Snell's Law of Refraction
When light passes from one transparent material into another, it bends at the boundary — this is called refraction. Light bends toward the normal (an imaginary line perpendicular to the surface) when entering a denser material, and away from the normal when entering a less dense one.
The formula
n₁ · sin θ₁ = n₂ · sin θ₂
- n₁ — refractive index (unitless): how much the first material slows down light
- θ₁ — angle of incidence (°): the angle of the incoming ray, measured from the normal
- n₂ — refractive index (unitless): how much the second material slows down light
- θ₂ — angle of refraction (°): the angle of the bent ray inside the second material, measured from the normal
Worked example
A ray of light grazes along the surface of an unknown liquid, entering at ninety degrees from the normal, and bends to thirty degrees from the normal inside the liquid. What is the liquid's refractive index?
- n₁ = 1 (air)
- θ₁ = 90°
- θ₂ = 30°
- n₂ = (n₁ · sin θ₁) / sin θ₂
- n₂ = (1 · sin 90°) / sin 30° = (1 × 1) / 0.5
n₂ = 2
Test yourself
This liquid has a refractive index of 2. Light traveling inside it hits the surface, trying to escape back into air. What is the sine of the critical angle for total internal reflection?
- Correct answer: 0.5
- 2
- 1
Right! sin θc = n₁ / n₂ = 1 / 2 = 0.5, which is a critical angle of 30 degrees.
Now suppose the liquid is even denser, with a refractive index of 4 instead of 2. Light again grazes in at ninety degrees. Compared to before, does the refracted ray bend closer to or further from the normal?
- Further from the normal
- Correct answer: Closer to the normal
- No change
Yes — a denser medium bends light more: sin θ₂ = 1 / 4 = 0.25, a smaller angle than the 30 degrees we found before.
Where you see this
A pencil in a glass of water looks broken at the surface, and a pool is deeper than it looks from the deck — both are light bending as it crosses from water into air. A straw in the drink does the same trick from the side: above the waterline it points one way, below it another.
Common mistakes
The direction slip: light entering the denser material bends TOWARD the normal — a bigger index pulls sin θ₂ down (entering a material of n = 4 at grazing incidence gives sin θ₂ = 1/4 = 0.25), so the ray hugs the normal more closely, not less. Then total internal reflection only runs one way: it happens to light trying to escape from dense to less dense, and the critical angle's sine is just the index ratio, sin θc = n₂ / n₁ — for index 2 into air, 1/2 = 0.5.
How it connects
Refraction is the workhorse of the whole optics module: a lens is nothing but glass shaped so its curved surfaces refract every ray by this law to meet at one point, and the same bend at a mirror's twin equation forms images next. Physically, light bends because it slows in the denser material — the wave relation from the waves module, speed traded for direction.