Summary 567

SUMMARY The goal of Chapter 17 has been to understand and apply the wave model of light.

GENERAL PRINCIPLES

The Wave Model The wave model considers li ght to be a wave propagating through space. Interference and diffraction arc importanl. The wave model is appropriate when light interacts with objects whose s ize is comparable to the wavelength of li ght or roughly less than about 0.1 mill.

IMPORTANT CONCEPTS

The index of refraction of a materiaJ determines the speed of light in that material: v = clll. The index of refraction of a material is always greater than 1, so that v is always less than c.

The wavelength A in a material w ith index of re fract ion II ;s shorter than the wavelength "-vue in a vacuum: A = Avajn.

The!reqlleflCY of light does not change as it moves from one materia l to another.

APPLICATIONS

Huygens' principle says that each point on a wave front is the source of a spheri cal wavelet. The wave front at a later lime is tangent to all the wavelets.

Diffraction is the spreading of a wave after it passes through an open ing.

Constructive and destructive interference are due to the overl ap of two or more waves as they spread behind openings.

Diffraction from a single slit Interference from mUltiple slits

A s ing le slit of width a has a bright central maximum of width

2AL

Waves overlap as they spread out behind s ljts. Bright frin ges are seen on the viewing screen aI positions where tbe path-length difference tJ.r between successive s lits is equal to mA, where m is an integer.

w= -­a

that is flan ked by weaker secondary maxima.

Secondary maxima Centml maximum

Dark fringes

Dark fringes are located at angles such that

asin0l' = pA p = 1,2,3, ..

If Ala « I , then from the small-angle approx imation,

pA e = -r a

pAL V = --• P a

Circular aperture of diameter D

A bright central max imum of diameter

2.44AL w= ---

D

is surrounded by circular secondary maxima. The first dark fringe is located at

l.22A e = -­, D l. 22AL

Y'= -D-

For an apermre of any shape, a smaller opening causes a greater spreading of the wave behind the opening.

Double slit with separation d

Equall y spaced bright fringes are located at 11111111111111 iliA

O"'= d mAL

Y"'= - d- m = 0, 1,2, ...

f ' . a AL The .rmge spacmg is )' = d

Diffraction grating wi th s lit spac ing d

Very bright and narrow fringes are located at angles and positions

Y,., = LtanO,.,

Thin-film interference

I

Interfere nce occurs between the waves reflected from the two surfaces of a thin film with index of refraction II. A wave that reflects from a sur­face at which the index of refract ion increases has a phase change.

Interference

Constructive

Destructi ve

o or 2 phase changes

A 2r=m­

II

21 = (III + -'-)~ 2 II

1 phase change

21 = (III + -'-)~ 2 II

A 2/=111-

II

568 CHAPTER 17 Wave Optics

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

QUESTIONS

Conceptual Questions

I. The frequency of a light wave in air is 5.3 X 10 14 Hz. Is the fre­quency of thi s wave higher, lower, or the same after the light enters a piece of glass?

2. Ra nk in order the following accord ing to the ir speeds, from slowest to fastest: (i) 425-nm-wavclength light th ro ugh a pane of glass, (ii) 500-nm-wavelength light through a ir, (iii ) 540-nm­wave length light thro ugh water, ( iv) 670-nm-wave1ength li ght through a diamond, and (v) 670-nm-wavelength light through a vacuum.

3. The wave length of a light wave is 700 nm in a ir; thi s light appears red. If thi s wave enters a pool of water, its wavelength becomes Aai/II = 530 nm. If you were swimmi ng underwate r, the light would still appear red. Gi ven thi s, what property of a wave detennines its color?

4. A do uble-slit interfe rence experiment shows fr inges on a screen. The entire experiment is then immersed in water. Do the fri nges on the screen get closer together, farther apart , remain the same, or disappear en ti rely? Explain.

5. Figure Q 17.5 shows the frin ges observed in a dou ble-sli t inter­ference experiment when the two sli ts are ilJuminated by white light. The central max i-

FIGURE 017 .5

mum is whi te, but as we move away from the central max imum, the fr inges become less di st inct and more colorful. What is spe­c ial about the centra l max imum that makes it white? Explain the presence of colors in the oullying fr inges.

6. In a double-sli t in terfe rence experiment, interference fringes are observed on a distan t screen. The width of both s!.i ts is then dou­bled without changing the distance between the ir centers. a . What happens to the spacing of the fringes? Explain. b. What happens to the in tensity of the bright fr inges? Expla in.

7. Figure Q 17.7 shows the view- ABC D E ing screen in a double-slit experiment with monochro-mat ic light. Fringe C is the central max imum. a. What wi ll happen to the

fringe spacing if the wave- FIGURE 017.7

length of the light is decreased?

b. What will happen [ 0 the fri nge spac ing if the spacing between the slits is decreased?

c. What will happen to the fri nge spac ing if the distance to the screen is decreased?

d. Suppose the wavelength of the light is 500 nm. How much fart her is it from the dot on the screen in the cen ter of fringe E to the le ft slit than it is from the dot to the ri ght sl it?

Problems labeled INT integrate significant material from earlier

chapters; BID are of biological or medical interest.

8. Figure Q 17.7 is the inte rfere nce pattern seen on a viewing screen behind 2 slits. Suppose the 2 slits were replaced by 20 sl its hav ing the same spacing d between adjacen t sli ts . a. Wo uld the num ber of fri nges on the screen increase,

decrease, or stay the same? b. Would the fringe spac ing increase, decrease , or stay the

same? c. Would the widlh of each fr inge inc rease, decrease, or Slay

the same? d. Would the brighlness of each frin ge increase, decrease, or

stay the same? 9. Figure Q 17.9 shows the light intensity on a viewing screen

behi nd a si ngle sli t of width a. The light's wavelength is A. Is A < n, ,.\ = n, A > a, or is it not possible to tell? Expla in.

FIGURE 017 .9

10. Figure Q I7.I O shows the light intensity on a viewing screen behi nd a c ircular aperture . What happens to the widt h of the cen­tral max imum if a. The wavelength is increased? b. The diamete r of the apertu re

is increased? c. How wi ll the screen appear if

the aperture diameter is less FIGURE 017 .10 than the light wavelength ?

11. Why does light re flec ted fro m peacock feathe rs change color BID when you see the fea thers at a different angle? 12. White light is inciden t on a diffract ion grating. What color is the

central max imum of the interference pattern? 13. A soap bubble usually pops because some part of it becomes too

thin due to evaporation or drainage of flu id. The change in thi ck­ness also changes the color of light the bubble refl ects. Why?

14 . An oil fil m on top of water has one patch that is much thi nner than the wave length of visible light. The index of refraction of the oil is less than that of water. Will the re fl ection from that extremely thin part of the film be bright or dark? Explain.

15. Should the antireflect ion coat ing of a microscope objective lens designed for use with ul trav iolet light be thi nner, thicker, or the same thickness as the coating on a lens des igned for visible light?

16. If the thin wedge of air be tween the two pl ates of glass in Figu re 17.21 were replaced by water, would the distance between the fringes inc rease, decrease, or remain the same? Explain .

17. Example 17.5 showed that a thin film whose thickness is one­quarter of the wavelength of light in the film serves as an antire­Ilec ti on coating when coated on glass . In Example 17.5,

" fi lm < " glass ' If a quarter-wave thickness film with " film> " glass were used instead, would the film still serve as an antireflection coat ing? Explain.

18. You are standing aga inst the wall near a corner of a large build­ing. A friend is standing against the wall that is around the cor­ner from you. You can' t see your friend. How is it that you can hear her when she talks to you?

Multiple-Choice Questions

19. I Light of wave length 500 nm in air enters a glass block with index of refraction n = 1.5. When the light enters the block, which of the followin g properties of the light will not change? A. The speed of the light B. The frequency of the light C. The wavelength of the light

20. j The frequency of a light wave in air is 4.6 X 1014 li z. What is the wavelength of thi s wave after it enters a pool of water? A. 300 nm B. 490 nm C. 650 nm D. 870 nm

2 1. I Light passes through a diffraction grating with a slit spacing of 0.001 mm. A viewing screen is 100 cm behind the grating. If the light is bille, with a wavelength of 450 nm, at about what di stance from the cenler of the interference pattern will the firsl­order max imum appear? A. 5 cm B. 25 cm C. 50cm

PROBLEMS

Section 17.1 What Is Light?

D. 100 cm

I. III a. How long does it take light to trave l through a 3.0-mm­thick piece of window glass?

b. Through what thickness of water could light travel in the same amount of time?

2. I a. How long ( in ns) does it take light to trave l 1.0 III in vacuum?

b. What di stance does li ght travel in water, glass, and diamond during the time that it travels 1.0 m in a vacuum?

3. 1111 A 5.0-cm-thick layer of oil (n = 1.46) is sandwiched between a 1.0-cm-thick sheet of glass and a 2.0-cm-thick sheet of poly­styrene plastic (II = 1.59). How long (in ns) does it take light incident perpendicular to the glass to pass through thi s S.O-cm­thick sandwich?

4. II A light wave has a 670 nm wavelength in air. Its wavelength in a transparent so lid is 420 nm. a. What is the speed of light in thi s sol id? b. What is the light's frequency in the sol id?

5. II How much time does it take a pulse of light to travel through 150 m of water?

6. II A helium-neon laser beam has a wavelength in air of633 nm. It takes 1.38 ns for the light to travel through 30.0 cm of an unknown liquid. What is the wave length of the laser beam in the liquid?

22.

23.

Problems 569

II Blue light of wavelength 450 nm passes through an interfer­ence grating with a slit spac ing of 0.001 mm and makes an interference pattern on the wall. How many bright fringes will be seen ? A. I B. 3 C. 5 D. 7 I Yellow light of wavelength 590 nm passes through a diffrac­tion grat ing and makes an interference pattern on a screen 80 em away. The first bright frin ges are 1.9 cm from the center of the pattern. How many lines per mm does thi s grating have? A. 20 B. 40 C. 80 D. 200

24. I Light passes through a ID-,um-wide slit and is viewed on a sc reen I m behind the sl it. If the width of the slit is narrowed,

25.

26.

the band of light on the screen wiU A. Become narrower. B. Become wider. C. Stay about the same. II Blue light of wavelength 450 nm passes through a 0.20-mm­wide slit and illuminates a screen 1.2 m away. How wide is the centralmaximllm of the diffraction pattern? A. 1.2 mm B. 2.0 mm C. 2.7 mm D. 5.4 mm II A green laser beam of wavelength 540 nm passes through a pinhole and illuminates a dartboard 3.0 m past the pinhole . The first minimum in the intensity coincides with the ring surround­ing the bull's-eye. 12 mm in diameter. What is the diameter of the pinhole? A. 0.14 mm B. 0.33 mm C. 0.59 mm D. 1.2 mm

Section 17.2 The Interrerence or Light

7. II Two narrow slit s 50,um apart are illuminated with light of wavelength 500 nm. What is the angle of the m = 2 bright fringe in radians? In degrees?

8. III Light from a sodium lamp (A = 589 nm) illuminates two nar­row slilS. The frin ge spacing on a screen ISO cm behind the slits is 4.0 mm. What is the spacing (in mm) between the two slits?

9. II Two narrow slits are illuminated by light of wavelength A. The slits are spaced 20 wavelengths apart. What is the angle, in radi­ans, between the central maximum and the m = I bright fringe?

10. II A double-slit experiment is performed with light of wave­length 600 nm. The bright intelference frin ges are spaced 1.8 mm apart on the viewing screen. What will the fringe spacing be if the light is changed to a wavelength of 400 nm?

11. 11111 Light fro lll a heliulll-neon laser (A = 633 nm) is used to illu­minate two narrow slits. The interference patte rn is observed on a sc reen 3.0 m behind the slilS. Twel ve bright frin ges are seen, spanning a di stance of 52 mm . What is the spacing (in mm) between the slits?

12. II Two narrow slits are 0.12 mm apart. Light of wave length 550 nm illuminates the slits, causing an interference pattern on a screen 1.0 m away. Light from each slit travels to the m = I max imum on the right side of the central max imum. How much farthe r did the light from the le ft slit travel than the light from the right slit?

570 CHAPTER 17 Wave Optics

13. III Consider a point P on the viewing screen of a double-slit interference experiment. This point is 75% of the way from the center of the 3rd bright fringe to the center of the 4th bright fringe. If the wavelength of the light is 600 nm, what is the extra di stance that the wave from one s lit traveled compared to the wave from the other?

Section 17.3 The DitTraction Grating

14. III A diffraction grat ing with 750 s lits/mm is illuminated by light that gives a first-order diffraction angle of 34.0°. What is the wavelength of the li ght?

15. III! A I .O-cm-wide diffraction grating has 1000 slits. It is illumi­nated by light of wavelength 550 nm. What are the angles of the first two diffraction orders?

16. III Light of wavelength 600 nm illuminates a diffraction grat ing. The second-order maximum is at angle 39.5°. How many lines per millimeter does thi s grating have?

17. III A lab technician uses laser light with a wavelength of 670 nm to test a diffraction grating. When the grating is 40.0 cm from the screen, the first-order maxima appear 6.00 cm from the center of the pattern. How many lines per millimeter docs this grating have?

18. II The human eye can readily detect wavelengths from about 400 nm to 700 nm. If white light illuminates a diffraction grat­ing having 750 lines/mm, over what range of angles does the vis ible 11/ = I spectrum extend?

19. III A diffraction grating with 600 lines/mm is illuminated with li ght of wavelength 500 nm. A very wide viewing screen is 2.0 m behind the grat ing. a. What is the distance between the two m = J fringes? b. How many bright fringes can be seen on the sc reen?

20. II A 500 line/mm diffraction grating is illuminated by light of wavelength 510 nm. How many diffraction orders are seen, and what is the angle of each?

Section 17.4 Thin-Film Interference

21. II What is the thinnes t film of MgF2 (n = 1.38) on g lass that produces a strong re fl ecti on for orange li ght with a wavelength

of 600 nm? 22. 1111 A very thin oil film (11 = 1.25) floats on water (11 = 1.33).

What is the thinnest film that produces a strong reflect ion for green light with a wavelength of 500 nm?

23. II A film with 1/ = 1.60 is deposited on g lass. What is the thinnest film that will produce construct ive interference in the reflection of light with a wavelength of 550 nm?

24. II Antireflec tion coat ings can be used on the inner surfaces of BID eyeglasses to reduce the reflection of stray light into the eye,

thus reducing eyestrain . a. A 90-nm-thick coating is applied to the lens. What must

be the coat ing's index of refract ion to be most effecti ve at 480 nm? Assume that the coating's index of refraction is less than that of the lens.

b. If the index of refraction of the coating is 1.38, what thick­ness should the coat ing be so as to be most effect ive at 480 nm? The thinnest possible coat ing is best.

25. II SolarceUs are given antireflection coat ings to maximize their efficiency. Conside r a s ilicon so lar ce ll (II = 3.50) coated with a layer of s ilicon dioxide (11 = 1.45). What is the minimum

coating thickness that will minimize the reOection at the wave­length of 700 nm, where so lar ce lls are most effic ien t?

26. III A thin film of MgF2 (II = 1.38) coats a piece of glass. Con­structi ve interference is observed for the reflect ion of light with wavelengths of 500 nm and 625 nm. What is the thinnest film for which thi s can occur?

27. I Looking stra ight downward into a rain puddle whose surface is covered with a thin film of gasoline, you notice a swirling pattern of colors caused by interference inside the gasol ine film . The point directly beneath YOll is colored a beautiful iridescent g reen. You happen to remember that the index of refraction of gaso line is 1.38 and that the wavelength of green light is about 540 nm. What is the minimum possible thickness o f the gaso­line layer directly beneath you?

Section 17.5 Single-Slit Interference

28. II A helium-neon laser (A = 633 nm) iLluminates a s ingle sl it and is observed on a screen 1.50 m behind the slit. The distance between the first and second minima in the diffraction pattern is 4.75 mm. What is the width (in mm) of the sl it?

29. III For a demonstration , a professor uses a razor blade to cut a thin sl it in a piece of aluminum fo il. When she shines a laser pointer (..\ = 680 nm) through the slit o nto a screen 5.5 m away, a diffraction pattern appears. The bright band in the cen ter of the pattern is 8.0 cm wide. What is the width of the slit?

30. II A 0.50-mm-wide sl it is illuminated by light o f wavelength 500 nm. What is tbe width of the central maximum on a screen 2.0 m behind the slit?

3 1. III The second minimum in the diffraction pattern of a 0.10-mm­wide sl it occurs at 0.700. What is the wavelength of the li ght?

32. II What is the width of a s lit for which the first minimum is at 45° when the s lit is illuminated by a helium-neon lase r (A ~ 633 nm)? Hint: The small-angle approximation is not valid at 45°.

Section 17.6 Circular-Aperture Diffraction

33 . III A 0.50-mm-diameter ho le is illuminated by light o f wave­length 500 nm. What is the width of the central maximum on a sc reen 2.0 m behind the slit?

34. II Light from a hel ium-neon laser (A = 633 nm) passes through a c ircular aperture and is observed on a screen 4.0 m behind the aperture. The width of the central maximum is 2.5 cm. What is the diameter (in mm) of the hole?

35. III You want to photograph a c ircular diffraction pattern whose central maximum has a diameter of 1.0 cm. You have a helium­neon laser (..\ = 633 nm) and a 0.12-mm-diameter pinhole. How far behind the pinhole should you place the viewing screen?

36. II Infrared light of wavelength 2.5 JLm iUuminates a 0.20-mm­diameter hole. What is the angle of the first dark fringe in rad i­ans? In degrees?

General Problems

37. III An advanced computer sends info rmation to its various parts via infrared light pulses traveling through s ilicon fibers (1/ = 3.50). To acquire data from memory, the central process­ing unit sends a light-pulse request to the me mory unit. The

memory unit processes the req uest, then sends a data pulse back to the central processing unit. The memory uni t takes 0.50 ns to process a request. If the information has to be obtained from memory in 2.00 ns, what is the maximum distance the memory unit can be from the cen tral processi ng uni t?

38. 1111 Figure PI7.38 shows the light Imensily (mW/m!)

39.

40.

41.

42.

intensity on a sc reen behind a double slit. The sl it spacing is 0.20 mm and the wavelength of the li ght is 600 nm. What is the distance from the slit s to the screen?

':1 /\1\/\1\/\ FtGURE P17 .38

11111 Figure P17.38 shows the light intens ity on a screen behind a doub le slit. The sl it spacing is 0.20 mm and the sc reen is 2.0 m behind the slits. What is the wave length of the light? II Your friend has been given a laser for her birthday. Unfonu­nate ly, she did not receive a manual with it and so she doesn' t know the wave length that it emits. You help her by performing a double-slit experiment, wi th sli ts separated by 0.36 mm. You find that the two brigh t fringes are 5.5 mm apart on a sc reen I .6 m from the slits. What is the wave length the laser emi ts? I A doub le sli t is illuminated simultaneously with orange Ught of wave length 600 nm and light of an unknown wavelength . The III = 4 bri ght fringe of the unknown wave length overl aps the 11/ = 3 bright orange fringe. What is the unknown wave-length? 11111 A laser beam, with a wavelength of 532 nm, is d irected exac tl y perpendicular to a sc reen hav ing two narrow slits spaced 0. 15 mm apart. Interference fr inges, including a centra l max imum, are observed on a sc reen 1.0 m away. The direct ion of the beam is then slowly rOLaLed around an axis parallel to the sli ts to an ang le of 1.0°. By what distance does the cent ral maxi­mum on the screen move?

43. I A laser beam of wavelength 670 nm shines through a diffrac­tion grat ing that has 750Iines/mm. Sketch the pallern that appears on a sc reen 1.0 m behind the grating, not ing distances on your drawing and expla ining where these numbers come from.

44. IU The two most prominen t wavelengths in the light emilled by a hydrogen discharge lamp are 656 nm (red) and 486 nm (b lue). Light from a hydrogen lamp illuminates a diffract ion grat ing wi th 500 lines/mOl, and the I.i ghl is observed on a sc reen 1.50 m behind the grat ing. What is the distance between the firSl-order red and blue frin ges?

45. 1111 A triple-s li t experiment illuminates three equally spaced, nar­row slits wi th light of wave length A. The intens ity of the wave from each slit is I I' Consider a point on a di stant sc reen at an ang le Stich that the path-length difference between any two adjacent slits is AI2. What is the intensity at thi s po in t? Give your answer as a multiple of I I'

46. II A diffrac tion grating consists of 100 slits. If the number of sl its is increased to 200, wi th the same spac ing, by what factor docs the maximum intensity of the bright fringes on the sc reen increase?

47. III A diffraction grating produces a first-order maximum at an angle of 20.0°. What is the angle of the second-order max imum?

48. I A diffraction grating is illuminated simultaneously with red light of wave length 660 nm and light of an unknown wave­length. The fifth-order maximum of the unk nown wave length exac tl y overlaps the third-order max imum of the red light. What is the unknown wavelength?

Problems 571

49. 1111 White light (400-700 nm) is inc ident on a 600 linelmm dif­fraction grating. What is the width of the fi rst-order rainbow on a screen 2.0 m behi nd the grating?

50. IIll1 For your sc ience fa ir project you need to des ign a diffraction grating that will di sperse the visib le spectrum (400-700 nm) over 30.0° in fi rst order. a. How many lines per millimeter docs your grati ng need? b. What is the first-order diffract ion angle of light from a

sodium lamp (A = 589 nm)? 5 1. III Figure P 17.5 1 shows the interference pattern on a screen 1.0

m behind an 800 li ne/mm diffract ion grating. What is the wave­length of the light?

89.7cmT\~ FIGURE P17.S1 43.6 em 43.6 em

52. III Figure PI 7.5 I shows the in terference pattern on a screen 1.0 m behind a diffraction grat ing. The wavelength of the light is 600 nm. How many lines per millimeter docs the grat ing have?

53. II Because sound is a wave, it is poss ible to make a diffraction INT grating for sound from a large board with several parallel slots

for the sound to go through. When 10 kHz sound waves pass through such a grating, li steners 10m from the grating repon "loud spotstl 1.4 m on both sides of center. What is the spacing between the s lo t~? Use 340 m/s for the speed of sound.

54. 1111 The shiny surface of a CO is im printed with millions of tiny pits, arranged in a pattern of thousands of essentially concentric c ircles that act like a rellcction grating when li ght shines on them. You decide to determi ne the distance between those c ir­cles by aiming a laser pointer (with A = 680 11m) perpendicular to the disk and measuring the diffraction pallern reflected on to a screen 1.5 m from the d isk. The central bright spot you expected to see is blocked by the laser pointe r itself. You do find two other bright spots separated by 1.4 m, one on either side of the missi ng central spot. The rest of the panem is appar­ently diffracted at angles too great to show on your screen. What is the distance between the c ircles on the CO's surface?

55. III If sun ligh t shines stra ight onto a peacock feather, the feather BID appears bright blue when viewed from 15° on e ither side of the

inc ident beam of sunl ight. The blue color is due to diffraction from the melan in bands in the feather barbules, as was shown in the photograph on page 558 . Blue light with a wavelength of 470 nm is diffracted at 15° by these bands (thi s is the first-order diffraction) while othe r wave lengths in the sunlight are dif­fracted at different ang les. What is the spacing o f the melanin bands in the feather?

56. III The wings of some beetles Bk) have closely spaced paralle l li nes

of melanin , causing the wing to act as a re nec tion grati ng. Sup­pose sun light shines straigh t onto a beetle wing. If the melanin lines on the wing are spaced 2.0 Mill apart , what is the fi rst-order diffraction angle for green light (A ~ 550 nm)?

57. 1111 A diffraction grating hav ing 500 li nes/mm di ffracts visible light at 30°. What is the light's wave length?

572 CHAPTER 17 Wave Optics

58. III Light emitted by Element X passes through a diffraction grat­ing hav ing 1200 lines/mm. The interference pattern is observed on a screen 75 .0 cm behind the grating. Bright fringes are seen on the screen at di stances of 56.2 cm, 65.9 cm, and 93.5 cm from the central maximum. No other fringes are seen. a. What is the value of 111 fo r each of these diffracted wave­

lengths? Ex plain why only one value is poss ible. b. What are the wavelengths of light emitted by Element X?

59. II Helium atoms em it light at several wavelengths. Light from a helium lamp illuminates a diffraction grat ing and is observed on a screen 50.00 cm behind the grat ing. The emiss ion at wave­length 501.5 nm c reates a first-order bright fringe 2 1.90 cm from the centra l maximum. What is the wavelength of the bri ght frin ge that is 3 1.60 cm from the central maximum ?

60. II A sheet of glass is coated with a 500-nm-thick layer of o il (/I = 1.42). a . For what visible wavelengths of light do the reflected waves

interfe re constructi vely? b. For what visible wavelengths of light do the reflected waves

interfere destructively? c . What is the color of reflected light? What is the color of

transmilled light? 61. A soap bubble is essentiall y a thin film of water surrounded

by air. The colors you see in soap bubbles are prod uced by inte rference. What visible wavelengths of light are strongly renec ted from a 390-nm-thick soap bubble? What color would such a soap bubble appear to be?

62. II In a single-sl it experiment, the slit width is 200 times the wave length of the light. What is the width of the central maxi­mum on a screen 2.0 m behind the slit?

63. III You need to use your cell phone, which broadcasts an 830 MHz signal , but you' re in an alley between two mass ive, rad io-wave­absorbing buildings that have onl y a 15 m space between them . What is the angular width , in degrees , of the electromagnetic wave after it emerges from between the buildings?

64. III Light from a sodium lamp (A = 589 nm) illuminates a nar­row sli t and is observed on a screen 75 cm behind the slit. The di stance between the fiIst and third dark fringes is 7.5 mm. What is the width (in mm) of the sl it?

65. 111 1 The open ing to a cave is a tall , 3D-cm-wide crack. A bat that INT is preparing to leave the cave emits a 30 kHz ultrasonic chirp.

How wide is the "sound beam" 100 m outside the cave open­ing? Use v sound = 340 m/s.

66. I For what slit-width-to-wavelength ratio does the first mini­mum of a s ingle-sl it diffraction pattern appear at (a) 300, (b) 60°, and (c) 90°? Hint: The small-angle approx imat ion is not valid .

67. III Figure P1 7.67 shows the light intensity on a screen behind a single slit. The wave length of t.he light is 500 nm and the screen is 1.0 m behind the slit. What is the width (i n mm) of the slit?

FIGURE P17 .67 o 2 x (cm)

3

68. III Figure P 17.67 shows the light intensity on a screen behind a single sli t. The wavelength of the light is 600 nm and the slit width is 0.15 mm. What is the di stance from the sl it to the screen?

69. II Figure P1 7.69 shows the light in tensity on a screen 2.5 m behind an aperture. The apertu re is illuminated with light of wavelength 600 nm. a. Is the aperture a single slit or a double slit? Explain. b. If the aperture is a single slit , what is its width? If it is a dou­

ble sl it , what is the spac ing between the slits?

' OIl~1\1\1\1\ x(cm)

o 2 4 5

FIGURE P17.69

'

OI

T"Y c-!\ c-O 2 3 4 5

FIGURE P17 .70

x (em) 6

70. II Figure P1 7.70 shows the l.ight intensity on a screen 2.5 m behind an aperture. The aperture is illuminated with light of wavelength 600 nm. a. Is the aperture a single slit or a double slit? Explain. b. If the aperture is a single slit, what is its width? If it is a dou­

ble slit , what is the spacing between the slits? 7 1. III One day, after pulling down yo ur window shade, you not ice

that sunlight is pass in g through a pinhole in the shade and making a small patch of light on the far wall. Hav ing recently studied optics in yo ur phys ics c lass, yo u' re not too surpri sed to see that the patch of light seems to be a c ircular diffraction pattern . It appears that the central max imum is about 3 cm across, and you estimate that the distance from the window shade to the wall is about 3 Ill. Knowing that the average wavelength of sunli ght is about 500 nm, estimate the diameter of the pinhole.

72. 1111 A radar for tracking airc raft broadcasts a 12 GHz microwave INT beam from a 2.0-m-diame ter c ircular radar antenna. From a

wave perspect ive , the antenna is a circular aperture through which the microwaves diffract. a. What is the diameter of the radar beam at a distance of30 km? b. If the antenna emits 100 kW of power, what is the average

microwave intensity at 30 km? 73. 11111 A helium-neon laser (A = 633 nm), shown in Figure PI 7 .73,

is built with a glass lube of inside diameter 1.0 mm. One mirror is partiaJly transmitti ng to allow the laser beam out. An electri ­cal discharge in the tube causes it to glow like a neon light. From an optical perspective, the laser beam is a light wave that dirfracts out through a I.O-mm-diameter circular opening. a. Expl ain why a laser beam can' t be pCIJeclly paralle l, with

no sprcading. b. The angle 01 to the first minimum is called the divergence

angle of a laser beam. What is the di vergence anglc of thi s laser beam?

c. What is the diameter (in mm) of the laser beam after it trav­els 3.0 m?

d. What is the diameter of the laser beam after it lravels 1.0 km?

FIGURE P17 .73

Mirror / Discharge 1.0mm

Laser beam

Eleclnxles __ mirror

Power supply

c 74. II In the laser range. findi ng experi ­

ments of Example 17.10, the laser beam fired toward the moon spreads Oll t as it trave ls because it d iffrac ts through a c ircular ex it as it leaves the laser. In order for the re flec ted li ght to be bright enough to detect, the laser spot on the moon must be no more than I km in d iameter. Staying w ithin thi s

di ameter is accomplished by us ing a spec ial Jarge-di ameter lase r. If A = 532 nm, what is the mini ­mum diameter of the c irc ular opening from which the laser beam emerges? The calth-moon distance is 384,000 km.

Passage Problems

The Blue Morpho Butterny rna

The bri ll iant blue color o f a blue morpho butterfl y is, like the colors of peacock feathers, due to inte rference. Figure P 17.7Sa shows an easy way to demonstrate thi s: If a drop of the clear so lvent acetone is placed on the wi ng of a blue morpho butterfl y, the color changes from a brilli ant bl ue to an equally bri ll ian t green-returni ng to blue once the acetone evaporates. There would be no change if the color were due to pigmen t.

FIGURE P17 .7S

Light refl ections from layers have

different pat h length s.

•

Stop to Think 17.1: IlJ > "l > 11 2' A = AvaJI/ , so a shorter wave· length corresponds to a hi gher index of refract ion.

Stop to Think 17.2: B. When the screen is closer, you don't have (0

move as far from the center to reach the poin t where the path ~1ength difference is one wavelength.

Stop to Think 17.3: Sma ller. The fr inge spacing ~y is di rect ly pro~ portional to the wavelength A.

Stop to T hink 17.4: D. Longer wavelengths have larger d iffracti on angles. Red light has a longer wave length than vio let light , so red light is d iffrac ted fill"lher from the cen ter.

Problems 573

A cross section of a scale from the wing of a blue morpho b Uller~

fl y reveals the sou rce of the butterfly 'S co lor. As Figure P1 7.75b shows, the sca les are covered wi th structures that look like small C hristmas trees. Light striking the wings reflects from d iffere nt lay~ ers of these structures, and the d iffer ing path lengths cause the re fl ec ted light to interfere construct ive ly or destruct ively, depend~ ing on the wave length. For light at normal inc idence, b lue light exper iences construc ti ve interfere nce whi le othe r colors undergo destructive interference a nd cancel. AcelOne fill s the spaces in the scales with a fl uid o f index of refract ion" = 1.38; thi s changes the conditions for construct ive interference and results in a c hange in color. 75. I The coloring of the blue morpho bu tterfl y is protecti ve. As

the butterfly flaps its wings, the ang le at which light strikes the wings cha nges. T hi s causes the butter fl y's color to change and makes it d ifficult for a predator to fo llow. This color change is because: A. A d iffract ion pattern appears on ly at ce rtain angles. B. The index of refraction of the wing ti ssues changes as the

wi ng fl exes. e. The mot ion of the wings causes a Do ppler shift in the

reflected light. D. As the a ngle changes. the d ifferences in paths among light

re flected fro m d ifferent surfaces change, resulti ng in con ~

stnlct ive interference fo r a d iffe rent color. 76. I T he change in colo r when acetone is p laced on the wing is

due to the d iffere nce be tween the ind ices of refrac tion of acetone and a ir. Consider li ght of some part ic ul ar color. In acetone, A. The freq uency oflhe light is less than in air. B. The freq uency of the li ght is greater than in a ir. e. The wavelength of the light is less than in air . D. The wavelength of the li ght is greater than in air.

77. I The scales on the butterfly wings are actuall y made of a transparen t material with index of re[ract ion 1.56. Light reflects from the surface of the scales because A. The scales' index of refrac ti on is d ifferent from that of air. B. The scales' index of refrac ti on is similar to that of glass. e. The scales ' density is di ffe re nt from that o f air. D. Differen t colors of light have d ifferent wavelengths.

Stop to T hink 17.5: B. An ex tra path d ifference of AI2 must be added to change from construc ti ve to destruct ive interference. In thin- fi lm in terference, one wave passes twice through the film . To inc rease the path length by Al2. the thi ckness needs to be increased by only one~ h alf thi s, or Al4.

Stop to Think 17.6: B or C . The width of the centra l max imum, whic h is propo rtional to Ala, has increased. T hi s could occur e ither because the wave length has increased or because the slit wid th has decreased.