Refraction
Snell’s law
Dip a pencil into a glass of water at a slant. It looks broken at the surface. Light changes direction as it crosses from one medium into another. This is refraction.
Draw the boundary and the normal at the point where the ray strikes. The angle between the incident ray and the normal is i; between the refracted ray and the normal it is r.
Two laws describe refraction. First, the incident ray, the refracted ray and the normal all lie in one plane.
Second, for a given pair of media, sin i over sin r is a constant. This is Snell’s law. The constant is n₂₁, the refractive index of medium 2 with respect to medium 1.
n₂₁ equals n₂ over n₁, the ratio of the refractive indices of the two media. So Snell’s law can also be written n₁ sin i = n₂ sin r.
If n₂₁ is greater than 1, medium 2 is optically denser, r is smaller than i, and the ray bends towards the normal. Going into a rarer medium, it bends away from the normal.
Example: light goes from air into glass of refractive index 1.5 at an angle of incidence of 45°. Then sin r equals sin 45° over 1.5, which is 0.707 over 1.5, or 0.471. So r is about 28°.
Indices also chain together. n₁₂ is 1 over n₂₁. And for three media, n₃₂ equals n₃₁ times n₁₂.
Now send a ray through a rectangular glass slab. It bends towards the normal entering the glass and away from the normal leaving it. The emergent ray is parallel to the incident ray, but shifted sideways. This is the lateral shift.
Apparent depth and the early sunrise
Look straight down into a bucket of water at a coin on the bottom. The bottom seems raised, and the coin looks closer than it really is.
Follow two rays from the coin. As they leave the water they bend away from the normal and spread out more. Your eye traces them back along straight lines, dashed here.
The dashed lines meet above the coin. That is where you see it: a virtual image at a smaller depth, called the apparent depth.
For viewing near the normal, the apparent depth equals the real depth divided by the refractive index of the water with respect to air: h′ = h/n.
A coin 12 centimetres deep in water of refractive index 1.33 appears at 12 over 1.33, about 9 centimetres. It seems raised by about 3 centimetres.
It works the other way too. If a needle at the bottom of a tank of water 12.5 centimetres deep appears to be 9.4 centimetres deep, the refractive index is 12.5 over 9.4, about 1.33.
Look at the coin from the side, at a slant, and it seems even shallower. The simple formula h/n holds only near the normal.
The same bending happens in the sky. The air gets denser closer to the ground, so sunlight bends gradually as it comes down through the atmosphere.
When the sun is actually just below the horizon, its bent rays still reach us. We see the sun about half a degree higher than it really is.
So the sun is visible about 2 minutes before it actually rises, and for about 2 minutes after it actually sets. Every day is a few minutes longer, thanks to refraction.
Try it: bend a ray
- Drag the amber dot. Check that n₁ sin i always equals n₂ sin r in the readouts.
- Air → glass: the ray bends towards the normal. Press Swap media: now it bends away.
- At i = 0 the ray goes straight through, whatever the media.
- A little light is always reflected too. Watch the brightness of the reflected ray as i grows.
Try it: the raised coin
- Looking straight down, the coin appears at h/n: 12/1.33 ≈ 9.0 cm.
- Change the liquid’s refractive index. A larger n raises the coin more.
- Drag the eye to the side. The coin seems shallower still, so h/n works only near the normal.
Air into glass
A ray of light passes from air into glass (n = 1.5) with an angle of incidence of 60°. What is the angle of refraction?
- 40.0°
- 35.3°
- 60.0°
- 54.7°
Answer
B. Snell’s law: 1 × sin 60° = 1.5 × sin r, so sin r = 0.866/1.5 = 0.577 and r ≈ 35.3°. The ray bends towards the normal. (40° comes from dividing the angle by 1.5; Snell’s law divides the sine.)
How much higher?
A coin lies at the bottom of a pond 18 cm deep. The water has refractive index 4/3. Seen from directly above, by how much does the coin appear to be raised?
- 13.5 cm
- 4.5 cm
- 24 cm
- 6 cm
Answer
B. Apparent depth = h/n = 18 ÷ (4/3) = 13.5 cm. The coin is raised by 18 − 13.5 = 4.5 cm, which is h(1 − 1/n). (13.5 cm is the apparent depth, not the shift; 24 cm comes from multiplying by n.)
Summary
- Snell’s law:
sin i / sin r = n₂₁ = n₂/n₁ - Same law, symmetric form:
n₁ sin i = n₂ sin r - Apparent depth (near normal):
h′ = h / n
- Light bends towards the normal entering a denser medium and away from it entering a rarer one.
- n₁₂ = 1/n₂₁ and n₃₂ = n₃₁ × n₁₂.
- A ray through a rectangular slab comes out parallel to the incident ray, with a lateral shift.
- Objects under water look raised: apparent depth = real depth / n when viewed near the normal.
- Atmospheric refraction makes the sun visible about 2 minutes before sunrise and after sunset.