Refraction and Snell's law
Refraction is the bending of light when it goes from one medium into another, because its speed changes. The refractive index n of a medium tells how much slower light is there: n = c / v. Air has n ≈ 1, water 1.33, glass about 1.5, diamond 2.42.
Snell's law: n₁ sin i = n₂ sin r. Here i is the angle in the first medium and r in the second, both from the normal. Also, the incident ray, refracted ray and normal lie in one plane.
- Rarer to denser (air → glass): bends towards the normal (r < i).
- Denser to rarer (glass → air): bends away from the normal (r > i).
The relative refractive index of medium 2 with respect to medium 1 is n₂₁ = n₂/n₁ = v₁/v₂. Note n₁₂ = 1/n₂₁.
Apparent depth
A coin at the bottom of a bucket of water looks raised. Rays from the coin bend away from the normal as they leave the water, and your eye traces them back to a higher point. When you look straight down:
n = real depth / apparent depth
The coin seems raised by d(1 − 1/n). For water, a 40 cm deep coin looks about 30 cm deep. The same bending lifts the Sun: we see it for about 2 minutes before real sunrise and after real sunset.
Total internal reflection
When light goes from a denser to a rarer medium, r is bigger than i. As i grows, r reaches 90° first. The angle i at which this happens is the critical angle C. Put r = 90° in Snell's law (n₁ = n, n₂ = 1):
sin C = 1/n (more generally sin C = n₂/n₁)
If i > C, no refracted ray can exist, so all the light reflects back into the denser medium. This is total internal reflection (TIR). Two conditions are needed:
- Light must travel from a denser to a rarer medium.
- The angle of incidence must be larger than the critical angle.
For glass (n = 1.5), C ≈ 41.8°; for water, C ≈ 48.8°; for diamond, C ≈ 24.4°. TIR reflects 100% of the light, better than any silvered mirror.
Optical fibres
An optical fibre is a very thin glass or plastic thread. Its core has a higher refractive index than the cladding around it. Light entering one end hits the core–cladding wall at an angle larger than C, so it undergoes TIR again and again and comes out at the far end with very little loss, even if the fibre bends gently.
Uses: internet and phone signals (a bundle carries huge data), endoscopes in hospitals, decorative lamps. Other TIR uses: sparkling of diamonds, totally reflecting prisms in binoculars and periscopes (turning light by 90° or 180°), and mirages.
Refraction through a prism
A prism has two flat refracting faces meeting at the angle of the prism A. A ray enters with angle i, bends to r₁, meets the second face at r₂ and leaves at angle e. The total turning of the ray is the angle of deviation δ.
Step 1: the angles inside
The two normals and the prism faces make a four-sided figure, which gives r₁ + r₂ = A.
Step 2: deviation
The ray turns by (i − r₁) at the first face and (e − r₂) at the second. So δ = (i − r₁) + (e − r₂) = i + e − (r₁ + r₂), giving δ = i + e − A.
Step 3: minimum deviation
As i increases, δ first decreases, reaches a smallest value δm, then increases. At δm the ray passes through the prism symmetrically: i = e and r₁ = r₂ = A/2. Then i = (A + δm)/2, so
n = sin((A + δm)/2) / sin(A/2)
Thin prism
For a small A (a few degrees), sin x ≈ x, so δ = (n − 1)A. Deviation does not depend on i for a thin prism.
Try it: make light disappear
Fill a clear glass with water and hold it a little above your eye level. Look up at the water surface from below, through the side. At a steep angle the surface shines like silver: that is total internal reflection. Or shine a phone torch into a stream of water flowing from a bottle hole in a dark room: the light follows the curved stream like a fibre. In the 3D, predict the angle where the ray stops leaving the glass, then tilt it to check.
Key formulas and definitions
- n = c / v
- n₁ sin i = n₂ sin r
- n = real depth / apparent depth
- sin C = 1/n (sin C = n₂/n₁)
- r₁ + r₂ = A
- δ = i + e − A
- n = sin((A + δm)/2) / sin(A/2)
- Thin prism: δ = (n − 1)A
Worked examples
1. Light enters glass (n = 1.5) from air at 45°. Find the angle of refraction.
Step 1: 1 × sin 45° = 1.5 × sin r. Step 2: sin r = 0.707/1.5 = 0.471. Step 3: r ≈ 28.1°. It bends towards the normal.
2. Find the speed of light in water (n = 1.33).
Step 1: v = c/n. Step 2: v = 3 × 10⁸ / 1.33. Step 3: v ≈ 2.26 × 10⁸ m/s.
3. A coin lies at the bottom of a tank of water 24 cm deep (n = 4/3). How deep does it appear from above?
Step 1: apparent depth = real depth / n. Step 2: = 24 ÷ (4/3) = 18 cm. Step 3: the coin looks raised by 6 cm.
4. Find the critical angle for glass of refractive index 1.5.
Step 1: sin C = 1/n = 1/1.5 = 0.667. Step 2: C = sin⁻¹(0.667) ≈ 41.8°. So any ray inside glass hitting the surface at more than 41.8° is totally reflected.
5. The critical angle for a medium is 30°. Find its refractive index and the speed of light in it.
Step 1: n = 1/sin C = 1/0.5 = 2. Step 2: v = c/n = 1.5 × 10⁸ m/s.
6. Light goes from glass (n = 1.5) to water (n = 1.33). Find the critical angle at the glass–water surface.
Step 1: sin C = n₂/n₁ = 1.33/1.5 = 0.887. Step 2: C ≈ 62.5°. It is larger than for glass–air because water is closer to glass in n.
7. A prism of angle 60° gives minimum deviation 30°. Find its refractive index.
Step 1: n = sin((A + δm)/2) / sin(A/2). Step 2: = sin 45° / sin 30° = 0.707/0.5. Step 3: n ≈ 1.414 (= √2).
8. A ray falls on a prism (A = 60°, n = √2). At minimum deviation, find r₁, the angle of incidence and δm.
Step 1: at δm, r₁ = r₂ = A/2 = 30°. Step 2: sin i = n sin r₁ = √2 × 0.5 = 0.707, so i = 45°. Step 3: i = e, so δm = 2i − A = 90° − 60° = 30°.
9. A thin prism of angle 5° and n = 1.6 is placed in air. Find the deviation. What if it is dipped in water (n = 1.33)?
Step 1: in air δ = (n − 1)A = 0.6 × 5° = 3°. Step 2: in water use the relative index n = 1.6/1.33 ≈ 1.2. Step 3: δ = 0.2 × 5° = 1°. The deviation becomes smaller in water.
Common mistakes
- Thinking TIR can happen when light goes from air into glass. It needs light moving from a denser to a rarer medium.
- Measuring angles from the surface instead of from the normal.
- Using δ = (n − 1)A for a big prism like 60°. It is only for thin prisms.
- Forgetting that at minimum deviation i = e and r₁ = r₂ = A/2.