Snell's Law
Find the refraction angle from n₁ sin θ₁ = n₂ sin θ₂, with total-internal-reflection detection.
n₂ > n₁: light bends toward the normal (θ₂ < θ₁).
Snell’s law is n₁ sin θ₁ = n₂ sin θ₂, so the refraction angle is θ₂ = asin(n₁ sin θ₁ ÷ n₂). Light passing from air (n = 1.00) into glass (n = 1.50) at θ₁ = 30° refracts to θ₂ = asin(sin 30° ÷ 1.50) = asin(0.3333) ≈ 19.5° — bending toward the normal as it enters the denser medium.
What Snell’s law describes
When light crosses the boundary between two transparent media it changes speed, and that change makes the ray bend. Snell’s law, n₁ sin θ₁ = n₂ sin θ₂, relates the angle of incidence (θ₁) to the angle of refraction (θ₂) through each medium’s refractive index (n), where every angle is measured from the normal — the line perpendicular to the surface, not the surface itself.
n₁, n₂ = refractive indices · θ₁ = angle of incidence · θ₂ = angle of refraction (both from the normal)
Worked example
A ray travels from air (n₁ = 1.00) into glass (n₂ = 1.50) striking the surface at θ₁ = 30°. Find the refraction angle:
- 1 Solve for the refraction angle. Rearrange n₁ sin θ₁ = n₂ sin θ₂ to θ₂ = asin(n₁ sin θ₁ ÷ n₂).
- 2 Substitute the values. θ₂ = asin(1.00 × sin 30° ÷ 1.50) = asin(0.5 ÷ 1.50) = asin(0.3333).
- 3 Take the inverse sine. θ₂ = asin(0.3333) ≈ 19.47° — the ray bends toward the normal entering the denser glass.
- 4 Check for total internal reflection. If n₁ sin θ₁ ÷ n₂ exceeds 1, asin has no solution: there is no refracted ray and the light reflects entirely.
Refractive indices (visible light)
Approximate indices used by this tool; exact values vary with wavelength and conditions.
| Medium | Refractive index (n) |
|---|---|
| Vacuum | 1.00 |
| Air | 1.0003 |
| Water | 1.33 |
| Glass (typical) | 1.50 |
| Diamond | 2.42 |
Bending, the critical angle, and total internal reflection
Which way light bends. Entering a denser medium (n₂ > n₁) the ray slows and bends toward the normal, so θ₂ < θ₁. Going the other way, into a less dense medium (n₂ < n₁), it bends away from the normal and θ₂ > θ₁.
The critical angle. When light moves from a denser to a rarer medium, there is an angle of incidence at which θ₂ would reach 90°. This critical angle is θ_c = asin(n₂ ÷ n₁) — for glass-to-air it is asin(1 ÷ 1.5) ≈ 41.8°.
Total internal reflection. Beyond the critical angle, n₁ sin θ₁ ÷ n₂ exceeds 1, the inverse sine is undefined, and no light escapes — it all reflects back inside. This is exactly why optical fibres trap light and why a submerged straw or a pool floor can look mirror-like from a shallow angle.