Refraction

The video lesson for this topic is coming soon.

Essential Question

Why does a straw look bent in a glass of water?

Put a straw in a clear glass of water and it looks broken right at the surface. The straw didn’t change, but the light coming from it did. By the end of this lesson, you should be able to explain why light bends, use the index of refraction n = c/v, and solve problems with Snell’s law.

What Is Refraction?

New material, new speed
The speed change bends light
Bending happens at the boundary

A ray diagram. Air is on top and water is on the bottom, shaded light blue. A dashed normal line crosses the boundary where a red ray hits. The ray comes in at 50 degrees from the normal in air and continues into the water at 35 degrees from the normal, bending toward the normal. Caption: slower in water, so it bends toward the normal.

Light travels at different speeds in different materials. When light crosses the boundary into a new material at an angle, the change in speed makes it change direction. This bending is called refraction. Here, light slows down as it enters water, so it bends toward the normal: 50° from the normal in air becomes about 35° in water. Light that hits the boundary straight along the normal changes speed but does not bend.

Index of Refraction

n = c / v

n = index of refraction (no unit)
c = speed of light in a vacuum, 3.00 × 10⁸ m/s
v = speed of light in the material, in m/s

The equation n = c / v in a yellow box. Below it, four cards: vacuum n = 1.00, water n = 1.33, glass n = 1.50, diamond n = 2.42. Caption: bigger n means slower light.

The index of refraction tells you how much a material slows light down. Water has n = 1.33, which means light is 1.33 times slower in water than in a vacuum. A bigger index means slower light. Since nothing is faster than light in a vacuum, n is never less than 1. It has no unit because it’s a ratio of two speeds.

Toward or Away from the Normal

Slows down: toward normal
Speeds up: away from normal
Along the normal: no bend

Two ray diagrams side by side. Left: light goes from air into glass, shaded green, and bends toward the dashed normal, from 45 degrees to 28 degrees. Right: light goes from glass into air and bends away from the normal, from 28 degrees to 45 degrees.

When light goes into a material where it travels slower (higher n), it bends toward the normal, so its angle gets smaller. When light goes into a material where it travels faster (lower n), it bends away from the normal, so its angle gets bigger. The path works the same in both directions: the right picture is just the left picture run backward. A ray that travels exactly along the normal changes speed but doesn’t bend.

Why Light Bends

Wave fronts slow down in glass
One side slows down first
Like a wagon rolling onto grass

Blue wave fronts, the lines of crests, travel down and to the right toward a boundary. In the air above, the wave fronts are widely spaced. Below the boundary, in glass shaded green, they are closer together and tilted, because the part of each wave front that enters the glass first slows down first. A red arrow shows the beam direction bending at the boundary.

Think of light as a set of wave fronts (lines of crests) moving forward. When a wave front hits the glass at an angle, one end enters the glass first and slows down, while the other end is still moving fast in the air. That makes the wave front pivot, so the whole wave turns. It’s like rolling a wagon off a sidewalk onto grass at an angle: the wheel that hits the grass first slows down first, and the wagon turns. Notice the wave fronts are closer together in the glass. The frequency stays the same, so a slower speed means a shorter wavelength.

Snell’s Law

n₁ sin θ₁ = n₂ sin θ₂

n₁, θ₁ = index and angle in the first medium (where the light starts)
n₂, θ₂ = index and angle in the second medium (where the light goes)
Both angles are measured from the normal

The equation n1 sin theta1 = n2 sin theta2 in a yellow box. Below it, four cards: n1 is the index of medium 1, theta1 is the angle in medium 1 measured from the normal, n2 is the index of medium 2, and theta2 is the angle in medium 2 measured from the normal.

Snell’s law tells you exactly how much light bends. To find θ₂, divide both sides by n₂ to get sin θ₂ = n₁ sin θ₁ / n₂, then take the inverse sine (sin⁻¹) of the result. Make sure your calculator is in degree mode.

The Bent Straw

Light bends leaving the water
Your brain traces rays straight back
Underwater part looks shallower

A glass of water with a straw in it. Above the water, the straw is a solid orange line. Below the surface, a dashed gray line shows where the straw really is, and a solid orange line, tilted upward, shows where the straw appears to be. The straw looks bent at the water's surface.

Light from the underwater part of the straw travels up through the water and into the air. At the surface it speeds up and bends away from the normal. Your brain assumes light travels in straight lines, so it traces those bent rays straight back and “sees” the underwater part of the straw higher up than it really is. The straw seems to bend right at the surface. The same effect makes pools look shallower than they are.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Light in Water

Water has an index of refraction of 1.33. How fast does light travel in water?

Knowns: n = 1.33, c = 3.00 × 10⁸ m/s
Equation: n = c/v, so v = c/n
Substitute: v = (3.00 × 10⁸ m/s) / 1.33
Answer: v = 2.26 × 10⁸ m/s

This is slower than c, as it should be, since n is greater than 1.

Example 2: Finding the Index

Light travels through a clear plastic at 2.00 × 10⁸ m/s. What is the plastic’s index of refraction?

Knowns: v = 2.00 × 10⁸ m/s, c = 3.00 × 10⁸ m/s
Equation: n = c/v
Substitute: n = (3.00 × 10⁸ m/s) / (2.00 × 10⁸ m/s). The units cancel.
Answer: n = 1.50 (no unit)

Example 3: Into Water

Light in air (n = 1.00) hits the surface of water (n = 1.33) at 40.0° from the normal. What is the angle of refraction?

A ray in air hits a water surface at 40.0 degrees from the dashed normal. The refracted ray in the water is marked with a question mark.

Knowns: n₁ = 1.00, θ₁ = 40.0°, n₂ = 1.33
Snell’s law: n₁ sin θ₁ = n₂ sin θ₂, so sin θ₂ = n₁ sin θ₁ / n₂
Substitute: sin θ₂ = (1.00)(sin 40.0°) / 1.33 = (0.6428) / 1.33 = 0.4833
Answer: θ₂ = sin⁻¹(0.4833) = 28.9° from the normal

Check: light slows down in water, so it should bend toward the normal. 28.9° is less than 40.0°, so it did.

Example 4: Out of Glass

Light inside a glass block (n = 1.50) hits the surface at 30.0° from the normal and passes into air (n = 1.00). What is the angle in the air?

A ray inside glass, shaded green on top, hits the boundary at 30.0 degrees from the dashed normal. Below the boundary is air, where the refracted ray is marked with a question mark.

Knowns: n₁ = 1.50 (glass), θ₁ = 30.0°, n₂ = 1.00 (air)
Snell’s law: sin θ₂ = n₁ sin θ₁ / n₂
Substitute: sin θ₂ = (1.50)(sin 30.0°) / 1.00 = (1.50)(0.500) / 1.00 = 0.750
Answer: θ₂ = sin⁻¹(0.750) = 48.6° from the normal

Light speeds up as it leaves the glass, so it bends away from the normal: 48.6° is bigger than 30.0°.

Big Ideas

Speed change at a boundary bends light
n = c/v; bigger n, slower light
Slower: toward normal. Faster: away
n₁ sin θ₁ = n₂ sin θ₂

Back to the essential question: light from the underwater part of the straw speeds up and bends away from the normal as it leaves the water. Your brain traces the bent rays straight back, so that part of the straw appears in the wrong place and the straw looks bent. Snell’s law, n₁ sin θ₁ = n₂ sin θ₂, tells us exactly how much light bends. Next up, we’ll see that light sometimes acts like a particle in The Photoelectric Effect.