Laws of Refraction at Plane Surfaces: Snell's Law in Depth

Refraction is the bending of light when it passes from one medium to another, such as from air into water or glass. At a plane surface, the bending follows precise laws. Snell law quantifies exactly how much the ray bends.

This bending explains why a straw looks broken in a glass of water, why pools appear shallower than they are, and how lenses focus light. Snell law is the foundation for designing glasses, cameras and fibre optics.

Key Concepts

Refractive index n: Ratio of speed of light in vacuum c to speed in medium v, n = c divided by v. It has no unit. For vacuum n = 1, for water about 1.33, for glass about 1.5.

Angle of incidence i: Angle between incident ray and normal in first medium. Angle of refraction r: Angle between refracted ray and normal in second medium. Both in degrees.

Normal: Perpendicular line at the point where ray crosses the surface.

• Light bends toward normal when it goes to optically denser medium, slower speed
• Light bends away from normal when it goes to rarer medium, faster speed
• Frequency stays same, wavelength and speed change

Formulas

First law: Incident ray, refracted ray and normal lie in same plane.

Snell law: n1 sin i = n2 sin r. Here n1, n2 are refractive indices of medium 1 and 2, dimensionless, i and r in degrees or radians for sine. Also written as sin i divided by sin r = n2 divided by n1 = constant for a pair of media. Assumes plane surface and monochromatic light.

n = c divided by v = real depth divided by apparent depth for near-normal viewing. Units cancel, n dimensionless.

Worked Example

A ray goes from air (n1 = 1.00) into water (n2 = 1.33) with incidence angle 45 degrees. Find refraction angle.

1. Write Snell law: n1 sin i = n2 sin r.
2. So sin r = n1 sin i divided by n2 = 1.00 times sin 45 degrees divided by 1.33.
3. sin 45 degrees = 0.707, so sin r = 0.707 divided by 1.33 = 0.532.
4. r = arcsin 0.532 = 32.1 degrees.

The ray bends toward the normal, from 45 degrees to about 32 degrees, because water is denser than air.

Common Misconceptions

• Light always bends same way: Direction depends on relative density. Air to glass bends toward normal, glass to air bends away.
• Refractive index can be less than 1: In ordinary materials n is 1 or more because light slows down. Vacuum is the fastest.
• Angle measured from surface: Like reflection, refraction angles are from normal, not surface. Using surface gives wrong Snell result.

Real-World Applications

• Corrective lenses bend light to focus on retina for clear vision
• Apparent depth effect used to estimate pool depth and in fishing, fish appear raised
• Design of camera lenses, microscopes and telescopes relies on precise refraction
• Atmospheric refraction makes the Sun visible a few minutes before actual sunrise

Refraction at a plane surface follows Snell law n1 sin i = n2 sin r. With refractive index and careful angle measurement, the path of any ray can be predicted.