Content check — College Physics (AP), Chapter 25: Geometric Optics

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 25: Geometric Optics

41 problems · 4 figures

Suppose a man stands in front of a mirror as shown in Figure 25.47. His eyes are 1.65 m above the floor, and the top of his head is 0.13 m higher. Find the height above the floor of the top and bottom of the smallest mirror in which he can see both the top of his head and his feet. How is this distance related to the man’s height?

A man standing in front of a mirror on a wall at a distance of several feet. The mirror’s top is at eye level, but its bottom is only waist high. Arrows illustrate how the man can see his reflection from head to toe in the mirror. — Figure 25.47 in the book ↗

Answer

Top

1 . 715 m

from floor, bottom

0 . 825 m

from floor. Height of mirror is

0 . 890 m

, or precisely one-half the height of the person.

25-5 §25.3 source ↗

What is the speed of light in water? In glycerine?

Answer

$2 . 25 \times 10^{8} m/s$ in water

$2 . 04 \times 10^{8} m/s$ in glycerine

25-7 §25.3 source ↗

Calculate the index of refraction for a medium in which the speed of light is

2.012 \times 10^{8} m/s

, and identify the most likely substance based on Table 25.1.

Table 25.1 Index of Refraction in Various Media in the book ↗

Medium	n
*Gases at $0ºC$ , 1 atm*
Air	1.000293
Carbon dioxide	1.00045
Hydrogen	1.000139
Oxygen	1.000271
Liquids at $20ºC$
Benzene	1.501
Carbon disulfide	1.628
Carbon tetrachloride	1.461
Ethanol	1.361
Glycerine	1.473
Water, fresh	1.333
Solids at $0ºC$
Diamond	2.419
Fluorite	1.434
Glass, crown	1.52
Glass, flint	1.66
Ice at $0ºC$	1.309
Magnesium Fluoride	1.38
Polystyrene	1.49
Plexiglas	1.51
Quartz, crystalline	1.544
Quartz, fused	1.458
Sodium chloride	1.544
Zircon	1.923

Answer

1.490

, polystyrene

25-9 §25.3 source ↗

There was a major collision of an asteroid with the Moon in medieval times. It was described by monks at Canterbury Cathedral in England as a red glow on and around the Moon. How long after the asteroid hit the Moon, which is

3.84 \times 10^{5} km

away, would the light first arrive on Earth?

Answer

1 . 28 s

25-11 §25.3 source ↗

Components of some computers communicate with each other through optical fibers having an index of refraction

n = 1 . 55

. What time in nanoseconds is required for a signal to travel 0.200 m through such a fiber?

Answer

1.03 ns

25-13 §25.3 source ↗

Suppose you have an unknown clear substance immersed in water, and you wish to identify it by finding its index of refraction. You arrange to have a beam of light enter it at an angle of

45 . 0º

, and you observe the angle of refraction to be

40 . 3º

. What is the index of refraction of the substance and its likely identity?

Answer

n = 1 . 46

, fused quartz

25-17 §25.3 source ↗

Unreasonable Results

Suppose light travels from water to another substance, with an angle of incidence of $10 . 0º$ and an angle of refraction of $14 . 9º$ . (a) What is the index of refraction of the other substance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Answer

(a) 0.898

(b) Can’t have $n < 1 . 00$ since this would imply a speed greater than $c$ .

25-19 §25.3 source ↗

Unreasonable Results

Light traveling from water to a gemstone strikes the surface at an angle of $80 . 0º$ and has an angle of refraction of $15 . 2º$ . (a) What is the speed of light in the gemstone? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Answer

(a) $\frac{c}{5.00}$

(b) Speed of light too slow, since index is much greater than that of diamond.

25-22 §25.4 source ↗

An optical fiber uses flint glass clad with crown glass. What is the critical angle?

Answer

66 . 3º

25-24 §25.4 source ↗

Suppose you are using total internal reflection to make an efficient corner reflector. If there is air outside and the incident angle is

45 . 0º

, what must be the minimum index of refraction of the material from which the reflector is made?

Answer

> 1.414

25-26 §25.4 source ↗

A ray of light, emitted beneath the surface of an unknown liquid with air above it, undergoes total internal reflection as shown in Figure 25.52. What is the index of refraction for the liquid and its likely identification?

A light ray travels from an object placed in a denser medium n1 at 15.0 centimeter from the boundary and on hitting the boundary gets totally internally reflected with theta c as critical angle. The horizontal distance between the object and the point of incidence is 13.4 centimeters. — Figure 25.52 in the book ↗

Answer

1.50, benzene

25-29 §25.5 source ↗

A beam of white light goes from air into water at an incident angle of

75 . 0º

. At what angles are the red (660 nm) and violet (410 nm) parts of the light refracted?

Answer

46 . 5º, red; 46 . 0º, violet

25-31 §25.5 source ↗

(a) A narrow beam of light containing yellow (580 nm) and green (550 nm) wavelengths goes from polystyrene to air, striking the surface at a

30 . 0º

incident angle. What is the angle between the colors when they emerge? (b) How far would they have to travel to be separated by 1.00 mm?

Answer

(a) $0 . 043º$

(b) $1 . 33 m$

25-33 §25.5 source ↗

A ray of 610 nm light goes from air into fused quartz at an incident angle of

55 . 0º

. At what incident angle must 470 nm light enter flint glass to have the same angle of refraction?

Answer

71.3º

25-35 §25.5 source ↗

A narrow beam of white light enters a prism made of crown glass at a

45 . 0º

incident angle, as shown in Figure 25.54. At what angles,

θ_{R}

and

θ_{V}

, do the red (660 nm) and violet (410 nm) components of the light emerge from the prism?

A blue incident light ray at an angle of incidence equal to 45 degrees falls on an equilateral triangular prism with angles each equal to 60 degrees. On falling onto the first surface, the ray refracts and splits into red and violet rays. These rays falling onto the second surface and emerge from the prism. Red with 660 nanometers and violet with 410 nanometers. — Figure 25.54 in the book ↗

Answer

53.5º, red;

55.2º, violet

25-37 §25.6 source ↗

Your camera’s zoom lens has an adjustable focal length ranging from 80.0 to 200 mm. What is its range of powers?

Answer

5.00 to 12.5 D

25-39 §25.6 source ↗

You note that your prescription for new eyeglasses is –4.50 D. What will their focal length be?

Answer

- 0.222 m

25-41 §25.6 source ↗

A certain slide projector has a 100 mm focal length lens. (a) How far away is the screen, if a slide is placed 103 mm from the lens and produces a sharp image? (b) If the slide is 24.0 by 36.0 mm, what are the dimensions of the image? Explicitly show how you follow the steps in the Problem-Solving Strategy for lenses.

Answer

(a) 3.43 m

(b) 0.800 by 1.20 m

25-42 §25.6 source ↗

A doctor examines a mole with a 15.0 cm focal length magnifying glass held 13.5 cm from the mole (a) Where is the image? (b) What is its magnification? (c) How big is the image of a 5.00 mm diameter mole?

Answer

(a) $- 1 . 35 m$ (on the object side of the lens).

(b) $+ 10 . 0$

25-43 §25.6 source ↗

How far from a piece of paper must you hold your father’s 2.25 D reading glasses to try to burn a hole in the paper with sunlight?

Answer

44.4 cm

25-45 §25.6 source ↗

A camera lens used for taking close-up photographs has a focal length of 22.0 mm. The farthest it can be placed from the film is 33.0 mm. (a) What is the closest object that can be photographed? (b) What is the magnification of this closest object?

Answer

(a) 6.60 cm

(b) –0.500

25-47 §25.6 source ↗

(a) What is the focal length of a magnifying glass that produces a magnification of 3.00 when held 5.00 cm from an object, such as a rare coin? (b) Calculate the power of the magnifier in diopters. (c) Discuss how this power compares to those for store-bought reading glasses (typically 1.0 to 4.0 D). Is the magnifier’s power greater, and should it be?

Answer

(a) $+ 7 . 50 cm$

(b) $13 . 3 D$

25-49 §25.6 source ↗

In Example 25.7, the magnification of a book held 7.50 cm from a 10.0 cm focal length lens was found to be 4.00. (a) Find the magnification for the book when it is held 8.50 cm from the magnifier. (b) Do the same for when it is held 9.50 cm from the magnifier. (c) Comment on the trend in m as the object distance increases as in these two calculations.

Example 25.7 Image Produced by a Magnifying Glass in the book ↗

Answer

(a) +6.67

(b) +20.0

(c) The magnification increases without limit (to infinity) as the object distance increases to the limit of the focal distance.

25-51 §25.6 source ↗

A camera with a 100 mm focal length lens is used to photograph the sun and moon. What is the height of the image of the sun on the film, given the sun is

1.40 \times 10^{6} km

in diameter and is

1.50 \times 10^{8} km

away?

Answer

- 0 . 933 mm

25-53 §25.7 source ↗

What is the focal length of a makeup mirror that has a power of 1.50 D?

Answer

+0.667 m

25-55 §25.7 source ↗

(a) Calculate the focal length of the mirror formed by the shiny back of a spoon that has a 3.00 cm radius of curvature. (b) What is its power in diopters?

Answer

(a) $–1.5 \times 10^{–2} m$

(b) $–66.7 D$

25-57 §25.7 source ↗

What is the focal length of a makeup mirror that produces a magnification of 1.50 when a person’s face is 12.0 cm away? Explicitly show how you follow the steps in the Problem-Solving Strategy for Mirrors.

Answer

+0.360 m (concave)

25-59 §25.7 source ↗

An object 1.50 cm high is held 3.00 cm from a person’s cornea, and its reflected image is measured to be 0.167 cm high. (a) What is the magnification? (b) Where is the image? (c) Find the radius of curvature of the convex mirror formed by the cornea. (Note that this technique is used by optometrists to measure the curvature of the cornea for contact lens fitting. The instrument used is called a keratometer, or curve measurer.)

Answer

(a) +0.111

(b) -0.334 cm (behind “mirror”)

25-61 §25.7 source ↗

Show that for a flat mirror

h_{i} = h_{o}

, knowing that the image is a distance behind the mirror equal in magnitude to the distance of the object from the mirror.

Answer

m = \frac{h_{i}}{h_{o}} = - \frac{d_{i}}{d_{o}} = - \frac{- d_{o}}{d_{o}} = \frac{d_{o}}{d_{o}} = 1 \Rightarrow h_{i} = h_{o}

25-63 §25.7 source ↗

Referring to the electric room heater considered in the first example in this section, calculate the intensity of IR radiation in

{W/m}^{2}

projected by the concave mirror on a person 3.00 m away. Assume that the heating element radiates 1500 W and has an area of

100 {cm}^{2}

, and that half of the radiated power is reflected and focused by the mirror.

Answer

6.82 k {W/m}^{2}

25-65 §25.7 source ↗

Critical Thinking Light composed of multiple frequencies passes through crown glass. (a) How much faster does the violet component of the light with a wavelength of 410 nm travel in the crown glass than the component of orange light with a wavelength of 610 nm? (b) In a mechanical situation with friction, the longer one surface moves relative to the other, the more the friction acts, slowing the relative motion during the process. Is this the same for light in a medium? (c) In a mechanical situation, it is possible for friction to stop the motion if it occurs over a long enough path. Is this the same for light in a medium? (d) In a mechanical friction situation, once the object slows it will not increase in speed once the friction no longer acts unless another force is present. If a light ray exits a medium into a vacuum, will it go back to a speed of c even though that would be an increase in speed not caused by an additional force?

Answer

(a) $\begin{array}{l} v_{610} = \frac{3.00 \times 10^{8}}{1.530} m/s, v_{410} = \frac{3.00 \times 10^{8}}{1.514} m/s \\ 3 .00 \times 10^{8} \frac{1}{1.514} - \frac{1}{1.530} m/s = 2.07 \times 10^{6} m/s \end{array}$

(b) No.

(d) Yes.

AP Test Prep 25.1 §25.1 source ↗

When light from a distant object reflects off of a concave mirror and comes to a focus some distance in front of the mirror, we model light as a _____ to explain and predict the behavior of light and the formation of an image.

a.
wave
b.
particle
c.
ray
d.
all of the above

Answer

C. ray

AP Test Prep 25.3 §25.2 source ↗

An object is 2 meters in front of a flat mirror. Ray 1 from the object travels in a direction toward the mirror and normal to the mirror’s surface. Ray 2 from the object travels at an angle of 5° from the direction of ray 1, and it also reflects off the mirror’s surface. At what distance behind the mirror do these two reflected rays appear to converge to form an image?

a.
0.2 m
b.
0.5 m
c.
2 m
d.
4 m

Answer

C. 2 m

AP Test Prep 25.5 §25.3 source ↗

When light travels from air into water, which of the following statements is accurate?

a.
The wavelength decreases, and the speed decreases.
b.
The wavelength decreases, and the speed increases.
c.
The wavelength increases, and the speed decreases.
d.
The wavelength increases, and the speed increases.

Answer

A. The wavelength decreases, and the speed decreases.

AP Test Prep 25.7 §25.3 source ↗

Two different potential paths from point A to point B are shown. Point A is in the air, and point B is in water. For which of these paths (upper or lower) would light travel from point A to point B faster? Which of the paths more accurately represents how a light ray would travel from point A to point B? Explain.

Diagram shows two arrows running from point A to the top of a shaded box. Where they meet the shaded box, two new arrows emerge inside the box. These arrows meet at a point B inside the box. — Figure 25.55 in the book ↗

Answer

Since light bends toward the normal upon entering a medium with a higher index of refraction, the upper path is a more accurate representation of a light ray moving from A to B.

AP Test Prep 25.9 §25.3 source ↗

Students in a lab group are given a plastic cube with a hollow cube-shaped space in the middle that fills about half the volume of the cube. The index of refraction of the plastic is known. The hollow space is filled with a gas, and the students are asked to collect the data needed to find the index of refraction of the gas. What information would you need to collect, and how would you use this information in order to deduce the index of refraction of the gas in the cube?

Answer

First, measure the angle of incidence and the angle of refraction for light entering the plastic from air. Since the two angles can be measured and the index of refraction of air is known, the student can solve for the index of refraction of the plastic.

Next, measure the angle of incidence and the angle of refraction for light entering the gas from the plastic. Since the two angles can be measured and the index of refraction of the plastic is known, the student can solve for the index of refraction of the gas.

AP Test Prep 25.11 §25.3 source ↗

A laser is fired from a submarine beneath the surface of a lake (n = 1.33). The laser emerges from the lake into air with an angle of refraction of 67°. How fast is the light moving through the water? What is the angle of incidence of the laser light when it crosses the boundary between the lake and the air?

Answer

The speed of light in a medium is simply c/n, so the speed of light in water is 2.25 × 10⁸ m/s. From Snell’s law, the angle of incidence is 44°.

AP Test Prep 25.13 §25.6 source ↗

An object is 25 cm in front of a converging lens with a focal length of 25 cm. Where will the resulting image be located?

a.
25 cm in front of the lens
b.
25 cm behind the lens
c.
50 cm behind the lens
d.
at infinity (either in front of or behind the lens)

Answer

D. at infinity (either in front of or behind the lens)

AP Test Prep 25.15 §25.6 source ↗

A student wishes to predict the magnification of an image given the distance from the object to a converging lens with an unknown index of refraction. What data must the student collect in order to make such a prediction for any object distance?

a.
A specific object distance and the image distance associated with that object distance.
b.
A specific image distance and a determination of whether the image formed is upright or inverted.
c.
The diameter and index of refraction of the lens.
d.
The radius of curvature of each side of the lens.

Answer

A. A specific object distance and the image distance associated with that object distance.

AP Test Prep 25.17 §25.7 source ↗

A student is testing the properties of a mirror with an unknown radius of curvature. The student notices that no matter how far an object is placed from the mirror, the image seen in the mirror is always upright and smaller than the object. What can the student deduce about this mirror?

a.
The mirror is convex.
b.
The mirror is flat.
c.
The mirror is concave.
d.
More information is required to deduce the shape of the mirror.

Answer

A. The mirror is convex.

AP Test Prep 25.19 §25.7 source ↗

A mirror shows an upright image twice as large as the object when the object is 10 cm away from the mirror. What is the focal length of the mirror?

a.
-10 cm
b.
10 cm
c.
20 cm
d.
40 cm

Answer

B. 10 cm

Loading chapter 26…

Problems and answers are adapted from College Physics for AP® Courses 2e by Gregg Wolfe, Erika Gasper, John Stoke, Julie Kretchman, David Anderson and others, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/college-physics-ap-courses-2e/pages/1-connection-for-ap-r-courses. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.