Physics

John D. Cutnell, Kenneth W. Johnson

Chapter 26

The Refraction Of Light: Lenses And Optical Instruments - all with Video Answers

Educators

Chapter Questions

Problem 1

Light travels at a speed of $2.201 \times 10^{8} \mathrm{~m} / \mathrm{s}$ in a certain substance. What substance from Table $26.1$ could this be? For the speed of light in a vacuum use $2.998 \times 10^{8} \mathrm{~m} / \mathrm{s}$; show your calculations.

Adnan Gill

Adnan Gill

Numerade Educator

Problem 2

In an ultra-low-temperature experiment, a collection of sodium atoms enter a special state called a Bose-Einstein condensate in which the index of refraction is $1.57 \times 10^{7}$. What is the speed of light in this condensate?

Adnan Gill

Numerade Educator

Problem 3

The refractive indices of materials $A$ and $B$ have a ratio of $n_{A} / n_{B}=1.33 .$ The speed of light in material $A$ is $1.25 \times 10^{8} \mathrm{~m} / \mathrm{s}$. What is the speed of light in material $B ?$

Adnan Gill

Numerade Educator

Problem 4

The frequency of a light wave is the same when the light travels in ethyl alcohol or in carbon disulfide. Find the ratio of the wavelength of the light in ethyl alcohol to that in carbon disulfide.

Adnan Gill

Numerade Educator

Problem 5

A plate glass window $(n=1.5)$ has a thickness of $4.0 \times 10^{-3} \mathrm{~m} .$ How long does it take light to pass perpendicularly through the plate?

Adnan Gill

Numerade Educator

Problem 6

Light has a wavelength of $340.0 \mathrm{~nm}$ and a frequency of $5.403 \times 10^{14} \mathrm{~Hz}$ when traveling through a certain substance. What substance from Table $26.1$ could this be? Show your calculations.

Adnan Gill

Numerade Educator

Problem 7

Interactive Solution 26.7 at www.wiley.com/college/cutnell offers one model for problems like this one. In a certain time, light travels $6.20 \mathrm{~km}$ in a vacuum. During the same time, light travels only $3.40 \mathrm{~km}$ in a liquid. What is the refractive index of the liquid?

Suzanne W.

Numerade Educator

Problem 8

A flat sheet of ice has a thickness of $2.0 \mathrm{~cm}$. It is on top of a flat sheet of crystalline quartz that has a thickness of $1.1 \mathrm{~cm} .$ Light strikes the ice perpendicularly and travels through it and then through the quartz. In the time it takes the light to travel through the two sheets, how far (in centimeters) would it have traveled in a vacuum?

Adnan Gill

Numerade Educator

Problem 9

A light ray in air is incident on a water surface at a $43^{\circ}$ angle of incidence. Find the angles of
(a) reflection and
(b) refraction.

Suzanne W.

Numerade Educator

Problem 10

The drawing shows four different situations in which a light ray is traveling from one medium into another. In some of the cases, the refraction is not shown correctly. For cases $(a),(b)$, and $(c)$, the angle of incidence is $55^{\circ}$; for case $(d)$, the angle of incidence is $0^{\circ}$.

Suzanne W.

Numerade Educator

Problem 11

A ray of light impinges from air onto a block of ice $(n=1.309)$ at a $60.0^{\circ}$ angle of incidence. Assuming that this angle remains the same, find the difference $\theta_{2, \text { ice }}-\theta_{2, \text { water }}$ in the angles of refraction when the ice turns to water $(n=1.333)$.

Narendra Kumar

Numerade Educator

Problem 12

A spotlight on a boat is $2.5 \mathrm{~m}$ above the water, and the light strikes the water at a point that is $8.0 \mathrm{~m}$ horizontally displaced from the spotlight (see the drawing). The depth of the water is $4.0 \mathrm{~m}$. Determine the distance $d$, which locates the point where the light strikes the bottom.

Sheh Lit Chang

University of Washington

Problem 13

13. ssm The drawing shows a coin resting on the bottom of a beaker filled with an unknown liquid. A ray of light from the coin travels to the surface of the liquid and is refracted as it enters into the air. A person sees the ray as it skims just above the surface of the liquid. How fast is the light traveling in the liquid?

Suzanne W.

Numerade Educator

Problem 14

A person working on the transmission of a car accidentally drops a bolt into a tray of oil. The oil is $5.00 \mathrm{~cm}$ deep. The bolt appears to be $3.40 \mathrm{~cm}$ beneath the surface of the oil, when viewed from directly above. What is the index of refraction of the oil?

Adnan Gill

Numerade Educator

Problem 15

A scuba diver, submerged under water, looks up and sees sunlight at an angle of $28.0^{\circ}$ from the vertical. At what angle, measured from the vertical, does this sunlight strike the surface of the water?

Adnan Gill

Numerade Educator

Problem 16

The drawing shows a ray of light traveling through three materials whose surfaces are parallel to each other. The refracted rays (but not the reflected rays) are shown as the light passes through each material. A ray of light strikes the $a-b$ interface at a $50.0^{\circ}$ angle of incidence. The index of refraction of material $a$ is $n_{a}=1.20$. The angles of refraction in materials $b$ and $c$ are, respectively, $45.0^{\circ}$ and $56.7^{\circ}$. Find the indices of refraction in these two media.

Adnan Gill

Numerade Educator

Problem 17

Light in a vacuum is incident on a transparent glass slab. The angle of incidence is $35.0^{\circ}$. The slab is then immersed in a pool of liquid. When the angle of incidence for the light striking the slab is $20.3^{\circ}$, the angle of refraction for the light entering the slab is the same as when the slab was in a vacuum. What is the index of refraction of the liquid?

Adnan Gill

Numerade Educator

Problem 18

A Stone held just beneath the surface of a swimming pool is released and sinks to the bottom at a constant speed of $0.48 \mathrm{~m} / \mathrm{s}$. What is the apparent speed of the stone, as viewed from directly above by an observer who is in air?

Suzanne W.

Numerade Educator

Problem 19

Refer to Figure $26.4 b$ and assume the observer is nearly above the submerged object. For this situation, derive the expression for the apparent depth: $d^{\prime}=d\left(n_{2} / n_{1}\right)$, Equation 26.3. (Hint: Use Snell's law of refraction and the fact that the angles of incidence and refraction are small, so tan $\theta \approx \sin \theta .$ )

Sheh Lit Chang

University of Washington

Problem 20

The drawing shows a rectangular block of glass $(n=1.52)$ surrounded by liquid carbon disulfide $(n=1.63)$. A ray of light is incident on the glass at point A with a $30.0^{\circ}$ angle of incidence. At what angle of refraction does the ray leave the glass at point $\mathrm{B}$ ?

Sheh Lit Chang

University of Washington

Problem 21

In Figure 26.6, suppose that the angle of incidence is $\theta_{1}=30.0^{\circ}$, the thickness of the glass pane is $6.00 \mathrm{~mm}$, and the refractive index of the glass is $n_{2}=1.52 .$ Find the amount (in $\mathrm{mm}$ ) by which the emergent ray is displaced relative to the incident ray.

Adnan Gill

Numerade Educator

Problem 22

The back wall of a home aquarium is a mirror that is a distance of $40.0 \mathrm{~cm}$ away from the front wall. The walls of the tank are negligibly thin. A fish, swimming midway between the front and back walls, is being viewed by a person looking through the front wall. The index of refraction of air is $n_{\text {air }}=1.000$ and that of water is $n_{\text {water }}=1.333 .$ (a) Calculate the apparent distance between the fish and the front wall. (b) Calculate the apparent distance between the image of the fish and the front wall.

Suzanne W.

Numerade Educator

Problem 23

Review Conceptual Example 4 as background for this problem. A man in a boat is looking straight down at a fish in the water directly beneath him. The fish is looking straight up at the man. They are equidistant from the air-water interface. To the man, the fish appears to be $2.0 \mathrm{~m}$ beneath his eyes. To the fish, how far above its eyes does the man appear to be?

Suzanne W.

Numerade Educator

Problem 24

A small logo is embedded in a thick block of crown glass $(n=1.52), 3.20 \mathrm{~cm}$ beneath the top surface of the glass. The block is put under water, so there is $1.50 \mathrm{~cm}$ of water above the top surface of the block. The logo is viewed from directly above by an observer in air. How far beneath the top surface of the water does the logo appear to be?

Adnan Gill

Numerade Educator

Problem 25

A beaker has a height of $30.0 \mathrm{~cm}$. The lower half of the beaker is filled with water, and the upper half is filled with oil $(n=1.48)$. To a person looking down into the beaker from above, what is the apparent depth of the bottom?

Narayan Hari

Numerade Educator

Problem 26

A glass is half-full of water, with a layer of vegetable oil $(n=1.47)$ floating on top. A ray of light traveling downward through the oil is incident on the water at an angle of $71.4^{\circ}$. Determine the critical angle for the oil-water interface and decide whether the ray will penetrate into the water.

Suzanne W.

Numerade Educator

Problem 27

One method of determining the refractive index of a transparent solid is to measure the critical angle when the solid is in air. If $\theta_{c}$ is found to be $40.5^{\circ}$, what is the index of refraction of the solid?

Suzanne W.

Numerade Educator

Problem 28

A point source of light is submerged $2.2 \mathrm{~m}$ below the surface of a lake and emits rays in all directions. On the surface of the lake, directly above the source, the area illuminated is a circle. What is the maximum radius that this circle could have?

Adnan Gill

Numerade Educator

Problem 29

Interactive Solution $26.29$ at www.wiley.com/college/cutnell provides one model for solving problems such as this. A glass block $(n=1.56)$ is immersed in a liquid. A ray of light within the glass hits a glass-liquid surface at a $75.0^{\circ}$ angle of incidence. Some of the light enters the liquid. What is the smallest possible refractive index for the liquid?

Suzanne W.

Numerade Educator

Problem 30

The drawing shows three materials, $a, b$, and $c$. A ray of light strikes the $a-b$ interface at an angle that just barely exceeds its critical angle of $40.0^{\circ}$ The reflected ray then strikes the $a-c$ interface at an angle of incidence that just barely exceeds its critical angle (which is not $40.0^{\circ}$ ). The index of refraction of material $a$ is $n_{a}=1.80$. Find the indices of refraction for the two other materials.

Adnan Gill

Numerade Educator

Problem 31

The drawing shows a crown glass slab with a rectangular cross section. As illustrated, a laser beam strikes the upper surface at an angle of $60.0^{\circ}$. After reflecting from the upper surface, the beam reflects from the side and bottom surfaces. (a) If the glass is surrounded by air, determine where part of the beam first exits the glass, at point $A, B$, or $C . \quad$ (b) Repeat part (a), assuming that the glass is surrounded by water instead of air.

Adnan Gill

Numerade Educator

Problem 32

The drawing shows three layers of different materials, with air above and below the layers. The interfaces between the layers are parallel. The index of refraction of each layer is given in the drawing. Identical rays of light are sent into the layers, and light zigzags through each layer, reflecting from the top and bottom surfaces. The index of refraction for air is $n_{\mathrm{air}}=1.00$. For each layer, the ray of light has an angle of incidence of $75.0^{\circ}$. For the cases in which total internal refection is possible from either the top or bottom surface of a layer, determine the amount by which the angle of incidence exceeds the critical angle.

Suzanne W.

Numerade Educator

Problem 33

The drawing shows a ray of light traveling from point $A$ to point $B$, a distance of $4.60 \mathrm{~m}$ in a material that has an index of refraction $n_{1} .$ At point $B$, the light encounters a different substance whose index of refraction is $n_{2}=1.63$. The light strikes the interface at the critical angle of $\theta_{\mathrm{c}}=48.1^{\circ} .$ How much time does it take for the light to travel from $A$ to $B$ ?

Adnan Gill

Numerade Educator

Problem 34

A layer of liquid $B$ floats on liquid $A$. A ray of light begins in liquid $A$ and undergoes total internal reflection at the interface between the liquids when the angle of incidence exceeds $36.5^{\circ} .$ When liquid $B$ is replaced with liquid $C$, total internal reflection occurs for angles of incidence greater than $47.0^{\circ}$. Find the ratio $n_{B} / n_{C}$ of the refractive indices of liquids $B$ and $C$.

Narayan Hari

Numerade Educator

Problem 35

Provide helpful background for this problem. The drawing shows a crystalline quartz slab with a rectangular cross section. A ray of light strikes the slab at an incident angle of $\theta_{1}=34^{\circ}$, enters the quartz, and travels to point $P$. This slab is surrounded by a fluid with a refractive index $n$. What is the maximum value of $n$ for which total internal reflection occurs at point $P$ ?

Suzanne W.

Numerade Educator

Problem 36

For light that originates within a liquid and strikes the liquidair interface, the critical angle is $39^{\circ} .$ What is Brewster's angle for this light?

Suzanne W.

Numerade Educator

Problem 37

Light is reflected from a glass coffee table. When the angle of incidence is $56.7^{\circ}$, the reflected light is completely polarized parallel to the surface of the glass. What is the index of refraction of the glass?

Adnan Gill

Numerade Educator

Problem 38

A Laser is mounted in air, at a distance of $0.476 \mathrm{~m}$ above the edge of a large, horizontal pane of crown glass, as shown in the drawing. The laser is aimed at the glass in such a way that the reflected beam is $100 \%$ polarized. Determine the distance $d$ between the edge of the pane and the point at which the laser beam is reflected.

Suzanne W.

Numerade Educator

Problem 39

When light strikes the surface between two materials from above, the Brewster angle is $65.0^{\circ} .$ What is the Brewster angle when the light encounters the same surface from below?

Narayan Hari

Numerade Educator

Problem 40

Light is incident from air onto the surface of a liquid. The angle of incidence is $53.0^{\circ}$, and the angle of refraction is $34,0^{\circ}$ At what angle of incidence would the reflected light be $100 \%$ polarized?

Suzanne W.

Numerade Educator

Problem 41

In Section $26.4$ it is mentioned that the reflected and refracted rays are perpendicular to each other when light strikes the surface at the Brewster angle. This is equivalent to saying that the angle of reflection plus the angle of refraction is $90^{\circ}$. Using Snell's law and Brewster's law, prove that the angle of reflection plus the angle of refraction is $90^{\circ}$.

Sheh Lit Chang

University of Washington

Problem 42

When red light in a vacuum is incident at the Brewster angle on a certain type of glass, the angle of refraction is $29.9^{\circ} .$ What are
(a) the Brewster angle and (b) the index of refraction of the glass?

Adnan Gill

Numerade Educator

Problem 43

A ray of sunlight is passing from diamond into crown glass; the angle of incidence is $35.00^{\circ}$. The indices of refraction for the blue and red components of the ray are: blue $\left(n_{\text {diamond }}=2.444, n_{\text {crown glass }}=1.531\right.$ ), and red $\left(n_{\text {diamond }}=2.410, n_{\text {crown glass }}=1.520\right.$ ). Determine the angle between the refracted blue and red rays in the crown glass.

Narayan Hari

Numerade Educator

Problem 44

Red light $(n=1.520)$ and violet light $(n=1.538)$ traveling in air are incident on a slab of crown glass. Both colors enter the glass at the same angle of refraction. The red light has an angle of incidence of $30.00^{\circ}$. What is the angle of incidence of the violet light?

Adnan Gill

Numerade Educator

Problem 45

A beam of sunlight encounters a plate of crown glass at a $45.00^{\circ}$ angle of incidence. Using the data in Table $26.2$, find the angle between the violet ray and the red ray in the glass.

Adnan Gill

Numerade Educator

Problem 46

Refer to Conceptual Example 8 as an aid in understanding this problem. The drawing shows a ray of light traveling through a gas $(n=1.00)$, a solid $(n=1.55)$, and a liquid $(n=1.55)$. At what angle $\theta$ does the light enter the liquid?

Narayan Hari

Numerade Educator

Problem 47

This problem relates to Figure 26.18, which illustrates the dispersion of light by a prism. The prism is made from glass, and its cross section is an equilateral triangle. The indices of refraction for the red and violet light are $1.662$ and 1.698, respectively. The angle of incidence for both the red and the violet light is $60.0^{\circ}$. Find the angles of refraction at which the red and violet rays emerge into the air from the prism.

Adnan Gill

Numerade Educator

Problem 48

The drawing shows a horizontal ray of white light incident perpendicularly on the vertical face of a prism (crown glass). The light enters the prism, and part of it undergoes refraction at the slanted face and emerges into the surrounding material. The rest of it is totally internally reflected and exits through the horizontal base of the prism. The colors of light that emerge from the slanted face may be chosen by altering the index of refraction of the material surrounding the prism. Find the required index of refraction of the surrounding material so that (a) only red light and (b) all colors except violet emerge from the slanted face. (See Table 26.2.)

Adnan Gill

Numerade Educator

Problem 49

An object is located $30.0 \mathrm{~cm}$ to the left of a converging lens whose focal length is $50.0 \mathrm{~cm} . \quad$ (a) Draw a ray diagram to scale and from it determine the image distance and the magnification. (b) Use the thin-lens and magnification equations to verify your answers to part (a).

Adnan Gill

Numerade Educator

Problem 50

The owner of a van installs a rear-window lens that has a focal length of $-0.300 \mathrm{~m}$. When the owner looks out through the lens at a person standing directly behind the van, the person appears to be just $0.240 \mathrm{~m}$ from the back of the van, and appears to be $0.34 \mathrm{~m}$ tall. (a) How far from the van is the person actually standing, and (b) how tall is the person?

Adnan Gill

Numerade Educator

Problem 51

A converging lens $(f=12.0 \mathrm{~cm})$ is held $8.00 \mathrm{~cm}$ in front of a newspaper that has a print size with a height of $2.00 \mathrm{~mm}$. Find (a) the image distance (in $\mathrm{cm}$ ) and (b) the height (in $\mathrm{mm}$ ) of the magnified print.

Narayan Hari

Numerade Educator

Problem 52

A tourist takes a picture of a mountain $14 \mathrm{~km}$ away using a camera that has a lens with a focal length of $50 \mathrm{~mm}$. She then takes a second picture when she is only $5.0 \mathrm{~km}$ away. What is the ratio of the height of the mountain's image on the camera's image sensor for the second picture to its height on the image sensor for the first picture?

Suzanne W.

Numerade Educator

Problem 53

A slide projector has a converging lens whose focal length is $105.00 \mathrm{~mm} .$ (a) How far (in meters) from the lens must the screen be located if a slide is placed $108.00 \mathrm{~mm}$ from the lens?
(b) If the slide measures $24.0 \mathrm{~mm} \times 36.0 \mathrm{~mm}$, what are the dimensions (in $\mathrm{mm}$ ) of its image?

Adnan Gill

Numerade Educator

Problem 54

To focus a camera on objects at different distances, the converging lens is moved toward or away from the image sensor, so a sharp image always falls on the sensor. A camera with a telephoto lens $(f=200.0 \mathrm{~mm})$ is to be focused on an object located first at a distance of $3.5 \mathrm{~m}$ and then at $50.0 \mathrm{~m}$. Over what distance must the lens be movable?

Adnan Gill

Numerade Educator

Problem 55

A diverging lens has a focal length of $-25 \mathrm{~cm} . \quad$ (a) Find the image distance when an object is placed $38 \mathrm{~cm}$ from the lens. (b) Is the image real or virtual?

Narayan Hari

Numerade Educator

Problem 56

An object is placed to the left of a lens, and a real image is formed to the right of the lens. The image is inverted relative to the object and is one-half the size of the object. The distance between the object and the image is $90.0 \mathrm{~cm} . \quad$ (a) How far from the lens is the object? (b) What is the focal length of the lens?

Adnan Gill

Numerade Educator

Problem 57

A camper is trying to start a fire by focusing sunlight onto a piece of paper. The diameter of the sun is $1.39 \times 10^{9} \mathrm{~m}$, and its mean distance from the earth is $1.50 \times 10^{11} \mathrm{~m}$. The camper is using a converging lens whose focal length is $10.0 \mathrm{~cm} . \quad$ (a) What is the area of the sun's image on the paper? (b) If $0.530 \mathrm{~W}$ of sunlight passes through the lens, what is the intensity of the sunlight at the paper?

Adnan Gill

Numerade Educator

Problem 58

Concept Simulation $26.4$ at www.wiley.com/college/cutnell provides the option of exploring the ray diagram that applies to this problem. The distance between an object and its image formed by a diverging lens is $49.0 \mathrm{~cm}$. The focal length of the lens is $-233.0 \mathrm{~cm}$. Find $\quad$ (a) the image distance and $\quad$ (b) the object distance.

Suzanne W.

Numerade Educator

Problem 59

An office copier uses a lens to place an image of a document onto a rotating drum. The copy is made from this image.
(a) What kind of lens is used, converging or diverging? If the document and its copy are to have the same size, but are inverted with respect to one another,(b) how far from the document is the lens located and (c) how far from the lens is the image located? Express your answers in terms of the focal length $f$ of the lens.

Adnan Gill

Numerade Educator

Problem 60

When a converging lens is used in a camera (as in Figure $26.26 b)$, the film must be at a distance of $0.210 \mathrm{~m}$ from the lens to record an image of an object that is $4.00 \mathrm{~m}$ from the lens. The same lens and film are used in a projector (see Figure $26.27 b$ ), with the screen $0.500 \mathrm{~m}$ from the lens. How far from the projector lens should the film be placed?

Narayan Hari

Numerade Educator

Problem 61

An object is $18 \mathrm{~cm}$ in front of a diverging lens that has a focal length of $-12 \mathrm{~cm}$. How far in front of the lens should the object be placed so that the size of its image is reduced by a factor of $2.0$ ?

Adnan Gill

Numerade Educator

Problem 62

An object is in front of a converging lens $(f=0.30 \mathrm{~m})$. The magnification of the lens is $m=4.0 .$ (a) Relative to the lens, in what direction should the object be moved so that the magnification changes to $m=-4.0$ ? (b) Through what distance should the object be moved?

Adnan Gill

Numerade Educator

Problem 63

A converging lens $(f=25.0 \mathrm{~cm})$ is used to project an image of an object onto a screen. The object and the screen are $125 \mathrm{~cm}$ apart, and between them the lens can be placed at either of two locations. Find the two object distances.

Adnan Gill

Numerade Educator

Problem 64

A filmmaker wants to achieve an interesting visual effect by filming a scene through a converging lens with a focal length of $50.0 \mathrm{~m}$. The lens is placed between the camera and a horse, which canters toward the camera at a constant speed of $7.0 \mathrm{~m} / \mathrm{s}$. The camera starts rolling when the horse is $40.0 \mathrm{~m}$ from the lens. Find the average speed of the image of the horse (a) during the first $2.0 \mathrm{~s}$ after the camera starts rolling and (b) during the following $2.0 \mathrm{~s}$.

Adnan Gill

Numerade Educator

Problem 65

Two identical diverging lenses are separated by $16 \mathrm{~cm}$. The focal length of each lens is $-8.0 \mathrm{~cm}$. An object is located $4.0 \mathrm{~cm}$ to the left of the lens that is on the left. Determine the final image distance relative to the lens on the right.

Adnan Gill

Numerade Educator

Problem 66

A converging lens $\left(f_{1}=24.0 \mathrm{~cm}\right)$ is located $56.0 \mathrm{~cm}$ to the left of a diverging lens $\left(f_{2}=-28.0 \mathrm{~cm}\right)$. An object is placed to the left of the converging lens, and the final image produced by the twolens combination lies $20.7 \mathrm{~cm}$ to the left of the diverging lens. How far is the object from the converging lens?66. (co A converging lens $\left(f_{1}=24.0 \mathrm{~cm}\right)$ is located $56.0 \mathrm{~cm}$ to the left of a diverging lens $\left(f_{2}=-28.0 \mathrm{~cm}\right)$. An object is placed to the left of the converging lens, and the final image produced by the twolens combination lies $20.7 \mathrm{~cm}$ to the left of the diverging lens. How far is the object from the converging lens?

Suzanne W.

Numerade Educator

Problem 67

Two converging lenses are separated by $24.00 \mathrm{~cm} .$ The focal length of each lens is $12.00 \mathrm{~cm}$. An object is placed $36.00 \mathrm{~cm}$ to the left of the lens that is on the left. Determine the final image distance relative to the lens on the right.

Adnan Gill

Numerade Educator

Problem 68

Two systems are formed from a converging lens and a diverging lens, as shown in parts $a$ and $b$ of the drawing. (The point labeled $" F_{\text {converging }} "$ is the focal point of the converging lens.) An object is placed inside the focal point of lens 1 at a distance of $10.00 \mathrm{~cm}$ to the left of lens 1 . The focal lengths of the converging and diverging lenses are $15.00$ and $-20.0 \mathrm{~cm}$, respectively. The distance between the lenses is $50.0 \mathrm{~cm}$. Determine the final image distance for each system, measured with respect to lens 2 .

Suzanne W.

Numerade Educator

Problem 69

A converging lens $(f=12.0 \mathrm{~cm})$ is located $30.0 \mathrm{~cm}$ to the left of a diverging lens $(f=-6.00 \mathrm{~cm})$. A postage stamp is placed $36.0 \mathrm{~cm}$ to the left of the converging lens. (a) Locate the final image of the stamp relative to the diverging lens. (b) Find the overall magnification. (c) Is the final image real or virtual? With respect to the original object, is the final image
(d) upright or inverted, and is it $\quad$ (e) larger or smaller?

Adnan Gill

Numerade Educator

Problem 70

Interactive LearningWare $26.2$ at www.wiley.com/college/ cutnell offers a review of the concepts that play roles in this problem. A diverging lens $(f=-10.0 \mathrm{~cm})$ is located $20.0 \mathrm{~cm}$ to the left of a converging lens $(f=30.0 \mathrm{~cm})$. A $3.00$ -cm-tall object stands to the left of the diverging lens, exactly at its focal point. (a) Determine the distance of the final image relative to the converging lens. (b) What is the height of the final image (including the proper algebraic sign)?

Suzanne W.

Numerade Educator

Problem 71

An object is placed $20.0 \mathrm{~cm}$ to the left of a diverging lens $(f=-8.00 \mathrm{~cm})$. A concave mirror $(f=12.0 \mathrm{~cm})$ is placed $30.0 \mathrm{~cm}$ to the right of the lens. (a) Find the final image distance, measured relative to the mirror. (b) Is the final image real or virtual? $\quad$ (c) Is the final image upright or inverted with respect to the original object?

Adnan Gill

Numerade Educator

Problem 72

Visitors at a science museum are invited to sit in a chair to the right of a full-length diverging lens $\left(f_{1}=-3.00 \mathrm{~m}\right)$ and observe a friend sitting in a second chair, $2.00 \mathrm{~m}$ to the left of the lens. The visitor then presses a button and a converging lens $\left(f_{2}=+4.00 \mathrm{~m}\right)$ rises from the floor to a position $1.60 \mathrm{~m}$ to the right of the diverging lens, allowing the visitor to view the friend through both lenses at once. Find (a) the magnification of the friend when viewed through the diverging lens only and (b) the overall magnification of the friend when viewed through both lenses. Be sure to include the algebraic signs ( $+$ or $-$ ) with your answers.

Adnan Gill

Numerade Educator

Problem 73

A coin is located $20.0 \mathrm{~cm}$ to the left of a converging lens $(f=16.0 \mathrm{~cm}) .$ A second, identical lens is placed to the right of the first lens in such a way that the image formed by the combination will have the same size and orientation as the original coin. Find the separation between the lenses.

Suzanne W.

Numerade Educator

Problem 74

A student is reading material written on a blackboard. Her contact lenses have a refractive power of $57.50$ diopters; the lens-to-retina distance is $1.750 \mathrm{~cm} . \quad$ (a) How far (in meters) is the blackboard from her eyes? $\quad(\mathbf{b})$ If the material written on the blackboard is $5.00 \mathrm{~cm}$ high, what is the size of the image on her retina?

Adnan Gill

Numerade Educator

Problem 75

Illustrates one approach to solving problems such as this one. A farsighted person has a near point that is $67.0 \mathrm{~cm}$ from her eyes. She wears eyeglasses that are designed to enable her to read a newspaper held at a distance of $25.0 \mathrm{~cm}$ from her eyes. Find the focal length of the eyeglasses, assuming that they are worn (a) $2.2 \mathrm{~cm}$ from the eyes and (b) $3.3 \mathrm{~cm}$ from the eyes.

Suzanne W.

Numerade Educator

Problem 76

A nearsighted patient's far point is $0.690 \mathrm{~m}$ from her eyes. She is able to see distant objects in focus when wearing glasses with a refractive power of $-1.50$ diopters. What is the distance between her eyes and the glasses?

Suzanne W.

Numerade Educator

Problem 77

Your friend has a near point of $138 \mathrm{~cm}$, and she wears contact lenses that have a focal length of $35.1 \mathrm{~cm}$. How close can she hold a magazine and still read it clearly?

Adnan Gill

Numerade Educator

Problem 78

A farsighted woman breaks her current eyeglasses and is using an old pair whose refractive power is 1.660 diopters. Since these eyeglasses do not completely correct her vision, she must hold a newspaper 42.00 cm from her eyes in order to read it. She wears the eyeglasses 2.00 cm from her eyes. How far is her near point from her eyes?

Adnan Gill

Numerade Educator

Problem 79

An optometrist prescribes contact lenses that have a focal length of $55.0 \mathrm{~cm} . \quad$ (a) Are the lenses converging or diverging, and $\quad(\mathbf{b})$ is the person who wears them nearsighted or farsighted? (c) Where is the unaided near point of the person located,

Suzanne W.

Numerade Educator

Problem 80

A farsighted man uses eyeglasses with a refractive power of $3.80$ diopters. Wearing the glasses $0.025 \mathrm{~m}$ from his eyes, he is able to read books held no closer than $0.280 \mathrm{~m}$ from his eyes. He would like a prescription for contact lenses to serve the same purpose. What is the correct contact lens prescription, in diopters?

Adnan Gill

Numerade Educator

Problem 81

The far point of a nearsighted person is $6.0 \mathrm{~m}$ from her eyes, and she wears contacts that enable her to see distant objects clearly. A tree is $18.0 \mathrm{~m}$ away and $2.0 \mathrm{~m}$ high. (a) When she looks through the contacts at the tree, what is its image distance? (b) How high is the image formed by the contacts?

Adnan Gill

Numerade Educator

Problem 82

Bill is farsighted and has a near point located $125 \mathrm{~cm}$ from his eyes. Anne is also farsighted, but her near point is $75.0 \mathrm{~cm}$ from her eyes. Both have glasses that correct their vision to a normal near point $(25.0 \mathrm{~cm}$ from the eyes), and both wear the glasses $2.0 \mathrm{~cm}$ from the eyes. Relative to the eyes, what is the closest object that can be seen clearly
(a) by Anne when she wears Bill's glasses and
(b) by Bill when he wears Anne's glasses?

Suzanne W.

Numerade Educator

Problem 83

A quarter (diameter $=2.4 \mathrm{~cm}$ ) is held at arm's length $(70.0 \mathrm{~cm})$. The sun has a diameter of $1.39 \times 10^{9} \mathrm{~m}$ and is $1.50 \times 10^{11} \mathrm{~m}$
from the earth. What is the ratio of the angular size of the quarter to that of the sun?

Narayan Hari

Numerade Educator

Problem 84

A quarter (diameter $=2.4 \mathrm{~cm}$ ) is held at arm's length $(70.0 \mathrm{~cm})$. The sun has a diameter of $1.39 \times 10^{9} \mathrm{~m}$ and is $1.50 \times 10^{11} \mathrm{~m}$ from the earth. What is the ratio of the angular size of the quarter to that of the sun?

Narayan Hari

Numerade Educator

Problem 85

An engraver uses a magnifying glass $(f=9.50 \mathrm{~cm})$ to examine some work, as in Figure $26.39 b$. The image he sees is located $25.0 \mathrm{~cm}$ from his eye, which is his near point. (a) What is the distance between the work and the magnifying glass? (b) What is the angular magnification of the magnifying glass?

Narayan Hari

Numerade Educator

Problem 86

A dentist is examining a dental filling in a patient's tooth. The diameter of the filling is $2.4 \mathrm{~mm}$, and the dentist's near point is $17.0 \mathrm{~cm}$. To get a better look at the filling, the dentist dons safety goggles fitted with magnifying glasses $(f=6.0 \mathrm{~cm})$. Find the greatest possible angular size (in radians) of the patient's filling when viewed by the $\begin{array}{lll}\text { dentist, both } & \text { (a) without and } & \text { (b) with the magnifying glasses. }\end{array}$

Suzanne W.

Numerade Educator

Problem 87

An object has an angular size of $0.0150$ rad when placed at the near point $(21.0 \mathrm{~cm})$ of an eye. When the eye views this object using a magnifying glass, the largest possible angular size of the image is $0.0380$ rad. What is the focal length of the magnifying glass?

Adnan Gill

Numerade Educator

Problem 88

A stamp collector is viewing a stamp with a magnifying glass held next to her eye. Her near point is $25 \mathrm{~cm}$ from her eye. $\quad$ (a) What is the refractive power of a magnifying glass that has an angular magnification of $6.0$ when the image of the stamp is located at the near point? (b) What is the angular magnification when the image of the stamp is $45 \mathrm{~cm}$ from the eye?

Adnan Gill

Numerade Educator

Problem 89

A farsighted person can read printing as close as $25.0 \mathrm{~cm}$ when she wears contacts that have a focal length of $45.4 \mathrm{~cm}$. One day, she forgets her contacts and uses a magnifying glass, as in Figure 26.39b. Its maximum angular magnification is $7.50$ for a young person with a normal near point of $25.0 \mathrm{~cm}$. What is the maximum angular magnification that the magnifying glass can provide for her?

Adnan Gill

Numerade Educator

Problem 90

An anatomist is viewing heart muscle cells with a microscope that has two selectable objectives with refracting powers of 100 and 300 diopters. When he uses the 100 -diopter objective, the image of a cell subtends an angle of $3 \times 10^{-3}$ rad with the eye. What angle is subtended when he uses the 300 -diopter objective?

Narayan Hari

Numerade Educator

Problem 91

A compound microscope has a barrel whose length is $16.0 \mathrm{~cm}$ and an eyepiece whose focal length is $1.4 \mathrm{~cm}$. The viewer has a near point located $25 \mathrm{~cm}$ from his eyes. What focal length must the objective have so that the angular magnification of the microscope will be $-320 ?$

Adnan Gill

Numerade Educator

Problem 92

An insect subtends an angle of only $4.0 \times 10^{-3}$ rad at the unaided eye when placed at the near point. What is the angular size (magnitude only) when the insect is viewed through a microscope whose angular magnification has a magnitude of $160 ?$

Narayan Hari

Numerade Educator

Problem 93

A microscope for viewing blood cells has an objective with a focal length of $0.50 \mathrm{~cm}$ and an eyepiece with a focal length of $2.5 \mathrm{~cm} .$ The distance between the objective and eyepiece is $14.0 \mathrm{~cm} .$ If a blood cell subtends an angle of $2.1 \times 10^{-5}$ rad when viewed with the naked eye at a near point of $25.0 \mathrm{~cm}$, what angle (magnitude only) does it subtend when viewed through the microscope?

Adnan Gill

Numerade Educator

Problem 94

In a compound microscope, the objective has a focal length of $0.60 \mathrm{~cm}$, while the eyepiece has a focal length of $2.0 \mathrm{~cm}$. The separation between the objective and the eyepiece is $L=12.0 \mathrm{~cm}$. Another microscope that has the same angular magnification can be constructed by interchanging the two lenses, provided that the distance between the lenses is adjusted to a value $L^{\prime} .$ Find $L^{\prime} .$

Adnan Gill

Numerade Educator

Problem 95

In a compound microscope, the focal length of the objective is $3.50 \mathrm{~cm}$ and that of the eyepiece is $6.50 \mathrm{~cm}$. The distance between the lenses is $26.0 \mathrm{~cm} . \quad$ (a) What is the angular magnification of the $\mathrm{mi}-$ croscope if the person using it has a near point of $35.0 \mathrm{~cm} ? \quad$ (b) If, as usual, the first image lies just inside the focal point of the eyepiece (see Figure 26.32), how far is the object from the objective? (c) What is the magnification (not the angular magnification) of the objective?

Suzanne W.

Numerade Educator

Problem 96

A stargazer has an astronomical telescope with an objective whose focal length is $180 \mathrm{~cm}$ and an eyepiece whose focal length is $1.20 \mathrm{~cm} .$ He wants to increase the angular magnification of a galaxy under view by replacing the telescope's eyepiece. Once the eyepiece is replaced, the barrel of the telescope must be adjusted to bring the galaxy back into focus. If the barrel can only be shortened by $0.50 \mathrm{~cm}$ from its current length, what is the best angular magnification the stargazer will be able to achieve?

Suzanne W.

Numerade Educator

Problem 97

Mars subtends an angle of $8.0 \times 10^{-5} \mathrm{rad}$ at the unaided eye. An astronomical telescope has an eyepiece with a focal length of $0.032 \mathrm{~m} .$ When Mars is viewed using this telescope, it subtends an angle of $2.8 \times 10^{-3}$ rad. Find the focal length of the telescope's objective lens.

Adnan Gill

Numerade Educator

Problem 98

An astronomical telescope has an angular magnification of $-132$. Its objective has a refractive power of $1.50$ diopters. What is the refractive power of its eyepiece?

Adnan Gill

Numerade Educator

Problem 99

An astronomical telescope has an angular magnification of $-184$ and uses an objective with a focal length of $48.0 \mathrm{~cm} .$ What is the focal length of the eyepiece?

Narayan Hari

Numerade Educator

Problem 100

An amateur astronomer decides to build a telescope from a discarded pair of eyeglasses. One of the lenses has a refractive power of 11 diopters, and the other has a refractive power of $1.3$ diopters. (a) Which lens should be the objective? (b) How far apart should the lenses be separated? (c) What is the angular magnification of the telescope?

Adnan Gill

Numerade Educator

Problem 101

A refracting telescope has an angular magnification of $-83.00$. The length of the barrel is $1.500 \mathrm{~m}$. What are the focal lengths of
(a) the objective and (b) the eyepiece?

Adnan Gill

Numerade Educator

Problem 102

The lengths of three telescopes are $L_{\mathrm{A}}=455 \mathrm{~mm}$, $L_{\mathrm{B}}=615 \mathrm{~mm}$, and $L_{\mathrm{C}}=824 \mathrm{~mm}$. The focal length of the eyepiece for each telescope is $3.00 \mathrm{~mm}$. Find the angular magnification of each telescope.

Adnan Gill

Numerade Educator

Problem 103

The telescope at Yerkes Observatory in Wisconsin has an objective whose focal length is $19.4 \mathrm{~m}$. Its eyepiece has a focal length of $10.0 \mathrm{~cm} .$ (a) What is the angular magnification of the telescope? (b) If the telescope is used to look at a lunar crater whose diameter is $1500 \mathrm{~m}$, what is the size of the first image, assuming that the surface of the moon is $3.77 \times 10^{8} \mathrm{~m}$ from the surface of the earth? (c) How close does the crater appear to be when seen through the telescope?

Adnan Gill

Numerade Educator

Problem 104

The angular magnification of a telescope is 32800 times as large when you look through the correct end of the telescope as when you look through the wrong end. What is the angular magnification of the telescope?

Suzanne W.

Numerade Educator

Problem 105

An object is located $9.0 \mathrm{~cm}$ in front of a converging lens $(f=6.0 \mathrm{~cm}) .$ Using an accurately drawn ray diagram, determine where the image is located.

Adnan Gill

Numerade Educator

Problem 106

The near point of a naked eye is $25 \mathrm{~cm}$. When placed at the near point and viewed by the naked eye, a tiny object would have an angular size of $5.2 \times 10^{-5} \mathrm{rad}$. When viewed through a compound microscope, however, it has an angular size of $-8.8 \times 10^{-3}$ rad. (The minus sign indicates that the image produced by the microscope is inverted.) The objective of the microscope has a focal length of $2.6 \mathrm{~cm}$, and the distance between the objective and the eyepiece is $16 \mathrm{~cm}$. Find the focal length of the eyepiece.

Adnan Gill

Numerade Educator

Problem 107

Concept Simulation $26.1$ at www.wiley.com/college/ cutnell illustrates the concepts that are pertinent to this problem. A ray of light is traveling in glass and strikes a glass-liquid interface. The angle of incidence is $58.0^{\circ}$, and the index of refraction of glass is $n=1.50 .$ (a) What must be the index of refraction of the liquid so that the direction of the light entering the liquid is not changed? (b) What is the largest index of refraction that the liquid can have, so that none of the light is transmitted into the liquid and all of it is reflected back into the glass?

Suzanne W.

Numerade Educator

Problem 108

As an aid in understanding this problem, refer to Conceptual Example 4. A swimmer, who is looking up from under the water, sees a diving board directly above at an apparent height of $4.0 \mathrm{~m}$ above the water. What is the actual height of the diving board?

Adnan Gill

Numerade Educator

Problem 109

A person has far points of $5.0 \mathrm{~m}$ from the right eye and $6.5 \mathrm{~m}$ from the left eye. Write a prescription for the refractive power of each corrective contact lens.

Adnan Gill

Numerade Educator

Problem 110

Amber $(n=1.546)$ is a transparent brown-yellow fossil resin. An insect, trapped and preserved within the amber, appears to be $2.5 \mathrm{~cm}$ beneath the surface when viewed directly from above. How far below the surface is the insect actually located?

Adnan Gill

Numerade Educator

Problem 111

A macroscopic (or macro) lens for a camera is usually a converging lens of normal focal length built into a lens barrel that can be adjusted to provide the additional distance needed between the lens and the image sensor when focusing at very close range. Suppose that a macro lens $(f=50.0 \mathrm{~mm})$ has a maximum distance of $275 \mathrm{~mm}$ between the lens and the image sensor. How close can the object be located in front of the lens?

Suzanne W.

Numerade Educator

Problem 112

Concept Simulation $26.3$ at www.wiley.com/college/cutnell reviews the concepts that play a role in this problem. A converging lens has a focal length of $88.00 \mathrm{~cm}$. An object $13.0 \mathrm{~cm}$ tall is located $155.0 \mathrm{~cm}$ in front of this lens. $\quad$ (a) What is the image distance? (b) Is the image real or virtual? (c) What is the image height? Be sure to include the proper algebraic sign.

Suzanne W.

Numerade Educator

Problem 113

A beam of light is traveling in air and strikes a material. The angles of incidence and refraction are $63.0^{\circ}$ and $47.0^{\circ}$, respectively. Obtain the speed of light in the material.

RG

Ryan Griffith

Numerade Educator

Problem 114

The near point of a naked eye is $32 \mathrm{~cm}$. When an object is placed at the near point and viewed by the naked eye, it has an angular size of $0.060$ rad. A magnifying glass has a focal length of $16 \mathrm{~cm}$, and is held next to the eye. The enlarged image that is seen is located $64 \mathrm{~cm}$ from the magnifying glass. Determine the angular size of the image.

Adnan Gill

Numerade Educator

Problem 115

A nearsighted person cannot read a sign that is more than $5.2 \mathrm{~m}$ from his eyes. To deal with this problem, he wears contact lenses that do not correct his vision completely, but do allow him to read signs located up to distances of $12.0 \mathrm{~m}$ from his eyes. What is the focal length of the contacts?

Narayan Hari

Numerade Educator

Problem 116

Violet light and red light travel through air and strike a block of plastic at the same angle of incidence. The angle of refraction is $30.400^{\circ}$ for the violet light and $31.200^{\circ}$ for the red light. The index of refraction for violet light in plastic is greater than that for red light by $0.0400$. Delaying any rounding off of calculations until the very end, find the index of refraction for violet light in plastic.

Adnan Gill

Numerade Educator

Problem 117

Consult Interactive Solution $26.117$ at www.wiley.com/ college/cutnell to review the concepts on which this problem depends. A camera is supplied with two interchangeable lenses, whose focal lengths are $35.0$ and $150.0 \mathrm{~mm}$. A woman whose height is $1.60 \mathrm{~m}$ stands $9.00 \mathrm{~m}$ in front of the camera. What is the height (including sign) of her image on the image sensor, as produced by (a) the $35.0$ -mm lens and (b) the $150.0-\mathrm{mm}$ lens?

Suzanne W.

Numerade Educator

Problem 118

(a) For a diverging lens $(f=-20.0 \mathrm{~cm})$, construct a ray diagram to scale and find the image distance for an object that is $20.0 \mathrm{~cm}$ from the lens. (b) Determine the magnification of the lens from the diagram.

Adnan Gill

Numerade Educator

Problem 119

The moon's diameter is $3.48 \times 10^{6} \mathrm{~m}$, and its mean distance from the earth is $3.85 \times 10^{8} \mathrm{~m}$. The moon is being photographed by a camera whose lens has a focal length of $50.0 \mathrm{~mm}$. (a) Find the diameter of the moon's image on the slide film. (b) When the slide is projected onto a screen that is $15.0 \mathrm{~m}$ from the lens of the projector $(f=110.0 \mathrm{~mm})$, what is the diameter of the moon's image on the screen?

Adnan Gill

Numerade Educator

Problem 120

A person using a magnifying glass as in Figure $26.39 b$ observes that for clear vision its maximum angular magnification is $1.25$ times as large as its minimum angular magnification. Assuming that the person has a near point located $25 \mathrm{~cm}$ from her eye, what is the focal length of the magnifying glass?

Suzanne W.

Numerade Educator

Problem 121

At age forty, a certain man requires contact lenses $(f=65.0 \mathrm{~cm})$ to read a book held $25.0 \mathrm{~cm}$ from his eyes. At age forty-five, while wearing these contacts he must now hold a book $29.0 \mathrm{~cm}$ from his eyes. $\quad$ (a) By what distance has his near point changed? (b) What focal-length lenses does he require at age fortyfive to read a book at $25.0 \mathrm{~cm}$ ?

Suzanne W.

Numerade Educator

Problem 122

An object is placed in front of a converging lens in such a position that the lens $(f=12.0 \mathrm{~cm})$ creates a real image located $21.0 \mathrm{~cm}$ from the lens. Then, with the object remaining in place, the lens is replaced with another converging lens $(f=16.0 \mathrm{~cm}) .$ A new, real image is formed. What is the image distance of this new image?

Adnan Gill

Numerade Educator

Problem 123

An astronomical telescope is being used to examine a relatively close object that is only $114.00 \mathrm{~m}$ away from the objective of the telescope. The objective and eyepiece have focal lengths of $1.500$ and $0.070 \mathrm{~m}$, respectively. Noting that the expression $M \approx-f_{\mathrm{o}} / f_{\mathrm{c}}$ is no longer applicable because the object is so close, use the thin-lens and magnification equations to find the angular magnification of this telescope. (Hint: See Figure $26.41$ and note that the focal points $F_{0}$ and $F_{\mathrm{c}}$ are so close together that the distance between them may be ignored.)

Suzanne W.

Numerade Educator

Problem 124

The equation $$\frac{1}{d_{\mathrm{o}}}+\frac{1}{d_{\mathrm{i}}}=\frac{1}{f}$$ is called the Gaussian form of the thin-lens equation. The drawing shows the variables $d_{0}, d_{i}$ and $f .$ The also shows the distances $x$ and
$x^{\prime}$, which are, respectively, the from the object to the focal point on the left of the lens and the distance from the focal point on the right of the lens to the image. An equivalent form of the thin-lens equation, involving $x, x^{\prime}$, and $f$, is called the Newtonian form. Show that the Newtonian form of the thin-lens equation can be written as $x x^{\prime}=f^{2}$.

Suzanne W.

Numerade Educator

Problem 125

The contacts worn by a farsighted person allow her to see objects clearly that are as close as $25.0 \mathrm{~cm}$, even though her uncorrected near point is $79.0 \mathrm{~cm}$ from her eyes. When she is looking at a poster, the contacts form an image of the poster at a distance of $217 \mathrm{~cm}$ from her eyes. (a) How far away is the poster actually located? $(\mathbf{b})$ If the poster is $0.350 \mathrm{~m}$ tall, how tall is the image formed by the contacts?

Adnan Gill

Numerade Educator