Law of Refraction
Figure 8.11 shows how a ray of light changes direction when it passes from one medium to another. As before, the angles are measured relative to a perpendicular to the surface at the point where the light ray crosses it. Some of the incident light will be reflected from the surface, but for now we will concentrate on the light that is transmitted. The change in direction of the light ray depends on how the speed of light changes. The change in the speed of light is related to the indices of refraction of the media involved. In the situations shown in Figure 8.11, medium 2 has a greater index of refraction than medium 1. This means that the speed of light is less in medium 2 than in medium 1. Note that as shown in Figure 8.11(a), the direction of the ray moves closer to the perpendicular when it slows down. Conversely, as shown in Figure 8.11(b), the direction of the ray moves away from the perpendicular when it speeds up. The path is exactly reversible. In both cases, you can imagine what happens by thinking about pushing a lawn mower from a footpath onto grass, and vice versa. Going from the footpath to grass, the front wheels are slowed and pulled to the side as shown. This is the same change in direction as for light when it goes from a fast medium to a slow one. When going from the grass to the footpath, the front wheels can move faster and the mower changes direction as shown. This, too, is the same change in direction as for light going from slow to fast.
The amount that a light ray changes its direction depends both on the incident angle and the amount that the speed changes. For a ray at a given incident angle, a large change in speed causes a large change in direction, and thus a large change in angle. The exact mathematical relationship is the law of refraction, or Snell’s Law, which is stated in equation form as
8.7 Here and are the indices of refraction for medium 1 and 2, and and are the angles between the rays and the perpendicular in medium 1 and 2, as shown in Figure 8.11. The incoming ray is called the incident ray and the outgoing ray the refracted ray, and the associated angles the incident angle and the refracted angle. The law of refraction is also called Snell’s law after the Dutch mathematician Willebrord Snell (1591–1626), who discovered it in 1621. Snell’s experiments showed that the law of refraction was obeyed and that a characteristic index of refraction could be assigned to a given medium. Snell was not aware that the speed of light varied in different media, but through experiments he was able to determine indices of refraction from the way light rays changed direction.
The Law of Refraction
8.8
Take-Home Experiment: A Broken Pencil
A classic observation of refraction occurs when a pencil is placed in a glass half filled with water. Do this and observe the shape of the pencil when you look at the pencil sideways, through air, glass, and water. Explain your observations. Draw ray diagrams for the situation.
Example 8.2 Determine the Index of Refraction from Refraction Data
Find the index of refraction for medium 2 in Figure 8.11(a), assuming medium 1 is air and given the incident angle is and the angle of refraction is
Strategy
The index of refraction for air is taken to be 1 in most cases (and up to four significant figures, it is 1.000). Thus here. From the given information, and With this information, the only unknown in Snell’s law is so that it can be used to find this unknown.
Solution
Snell’s law is
8.9 Rearranging to isolate gives
8.10 Entering known values,
8.11 Discussion
This is the index of refraction for water, and Snell could have determined it by measuring the angles and performing this calculation. He would then have found 1.33 to be the appropriate index of refraction for water in all other situations, such as when a ray passes from water to glass. Today we can verify that the index of refraction is related to the speed of light in a medium by measuring that speed directly.
Example 8.3 A Larger Change in Direction
Suppose that in a situation like that in Example 8.2, light goes from air to diamond and that the incident angle is Calculate the angle of refraction in the diamond.
Strategy
Again the index of refraction for air is taken to be and we are given We can look up the index of refraction for diamond in Table 8.1, finding The only unknown in Snell’s law is which we wish to determine.
Solution
Solving Snell’s law for sin yields
8.12 Entering known values,
8.13 The angle is thus
8.14 Discussion
For the same angle of incidence, the angle of refraction in diamond is significantly smaller than in water rather than —see the preceding example). This means there is a larger change in direction in diamond. The cause of a large change in direction is a large change in the index of refraction, or speed. In general, the larger the change in speed, the greater the effect on the direction of the ray.