Refraction, Snell’s Law & Total Internal Reflection

You’ll be able to

Relate the index of refraction to the speed of light in a medium
Apply Snell’s law to find the direction of a refracted ray
Explain the condition for total internal reflection

Why light bends: the index of refraction

Light travels fastest in a vacuum (c = 3.0 × 10⁸ m·s⁻¹) and slower in any material. The index of refraction n = c/v measures that slowdown: n = 1 in vacuum, about 1.33 in water, 1.5 in glass. When light crosses from one medium into another, the change in speed makes it bend at the boundary. Entering a denser (higher-n) medium, light slows and bends toward the normal; entering a less dense medium, it speeds up and bends away from the normal.

Index of refraction and Snell’s law

n = c / v · n₁ · sinθ₁ = n₂ · sinθ₂

All angles are measured from the normal. The product n·sinθ is the same on both sides of the boundary, so a bigger n forces a smaller angle.

Total internal reflection

When light tries to pass from a denser medium into a less dense one (say glass to air), it bends away from the normal. Increase the angle of incidence and the refracted ray bends further, until at the critical angle it grazes along the boundary at 90°. Beyond that angle the light cannot escape at all — it is entirely reflected back inside. This total internal reflection (TIR) is what pipes light down optical fibers and makes diamonds sparkle. It requires going from high n to low n, at an angle past critical.

Worked example

Light travels from air (n = 1.0) into glass (n = 1.5), striking the surface at 30° from the normal. Find the angle of refraction. (sin 30° = 0.50)

1.Apply Snell’s law: n₁ sinθ₁ = n₂ sinθ₂.
2.Substitute: (1.0)(sin 30°) = (1.5)(sinθ₂), so (1.0)(0.50) = 1.5 sinθ₂.
3.Solve for sinθ₂: sinθ₂ = 0.50 / 1.5 = 0.33.
4.Take the inverse sine: θ₂ = sin⁻¹(0.33) ≈ 19.5°.

Answer: ≈ 19.5°. The ray bends toward the normal (from 30° to about 19.5°) because it enters the denser glass.

Checkpoint

Light passes from air (n = 1.0) into glass (n = 1.5), hitting the surface at 30° from the normal. What is the angle of refraction? (sin 30° = 0.50)

Tip

Sanity-check every refraction by direction: entering a denser (higher-n) medium, the ray bends toward the normal, so the angle gets smaller. If your Snell’s-law answer bends the wrong way, you likely swapped n₁ and n₂.

Checkpoint

Total internal reflection can occur only when light travels:

On the exam

Total internal reflection has two requirements, and both must hold: (1) light moving from high n to low n, and (2) an angle of incidence beyond the critical angle. If either fails, some light refracts through and it is not total.

Answer the 2 checkpoints as you read.