Refraction of Light

To further our understanding of light and the rainbow, we now consider what happens to light as it travels through water. This requires relaxing our previous assumption that light travels at a constant speed. It is known that the speed of light traveling through water is less than the speed of light through air.

Note that Fermat's Principle would still hold if the mirror in Figure 1 were completely submerged in water. Consider, then, what happens to the path of a light ray when a portion of the light path is in the water and a portion in the air. In Figure 3, a light ray leaves point A and passes through B. Instead of a mirror, consider what happens as the light ray passes from air into water. Now, the air/water interface, like the mirror, occurs at the point O. Recall that Fermat's Principle states that light follows a path that minimizes total travel time---regardless of the speed of the light ray. Therefore, unlike our investigation with the mirror, our analysis of where the point O occurs must consider both the speed of light in the air and in the water.

Figure 3: A light ray passing from air into water.

Question 2

Let c_a be the speed of light in air and c_w the speed of light in water. Remembering that distance equals speed times time,
Referring to Figure 3 the angle that the path AO makes with the line perpendicular to the water's surface is called the angle of incidence and is represented by the angle a. The corresponding angle between the path OB and the perpendicular is called the angle of refraction and is represented by the angle b. In Questions 2 you deduced Snell's Law or the Law of Refraction which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant.

The speed of light in air depends on the temperature and pressure of the air, and similarly for water and other substances. By contrast the speed of light in a vacuum is an absolute constant, which we represent by c.

The index of refraction for substances is the ratio c/v where v is the speed of light in that substance. Tables have been compiled for the ratio of the speed of light in a vacuum to that of various other substances.

Question 3

Note that the Law of Reflection and Snell's Law do not depend on the direction of the light ray. That is, our results would have been identical if we had assumed that the source of light was the point B instead of A. With this observation, answer the following questions:

Question 4

Experimentally it may be difficult to measure the angle of refraction for a crystal with an unknown index of refraction.

Question 5

Suppose we experimentally determine the index of refraction for some crystal by accurately measuring the angle of refraction, b0, that occurs when incoming light travels through a vacuum and strikes the crystal at an angle of incidence equal to a0.
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Frederick J. Wicklin <>
Paul Edelman <>

This lab is based on a module developed by Steven Janke and published in Modules in Undergraduate Mathematics and its Applications, 1992.

Last modified: Tue Oct 24 15:00:24 1995