Snell's Law — n₁sinθ₁ = n₂sinθ₂

Snell's law n₁sinθ₁ = n₂sinθ₂ explained: refraction at boundaries, why light bends toward the normal, and a worked air-to-water example.

What it means

Snell's law governs every bend of light at a boundary. Light slows down in denser media, and to keep its wavefronts continuous across the interface, the ray must change direction — bending toward the normal when entering a slower medium, away when entering a faster one. When going from slow to fast at a steep enough angle, no real solution exists: that is total internal reflection, the principle behind fiber optics. Every lens, prism, rainbow and mirage is Snell's law in action.

Variables

Assumptions

• Sharp boundary between two uniform transparent media
• Monochromatic light (single wavelength — n varies slightly with color)
• Angles measured from the normal to the surface, not the surface itself

Derivation

Fermat's principle — light takes the path of least time — applied to a ray crossing an interface gives $n_1\sin\theta_1 = n_2\sin\theta_2$ after minimizing the travel time. Equivalently, it follows from matching the wave's phase along the boundary.

Worked example

Given:

• n₁ = 1.00 (air)
• θ₁ = 45°
• n₂ = 1.33 (water)

1. Write Snell's law: $n_1\sin\theta_1 = n_2\sin\theta_2$

2. Rearrange: $\sin\theta_2 = (n_1/n_2)\sin\theta_1$

3. Substitute: $\sin\theta_2 = (1.00/1.33) \times \sin 45°$

4. Calculate: $\sin\theta_2 = 0.7519 \times 0.7071 = 0.5317$

5. Take arcsine: $\theta_2 = \sin^{-1}(0.5317) = 32.1°$