Snell–Descartes’ law of refraction

The statement of the law

When a ray of light passes from medium 1 (index n1) to medium 2 (index n2), the angles of incidence i1 and refraction i2, measured relative to the normal to the dioptre (the line perpendicular to the surface at the point of incidence), satisfy:

n1 * sin(i1) = n2 * sin(i2)

This relationship is Snell–Descartes’ law of refraction. It allows one angle to be calculated if the other angle and both indices are known.

Step-by-step method

1. Identify the two media and their indices n1 and n2.
2. Measure or identify the angle of incidence i1 (always relative to the normal, never relative to the surface).
3. Apply the formula to isolate sin(i2): sin(i2) = (n1 / n2) * sin(i1).
4. Calculate i2 = arcsin((n1/n2) * sin(i1)).

Numerical example

A ray of light travels from air (n1 = 1) into water (n2 = 1.33) at an angle of incidence i1 = 30 degrees.

sin(i2) = (1 / 1.33) * sin(30) = 0.75 * 0.5 = 0.375

i2 = arcsin(0.375) = approximately 22 degrees.

The ray approaches the normal as it enters the more refractive medium (higher refractive index): this makes sense, as when n2 > n1, then i2 < i1.

Special cases and pitfalls

• If the ray strikes the dioptre perpendicularly (i1 = 0), then i2 = 0: there is no deviation, even if the indices are different.
• Be careful to use sin and not the angle directly: the law is NOT n1 * i1 = n2 * i2.
• The normal is always perpendicular to the surface; it must never be confused with the surface itself: a common mistake is to measure the angle relative to the dioptre rather than the normal.