Critical angle and total internal reflection
When light goes from a denser to a rarer medium at more than the critical angle C, where sin C = 1/n, all of it reflects back. For water C = 48.75°.
Refraction
Snell's law: n₁ sin i = n₂ sin r, so sin i / sin r = n₂₁ = n₂/n₁. Angles are measured from the normal. Going from a denser to a rarer medium beyond the critical angle C (sin C = n₂/n₁), there is no refracted ray. Drag the incoming ray or type an angle.
Light from n = 1.33 meets n = 1 at 55°, past the critical angle of 48.75°, so it is totally reflected.
Show the working
- n₁ sin i = n₂ sin r, so sin r = n₁ sin i / n₂ = 1.33 × sin 55° / 1 = 1.0895
- sin r would be more than 1, so there is no refracted ray: total internal reflection. The critical angle is sin C = n₂/n₁ = 0.7519, C = 48.75°.
- n₂₁ = n₂/n₁ = 1/1.33 = 0.7519. Light travels at c/n: 2.26 × 10⁸ m/s in medium 1 and 3 × 10⁸ m/s in medium 2.
When does it happen?
- Light must go from an optically denser medium into a rarer one, for example from glass or water into air.
- The angle of incidence must be greater than the critical angle C, where sin C = n₂/n₁. Into air (n taken as 1), sin C = 1/n.
- At exactly C the refracted ray grazes along the surface (r = 90°). Beyond it there is no refracted ray at all: every bit of the light reflects back.
- Uses: optical fibres carry light by repeated total internal reflection, diamonds sparkle because C is only about 24°, and mirages and totally reflecting prisms work the same way.
Into air, sin C = 1/n
| Medium | n | Critical angle C |
|---|---|---|
| Water | 1.33 | 48.75° |
| Crown glass | 1.52 | 41.14° |
| Dense flint glass | 1.65 | 37.31° |
| Diamond | 2.42 | 24.41° |
Water into air at 55°
- Light inside water (n = 1.33) meets the surface with air at 55°.
- n₁ sin i = n₂ sin r, so sin r = n₁ sin i / n₂ = 1.33 × sin 55° / 1 = 1.0895
- sin r would be more than 1, so there is no refracted ray: total internal reflection. The critical angle is sin C = n₂/n₁ = 0.7519, C = 48.75°.
- n₂₁ = n₂/n₁ = 1/1.33 = 0.7519. Light travels at c/n: 2.26 × 10⁸ m/s in medium 1 and 3 × 10⁸ m/s in medium 2.
Quick answers
What is the critical angle?
The angle of incidence in the denser medium for which the angle of refraction is 90°. For light going into air, sin C = 1/n.
What are the two conditions for total internal reflection?
Light must go from an optically denser medium to a rarer one, and the angle of incidence must be greater than the critical angle.
Where is total internal reflection used?
Optical fibres, totally reflecting prisms in binoculars and periscopes, and the sparkle of diamonds. Mirages are a natural example.