602 CHAPTER 18 Ray Optics

SUMMARY The goal of Chapter 18 has been to understand and apply the ray model of light .

GENERAL PRINCIPLES

Reflection Law of reflection: Or = OJ

Reflection can be specular (mirror-l ike) or diffuse (from rough surfaces).

,

~ s / = s

Plane min'ors: A virtual image is formed al pi with s' = s, where s is the object distance and s' is the image distance.

IMPORTANT CONCEPTS

The ray model of light

Lighllravels along slraightlines, called light rays, at speed v = clll.

A light ray continues forever unless an interaction with mattcr causes it to reflect, refract, scatler, or be absorbed.

Light rays come from self-luminous or reflective objects. Each point on the object sends rays in all directions.

Ray diagrams represent all the rays emilled by an object by only a few se lect rays.

In order for the eye to see an object (or image), rays from the object or image must enter the eye.

APPLICATIONS

Ray tracing for lenses

Three spec ial rays in three basic s ituations:

Converging lens Real image

Converging lens Virtual image

Ray tracing for mirrors

Three speciaJ rays in three bas ic s ituations:

Concave mirror Real image

Concave mirror Virtual image

Di verging lens Virtual image

Convex mirror Virtual image

Refraction Snell's law of refraction:

Index of refraction is 11 = elv. The ray is closer to the normal on the side with the larger index of refraction .

::~i,,\d,"l 0 , ~No'm'l II I I

", - : 0l Refr.J.cted

I ray ,

If II 2 < n I , total internal reflection (TIR) occurs when the angle of incidence 81 is greater than 8" = s in- I(1l 2/1l 1) '

Image formation

If rays diverge from P and, after interact ing with a lens or mirror. appear to diverge from pi without actuall y pass ing through P', then P' is a virtual jrnage of P.

These rays appear 10

p ' ha\'c come frOl

J pt.

.;:

imagc

If rays diverge from P and interact with a lens or mirror so that the refracted rays converge al pi , then pi is a real image of P. Rays actually pass through a real image.

p Thc;;e r.lYS <lctually

r~~~::~;::;~,""~~dO come, from p ' J Real image 1

j

The thin-lens equation

For a lens or curved mirror, the object distance s, the image di stance s' , and the focal length f are related by the thin -lens equation:

I I I - + - ~ ­s ::;' J

The magnification of a lens or mirror is 111 = - s'/S.

Sign conventions for the thin-lens equation:

Quantity Positive when Negative when

s Always Not treated here

" Rea/ image; on opposite Virtual image; on same s ide o f a lens from object, side of a lens as object, or in front of a mirror or behind a mirror

f Converging le ns or Diverging le ns or concave mirror convex mirror

m Image is upright. Image is in verted.

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Problem difficulty is labeled as I (straightforward) to 11111 (challenging).

o QUESTIONS

Conceptual Questions

1. The idea of light rays goes back to the a nc ient Greeks. How~ ever, they believed that "visual rays" were emiffed by eyes. If

yo u were transported back in time, what arguments would you

present to those cillly scien ti sts to conv ince- them that vis ion has something 10 do with rays going into, rather than out of, eyes?

2. Is there any property that di stingui shes a light ray emitted by a

li ght bulb and one that has been diffusely re fl ected by the page of a book? Explain.

3. If you turn o n your car headl ights during the day, the road ahead of you doesn ' t appear to get brighter. Why not?

4. Can you see th e rays from the sun on a clear day? Why or why

not? How about when they stream through a forest o n a foggy

morning? Why or why not? 5. If you take a walk on a summe r night alo ng a dark, unpaved

road in the woods, with a flashlight po inting at the ground sev­eral yards ahead to guide your steps , any water-filled potholes

are not iceable because they appear much darker than the sur­

rounding dry road . Explain why. 6. You are looking at the image of a pencil in a mirror, as shown in

Figure QI 8.6. a . What happens to the image

you see if the top half of the mirror, down to the mid­

point, is covered with a piece of cardboard? Explain.

b. What happens to the image

you see if the bottom half of the mirror is cove red with a piece o f cardboard?

7. In Tlte Toilet oj Vel/its by Velazquez (see Figure Q I S. 7),

.-- Midpoint

FIGURE Q18.6

we can see the face of Venus in the mirror. Can she see her own

face in the mirror, when the mirror is held as shown in the pic­ture? If yes, explain why; if not, what does she see instead?

FIGURE 018.7

Diego de Silva Velazquez (1599-1660), "Venus and Cupid," 1650. Oil on canvas. National Gallery, London. Erich Lessing/Art Resource, N.Y.

Questions 603

Problems labeled !NT integrate signif icant material from earlier

chapters; BKl are of biological or medical interest.

S. In Manet 's A Bar at the Folies-Bergere (see Figure QI8.S) the re fl ect io n of the barmaid is visible in the mirror behind he r. Is thi s th e reflection you would expect if th e mirror 's surface is

parallel to the bar? Where is the man seen fac ing her in the mir­

ror actually standing?

FIGURE 018.8

Edouard Manet 1832·1883. "Bar at the Folies-Bergere'·. 1881182, Oil on C:mvas. 37 13116" X 51" (90 X 130 cm). Courtauld Institute Galleries. London. AKG­Images.

9. Explain why am bulances have the word ;;AMBULANCE" written backward on the front of them,

10. a. Cons ider aile point o n an object near a lens. What is the mini­

mum number of rays needed to locate its image point? b, For each po int on the object, how many rays from thi s point

actually strike the le ns and refract to the image point? II. When you look at yo ur re necti on in the bowl of a spoon, it is

ups ide down. Why is this? 12, A concave mi rror brings the sun 's rays to a foclls at a di stance

of 30 em from the mirror. If the mirro r were submerged in a

swimming pool, would the sun 's rays be focused nearer to, fur­ther from, or at the same di stance from the mirror?

13. A studen t draws the ray diagram shown in

Figure Q IS. 13 but forgets

to label the object , the image, or the type of lens used, Us ing the diagram ,

f:is: FIGURE 018.13 ~

explain whether the lens is converg ing or diverging, which

arrow represents the object, and which represents the image, 14. An objecL at distance is fro m a concave mirror of foca l length

j produces a rea l image at distance s' fro m the mirror. Sup­pose the mirror is replaced by a new mirror, at the same loca­tion , with focal le ng th 1f. Will the new image be real or virtual? Will it s distance from the mirro r be more or less than s'? Ex plain .

15 . A lens can be used to start a fire by focusing an image of the sun onto a piece of flamm able material. All other things being equal , would a lens with a sho rt focal length or a long focal

length be better as a fire starter? Explain .

604 CHAPTER 18 Ray Optics

Multiple-Choice Questions

Questio ns 16 thro ugh 18 are con­cerned with the situation sketched in Figure QI8. 16, in which a beam of light in the air encounters a trans­

parent block with index of refrac­tion 1/ = 1.53. Some of the light is reflected and some is refracted. 16. I What is Ol ?

A. 40° B. 45° C. 50° D. 90°

17. j What is 6l1? A. 200 B. 30° C. 50°

FIGURE 018.16

8,

0,

18. I Is there an ang le of incidence between 0° and 90° such that all of the light will be reflected? A. Yes, al an angle greater than 50° B. Yes, at an angle less than 50° C. No

19. I A 2.0-m-lall man is 5 .0 m from the converging lens o f a cam­era. His image appears on a detector that is 50 mm behind the lens. How lall is hi s image on the detec tor? A. IOmm B.20mm C. 25 mm D.50mm

20. II You are 2.4 m from a plane mirror, and you would like to take a picture of yourself in the mirror. You need to manuall y adjusl the focus of the camera by dialing in the di slance 10 what you are photographing. What di stance do you dial in? A. 1.2 m B. 2.4 m C. 3 .6 m D. 4.S m

2 1. II Fig ure Q18 .2 l shows an object and lens pos itioned to fonn a well-focused, inverted image on a view ing sc reen . Then a piece of cardboard is lowered just in front of the lens to cover the top Iwlf of the lens. What happe ns to the image on Ihe sc reen?

FIGURE 018 .21

PROBLEMS

Section 18.1 The Ray Model of Light

Lens

1. II A 5.0-ft-tall girl stands on level ground . The sun is 25° above the hori zon. How lo ng is her shadow?

2. 1111 A 10-cm-diameter di sk em its light uniforml y from its sur­face. 20 cm from thi s di sk, along its ax is, is an S.O-cm-diameter opaque black d isk; the faces o f the two di sks are parallel. 20 em beyond the black di sk is a white v iewi ng screen. The lighted disk illumi nates the sc reen. but there' s a shadow in the center due to the black disk. What is the diameter of the COli/pIe/ely

dark part of thi s shadow? 3. 11111 A point source of light illuminates an apertu re 2 .00 m away.

A 12.0-cm-wide bright patch of li ght appears o n a sc reen 1.00 m behind the aperture. How wide is the aperture?

A. Nothing . B. The upper half of the image will van ish. C. The lower half of the image will vani sh. D. The image will become fuzzy and o ut offocus. E. The image will become dimmer.

22. II A rea l image of an objec t can be formed by A. A converg ing lens. B. A plane mirror. C. A convex mirror. D. Any o f the above.

23. I An objec t is 40 cm from a converg ing lens with a focal length of 30 cm. A real image is Formed o n the other s ide of the le ns, 120 cm from the lens. What is the magn ificat ion? A. 2D B. 3.0 C. 4D D. 1.33 E. 0.33

24. I The lens in Figure QIS.24 is used to produce a real image of a candle flame. What is the 12 em 36 em

focal length of the lens? A. 9.0cm B. 12 cm C. 24 c m D. 36cm E. 48cm

FIGURE 018.24

25. I A converg ing lens of focal length 20 cm is used to form a real image 1.0 m away fro m the lens. How far from the lens is the o bject? A. 20cm B. 25em C. 50cm D. 100cm

26. I You look at yourself in a convex mirror. Your image is A. Erect. B. Inverted. C. It 's impossible 10 teU withollt knowing how far you are from

the mirror and its focal length. 27. II An objec t is 50 em from a diverging lens with.a focal length

of - 20 em. How far from the lens is the image, and on which side o f the lens is it? A. 14 cm, on the same side as the object B. 14 cm, on the opposite s ide fro m the objec t C. 30 cm, on the same s ide as the object D. 33 em, on the same side as the object E. 33 em, on the oppos ite s ide from the objec t

Section 18.2 Reflection

4 . I The mirror in Figure PISA deflects a horizo ntal laser beam by 60°. What is the angle ¢? FIGURE P18.4

5 . I Figure PI S.5 shows an objec t 0 in front o f a plane mirro r. Usc ray trac ing to determine from which locations A- D the object's image is vis ible.

Mirror

FIGURE P18.5 · 0

6. III A light ray leaves point A in Figure PI 8.6, reflects from the mi rror, and reaches point B. How far below the top edge does the ray strike the mirror?

7. 1111 II is 165 cm from your eyes to your toes. You 're standing

15 c111

j 5em • IDcm

A

• B 15 em

200 cm in front of a taU mi rror. FIGURE P18.6

How far is it from your eyes to the image of your toes?

Mi rror

S. HI A ray of light impinges on a mirror as shown in Figure PIS.S. A second milTor is fastened at 90° to the first. a. After ~tJ'iki n g both mirrors, at what angle relative to the

incomi ng ray does the outgoing ray emerge? b. What is the answer if the incoming angle is 30°?

3m

2m

1 m

o·

FIGURE P18.8 FIGURE P18.9

9. III A red ball is placed at point A in Figure PI S.9. a. How many images are seen by an observer aI point O? b. Where is each image located? c. Draw a ray diagram showing the formati on of each image.

Section 18.3 Refraction

10. III An underwater diver sees the sun 50° above horizontal. How high is the sun above the horizon to a fisherman in a boat above the di ver?

II . II A laser beam in air is incident on a liquid at an angle of 37° wilh respect to the normal. The laser beam's ang le in the liquid is 26°. What is the liquid 's index of refraction?

12. III A I .O-cm-thick layer of water stands on a horizon tal slab of glass. A light ray in the air is incident on the water 60° from the normal. Arterentering the glass, what is the ray's angle from the normal?

13. III A 4 .0-m-wide swimming pool is fiUed to the top. The bottom of the pool becomes completely shaded in the afternoon when the sun is 20° above the horizon. How deep is the pool?

14. II A diamond is underwate r. A li ght ray enters one face of the diamond. then travels at an angle of 30° with respect to the nor­mal. What was the ray 's angle of inc idence on the diamond?

15. II A thin glass rod is submerged in oi l. What is the critical ang le for light trave ling inside the rod?

Section 18.4 Image Formation by Refraction

16. III A biologist keeps a specimen of hi s favor ite beetle embedded in a cube of polystyrene plastic. The hapless bug appears to be 2.0 cm withi n the plasti c. What is the beetle 's ac tual di stance beneath the surface?

17. II A fi sh in a fl at-sided aquarium sees a can offish food on the counter. To the fi sh's eye, the can looks to be 30 em outside the aquarium. What is the ac tua l di stance between the can and the aquarium? (You can ignore the thin glass wall of the aquari um.)

IS. I A swim mask has a pocket of air between your eyes and the fl at glass front. a. If you look at a fi sh while

swimming underwater with a swim mask on, does the fi sh appear closer or farther than it

Problems 605

really is? Draw a ray diagram to explain. b. Does the fi sh see your face closer or farther than it really is?

Draw a ray diagram to explain.

Section 18.5 Thin Lenses: Ray Tracing

19. I An object is 30 cm in front of a converg ing lens with a focal length of 10 cm. Use ray trac ing to determine the location of the image. Is the image upright or inverted? Is it real or virtual?

20. I An object is 6.0 cm in fron t of a converging lens with a focal length of 10 cm. Use ray trac ing to determine the location of the image. Is the image upright or inverted? Is it real or virtua l?

2 1. II An object is 20 cm in front of a diverging lens with a focal length of 10 cm. Use ray tracing to determine the location of the image. Is the image upright or inverted? Is it real or virtual?

22. I An object is IS cm in front of a di verging lens with a focal length of 10 em. Use ray trac ing to determine the locati on of the image, Is the image upright or inverted? Is it real or virtual?

Section 18.6"hnage Formation with Spherical Mirrors

23. I A concave cosmetic mi rror has a focal length of 40 cm. A 5-cm-Iong mascara brush is held upright 20 cm from the mirror. Use ray trac ing to determine the location and he ight of its image. Is the image upright or inverted? Is it real or virtual?

24. I A light bulb is 60 cm from a concave mirror wi th a focaJ length of 20 cm. Use ray tracing to determine the location of its image. Is the image upright or inve rted? Is it real or virtual?

25. I The illuminat ion lights in an BID operating room use a concave

mi rror to focus an image of a bright lamp onto the surgical site. One such light has a mirror with a focal length of 15.0 cm. Use ray trac ing to find the position of its lamp when the patient is pos itioned 1.0 m from the mirror (you ' ll need a careful drawing to get a good answer).

26. II A denti st uses a curved mirror to view the back side of BID tee th on the upper jaw. Suppose she wan ts an erect image

with a magni ficat ion o f 2.0 when the mirror is 1.2 em from a tooth. (Treat thi s problem as though the object and image li e along a strai ght line.) Use ray trac ing to dec ide whether a concave or convex mirror is needed, and to est imate its focal length.

27. I A convex mi rror, like the passenger-side rearview mirror on a car, has a focal length of2.0 m. An object is 4.0 m from the mir­ror. Use ray trac ing to determine the locat ion of its image. Is the image upright or inverted? Is it real or virtual?

2S. I An object is 6 em in front of a convex mirror with a focal length of 10 cm. Use ray trac ing to determine the location of the image. Is the image upright or inverted? Is it real or virtual?

606 CHAPTER 18 Ray Optics

Section 18.7 The Thin-Lens Equation

For Problems 29 through 38, calcu late the image position and he ight. 29. I A 2.0-em-tall objec t is 40 em in front of a convergi ng lens

that has a 20 em focal length. 30. II A I .D-em-taU obj ec t is 10 em in front of a converg ing lens

that has a 30 em focal length. 3 1. II A 2.0-em-lall objec t is 15 em in front of a converg ing lens

that has a 20 em focal length . 32. 1 A l.O-em-taU object is 75 em in front of a converg ing lens

that has a 30 em focal length . 33. I A 2.0-em-tall object is 15 em in front of a diverging lens that

has a - 20 em focal length. 34. I A I.D-em-tall object is 60 e m in front of a diverging lens that

has a -30 em focal length. 35. II A 3.0-em-tall objec t is 15 em in front of a convex mirror that

has a - 25 em focal length. 36. II A 3.0-cm-tall object is 45 em in front of a convex mirror that

has a -25 em focal length. 37. I A 3.0-em-tall object is 15 em in front of a concave mirror that

has a 25 em focallenglh. 3S. I A 3.0-em-tall object is 45 em in front of a concave mirror that

has a 25 em focal length.

General Problems

39. III Starting 3.5 m from a department store mirror, Suzanne INT walks toward the mirror at 1.5 mls for 2.0 s. How far is Suzanne

from her image in the mirror after 2 .0 s? 40. IN!

4 1.

42.

43.

44.

II You slowly back away from a plane mirror at a speed of 0.10 m/s . With what speed does your image appear to be mov­ing away from you? II At what angle cp should the laser Mirror beam in FigurePIS.41 be aimed at the mirrored ceil ing in orde r to hi t the midpo int of the far waU? 1111 You ' re he lping with an experi ­me nt in which a verti ca l cy linder will rotate about its ax is by a very

E 8 Wall <"l ~_Laser beam

5.00 m

small angle. You need to dev ise a FIGURE P18 .41

way to measure thi s angle. You dec ide to use what is call ed an optical leVel: You begin by mo unting a small mirror on top of the cy linder. A lase r 5.0 m away shoots a laser beam at the mirror. Before the ex periment starts, the mi rror is adjusted to renect the lase r beam directly back to the laser. Later, you measure that the reflected laser beam, when it returns to the laser, has been deflected sideways by 2.0 mm. How many degrees has the cy linder rotated? II Figure P 18.43 shows a li ght ray incident o n a polished metal cy linder. At what angle 0 will the ray be reflected?

FIGURE P18 .43

R 2

III The place you get your ha ir c ut has two nearly parallel mir­rors 5.0 m apart. As you sit in the chair, your head is 2.0 m from

the nearer mirror. Looking toward thi s mirror, YOll first see YOllr face and then, farther away, the back of your head. (The mirrors need to be s li ghtl y nonparallel for you to be able to see the back o f your head, but you can treat them as parallel in thi s problem.) How far away does the back of your head appear to be? Neglect the Ihickness o f your head.

45. I You shine your laser po in te r through the fl at g lass s ide of a rectangular aquarium at an angle of incidence o f 45". The index of refract ion of thi s Iype of glass is 1.55. a. At what angle from the normal does the beam from the laser

pointer enter the water inside the aquarium? b. Does your answer to patl a depend on the index of refraction

of the g lass? 46. II A ray of li ght traveling lhrough ai r encounters a 1.2-cm-thi ck

sheet of glass al a 35° ang le of incidence. How far does the li ght ray travel in the glass before e merg ing on the far side?

47. II What is the angle of inc idence in a ir of a light ray whose angle of refracti on in glass is half the angle of inc idence?

48. 1111 F igure P 18.48 shows a light ray inc iden t on a glass cyl inder. What is the ang le a of the ray after it has en tered the cy linder?

FIGURE P18 .48 FIGURE P18.49

49. I if you look at a fi sh through the corner of a rectangular aquar­ium you somet imes see two fi sh, o ne on each side of the corner, as shown in Figure PI S.49. S ketch some of the li g ht rays that reach your eye from the fish to show how this can happen.

50. 11111 It 's nighttime, and you've dropped your gogg les in to a swim­ming pool that is 3.0 m deep. If you ho ld a laser pointer 1.0 m d irect ly above the edge of the poo l, you can illuminate the gog­gles if the laser beam enters the water 2.0 m from the edge. How far are the goggles from the edge of the pool?

5 1. 1111 One of the contests at the school carn ival is to throw a spear at an underwater target lying flat on the bOllom of a pool. The water is 1.0 m deep. You ' re standing on a small stool that places your eyes 3.0 m above the bottom of the pool. As you look at the target, your gaze is 30° be low horizontal. At what angle below hori zontal should you throw the spear in order to hil the target? Your rai sed arm brings the spear point to the level of your eyes as you throw it , and over thi s sho rt di stance you ean assume that the spear travels in a straight line rather than a para-

52. bolic trajectory. II I Figure P 18 .52 shows a meter sti ck lying on the bot­lo rn o f a 100-em-Iong tank with its zero mark aga inst the left edge. You look into the tank at a 30° angle, with

your li ne of sight just graz ing

~ine of sight

lStb

Zero I Meter stick

loocm

the upper left edge of the FIGURE P18 .52

50cm

tank. What mark do you see o n the meter sti ck if the tank is (a) e mpty. (b) half full o f water, and (c) complete ly full o f water?

c 53. III T he re is j ust one angle o f inci dence {3 on lo a pr ism for

whi ch the light ins ide an isosceles pri sm travels para lle l to the base and emerges at thaL same ang le {3, as shown in Figure P I8.53 . a. Fi nd an express ion for f3 in terms of the prism's apex ang le

a and index c rrcfracti on II. b. A laboratory measure ment fin ds that {3 = 52.2° fo r a prism

that is shaped as an equ ilatcraltri angle . What is the prism's index of refract ion?

" FIGURE P18.53

60"(

FIGURE P18 .54

54. 1111 W hat is the small est angle 8 1 fo r wh ich a laser beam wi ll

undergo totaJ internal refl ec tion on the hypotenuse of the glass prism in Figure PI 8.54?

55. III A I .O-em-thick layer of water stands on a hQ[izontal slab of glass. Light from wi thi n the glass is inciden t on the glass-water boundary. What is the maximum angle of inc idence for which a li ght ray can emerge in to the air above the water?

56. 111 11 The glass core of an optical fiber has index of refraction 1.60. The index of refract ion of the cladding is 1.48. What is the max­imum angle between a light ray and the wa ll of the core if the ray is to remain ins ide the core?

57. II A swimmer looks upward fro m the bottom of a 3.0-m-deep swimming pool. T he e nd of the d iving board is d irectl y above him, 2.0 m above the waler's surface . How far from the swim­mer does the board appear to be?

58. 111 1 A 150-cm-tall diver is standing completely submerged on Ihe bottom of a swimming pool full of waler. You are sitt ing on the e nd of the di ving board, almost d irectly over her. How tall does the di ver appear to be?

59. II To a fish, the 4 .00-mm-t hick aq uarium wa ll s appear on ly 3.50 mm thick. What is the index of refraction of the walls?

60. 11 A microscope is focllsed on an amoeba. When a 0. 15-mm­BIO thi ck cover glass (1/ = 1.50) is placed over the amoeba, by how

far must the microscope objecti ve be moved to bri ng the organ­ism back inlo focus? Must il be raised or lowered?

6 1. II A ray di agram can be ll sed to find the location of an objec t if yo u are g iven the location of its image and the foeal length of the mirro r. Draw a ray d iagram to find the he ight and pos ition of an objec t that makes a 2.0-cm-high upright virtual image that appears 8.0 cm behi nd a convex mi n'or of focal length 20 cm.

62. 1 A 2.0-cm-tall object is located 8 .0 cm in front of a converg­ing lens with a focal length of 10 e m. Use ray traci ng to deter­mine the locati on and height of the image. Is the image upri ght or inverted? Is it real or virtual?

63. II T he image produced by a converg ing lens is ty pica lly a d if­ferent s ize from the object itsel f. However, for a lens with focal length f there is one object di stance that will yield an image the same size as the object. What is that object d istance?

64. 1111 A near-s ighted person mig ht correc t hi s vision by wearing BIO diverging lenses with focal lengthf= -50 cm. When weari ng hi s

glasses, he looks not at actual objects but at the virtual images of those objects formed by hi s g lasses . Suppose he looks at a 12-cm-long penc il held verti cal.ly 2.0 m from hi s glasses. Use ray trac ing to determi ne the location and he ight of the image.

65. I A I .O-cm-tall object is 20 cm in front of a converg ing lens that has a 10 cm focal length . Use ray trac ing to fin d the posi-

66.

67.

68.

Problems 607

tion and he ight of the image. To do thi s accurately, use a ruler or paper with a grid . Determi ne the image d istance and image he ight by maki ng measurements on your d iagram. II A 2.0-cm-ta ll object is 20 cm in fro nt of a converging lens that has a 60 c m focal length. Use ray traci ng to fi nd the pos i­ti on and he ight of the image. To do thi s accurate ly, use a ruler or paper wi th a grid. Determi ne the image d istance and image he ight by making measurements on your d iagram. II A I.O-cm-tall object is 7.5 cm in fro nt of a d ive rging lens that has a 10 cm foca l length. Use ray trac ing to find the pos ition and height of the image. To do thi s acc urately, use a ruler or paper wi th a g rid. Determi ne the image d istance and image he ight by making measurements on your d iagram. II A 1.5-cm-tall Object is 90 em in fron l of a d ive rging le ns that has a 45 cm focal length. Use ray trac ing to fi nd the pos ition and he ight of the image. To do thi s accurately, use a ru ler or paper with a grid. Determi ne the image d istance and image he ight by making measurements on your diagram.

69. I A 1.6-m-tall woman stands 2.0 m in fro lll o r a convex fun­house mirror with a focal le ngth of 2/3 m. Use ray trac ing to determine the locat ion and he ight o rher image.

70. III A 2.0-cm-tall candle n ame is 2.0 m rrom a wa l.l. You happe n to have a lens with a focal length of 32 c m. How many places can you put the lens to rorm a well-focused image of the candle flame on the wal l? For each locat ion, what are the he ight and orientat ion of the image?

7 1. II A 2.0-c rn-d ia meter sp ider is 2.0 m fro m a wal l. Determi ne the focal length and pos ition (measured rrorn the wall) of a lens that will make-a half-s ize image of lhe sp ider on the wall.

72. III Figure P 18.72 shows a meter stick he ld lengthwise along the optical ax is of a conoave mi rror. I-Iow long is the image of the meter st ick?

FtGURE P18.72

Meter stick , lOOcm

73. II A slide projector needs to create a 98-cm-high image of a 2.0-cm-taU sl ide . The sc reen is 300 em rrom the sl ide. a. W hat focal length does the lens need? Assume that it is a

thin lens. b. I-I ow far should you place the lens from the slide?

74. IIJ The wr iti ng on the passenger-s ide mirror of yo ur car says "Warn ing! Objec ts in mirror are closer than they appear." There is no such warn­ing on the dri ver 's mirro r. Conside r a typical convex pas­senger-side mirror wi th a foca l length of -80 cm. A 1.5-m-lalJ cycl ist on a bicycle is 25 m from the mi rror. You are 1.0 m from the mi rror, and suppose, for s im­pl ic ity, that the mi rror, you, and the cycl ist all li e along a li ne. a. How fa r are you from the image of the cycl ist? b. How far wo uld you have been from the image if the mi rror

were flat? c. What is the image height? d. W hat would the image height have been if Ihe mirror were

fl at? e. W hy is there a label on the passenger-side mirror?

608 CHAPTER 18 Ray Optics

Passage Problems

Mirages

There is an interest ing opt ical effect you have likely noticed while driving along a flat stretch of road on a sunny day. A smaJI, distant dip in the road appears to be filled with water. You may even see the reflect ion of an oncoming car. But, as you gel closer, YOll rind no puddle of water after all; the shimmering surface van ishes, and you see nothing but empty road. It was onl y a mirage, the name for thi s phenomenon.

The mirage is due to the different index of refraction of hOI and cool air. The ac tual bending of the light rays that produces the mirage is subtle, but we can make a simple model as follows. When air is heated, its density decreases and so does its index of re fraction. Consequently, a pocket of hot air in a dip in a road has a lower index of refraction than the cooler air above it. Inciden t light rays with large angles of inci­dence (that is, nearly parallel to the road, as shown in Figure PI8.75) expe ri e nce total internal reflec­tion. The mirage that you see is

~ - \ Pocket or hot air

FIGURE P18.75

Stop to Think 18.1: C. There's one image behind the vertical mirror and a second behind the horizontal mirror. A third image in the cor­ner ari ses from rays that reflect twice , once off each mirror.

Stop to Think 18.2: A. The ray travels closer to the normal in both media I and 3 than in medium 2, so 1/ I and 1/ 3 are both larger than 1/ 2.

The angle is smaller in medium 3 than in medium 1, so 11 3 > " I '

Stop to Think 18.3: E. The rays from the object are diverging. With­out a lens, the rays cannol converge to form any kind of image on the screen.

due to thi s renect ion. As yo u gel nearer, the angle goes below the criti ca l angle and the re is no more total internal refl ect ion; the "water" di sappears! 75. I The pocket of hot air appears to be a pool of water because

A. Light reflects at the boundary between the hot and cool air. B. ILs density is close to that of water. C. Light refracts at the boundary between the hot and cool

air. D. The hot a ir emits blue light that is the same color as the day­

time sky. 76. I Which of these changes would allow you to get closer to the

mirage before it van ishes? A. Making the pocket of hot air nearer in temperature to the air

above it B. Looking for the mirage on a windy day, which mixes the air

layers C. Increasing the difference in temperature between the pocke t

of hot air and the air above it D. Looking at it from a greater height above the ground

77. I If you could clearly see the image of an object that was reflected by a mirage, the image would appear A. Magnified. B. With lip and down reversed. C. Farther away than the object. D. With right and left reversed.

Stop to Think 18.4: C . For a converging mirror. the focal length f is the di stance from the mirror at which incoming paralle l rays meel. The moon is so di stant that rays from any point on the moon are very nearl y parallel. Thus the image of the moon would be very nearl y at a di stance f in front of the mirror.

Stop to Think 18.5 : A. The thin-lens equat ion is l Is + l Is' = I If. The focal length of the lens is fixed. Because l Is gets smaller as s is increased, lis' must get larger to compensate. Thus s' must decrease.