bent least and violet on the side bent most.
Your turn
1State what happens to the speed, wavelength and frequency of water waves that move from deep water into shallow water.
Speed decreases, wavelength decreases, frequency stays the same.
2How are rays and wavefronts related?
Rays are perpendicular to wavefronts and point in the direction the wave travels.
3Find the speed of light in glass with n = 1.50.
v = c/n = 3.00 × 10⁸ ÷ 1.50 = 2.00 × 10⁸ m/s.
4Yellow light from a sodium lamp has λ = 589 nm in air. Find its wavelength in glass with n = 1.50.
λₙ = 589 ÷ 1.50 = 393 nm.
5Light passes from air into glass (n = 1.50) at an angle of incidence of 35°. Find the angle of refraction.
sin θ₂ = sin 35°/1.50 = 0.5736/1.50 = 0.3824, so θ₂ = 22.5°.
6Define dispersion and name one everyday example.
Dispersion is the separation of white light into colours because n depends on wavelength. Examples: a prism, a rainbow, coloured flashes from a cut-glass ornament.
7In glass, which colour has the larger index of refraction, red or violet? Which bends more?
Violet has the larger n and bends more.
More practice: warm-up
1What sets the frequency of a wave?
The source (for example, how fast the dipper vibrates).
2Write the equation that links speed, frequency and wavelength.
3Water has n = 1.33. Is light faster in air or in water?
Faster in air (v = c/n is larger when n is smaller).
4When light slows down entering a medium at an angle, does it bend toward or away from the normal?
A line joining neighbouring points of a wave that are in the same phase, such as one crest.
More practice: on your own
1Water waves in a ripple tank have λ = 3.0 cm in deep water and 2.0 cm in shallow water. The speed in deep water is 18 cm/s. Find the frequency and the speed in shallow water.
f = 18 ÷ 3.0 = 6.0 Hz (unchanged). v₂ = fλ₂ = 6.0 × 2.0 = 12 cm/s.
2Light travels from water (n = 1.33) into glass (n = 1.50) at an angle of incidence of 30°. Find the angle of refraction.
sin θ₂ = 1.33 sin 30°/1.50 = 0.665/1.50 = 0.4433, so θ₂ = 26.3°. It bends toward the normal because the glass has the larger n.
3In a ripple tank, waves meet a boundary at 30° and refract at 22°. Find the ratio v₁/v₂. Which region is shallower?
v₁/v₂ = sin 30°/sin 22° = 0.500/0.3746 = 1.33. The waves are slower in region 2, so region 2 is shallower.
4Green light of frequency 5.66 × 10¹⁴ Hz travels through a material where its wavelength is 400 nm. Find the index of refraction of the material.
Speed in material v = fλ = 5.66 × 10¹⁴ × 400 × 10⁻⁹ = 2.264 × 10⁸ m/s. n = c/v = 3.00 × 10⁸ ÷ 2.264 × 10⁸ = 1.33 (it is water).
5Explain, using wavefronts, why a wave that hits a boundary head-on (angle of incidence 0°) slows down but does not change direction.
The whole wavefront reaches the boundary at the same moment, so every part of it slows at once. No end gets ahead of the other, so the wavefront does not swing around: λ shortens but the direction stays the same.
Challenge
1Find the critical angle for light going from water (n = 1.33) into air, and explain in wave terms why total internal reflection can happen only when light travels toward a medium where it moves faster.
sin θc = 1.00/1.33 = 0.752, so θc = 48.8°. Light speeds up and bends away from the normal when it enters a faster medium, so the refracted angle reaches 90° before the incident angle does. Going into a slower medium, the refracted angle is always smaller than the incident angle and never reaches 90°, so there is always a refracted ray.
2A ray of white light enters a glass block at 60.0°. In this glass n_red = 1.510 and n_violet = 1.530. The block is 5.00 cm thick. How far apart are the red and violet rays where they reach the bottom face?
Red: sin θ = sin 60.0°/1.510 = 0.8660/1.510 = 0.5735, θ = 35.00°. Violet: 0.8660/1.530 = 0.5660, θ = 34.47°. Horizontal distances travelled: 5.00 tan 35.00° = 3.501 cm and 5.00 tan 34.47° = 3.433 cm. Separation ≈ 0.068 cm (0.68 mm).
3The particle model of light (Newton's idea, which you will meet in a later lesson) explained refraction toward the normal by saying light speeds up in water. What does the wave model predict instead, and how could a measurement decide between them?
The wave model says sin θ₁/sin θ₂ = v₁/v₂, so bending toward the normal means v₂ < v₁: light must slow down in water. Measuring the speed of light in water decides it. When this was done in the mid-1800s, light was found to be slower in water, supporting the wave model.
Quiz
1. Water waves travel from deep water into shallow water. Which quantity stays the same?
2. Light has a wavelength of 600 nm in air. What is its wavelength in a plastic with n = 1.50?
3. Which relation is Snell's law written for waves?
4. What is the speed of light in diamond (n = 2.42)?
5. Why does a glass prism separate white light into colours?
6. A ray in a ripple tank is drawn at 90° to…
Big idea: this lesson reconnects Grade 10 ray optics to the wave model (expectations E2.1, E3.1 to E3.3, with the refractive-index measurement of E2.3). The key fact students must hold on to is that frequency never changes when a wave enters a new medium; speed and wavelength change together.
How to help: do the apparent-depth activity together and talk about sources of error. If your teen has a computer, a free ripple-tank or wave-on-a-string simulation (such as PhET's Wave Interference or Bending Light) is a great substitute for a real ripple tank.
Common difficulties:
- Thinking the colour of light changes in water because the wavelength does. Colour is tied to frequency.
- Mixing up which angle to use: measure angles from the normal (or, equivalently, between the wavefront and the boundary).
- Calculator in radian mode. Check that sin 30° = 0.5.
- Rounding too early in multi-step problems. Keep extra digits until the end.