5.8 Refractive Index and Snell’s Law

For physicists, it is good to know that waves react, thought it is more helpful to know when, and by how much. Refraction can be quantified by relating the angle of incidence (to the boundary between the two media in question) to the angle of refraction.. These, and the refractive indexes of the two media can be used to precisely calculate the change in direction of a wave.

The refractive index of a medium (for a certain wave) is the ratio of the speed of the wave in unrestrained conditions (the absolute fastest speed) to the speed of the wave in that medium. The refractive index has symbol n, and, being a ratio, has no unit. In some cases, a single refractive index is given for the two materials involved, but this is simply the combined ratios of their two n’s. However, in this unit, we will discuss refractive indexes for individual materials.

The following relates the refractive indices, n₁ and n₂, of two media with two more familiar terms, the angle of incidence i, and the angle of refraction, r:

This is known as Snell’s Law. However, since n, the refractive index is a ratio of the fastest possible speed of the wave to the speed in the medium, we can simplify to get one more equation:

If u is the maximum speed of the wave (e.g speed of light in a vacuum), and c₁ and c₂ are the speeds of the wave in their respective media 1 and 2,

and

Where c is the speed of the wave in the medium.

A special case of Snell’s Law is applied for Total Internal Reflection (T.I.R.), Unit 8.