Snell's Law Formula
Snell's Law calculates refraction of light between media: n1 sin(a1) = n2 sin(a2).
Includes total internal reflection and examples.
The Formula
Snell's Law describes how light bends when passing from one medium to another. The amount of bending depends on the refractive indices of the two materials.
Variables
| Symbol | Meaning |
|---|---|
| n₁ | Refractive index of the first medium |
| θ₁ | Angle of incidence (measured from the normal) |
| n₂ | Refractive index of the second medium |
| θ₂ | Angle of refraction (measured from the normal) |
Common Refractive Indices
| Material | Refractive Index (n) |
|---|---|
| Vacuum | 1.000 |
| Air | 1.0003 |
| Water | 1.333 |
| Glass (crown) | 1.52 |
| Diamond | 2.42 |
Example 1
Light enters water from air at a 45° angle. What is the angle of refraction?
n₁ = 1.0003 (air), θ₁ = 45°, n₂ = 1.333 (water)
sin(θ₂) = n₁ × sin(θ₁) / n₂
sin(θ₂) = 1.0003 × sin(45°) / 1.333 = 1.0003 × 0.7071 / 1.333
sin(θ₂) = 0.5306
θ₂ ≈ 32.0° (the light bends toward the normal as it enters the denser medium)
Example 2
Find the critical angle for total internal reflection when light goes from glass (n = 1.52) to air.
At the critical angle, θ₂ = 90°, so sin(θ₂) = 1
sin(θc) = n₂ / n₁ = 1.0003 / 1.52
sin(θc) = 0.6581
θc ≈ 41.1° (any angle greater than this causes total internal reflection)
When to Use It
Use Snell's Law for optics and light refraction problems:
- Designing lenses for glasses, cameras, and telescopes
- Understanding why objects look bent or distorted in water
- Calculating fiber optic light transmission (total internal reflection)
- Analyzing prisms and rainbows (light dispersion)
Key Notes
- Formula: n₁sinθ₁ = n₂sinθ₂: When light crosses from medium 1 (refractive index n₁) to medium 2 (n₂), the angle changes to satisfy this equation. Higher n means slower light and more bending toward the normal. The angle is always measured from the perpendicular (normal) to the surface.
- Refractive index: n = c/v: The ratio of the speed of light in vacuum (c ≈ 3×10⁸ m/s) to the speed in the medium. Air ≈ 1.0003; water ≈ 1.33; glass ≈ 1.5; diamond ≈ 2.42. Higher n = slower light = more bending when entering from a lower-n medium.
- Total internal reflection: θ_c = arcsin(n₂/n₁) for n₁ > n₂: When light in a denser medium hits the boundary at an angle greater than the critical angle, it reflects entirely — no transmitted ray. This is the operating principle of optical fiber (glass core, lower-n cladding).
- Dispersion: The refractive index n varies slightly with wavelength (color). Blue light bends more than red light in glass. This is why prisms split white light into spectra, and why rainbows form — each raindrop acts as a tiny prism.
- Applications: Snell's law governs optical fiber communications, camera and microscope lens design, eyeglass prescriptions (lens curvature depends on n), mirages (temperature gradient bends light in air), and gemstone cutting angles (exploiting total internal reflection for sparkle).