All calculatorsSnell’s Law Calculator
Solve refraction fast using Snell’s Law: n₁·sin(θ₁) = n₂·sin(θ₂). Choose what you want to solve for, pick materials (or enter custom indices), and get a clean answer with Total Internal Reflection detection, critical angle, step-by-step, and a mini ray diagram.
Background
When light crosses a boundary between two media (like air → water), it usually bends. Angles θ₁ and θ₂ are measured from the normal (the line perpendicular to the surface), not from the surface itself.
Snell’s Law links the refractive indices n₁, n₂ and the angles: n₁·sin(θ₁) = n₂·sin(θ₂). If light goes from a higher index to a lower index (like glass → air), there’s a maximum incident angle called the critical angle. Above it, refraction stops and you get Total Internal Reflection (TIR).
How to use this calculator
- Pick what you want to solve for: θ₂, θ₁, n₁, n₂, or critical angle.
- Choose your media (quick picks) or type custom refractive indices.
- Enter the angles you know (measured from the normal).
- Click Calculate to get the missing value, plus TIR/critical angle checks.
How this calculator works
- Snell’s Law: n₁·sin(θ₁) = n₂·sin(θ₂)
- Solve for θ₂: sin(θ₂) = (n₁/n₂)·sin(θ₁) then θ₂ = asin(…)
- Total Internal Reflection: happens when n₁ > n₂ and sin(θ₂) would exceed 1.
- Critical angle: if n₁ > n₂, then sin(θc) = n₂/n₁ and θc = asin(n₂/n₁).
- Extra insight: speed ratio v₂/v₁ = n₁/n₂ and wavelength ratio λ₂/λ₁ = n₁/n₂ (frequency stays constant).
Formula & Equation Used
Snell’s Law: n₁·sin(θ₁) = n₂·sin(θ₂)
Solve for refraction angle: θ₂ = asin((n₁/n₂)·sin(θ₁))
Solve for incident angle: θ₁ = asin((n₂/n₁)·sin(θ₂))
Solve for refractive index: n₁ = n₂·sin(θ₂)/sin(θ₁), n₂ = n₁·sin(θ₁)/sin(θ₂)
Critical angle (if n₁ > n₂): θc = asin(n₂/n₁)
Example Problems & Step-by-Step Solutions
Example 1 — Air → Water (find θ₂)
Light goes from air (n₁≈1.0003) into water (n₂≈1.333) with θ₁=30°. Find θ₂.
- Use Snell’s Law: sin(θ₂)=(n₁/n₂)·sin(θ₁)
- sin(θ₂)=(1.0003/1.333)·sin(30°)≈0.375
- θ₂=asin(0.375)≈22.0°
Example 2 — Glass → Air (TIR check)
Light goes from glass (n₁≈1.52) to air (n₂≈1.0003) with θ₁=50°. What happens?
- Compute sin(θ₂)=(n₁/n₂)·sin(θ₁).
- If sin(θ₂)>1, there is no real θ₂ → Total Internal Reflection.
- Find critical angle: θc=asin(n₂/n₁)=asin(1.0003/1.52)≈41.1°.
- Since 50°>41.1°, TIR occurs.
Example 3 — Solve for n₂
Light goes from air (n₁≈1.0003) into an unknown medium with θ₁=30° and θ₂=19.47°. Find n₂.
- Rearrange: n₂ = n₁·sin(θ₁)/sin(θ₂)
- n₂ ≈ 1.0003·sin(30°)/sin(19.47°) ≈ 1.52
Frequently Asked Questions
Q: Are angles measured from the surface or the normal?
From the normal (the perpendicular line). If you measured from the surface, convert first: angle-from-normal = 90° − angle-from-surface.
Q: When does Total Internal Reflection happen?
When light travels from a higher index to a lower index (n₁ > n₂) and the incident angle exceeds the critical angle θc.
Q: Why can’t θ₂ be computed during TIR?
Snell’s Law would require sin(θ₂) > 1, which is impossible for real angles.
Q: Does frequency change when light enters a new medium?
Frequency stays the same. The speed and wavelength change, with ratios tied to refractive index.
