reactance X Ω ) The proportionality constant between
33.7 When the ray reflected from the bottom surface
What range of refraction angles is possible? To answer this question, let’s first consider the case where the ray
ConCepts 33.3 refraCtiOn and dispersiOn 901
Because the speed of light in any given medium depends slightly on the frequency of the light, the angle of refrac- tion also depends on frequency. This phenomenon is called dispersion because it causes rays of different colors to separate—to be dispersed—when refracted. Prisms like the one shown in Figure 33.22 are designed to separate colors by the frequency dependence of the angle of refraction. In most media, high-frequency light travels more slowly than low-frequency light, and so high-frequency light bends more strongly toward the normal. The lowest frequency of visible light is red and the highest is violet, which means violet light bends the most, as the rainbow of Figure 33.22c shows.
Both rainbows and the brilliance of gems result from a combination of total internal reflection and dispersion. In a rainbow (Figure 33.22c), the combination of total internal reflection and dispersion means that we see different col- ors coming from water droplets at different viewing angles.
Gems are cut with many internal surfaces from which total internal reflection takes place. Because the light is also dis- persed, colorless gems such as diamonds shine with many distinct colors.
33.8 Because of dispersion, the critical angle for total in- ternal reflection in a given medium varies with frequency. Is the critical angle for a violet ray greater or less than that for a red ray?
reflects light just as a mirror would. A light ray enters the prism’s front surface at normal incidence. Because the back surface is slanted relative to the front surface, the angle at which the ray hits the back surface is less than 90°. This back-surface angle of incidence is greater than the criti- cal angle for the glass, however, and so the light is totally reflected from the back surface. Such prisms are actually better mirrors than most regular mirrors; they reflect very close to 100% of the incident light, whereas mirrors are less reflective due to imperfections in the reflecting surface.
Optical fibers also guide light by means of total inter- nal reflection. An optical fiber is a long, thin fiber made of a transparent material such as glass. If light shines into one end of the fiber at an angle greater than the critical angle, the light travels along the fiber through repeated total internal reflections, and essentially all of the light that entered the fiber emerges at the other end (Figure 33.21). Because very little light is lost as the light travels, only a faint glow comes from the rest of the fiber.
Figure 33.20 A prism can act as a perfect mirror by means of total inter- nal reflection.
Light that is incident
normal to a prism face c cundergoes total
internal reflection.
uc uc
Figure 33.21 How optical fibers work.
Optical fiber guides light by means of total internal reflection.
Figure 33.22 The phenomenon of dispersion, which results from the fact that the speed of light in a given medium (and hence the angle of refrac- tion) depends slightly on the frequency of the light.
(a) Prism refracts light of single frequency
(c) Rainbows result from dispersion of sunlight by raindrops
(b) Dispersion: different colors have different angles of refraction
red violet white
ConCepts
line perpendicular to the lens through its center—converge through such a lens onto a single point called either the focus or the focal point. A lens with convex surfaces is therefore called a converging lens. The distance from the center of the lens to the focus is called the focal length f.
33.4 Forming images
As shown in Figure 33.24, by combining two prisms and a glass slab we can create a device that steers parallel light rays toward each other. The rays through the center of the device pass straight through, those through the top prism are refracted downward, and those through the bottom prism are refracted upward.
To bring all parallel incident rays to a single point, a structure called a lens is used. A lens is designed with curved surfaces so that the refraction of incident rays in- creases gradually as we move away from the center. To ac- complish this, lenses are typically made with spherical sur- faces, which are easy to manufacture.
Figure 33.25a shows a lens with convex spherical sur- faces, where a convex surface is defined as one that curves like the outside of a sphere. Rays parallel to the lens axis—a
increases the length of the path. Thus, Fermat’s principle implies the law of reflection.
When the ray must travel through some air and some glass, as in Figure 33.22c and d, the quickest path is not a straight line because the ray’s speed in the glass is only two-thirds of its speed in air. To minimize the time interval needed to travel from A to B, the ray bends on entering and exiting the glass. Such a bent path reduces the distance traveled through the glass without increas- ing the distance traveled in air so much that it offsets the amount of time saved. In Example 33.7, we shall see that calculating the bending angles with Fermat’s principle gives the same result as with ray optics.
Figure 33.23 shows four ways in which a light ray can travel between two locations A and B: directly, reflected from a mirror, refracted through a glass slab, and re- fracted through a prism.* You could say that in each case the ray reaches B because it is aimed properly from A.
However, an entirely different way of looking at the path followed by the light was suggested by the French mathe- matician Pierre de Fermat (1601–1665) in a formulation today known as Fermat’s principle:
The path taken by a light ray between two loca- tions is the path for which the time interval needed to travel between those locations is a minimum.
This principle may seem to imply that light always travels in a straight line. However, the quickest path between two locations is not necessarily the shortest distance when the speed of light differs in different regions.
Let’s consider the four paths in Figure 33.23 using Fermat’s principle. In Figure 33.23a, the ray does follow a straight path because the medium in which the ray trav- els is uniform. As a result, the quickest path is indeed the shortest distance: a straight line from A to B.
In Figure 33.23b, the fact that the straight-line path from A to B is blocked means that the ray must reflect somewhere off the mirror in order to travel from A to B.
The path shown, which satisfies the law of reflection, is the shortest distance from A to B involving reflection from the mirror. Because the distance from A to the reflection location P equals the distance between the image location I and P, the straight line IB is equal in length to the path traveled by the ray from A to B. Mov- ing the reflection location to either side of P, so that the angle of incidence does not equal the angle of reflection, Fermat’s principle
*Note that what distinguishes a glass slab from a glass prism is the way I use the terms: In a slab, the two opposite surfaces are parallel to each other;
in a prism, they are not.
Figure 33.23 Ray diagrams illustrating the quickest path for a light ray traveling from A to B for four situations.
(a)
(b)
(c)
(d)
A B
A B
A A
P I
B
B
Figure 33.24 A device that redirects parallel light rays toward each other.
Prism redirects ray. Two prisms and a slab cause parallel rays to converge.
ConCepts 33.4 fOrming images 903
the image.) An image of the entire object is made up of the images of all the individual points on the object.
To determine where the rays emanating from a point on an object converge, we don’t need to draw all the rays.
Instead, we draw three special ones, called principal rays, and see where they converge: