Snell's Law

Core Idea

Snell's law gives the quantitative relationship between incident and refracted angles at a boundary: n₁sinθ₁ = n₂sinθ₂. Here n₁ and n₂ are the refractive indices of the two media, and the angles are measured from the normal. When light moves from a lower-index medium to a higher-index medium, it bends toward the normal (θ₂ < θ₁). Snell's law is derived from the requirement that frequency is preserved at the boundary while speed changes.

How It's Best Learned

Trace a light ray through a glass block with known index (~1.5), measure both angles, and verify n₁sinθ₁ = n₂sinθ₂. Then solve a progression of problems: air-to-water, water-to-glass, and multi-layer systems.

Common Misconceptions

• Students sometimes swap n₁ and n₂ or θ₁ and θ₂; draw a clear diagram labeling which side is which before substituting.
• Snell's law applies to the transmitted (refracted) ray, not the reflected ray.

Explainer

From your introduction to refraction, you know that light changes direction when it crosses a boundary between two media. Snell's law gives you the precise quantitative rule for *how much* it bends: n₁ sinθ₁ = n₂ sinθ₂. To use it correctly, you need to be clear on two things — what n means and how to measure the angles.

The index of refraction n = c/v tells you how much slower light travels in a medium compared to a vacuum. Air has n ≈ 1.00 (light barely slows down), water has n ≈ 1.33, and glass has n ≈ 1.5. The higher the index, the slower the speed. Angles in Snell's law are always measured from the normal — the imaginary line perpendicular to the surface at the point of contact. Drawing this normal explicitly in every diagram is the single most reliable way to avoid sign and labeling errors.

The bending direction follows a simple rule: when light enters a *denser* medium (higher n), it bends *toward* the normal, so the transmitted angle is smaller than the incident angle. When light enters a *less dense* medium (lower n), it bends *away* from the normal. You can verify this directly from the equation: if n₂ > n₁, then sinθ₂ < sinθ₁, so θ₂ < θ₁. The physical reason is that light slows down in the denser medium, and the wavefront has to pivot — like a row of marching soldiers where one end hits mud before the other.

To solve Snell's law problems, the workflow is straightforward: (1) draw the boundary and the normal, (2) label n₁, n₂, θ₁ on your diagram, (3) substitute into n₁ sinθ₁ = n₂ sinθ₂ and solve for the unknown. The most common mistakes are swapping which side is 1 and which is 2, and forgetting that the angles are measured from the normal rather than the surface itself.

Snell's law only describes the *refracted* ray. There is always a reflected ray as well (at the same angle as incidence, on the same side of the boundary), but that is governed by the law of reflection — a separate rule. A consequence of Snell's law you will explore soon is total internal reflection: when light tries to exit a denser medium at a steep enough angle, sinθ₂ would have to exceed 1, which is impossible, so no refracted ray exists and all the light reflects back internally. This is the principle behind optical fibers.