Refractive index and the speed of light
Light travels at c = 3.00 × 10⁸ m/s in a vacuum. In any material it is slower. The refractive index tells how much slower:
n = c / v (no unit, always ≥ 1)
Air ≈ 1.00, water 1.33, glass about 1.5, diamond 2.42. A larger n is called optically denser.
What changes and what stays
- Frequency f stays the same: the wave crests arrive at the boundary at the same rate they leave.
- Speed falls to v = c/n.
- So wavelength shrinks: λ = v/f = λ₀/n.
The colour we see depends on frequency, so light does not change colour inside water.
Snell's law
Measure both angles from the normal (not from the surface).
n₁ sin θ₁ = n₂ sin θ₂
- n₂ > n₁: θ₂ < θ₁, bends toward the normal.
- n₂ < n₁: θ₂ > θ₁, bends away from the normal.
- θ₁ = 0 (along the normal): no bending at all, only a change of speed.
Some light is also reflected at every boundary (the faint orange ray in the 3D). The incident, refracted and reflected rays and the normal all lie in one plane.
Why it bends
A wavefront hitting the surface at an angle has one end enter the slow medium first. That end slows down while the other end is still fast, so the whole front turns.
Critical angle and total internal reflection
When light goes from a higher n to a lower n, θ₂ is bigger than θ₁. At one special angle, θ₂ reaches 90°. That angle is the critical angle:
sin θc = n₂ / n₁ (n₁ > n₂)
Glass to air: sin θc = 1/1.5, θc ≈ 41.8°. Water to air: θc ≈ 48.8°. Diamond to air: θc ≈ 24.4°.
If θ₁ > θc, Snell's law would need sin θ₂ > 1, which is impossible. No light leaves; it all reflects. This is total internal reflection (TIR).
Two conditions: (1) light goes from denser to rarer medium, (2) angle of incidence is greater than the critical angle.
Uses: optical fibres, endoscopes, prism binoculars, the sparkle of diamonds, road reflectors.
Apparent depth and dispersion
Apparent depth: looking straight down into water, an object at real depth d seems to be at d/n. A 2 m pool looks about 1.5 m deep.
Dispersion: n is slightly larger for violet than for red light, so violet bends more. A prism spreads white light into a spectrum, and raindrops make rainbows.
Try it: put a coin in an empty cup, move back until it just hides behind the rim, then slowly pour water in. The coin appears! Refraction bends its light over the rim to your eye.
Key formulas and definitions
- n = c / v
- n₁ sin θ₁ = n₂ sin θ₂
- λ_medium = λ₀ / n, f unchanged
- sin θc = n₂ / n₁ (n₁ > n₂)
- Relative index n₂₁ = n₂ / n₁ = v₁ / v₂
- Apparent depth = real depth / n (near-normal view)
Worked examples
1. Light travels at 2.25 × 10⁸ m/s in water. Find the refractive index of water.
n = c/v = 3.00 × 10⁸ ÷ 2.25 × 10⁸ = 1.33.
2. A ray enters glass (n = 1.5) from air at 30°. Find the angle of refraction.
1 × sin 30° = 1.5 sin θ₂ → sin θ₂ = 0.5 ÷ 1.5 = 0.333 → θ₂ ≈ 19.5°.
3. Red light of wavelength 650 nm enters water (n = 1.33). Find its wavelength and frequency in water.
λ = 650 ÷ 1.33 ≈ 489 nm. f = c/λ₀ = 3 × 10⁸ ÷ 650 × 10⁻⁹ ≈ 4.6 × 10¹⁴ Hz, the same as in air.
4. Find the critical angle for light going from glass (n = 1.5) into water (n = 1.33).
sin θc = 1.33 ÷ 1.5 = 0.887 → θc ≈ 62.5°.
5. A ray in water hits the water–air surface at 55°. Does it come out?
θc for water = sin⁻¹(1/1.33) ≈ 48.8°. Since 55° > 48.8°, it is totally internally reflected and does not come out.
6. A fish is 1.2 m below the surface. How deep does it look to someone directly above? (n = 1.33)
Apparent depth = 1.2 ÷ 1.33 ≈ 0.90 m. It looks about 30 cm closer.
Common mistakes
- Measuring angles from the surface instead of the normal. Always use the angle with the normal.
- Thinking the frequency (colour) changes in glass. Only speed and wavelength change.
- Using sin θc = n₁/n₂ upside down. The answer must be less than 1: put the smaller index on top.
- Expecting total internal reflection when light goes from air into glass. TIR needs dense → rare.