The Law of Refraction

reflecting material such that the reflected ray makes the same angle with the
normal as the incident ray. If the reflecting surface is a boundary between two
different transparent mediums, such as air and glass, some of the incident light is
also transmitted into the glass, as shown in figure 31.1. However, it is observed
Figure 31.1 Reflection and refraction of light.
experimentally that this transmitted ray of light is bent as it enters the second
medium. The bending of light as it passes from one medium into another is called
refraction. Refraction of light occurs because light travels at different speeds in
different mediums. Light traveling through a vacuum travels at the speed c = 3.00 ×
108 m/s. But when light enters a medium there is a complex interaction between the
electromagnetic wave (light) and the atomic configuration of the medium. This
interaction causes the electromagnetic wave to slow down in the medium. This
slowing down of the wave as it goes from a vacuum into the medium causes it to
bend. We will use Huygens’ principle to show how this is accomplished in section
31.2.
31-1
31.2 The Law of Refraction
Let us consider a wave front B1B2 of a plane parallel monochromatic wave
impinging on the boundary of two different mediums, as shown in figure 31.2. The
incident ray makes an angle of incidence i with the normal N. The incident light
Figure 31.2 The law of refraction by Huygens’ principle.
moves at a speed v1 in medium 1 and v2 in medium 2, and we assume that v1 is
greater than v2. The incident wave has just touched the boundary at B1. In a time
?t, B2, the upper portion of the initial wave front, travels a distance v1?t, and
impinges at the boundary of the interface at B’2.
In this same time interval ?t, the wave front at B1 enters the second medium.
By Huygens’ principle, a secondary wavelet can be drawn emanating from the point
B1. This wave moves a radial distance v2?t in the second medium in the time
interval ?t, and is shown as the circle of radius v2?t in the figure. The radial
distance v2?t is less than the distance v1?t because v2 is less than v1. By Huygens’
principle, the line drawn from B’2 that is tangent to the secondary wavelet is the
new wave front. The point of tangency is denoted by B’1 and the new wave front in
medium 2 is B’1B’2 .The radius from B1 to B’1, when extended, becomes the refracted
ray B1C. The other refracted rays are drawn parallel to B1C, as shown in figure 31.2.
The angle that the refracted ray makes with the normal is called the angle of
refraction r.
We can obtain the relation between the angles i and r from the geometry of
figure 31.2. Since line B2B’2 makes an angle i with the dashed normal, angle B2B’2B1
is equal to (900 ? i), and since the sum of the angles in triangle B1B2B’2 must equal
1800, it follows that angle B2B1B’2 is also equal to the angle i. Using similar
reasoning, angle B1B’2B’1 is equal to r. Hence, from the trigonometry of figure 31.2,
Chapter 31 The Law of Refraction
31-2
sin i = v1?t (31.1)
B1B’2
and
sin r = v2?t (31.2)
B1B’2
Let us divide equation 31.1 by equation 31.2 and obtain
sin i
sin r =
v1?t
B1B2 ?
v2?t
B1B2 ?
and
sini (31.3)
sin r = v1
v2 = constant = n21
Equation 31.3 is the law of refraction. It says that the ratio of the sine of the angle
of incidence to the sine of the angle of refraction is equal to the ratio of the speed of
light in medium 1 to the speed of light in medium 2. Because the speed of light in
medium 1 v1 is a constant and the speed of light in medium 2 v2 is a constant, then
their ratio v1/v2, must also be a constant. This constant is called the index of
refraction of medium 2 with respect to medium 1 and is denoted by n21.
If medium 1 is a vacuum, then v1 = c, and the index of refraction of the
medium with respect to a vacuum is
n = c (31.4)
v
Since the speed v in any medium is always less than c, the index of refraction, n =
c/v, is always greater than 1, except for in a vacuum where it is equal to 1. Indices of
refraction for various substances are given in table 31.1. Notice that the index of
refraction of air is so close to the value 1, the index of refraction of a vacuum, that in
many practical situations, air is used in place of a vacuum.
The law of refraction can be put in a more convenient form by using equation
31.4. We can write the index of refraction of medium 1 with respect to a vacuum as
n1 = c (31.5)
v1
whereas we can write the index of refraction of medium 2 with respect to a vacuum
as
n2 = c (31.6)
v2
Chapter 31 The Law of Refraction
31-3
1.00029
1.5
2.42
1.52
1.57-1.72
1.46
1.47
1.31
1.49
1.54
1.33
Air
Benzene
Diamond
Glass, crown
Glass, flint
Glass, fused quartz
Glycerine
Ice
Plexiglass
Quartz crystal
Water
Substance n
Table 31.1
Index of Refraction for Various Materials (? = 589.2 nm, the D line of sodium)
Solving for the speeds v1 and v2 from equations 31.5 and 31.6, respectively, and
substituting them into equation 31.3, gives
n21 = v1 = c/n1 = n2 (31.7)
v2 c/n2 n1
Using equation 31.7, we can write the law of refraction, equation 31.3, as
sin i = n21 = n2
sin r n1
or
n1 sin i = n2 sin r (31.8)
Equation 31.8 is the form of the law of refraction that we will use in what follows. It
is also called Snell’s law after its discoverer, Willebrord Snell (1591-1626) a Dutch
mathematician who discovered it in 1620, the same year the Pilgrims landed at
Plymouth Rock. Note that if a ray lies along the normal, then the angle of incidence
i is equal to zero, and hence the angle of refraction r must also be zero, and there is
no refraction of this ray.
The fact that the speed of light varies from medium to medium has an
important effect on the wavelength of light. When an initial wave enters a second
medium, its wavelength changes. We can see this from equation 31.3, w