College Physics

lan Giambattista, Betty McCarthy Richardson, Robert C. Richardson

Chapter 23

Reflection and Refraction of Light - all with Video Answers

Educators

DM

Chapter Questions

Problem 1

Sketch the wavefronts and rays for the light emitted by an isotropic point source (isotropic $=$ same in all directions). Use Huygens's principle to illustrate the propagation of one of the wavefronts.

Samantha Baker

Samantha Baker

Numerade Educator

Problem 2

Apply Huygens's principle to a 5-cm-long planar wavefront approaching a reflecting wall at normal incidence. The wavelength is $1 \mathrm{~cm}$ and the wall has a wide opening (width $=4 \mathrm{~cm}$ ). The center of the incoming wavefront approaches the center of the opening. Repeat the procedure until you have wavefronts on both sides of the wall. Without worrying about the details of edge effects, what are the general shapes of the wavefronts on each side of the reflecting wall?

Samantha Baker

Numerade Educator

Problem 3

Repeat Problem 2 for an opening of width $0.5 \mathrm{~cm}$.

Samantha Baker

Numerade Educator

Problem 4

A plane wave reflects from the surface of a sphere. Draw a ray diagram and sketch some wavefronts for the reflected wave.

Prabhat Tyagi

Numerade Educator

Problem 5

A spherical wave (from a point source) reflects from a planar surface. Draw a ray diagram and sketch some wavefronts for the reflected wave.

Samantha Baker

Numerade Educator

Problem 6

Light rays from the Sun, which is at an angle of $35^{\circ}$ above the western horizon, strike the still surface of a pond. (a) What is the angle of incidence of the Sun's rays on the pond? (b) What is the angle of reflection of the rays that leave the pond surface? (c) In what direction and at what angle from the pond surface are the reflected rays traveling?

Samantha Baker

Numerade Educator

Problem 7

A light ray reflects from a plan mirror as shown in the figure What is the angle of deviation $\delta$ ?

Linda Winkler

Numerade Educator

Problem 8

Two plane mirrors form a $70.0^{\circ}$ angle as shown. For what angle $\theta$ is the final ray horizontal?

Samantha Baker

Numerade Educator

Problem 9

Choose two rays in Fig. $23.7$ and use them to prove that the angle of incidence is equal to the angle of reflection. [Hint: Choose a wavefront at two different times, one before reflection and one after. The time for light to travel from one wavefront to the other is the same for the two rays.]

Samantha Baker

Numerade Educator

Problem 10

Sunlight strikes the surface of a lake at an angle of incidence of $30.0^{\circ}$. At what angle with respect to the normal would a fish see the Sun?

Samantha Baker

Numerade Educator

Problem 11

Sunlight strikes the surface of a lake. A diver sees the Sun at an angle of $42.0^{\circ}$ with respect to the vertical. What angle do the Sun's rays in air make with the vertical?

Samantha Baker

Numerade Educator

Problem 12

A beam of light in air is incident upon a stack of four fla transparent materials with indices of refraction $1.20,1.40$, 1.32, and $1.28$. If the angle of incidence for the beam on the first of the four materials is $60.0^{\circ}$, what angle does the beam make with the normal when it emerges into the ain after passing through the entire stack?

Linda Winkler

Numerade Educator

Problem 13

At a marine animal park, Alison is looking through a glass window and watching dolphins swim underwater. If the dolphin is swimming directly toward her at $15 \mathrm{~m} / \mathrm{s}$, how fast does the dolphin appear to be moving?

Samantha Baker

Numerade Educator

Problem 14

A light ray in the core $(n=1.40)$ of a cylindrical optical fiber travels at an angle $\theta_{1}=49.0^{\circ}$ with respect to the axis of the fiber. A ray is transmitted through the cladding $(n=$ $1.20$ ) and into the air. What angle $\theta_{2}$ does the exiting ray make with the outside surface of the cladding?

Narayan Hari

Numerade Educator

Problem 15

A light ray in the core $(n=1.40)$ of a cylindrical optical fiber is incident on the cladding. See the figure with Problem $14 .$ A ray is transmitted through the cladding $(n=1.20)$ and into the air. The emerging ray makes an angle $\theta_{2}=5.00^{\circ}$ with the outside surface of the cladding. What angle $\theta_{1}$ did the rav in the core make with the axis?

Linda Winkler

Numerade Educator

Problem 16

A glass lens has a scratch-resistant plastic coating on it. The speed of light in the glass is $0.67 c$ and the speed of light in the coating is $0.80 c$. A ray of light in the coating is incident on the plastic-glass boundary at an angle of $12.0^{\circ}$ with respect to the normal. At what angle with respect to the normal is the ray transmitted into the glass?

Samantha Baker

Numerade Educator

Problem 17

In Figure $23.12$, a coin is right up against the far edge of the mug. In picture (a) the coin is just hidden from view and in picture (b) we can almost see the whole coin. If the mug is $6.5 \mathrm{~cm}$ in diameter and $8.9 \mathrm{~cm}$ tall, what is the diameter of the coin?

Samantha Baker

Numerade Educator

Problem 18

A horizontal light ray is incident on a crown glass prism as shown in the figure where $\beta=30.0^{\circ} .$ Find the angle of deviation $\delta$ of the ray $-$ the angle that the ray emerging from the prism makes with the incident ray.

Samantha Baker

Numerade Educator

Problem 19

A horizontal light ray is incident on a prism as shown in the figure with Problem 18 where $\beta$ is a small angle (exaggerated in the figure). Find the angle of deviation $\delta$ of the ray-the angle that the ray emerging from the prism makes with the incident ray-as a function of $\beta$ and $n$, the index of refraction of the prism and show that $\delta$ is proportional to $\beta$.

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 20

A diamond in air is illuminated with white light. On one particular facet, the angle of incidence is $26.00^{\circ}$. Inside the diamond, red light $(\lambda=660.0 \mathrm{~nm}$ in vacuum) is refracted at $10.48^{\circ}$ with respect to the normal; blue light $\left(\lambda=470.0 \mathrm{~nm}\right.$ in vacuum) is refracted at $10.33^{\circ}$.
(a) What are the indices of refraction for red and blue light in diamond? (b) What is the ratio of the speed of red light to the speed of blue light in diamond? (c) How would a diamond look if there were no dispersion?

Samantha Baker

Numerade Educator

Problem 21

The prism in the figure is made of crown glass. Its index of refraction ranges from $1.517$ for the longest visible wavelengths to $1.538$ for the shortest. Find
the range of refraction angles for the light transmitted into air through the right side of the prism.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 22

Calculate the critical angle for a sapphire surrounded by air.

Samantha Baker

Numerade Educator

Problem 23

(a) Calculate the critical angle for a diamond surrounded by air. (b) Calculate the critical angle for a diamond under water. (c) Explain why a diamond sparkles less under water than in air.

Samantha Baker

Numerade Educator

Problem 24

Is there a critical angle for a light ray coming from a medium with an index of refraction $1.2$ and incident on a medium that has an index of refraction $1.4 ?$ If so, what is the critical angle that allows total internal reflection in the first medium?

Samantha Baker

Numerade Educator

Problem 25

The figure shows some light rays reflected from a small defect in the glass toward the surface of the glass. (a) If $\theta_{c}=40.00^{\circ}$, what is the index of refraction of the glass? (b) Is there any point above the glass at which a viewer would not be able to see the defect? Explain.

Samantha Baker

Numerade Educator

Problem 26

A $45^{\circ}$ prism has an index of refraction of $1.6 .$ Light is normally incident on the left side of the prism. Does light exit the back of the prism (for example, at point $P$ )?
If so, what is the angle of refraction with respect to the normal at point $P$ ? If not, what happens to the light?

Samantha Baker

Numerade Educator

Problem 27

Light incident on a $45.0^{\circ}$ prism as shown in the figure undergoes total internal reflection at point $P$. $\mathbf{W}$ What can you conclude about the index of refraction of the prism? (Determine either a minimum or maximum possible value.)

Linda Winkler

Numerade Educator

Problem 28

The angle of incidence $\theta$ of a ray of light in air is adjusted gradually as it enters a shallow tank made of Plexiglas and filled with carbon disulfide. Is there an angle of incidence for which light is transmitted into the carbon disulfide but not into the Plexiglas at the bottom of the tank? If so, find the angle. If not, explain why not.

Samantha Baker

Numerade Educator

Problem 29

Repeat Problem 28 for a Plexiglas tank filled with carbon tetrachloride instead of carbon disulfide.

Samantha Baker

Numerade Educator

Problem 30

What is the index of refraction of the core of an optical fiber if the cladding has $n=1.20$ and the critical angle at the core-cladding boundary is $45.0^{\circ} ?$

Samantha Baker

Numerade Educator

Problem 31

Some glasses used for viewing $3 \mathrm{D}$ movies are polarized, one lens having a vertical transmission axis and the other horizontal. While standing in line on a winter afternoon for a $3 \mathrm{D}$ movie and looking through his glasses at the road surface, Maurice notices that the left lens cuts down reflected glare significantly, but the right lens does not. The glare is minimized when the angle between the reflected light and the horizontal direction is $37^{\circ}$. (a) Which lens has the transmission axis in the vertical direction? (b) What is Brewster's angle for this case? (c) What is the index of refraction of the material reflecting the light?

Samantha Baker

Numerade Educator

Problem 32

(a) Sunlight reflected from the still surface of a lake is totally polarized when the incident light is at what angle with respect to the horizontal? (b) In what direction is the reflected light polarized? (c) Is any light incident at this angle transmitted into the water? If so, at what angle below the horizontal does the transmitted light travel?

Samantha Baker

Numerade Educator

Problem 33

(a) Sunlight reflected from the smooth ice surface of a frozen lake is totally polarized when the incident light is at what angle with respect to the horizontal? (b) In what direction is the reflected light polarized? (c) Is any light incident at this angle transmitted into the ice? If so, at what angle below the horizontal does the transmitted light travel?

Samantha Baker

Numerade Educator

Problem 34

Light travels in a medium with index $n_{1}$ toward a boundary with another material of index $n_{2}<n_{1}$.
(a) Which is larger, the critical angle or Brewster's angle? Does the answer depend on the values of $n_{1}$ and $n_{2}$ (other than assuming $n_{2}<n_{1}$ )? (b) What can you say
about the critical angle and Brewster's angle for light coming the other way (from the medium with index $n_{2}$ toward the medium with $n_{1}$ )?

Samantha Baker

Numerade Educator

Problem 35

A defect in a diamond appears to be $2.0 \mathrm{~mm}$ below the surface when viewed from directly above that surface. How far beneath the surface is the defect?

Samantha Baker

Numerade Educator

Problem 36

An insect is trapped inside a piece of amber $(n=$ 1.546). Looking at the insect from directly above, it appears to be $7.00 \mathrm{~mm}$ below a smooth surface of the amber. How far below the surface is the insect?

Samantha Baker

Numerade Educator

Problem 37

A penny is at the bottom of a bowl full of water. When you look at the water surface from the side, with your eyes at the water level, the penny appears to be just barely under the surface and a horizontal distance of $3.0 \mathrm{~cm}$ from the edge of the bowl. If the penny is actually $8.0 \mathrm{~cm}$ below the water surface, what is the horizontal distance between the penny and the edge of the bowl? [Hint: The rays you see pass from water to air with refraction angles close to $90^{\circ} .$ ]

Samantha Baker

Numerade Educator

Problem 38

Norah wants to buy a mirror so that she can check on her appearance from top to toe before she goes off to work. If Norah is $1.64 \mathrm{~m}$ tall, how tall a mirror does she need?

Willis James

Numerade Educator

Problem 39

Daniel's eyes are $1.82 \mathrm{~m}$ from the floor when he is wearing his dress shoes, and the top of his head is $1.96 \mathrm{~m}$ from the floor. Daniel has a mirror that is $0.98 \mathrm{~m}$ in length. How high from the floor should the bottom edge of the mirror be located if Daniel is to see a full-length image of himself? Draw a ray diagram to illustrate your answer.

Samantha Baker

Numerade Educator

Problem 40

A rose in a vase is placed $0.250 \mathrm{~m}$ in front of a plane mirror. Nagar looks into the mirror from $2.00 \mathrm{~m}$ in front of it. How far away from Nagar is the image of the rose?

Samantha Baker

Numerade Educator

Problem 41

Entering a darkened room, Gustav strikes a match in an attempt to see his surroundings. At once he sees what looks like another match about $4 \mathrm{~m}$ away from him. As it turns out, a mirror hangs on one wall of the room. How far is Gustav from the wall with the mirror?

Samantha Baker

Numerade Educator

Problem 42

In an amusement park maze with all the walls covered with mirrors, Pilar sees Hernando's reflection from a series of three mirrors. If the reflected angle from mirror 3 is $55^{\circ}$ for the mirror arrangement shown in the figure, what is the angle of incidence on mir. ror $1 ?$

Linda Winkler

Numerade Educator

Problem 43

Maurizio is standing in a rectangular room with tw adjacent walls and the ceiling all covered by plane min rors. How many images of himself can Maurizio see?

DM

Debra Mangion

Numerade Educator

Problem 44

Hannah is standing in the middle of a room with two opposite walls that are separated by $10.0 \mathrm{~m}$ and covered by plane mirrors. There is a candle in the room $1.50 \mathrm{~m}$ from one mirrored wall. Hannah is facing the opposite mirrored wall and sees many images of the candle. How far from Hannah are the closest four images of the candle that she can see?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 45

A point source of light is in front of a plane mirror. (a) Prove that all the reflected rays, when extended back behind the mirror, intersect in a single point. [Hint: See Fig. 23.27a and use similar triangles.] (b) Show that the image point lies on a line through the object and perpendicular to the mirror, and that the object and image distances are equal. [Hint: Use any pair of rays in Fig. 23.27a.]

Linda Winkler

Numerade Educator

Problem 46

An object $2.00 \mathrm{~cm}$ high is placed $12.0 \mathrm{~cm}$ in front of $\mathrm{a}$ convex mirror with radius of curvature of $8.00 \mathrm{~cm}$. Where is the image formed? Draw a ray diagram to illustrate.

Samantha Baker

Numerade Educator

Problem 47

A $1.80-\mathrm{cm}$ -high object is placed $20.0 \mathrm{~cm}$ in front of a concave mirror with a $5.00-\mathrm{cm}$ focal length. What is the position of the image? Draw a ray diagram to illustrate.

Samantha Baker

Numerade Educator

Problem 48

In her job as a dental hygienist, Kathryn uses a convex mirror to see the back of her patient's teeth. When the mirror is $1.20 \mathrm{~cm}$ from a tooth, the image is upright and $3.00$ times as large as the tooth. What are the focal length and radius of curvature of the mirror?

Linda Winkler

Numerade Educator

Problem 49

An object is placed in front of a concave mirror with a $25.0-\mathrm{cm}$ radius of curvature. A real image twice the size of the object is formed. At what distance is the object from the mirror? Draw a ray diagram to illustrate.

Samantha Baker

Numerade Educator

Problem 50

An object is placed in front of a convex mirror with a $25.0-\mathrm{cm}$ radius of curvature. A virtual image half the size of the object is formed. At what distance is the object from the mirror? Draw a ray diagram to illustrate.

Samantha Baker

Numerade Educator

Problem 51

The right-side rearview mirror of Mike's car says that objects in the mirror are closer than they appear. Mike decides to do an experiment to determine the focal length of this mirror. He holds a plane mirror next to the rearview mirror and views an object that is $163 \mathrm{~cm}$ away from each mirror. The object appears $3.20 \mathrm{~cm}$ wide in the plane mirror, but only $1.80 \mathrm{~cm}$ wide in the rearview mirror. What is the focal length of the rearview mirror?

Samantha Baker

Numerade Educator

Problem 52

A concave mirror has a radius of curvature of $5.0 \mathrm{~m}$. An object, initially $2.0 \mathrm{~m}$ in front of the mirror, is moved back until it is $6.0 \mathrm{~m}$ from the mirror. Describe how the image location changes.

Samantha Baker

Numerade Educator

Problem 53

Derive the magnification equation, $m=h^{\prime} / h=-q / p$, for a convex mirror. Draw a ray diagram as part of the solution. [Hint: Draw a ray that is not one of the three principal rays, as was done in the derivation for a concave mirror.]

Samantha Baker

Numerade Educator

Problem 54

In a subway station, a convex mirror allows the attendant to view activity on the platform. A woman $1.64 \mathrm{~m}$ tall is standing $4.5 \mathrm{~m}$ from the mirror. The image formed of the woman is $0.500 \mathrm{~m}$ tall. (a) What is the radius of curvature of the mirror? (b) The mirror is $0.500 \mathrm{~m}$ in diameter. If the woman's shoes appear at the bottom of the mirror, does her head appear at the top in other words, does the image of the woman fill the mirror from top to bottom? Explain.

Samantha Baker

Numerade Educator

Problem 55

Show that when rays parallel to the principal axis reflect from a concave mirror, the reflected rays all pass through the focal point at a distance $R / 2$ from the vertex. Assume that the angles of incidence are small. [Hint: Follow the similar derivation for a convex mirror in the text.]

Sophie S

Numerade Educator

Problem 56

Starting with Fig. $23.39$, perform all the algebraic steps to obtain the mirror equation in the form of Eq. $(23-10)$.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 57

(a) For a converging lens with a focal length of $3.50 \mathrm{~cm}$, find the object distance that will result in an inverted image with an image distance of $5.00 \mathrm{~cm}$. Use a ray diagram to verify your calculations. (b) Is the image real or virtual? (c) What is the magnification?

Samantha Baker

Numerade Educator

Problem 58

Sketch a ray diagram to show that when an object is placed more than twice the focal length away from a converging lens, the image formed is inverted, real, and diminished in size.

JG

Jonelle Guzman

Numerade Educator

Problem 59

Sketch a ray diagram to show that when an object is placed at twice the focal length from a converging lens, the image formed is inverted, real, and the same size as the object.

Samantha Baker

Numerade Educator

Problem 60

Sketch a ray diagram to show that when an object is placed between twice the focal length and the focal length from a converging lens, the image formed is inverted, real, and enlarged in size.

Samantha Baker

Numerade Educator

Problem 61

Sketch a ray diagram to show that when an object is a distance equal to the focal length from a converging lens, the emerging rays from the lens are parallel to each other, so the image is at infinity.

Samantha Baker

Numerade Educator

Problem 62

When an object is placed $6.0 \mathrm{~cm}$ in front of a converging lens, a virtual image is formed $9.0 \mathrm{~cm}$ from the lens. What is the focal length of the lens?

Samantha Baker

Numerade Educator

Problem 63

An object of height $3.00 \mathrm{~cm}$ is placed $12.0 \mathrm{~cm}$ from a diverging lens of focal length $-12.0 \mathrm{~cm}$. Draw a ray diagram to find the height and position of the image.

Samantha Baker

Numerade Educator

Problem 64

A diverging lens has a focal length of $-8.00 \mathrm{~cm}$.
(a) What are the image distances for objects placed at these distances from the lens: $5.00 \mathrm{~cm}, 8.00 \mathrm{~cm}$, $14.0 \mathrm{~cm}, 16.0 \mathrm{~cm}, 20.0 \mathrm{~cm} ?$ In each case, describe the
image as real or virtual, upright or inverted, and enlarged or diminished in size. (b) If the object is $4.00 \mathrm{~cm}$ high, what is the height of the image for the object distances of $5.00 \mathrm{~cm}$ and $20.0 \mathrm{~cm} ?$

Linda Winkler

Numerade Educator

Problem 65

A converging lens has a focal length of $8.00 \mathrm{~cm} .$ (a) What are the image distances for objects placed at these distances from the thin lens: $5.00 \mathrm{~cm}, 14.0 \mathrm{~cm}, 16.0 \mathrm{~cm}$
$20.0 \mathrm{~cm} ?$ In each case, describe the image as real or virtual, upright or inverted, and enlarged or diminished in size. (b) If the object is $4.00 \mathrm{~cm}$ high, what is the height of the image for the object distances of $5.00 \mathrm{~cm}$ and $20.0 \mathrm{~cm} ?$

Linda Winkler

Numerade Educator

Problem 66

Sketch a ray diagram to show that if an object is placed less than the focal length from a converging lens, the image is virtual and upright.

JG

Jonelle Guzman

Numerade Educator

Problem 67

For each of the lenses in the figure, list whether the lens is converging or diverging.

Linda Winkler

Numerade Educator

Problem 68

In order to read his book, Stephen uses a pair of reading glasses. When he holds the book at a distance of $25 \mathrm{~cm}$ from his eyes, the glasses form an upright image a distance of $52 \mathrm{~cm}$ from his eyes. (a) Is this a converging or diverging lens? (b) What is the magnification of the lens? (c) What is the focal length of the lens?

Sophie S

Numerade Educator

Problem 69

A standard " $35-\mathrm{mm}$ " slide measures $24.0 \mathrm{~mm}$ by $36.0 \mathrm{~mm}$. Suppose a slide projector produces a $60.0-\mathrm{cm}$ by $90.0-\mathrm{cm}$ image of the slide on a screen. The focal length of the lens is $12.0 \mathrm{~cm}$. (a) What is the distance between the slide and the screen? (b) If the screen is moved farther from the projector, should the lens be moved closer to the slide or farther away?

Linda Winkler

Numerade Educator

Problem 70

An object that is $6.00 \mathrm{~cm}$ tall is placed $40.0 \mathrm{~cm}$ in front of a diverging lens. The magnitude of the focal length of the lens is $20.0 \mathrm{~cm}$. Find the image position and size. Is the image real or virtual? Upright or inverted?

Sophie S

Numerade Educator

Problem 71

Samantha puts her face $32.0 \mathrm{~cm}$ from a makeup mirror and notices that her image is magnified by $1.80$ times.
(a) What kind of mirror is this?
(b) Where is her facerelative to the radius of curvature or focal length?
(c) What is the radius of curvature of the mirror?

Linda Winkler

Numerade Educator

Problem 72

A converging lens made of glass $(n=1.5)$ is placed under water $(n=1.33)$. Describe how the focal length of the lens under water compares to the focal length in air? Draw a diagram to illustrate your answer.

Sophie S

Numerade Educator

Problem 73

An object $8.0 \mathrm{~cm}$ high forms a virtual image $3.5 \mathrm{~cm}$ high located $4.0 \mathrm{~cm}$ behind a mirror. (a) Find the object distance. (b) Describe the mirror: is it plane, convex, or concave? (c) What are its focal length and radius of curvature?

Sophie S

Numerade Educator

Problem 74

$\mathrm{A}$ point source of light is placed $10 \mathrm{~cm}$ in front of a concave mirror; the reflected rays are parallel. What is the radius of curvature of the mirror?

Sophie S

Numerade Educator

Problem 75

A radar station is located at a height of $24.0 \mathrm{~m}$ above the shoreline. When the radar is aimed at a spot $150.0 \mathrm{~m}$ out to sea, it detects a whale at the bottom of the ocean. If it takes $2.10 \mu \mathrm{s}$ for the radar to send out a beam and receive it again, how deep is the ocean where the whale is swimming?

Linda Winkler

Numerade Educator

Problem 76

A ray of light in air is incident at an angle of $60.0^{\circ}$ with the surface of some benzene con-
tained in a shallow tank made of crown glass.
What is the angle of refraction of the light ray when it enters the glass at the bottom of the tank?

Linda Winkler

Numerade Educator

Problem 77

A ray of light passes from air through dense flint glass and then back into air. The angle of incidence on the first glass surface is $60.0^{\circ} .$ The thickness of the glass is $5.00 \mathrm{~mm}$; its front and back surfaces are parallel. How far is the ray displaced as a result of traveling through the glass?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 78

A glass prism bends a ray of blue light more than a ray of red light since its index of refraction is slightly higher for blue than for red. Does a diverging glass lens have the same focal point for blue light and for red light? If not, for which color is the focal point closer to the lens?

Linda Winkler

Numerade Educator

Problem 79

A laser beam is traveling through an unknown substance. When it encounters a boundary with air, the angle of reflection is $25.0^{\circ}$ and the angle of refraction is $37.0^{\circ}$. (a) What is the index of refraction of the substance? (b) What is the speed of light in the substance?
(c) At what minimum angle of incidence would the light be totally internally reflected?

Sophie S

Numerade Educator

Problem 80

In many cars the passenger's side mirror says: "Objects in the mirror are closer than they appear." (a) Does this mirror form real or virtual images? (b) Since the image is diminished in size, is the mirror concave or convex? Why? (c) Show that the image must actually be closer to the mirror than is the object. (d) How then can the image seem to be farther away?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 81

A scuba diver in a lake aims her underwater spotlight at the lake surface. (a) If the beam makes a $75^{\circ}$ angle of incidence with respect to a normal to the water surface, is it reflected, transmitted, or both? Find the angles of the reflected and transmitted beams (if they exist).
(b) Repeat for a $25^{\circ}$ angle of incidence.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 82

Laura is walking directly toward a plane mirror at a speed of $0.8 \mathrm{~m} / \mathrm{s}$ relative to the mirror. At what speed is her image approaching the mirror?

Sophie S

Numerade Educator

Problem 83

$\mathrm{Xi}$ Yang is practicing for his driver's license test. He notices in the rearview mirror that a tree, located directly behind the automobile, is approaching his car as he is backing up. If the car is moving at $8.0 \mathrm{~km} / \mathrm{h}$ in reverse, how fast relative to the car does the image of the tree appear to be approaching?

Sophie S

Numerade Educator

Problem 84

A plane mirror reflects a beam of light. Show that the rotation of the mirror by an angle $\alpha$ causes the beam to rotate through an angle $2 \alpha$.

Sophie S

Numerade Educator

Problem 85

A $3.00$ -cm-high pin, when placed at a certain distance in front of a concave mirror, produces an upright image $9.00 \mathrm{~cm}$ high, $30.0 \mathrm{~cm}$ from the mirror. Find the position of the pin relative to the mirror and the image. Draw a ray diagram to illustrate.

Sophie S

Numerade Educator

Problem 86

A dentist holds a small mirror $1.9 \mathrm{~cm}$ from a surface of â patient's tooth. The image formed is upright and $5.0$ times as large as the object. (a) Is the image real or virtual? (b) What is the focal length of the mirror? Is it concave or convex? (c) If the mirror is moved closer to the tooth, does the image get larger or smaller? (d) For what range of object distances does the mirror produce an upright image?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 87

An object of height $5.00 \mathrm{~cm}$ is placed $20.0 \mathrm{~cm}$ from a converging lens of focal length $15.0 \mathrm{~cm} .$ Draw a ray diagram to find the height and position of the image.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 88

A letter on a page of the compact edition of the $O x$ ford English Dictionary is $0.60 \mathrm{~mm}$ tall. A magnifying glass (a single thin lens) held $4.5 \mathrm{~cm}$ above the page forms an image of the letter that is $2.4 \mathrm{~cm}$ tall. (a) Is the image real or virtual? (b) Where is the image? (c) What is the focal length of the lens? Is it converging or diverging?

Linda Winkler

Numerade Educator

Problem 89

An object is placed $10.0 \mathrm{~cm}$ in front of a lens. An upright, virtual image is formed $30.0 \mathrm{~cm}$ away from the lens. What is the focal length of the lens? Is the lens converging or diverging?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 90

A manufacturer is designing a shaving mirror, which is intended to be held close to the face. If the manufacturer wants the image formed to be upright and as large as possible, what characteristics should he choose? (type of mirror? long or short focal length relative to the object distance of face to mirror?)

William Dunkerton

William Dunkerton

Numerade Educator

Problem 91

Show that the deviation angle $\delta$ for a ray striking a thin converging lens at a distance $d$ from the principal axis is given by $\delta=d / f$. Therefore, a ray is bent through an angle $\delta$ that is proportional to $d$ and does not depend on the angle of the incident ray (as long as it is parax-ial). [Hint: Look at the figure and use the small-angle approximation $\sin \theta=\tan \theta=\theta$ (in radians)].

Linda Winkler

Numerade Educator

Problem 92

The focal length of a thin lens is $-20.0 \mathrm{~cm}$. A screen is placed $160 \mathrm{~cm}$ from the lens. What is the $y$ -coordinate of the point where the light ray shown hits the screen? The incident ray is parallel to the central axis and is $1.0 \mathrm{~cm}$ from that axis.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 93

The angle of deviation through a triangular prism is defined as the angle between the incident ray and the emerging ray (angle $\delta$ ). It can be shown that when the angle of incidence $i$ is equal to the angle of refraction $r^{\prime}$ for the emerging ray, the angle of deviation is at a minimum. Show that the minimum deviation angle ( $\delta_{\min }=D$ ) is related to the prism angle $A$ and the index of refraction $n$, by
$$
n=\frac{\sin \frac{1}{2}(A+D)}{\sin \frac{1}{2} A}
$$
[Hint: For an isosceles triangular prism, the minimum angle of deviation occurs when the ray inside the prism is parallel to the base, as shown in the figure.]

Ajay Singhal

Numerade Educator

Problem 94

A ray of light is reflected from two mirrored surfaces as shown in the figure. If the initial angle of incidence is $34^{\circ}$, what are the values of angles $\alpha$ and $\beta$ ? (The figure is not to scale.)

William Dunkerton

William Dunkerton

Numerade Educator

Problem 95

A beam of light consisting of a mixture of red, yellow, and blue light originates from a source submerged in some carbon disulfide. The light beam strikes an interface between the carbon disulfide and air at an angle of incidence of $37.5^{\circ}$ as shown in the figure. The carbon disulfide has the following indices of refraction for the wavelengths present: red $(656.3 \mathrm{~nm}), n=1.6182 ;$ yellow $(589.3 \mathrm{~nm}), 1.6276 ;$ blue $(486.1 \mathrm{~nm}), 1.6523 .$
Which color(s) is/are recorded by the detector located above the surface of the carbon disulfide?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 96

A ray of light is incident normally from air onto a glass $(n=1.50)$ prism as shown in the figure with Problem
26. (a) Draw all of the rays that emerge from the prism and give angles to represent their directions. (b) Repeat part (a) with the prism immersed in water $(n=1.33)$.
(c) Repeat part (a) with the prism immersed in a sugar solution $(n=1.50)$.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 97

A concave mirror has a radius of curvature of $14 \mathrm{~cm}$. If a pointlike object is placed $9.0 \mathrm{~cm}$ away from the mirror on its principal axis, where is the image?

Narayan Hari

Numerade Educator

Problem 98

A glass block $(n=1.7)$ is submerged in an unknown liquid. A ray of light inside the block undergoes total internal reflection. What can you conclude concerning the index of refraction of the liquid?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 99

Ray diagrams often show objects that conveniently have one end on the principal axis. Draw a ray diagram and locate the image for the object shown in the figure that extends beyond the principal axis.

William Dunkerton

William Dunkerton

Numerade Educator

Problem 100

A $5.0-\mathrm{cm}$ -tall object is placed $50.0 \mathrm{~cm}$ from a lens with focal length $-20.0 \mathrm{~cm}$. (a) How tall is the image? (b) Is the image upright or inverted?

William Dunkerton

William Dunkerton

Numerade Educator

Problem 101

The vertical displacement $d$ of light rays parallel to the axis of a lens is measured as a function of the vertical displacement $h$ of the incident ray from the principal axis as shown in part (a) of the figure. The data are graphed in part (b) of the figure. The distance $D$ from the lens to the screen is $1.0 \mathrm{~m}$. What is the focal length of the lens for paraxial rays?

William Dunkerton

William Dunkerton

Numerade Educator