To Snellius and his contemporaries, $n$ was simply an “index of the refractive power” of a transparent substance. The relationship between the index of refraction and the speed of light was not recognized until the development of a wave theory of light in the 19th century. Theory predicts, and experiment confirms, that light travels through a transparent medium, such as glass or water, at a speed less than its speed $c$ in vacuum. In Section 16.5, we defined the index of refraction n of a transparent medium as

where $v_{\text{medium}}$ is the light speed in the medium. This implies, of course, that $v_{\text{medium}}=c/n$. The index of refraction of a medium is always $n>1$ except for vacuum.

TABLE 34.1 shows measured values of $n$ for several materials. There are many types of glass, each with a slightly different index of refraction, so we will keep things simple by accepting $n=1.50$ as a typical value. Notice that cubic zirconia, used to make costume jewelry, has an index of refraction much higher than glass.

We can accept Snell’s law as simply an empirical discovery about light. Alternatively, and perhaps surprisingly, we can use the wave model of light to justify Snell’s law. The key ideas we need are:

• Wave fronts represent the crests of waves. They are spaced one wavelength apart.

• The wavelength in a medium with index of refraction $n$ is $\text{λ}={\text{λ}}_{\text{vac}}/n$, where ${\text{λ}}_{\text{vac}}$ is the vacuum wavelength.

• Wave fronts are perpendicular to the wave’s direction of travel.

• The wave fronts stay lined up as a wave crosses from one medium into another.

FIGURE 34.15 shows a wave crossing the boundary between two media, where we’re assuming $n_2>n_1$. Because the wavelengths differ on opposite sides of the boundary, the wave fronts can stay lined up only if the waves in the two media are traveling in different directions. In other words, the wave must refract at the boundary to keep the crests of the wave aligned.

FIGURE 34.15 Snell’s law is a consequence of the wave model of light.

To analyze Figure 34.15, consider the segment of boundary of length $l$ between the two wave fronts. This segment is the common hypotenuse of two right triangles. From the upper triangle, which has one side of length ${\text{λ}}_1$, we see

where ${\theta }_1$ is the angle of incidence. Similarly, the lower triangle, where ${\theta }_2$ is the angle of refraction, gives

Equating these two expressions for $l$, and using ${\text{λ}}_1={\text{λ}}_{\text{vac}}/n_1$ and ${\text{λ}}_2={\text{λ}}_{\text{vac}}/n_2$, we find

Equation 34.7 can be true only if

which is Snell’s law.