Problems & Exercises

Show that when light reflects from two mirrors that meet each other at a right angle, the outgoing ray is parallel to the incoming ray, as illustrated in the following figure.

Figure 25.48 A corner reflector sends the reflected ray back in a direction parallel to the incident ray, independent of incoming direction.

Light shows staged with lasers use moving mirrors to swing beams and create colorful effects. Show that a light ray reflected from a mirror changes direction by $2\theta$ when the mirror is rotated by an angle $\theta$.

A flat mirror is neither converging nor diverging. To prove this, consider two rays originating from the same point and diverging at an angle $\theta$. Show that after striking a plane mirror, the angle between their directions remains $\theta$.

Figure 25.49 A flat mirror neither converges nor diverges light rays. Two rays continue to diverge at the same angle after reflection.

What is the speed of light in air? In crown glass?

In what substance in Table 25.1 is the speed of light $2\text{.}\text{290}\times {\text{10}}^8\text{m/s}$?

A scuba diver training in a pool looks at his instructor as shown in Figure 25.50. What angle does the ray from the instructor’s face make with the perpendicular to the water at the point where the ray enters? The angle between the ray in the water and the perpendicular to the water is $\text{25}\text{.}\mathrm{0º}$.

Figure 25.50 A scuba diver in a pool and his trainer look at each other.

(a) Given that the angle between the ray in the water and the perpendicular to the water is $\text{25}\text{.}\mathrm{0º}$, and using information in Figure 25.50, find the height of the instructor’s head above the water, noting that you will first have to calculate the angle of refraction. (b) Find the apparent depth of the diver’s head below water as seen by the instructor. Assume the diver and the diver's image are the same horizontal distance from the normal.

On the Moon’s surface, lunar astronauts placed a corner reflector, off which a laser beam is periodically reflected. The distance to the Moon is calculated from the round-trip time. What percent correction is needed to account for the delay in time due to the slowing of light in Earth’s atmosphere? Assume the distance to the Moon is precisely $3\text{.}\text{84}\times {\text{10}}^8\text{m}$, and Earth’s atmosphere (which varies in density with altitude) is equivalent to a layer 30.0 km thick with a constant index of refraction $n=1\text{.}\text{000293}$.

Suppose Figure 25.51 represents a ray of light going from air through crown glass into water, such as going into a fish tank. Calculate the amount the ray is displaced by the glass ($\mathrm{\Delta }x$), given that the incident angle is $\text{40}\text{.}\mathrm{0º}$ and the glass is 1.00 cm thick.

Figure 25.51 shows a ray of light passing from one medium into a second and then a third. Show that ${\theta }_3$ is the same as it would be if the second medium were not present (provided total internal reflection does not occur).

Figure 25.51 A ray of light passes from one medium to a third by traveling through a second. The final direction is the same as if the second medium were not present, but the ray is displaced by $\mathrm{\Delta }x$ (shown exaggerated).

Construct Your Own Problem

Consider sunlight entering the Earth’s atmosphere at sunrise and sunset—that is, at a $\text{90º}$ incident angle. Taking the boundary between nearly empty space and the atmosphere to be sudden, calculate the angle of refraction for sunlight. This lengthens the time the Sun appears to be above the horizon, both at sunrise and sunset. Now construct a problem in which you determine the angle of refraction for different models of the atmosphere, such as various layers of varying density. Your instructor may wish to guide you on the level of complexity to consider and on how the index of refraction varies with air density.

Verify that the critical angle for light going from water to air is $\text{48.6º}$, as discussed at the end of Example 25.4, regarding the critical angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air.

(a) At the end of Example 25.4, it was stated that the critical angle for light going from diamond to air is $\text{24}\text{.}\mathrm{4º}$. Verify this. (b) What is the critical angle for light going from zircon to air?

At what minimum angle will you get total internal reflection of light traveling in water and reflected from ice?

You can determine the index of refraction of a substance by determining its critical angle. (a) What is the index of refraction of a substance that has a critical angle of $\text{68}\text{.}\mathrm{4º}$ when submerged in water? What is the substance, based on Table 25.1? (b) What would the critical angle be for this substance in air?

A light ray entering an optical fiber surrounded by air is first refracted and then reflected as shown in Figure 25.53. Show that if the fiber is made from crown glass, any incident ray will be totally internally reflected.

Figure 25.53 A light ray enters the end of a fiber, the surface of which is perpendicular to its sides. Examine the conditions under which it may be totally internally reflected.

(a) What is the ratio of the speed of red light to violet light in diamond, based on Table 25.2? (b) What is this ratio in polystyrene? (c) Which is more dispersive?

By how much do the critical angles for red (660 nm) and violet (410 nm) light differ in a diamond surrounded by air?

A parallel beam of light containing orange (610 nm) and violet (410 nm) wavelengths goes from fused quartz to water, striking the surface between them at a $\text{60}\text{.}\mathrm{0º}$ incident angle. What is the angle between the two colors in water?

A narrow beam of light containing red (660 nm) and blue (470 nm) wavelengths travels from air through a 1.00 cm thick flat piece of crown glass and back to air again. The beam strikes at a $\text{30}\text{.}\mathrm{0º}$ incident angle. (a) At what angles do the two colors emerge? (b) By what distance are the red and blue separated when they emerge?

What is the power in diopters of a camera lens that has a 50.0 mm focal length?

What is the focal length of 1.75 D reading glasses found on the rack in a pharmacy?

How far from the lens must the film in a camera be, if the lens has a 35.0 mm focal length and is being used to photograph a flower 75.0 cm away? Explicitly show how you follow the steps in the Problem-Solving Strategy for lenses.

A camera with a 50.0 mm focal length lens is being used to photograph a person standing 3.00 m away. (a) How far from the lens must the film be? (b) If the film is 36.0 mm high, what fraction of a 1.75 m tall person will fit on it? (c) Discuss how reasonable this seems, based on your experience in taking or posing for photographs.

Suppose your 50.0 mm focal length camera lens is 51.0 mm away from the film in the camera. (a) How far away is an object that is in focus? (b) What is the height of the object if its image is 2.00 cm high?

What magnification will be produced by a lens of power –4.00 D (such as might be used to correct myopia) if an object is held 25.0 cm away?

Suppose a 200 mm focal length telephoto lens is being used to photograph mountains 10.0 km away. (a) Where is the image? (b) What is the height of the image of a 1000 m high cliff on one of the mountains?

Combine thin lens equations to show that the magnification for a thin lens is determined by its focal length and the object distance and is given by $m=f/\left(f-d_{\text{o}}\right)$.

Some telephoto cameras use a mirror rather than a lens. What radius of curvature mirror is needed to replace a 800 mm focal length telephoto lens?

Find the magnification of the heater element in Example 25.9. Note that its large magnitude helps spread out the reflected energy.

A shopper standing 3.00 m from a convex security mirror sees his image with a magnification of 0.250. (a) Where is his image? (b) What is the focal length of the mirror? (c) What is its radius of curvature? Explicitly show how you follow the steps in the Problem-Solving Strategy for Mirrors.

Ray tracing for a flat mirror shows that the image is located a distance behind the mirror equal to the distance of the object from the mirror. This is stated $d_{\text{i}}={\mathrm{–d}}_{\text{o}}$, since this is a negative image distance (it is a virtual image). (a) What is the focal length of a flat mirror? (b) What is its power?

Use the law of reflection to prove that the focal length of a mirror is half its radius of curvature. That is, prove that $f=R/2$. Note this is true for a spherical mirror only if its diameter is small compared with its radius of curvature.

Consider a 250-W heat lamp fixed to the ceiling in a bathroom. If the filament in one light burns out then the remaining three still work. Construct a problem in which you determine the resistance of each filament in order to obtain a certain intensity projected on the bathroom floor. The ceiling is 3.0 m high. The problem will need to involve concave mirrors behind the filaments. Your instructor may wish to guide you on the level of complexity to consider in the electrical components.