16.5.3 The Index of Refraction

Light waves travel with speed c in a vacuum, but they slow down as they pass through transparent materials such as water or glass or even, to a very slight extent, air. The slowdown is a consequence of interactions between the electromagnetic field of the wave and the electrons in the material. The speed of light in a material is characterized by the material’s index of refraction n, defined as

n = \frac{speed of light in a vacuum}{s p e e d o f l i g h t i n t h e m a t e r i a l} = \frac{c}{v}

(16.37)

The index of refraction of a material is always greater than 1 because $v < c$ . A vacuum has $n = 1$ exactly. TABLE 16.2 shows the index of refraction for several materials. You can see that liquids and solids have larger indices of refraction than gases.

TABLE 16.2 Typical indices of refraction

Material	Index of refraction
Vacuum	1 exactly
Air	1.0003
Water	1.33
Glass	1.50
Diamond	2.42

NOTE An accurate value for the index of refraction of air is relevant only in very precise measurements. We will assume $n_{air} = 1.00$ in this text.

If the speed of a light wave changes as it enters into a transparent material, such as glass, what happens to the light’s frequency and wavelength? Because $v = λ f$ , either $λ$ or f or both have to change when v changes.

As an analogy, think of a sound wave in the air as it impinges on the surface of a pool of water. As the air oscillates back and forth, it periodically pushes on the surface of the water. These pushes generate the compressions of the sound wave that continues on into the water. Because each push of the air causes one compression of the water, the frequency of the sound wave in the water must be exactly the same as the frequency of the sound wave in the air. In other words, the frequency of a wave is the frequency of the source. It does not change as the wave moves from one medium to another.

The same is true for electromagnetic waves; the frequency does not change as the wave moves from one material to another.

FIGURE 16.21 shows a light wave passing through a transparent material with index of refraction n. As the wave travels through vacuum it has wavelength $λ_{vac}$ and frequency $f_{vac}$ such that $λ_{vac} f_{vac} = c$ . In the material, $λ_{mat} f_{mat} = v = c / n$ . The frequency does not change as the wave enters $(f_{mat} = f_{vac})$ , so the wavelength must. The wavelength in the material is

λ_{mat} = \frac{v}{f_{mat}} = \frac{c}{n f_{mat}} = \frac{c}{n f_{vac}} = \frac{λ_{vac}}{n}

(16.38)

The figure illustrates light passing through a transparent material with an index of refraction n.

FIGURE 16.21 Full Alternative Text

The wavelength in the transparent material is less than the wavelength in vacuum. This makes sense. Suppose a marching band is marching at one step per second at a speed of $1 m / s$ . Suddenly they slow their speed to $\frac{1}{2} m / s$ but maintain their march at one step per second. The only way to go slower while marching at the same pace is to take smaller steps. When a light wave enters a material, the only way it can go slower while oscillating at the same frequency is to have a smaller wavelength.

EXAMPLE 16.6 Light traveling through glass

Orange light with a wavelength of 600 nm is incident upon a 1.00-mm-thick glass microscope slide.

What is the light speed in the glass?
How many wavelengths of the light are inside the slide?

SOLVE

From Table 16.2 we see that the index of refraction of glass is $n_{glass} = 1.50$ . Thus the speed of light in glass is

$v_{glass} = \frac{c}{n_{glass}} = \frac{3.00 \times 10^{8} m/s}{1.50} = 2.00 \times 10^{8} m / s$
The wavelength inside the glass is

$λ_{glass} = \frac{λ_{vac}}{n_{glass}} = \frac{600 n m}{1.50} = 400 n m = 4.00 \times 10^{- 7} m$

N wavelengths span a distance $d = N λ$ , so the number of wavelengths in $d = 1.00 mm$ is

$N = \frac{d}{λ} = \frac{1.00 \times 10^{- 3} m}{4.00 \times 10^{- 7} m} = 2500$

REVIEW The fact that 2500 wavelengths fit within 1 mm shows how small the wavelengths of light are.