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chapter 22

Problems 747

2. Figure P22.2 shows the apparatus used by Armand H. L.Fizeau (1819–1896) to measure the speed of light. Thebasic idea is to measure the total time it takes light totravel from some point to a distant mirror and back. If d isthe distance between the light source and the mirror, andif the transit time for one round-trip is t, then the speedof light is c ! 2d/t. To measure the transit time, Fizeauused a rotating toothed wheel, which converts an other-wise continuous beam of light to a series of light pulses.The rotation of the wheel controls what an observer at thelight source sees. For example, assume that the toothedwheel of the Fizeau experiment has 360 teeth and is rotat-ing at a speed of 27.5 rev/s when the light from thesource is extinguished— that is, when a burst of light pass-ing through opening A in Figure P22.2 is blocked bytooth B on return. If the distance to the mirror is 7 500 m,find the speed of light.

have rotated to position B, causing the light to bereflected to the eye? (b) What is the next-higher angularvelocity that will enable the source of light to be seen?Figure P22.5 shows an apparatus used to measure the dis-tribution of the speeds of gas molecules. The device con-sists of two slotted rotating disks separated by a distance d,with the slots displaced by the angle ". Suppose the speedof light is measured by sending a light beam toward theleft disk of this apparatus. (a) Show that a light beam willbe seen in the detector (that is, will make it through bothslots) only if its speed is given by c ! #d/", where # is theangular speed of the disks and " is measured in radians.(b) What is the measured speed of light if the distancebetween the two slotted rotating disks is 2.500 m, the slot inthe second disk is displaced 1/60 of 1 degree from the slotin the first disk, and the disks are rotating at 5 555 rev/s?

5.

d

A

BC

Toothedwheel

Mirror

Figure P22.2 (Problems 2 and 3)

3. In an experiment designed to measure the speed of lightusing the apparatus of Fizeau described in the precedingproblem, the distance between light source and mirrorwas 11.45 km and the wheel had 720 notches. The experi-mentally determined value of c was 2.998 $ 108 m/s.Calculate the minimum angular speed of the wheel forthis experiment.

4. Albert A. Michelson very carefully measured the speed oflight using an alternative version of the technique devel-oped by Fizeau. (See Problem 22.2.) Figure P22.4 showsthe approach Michelson used. Light was reflected fromone face of a rotating eight-sided mirror towards a station-ary mirror 35.0 km away. At certain rates of rotation, thereturning beam of light was directed toward the eye of anobserver as shown. (a) What minimum angular speedmust the rotating mirror have in order that side A will

Observer

Rotatingmirror

Lightsource

Stationarymirror

35.0 km

AB

Figure P22.4

Motor

Detector

Beam

d

uv

Figure P22.5

Section 22.2 Reflection and RefractionSection 22.3 The Law of Refraction

6. The two mirrors in Figure P22.6 meet at a right angle. Thebeam of light in the vertical plane P strikes mirror 1 asshown. (a) Determine the distance the reflected light beamtravels before striking mirror 2. (b) In what direction doesthe light beam travel after being reflected from mirror 2?

Mirror2

Mirror1

Lightbeam

P

40.0°

1.25 m

Figure P22.6

7. An underwater scuba diver sees the Sun at an apparentangle of 45.0° from the vertical. What is actual directionof the Sun?

8. Light is incident normal to a 1.00-cm layer of water that lieson top of a flat Lucite® plate with a thickness of 0.500 cm.How much more time is required for light to passthrough this double layer than is required to traverse thesame distance in air (nLucite ! 1.59)?

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748 Chapter 22 Reflection and Refraction of Light

A laser beam is incident at an angle of30.0° to the vertical onto a solution of corn syrup in water. Ifthe beam is refracted to 19.24° to the vertical, (a) what isthe index of refraction of the syrup solution? Supposethe light is red, with wavelength 632.8 nm in a vacuum.Find its (b) wavelength, (c) frequency, and (d) speed in thesolution.

10. Light containing wavelengths of 400 nm, 500 nm, and650 nm is incident from air on a block of crown glass at anangle of 25.0°. (a) Are all colors refracted alike, or is onecolor bent more than the others? (b) Calculate the angle ofrefraction in each case to verify your answer.

11. Light of wavelength !0 in a vacuum has a wavelength of438 nm in water and a wavelength of 390 nm in benzene.(a) What is the wavelength !0? (b) Using only the givenwavelengths, determine the ratio of the index of refrac-tion of benzene to that of water.

12. Light of wavelength 436 nm in air enters a fishbowl filledwith water, then exits through the crown-glass wall of thecontainer. Find the wavelengths of the light (a) in thewater and (b) in the glass.

13. A ray of light is incident on the surface of a block of clearice at an angle of 40.0° with the normal. Part of the lightis reflected and part is refracted. Find the angle betweenthe reflected and refracted light.

14. The laws of refraction and reflection are the same forsound as for light. The speed of sound is 340 m/s in airand 1 510 m/s in water. If a sound wave traveling in airapproaches a plane water surface at an angle of incidenceof 12.0°, what is the angle of refraction?

15. The light emitted by a helium–neon laser has a wave-length of 632.8 nm in air. As the light travels from air intozircon, find (a) its speed in zircon, (b) its wavelength inzircon, and (c) its frequency.

16. A flashlight on the bottom of a 4.00-m-deep swimmingpool sends a ray upward and at an angle so that the raystrikes the surface of the water 2.00 m from the pointdirectly above the flashlight. What angle (in air) does theemerging ray make with the water’s surface?How many times will the incident beam shown in FigureP22.17 be reflected by each of the parallel mirrors?

17.

9.

20. Find the time required for the light to pass through theglass block described in Problem 19.The light beam shown in Figure P22.21 makes an angleof 20.0° with the normal line NN " in the linseed oil.Determine the angles # and #". (The refractive index forlinseed oil is 1.48.)

21.

Mirror Mirror

1.00 m

1.00 m

Incident beam5.00°

Figure P22.17

18. A ray of light strikes a flat 2.00-cm-thick block of glass (n $1.50) at an angle of 30.0° with the normal (Fig. P22.18).Trace the light beam through the glass, and find the anglesof incidence and refraction at each surface.

19. When the light ray in Problem 18 passes through the glassblock, it is shifted laterally by a distance d (Fig. P22.18).Find the value of d.

2.00 cm

d

30.0°

Figure P22.18 (Problems 18, 19, and 20)

Linseed oil

Water

20.0°

N "

N

Air

u"

u

Figure P22.21

22. A submarine is 300 m horizontally out from the shore and100 m beneath the surface of the water. A laser beam issent from the sub so that it strikes the surface of the waterat a point 210 m from the shore. If the beam just strikesthe top of a building standing directly at the water’s edge,find the height of the building.

23. Two light pulses are emitted simultaneously from asource. The pulses take parallel paths to a detector 6.20 maway, but one moves through air and the other through ablock of ice. Determine the difference in the pulses’ timesof arrival at the detector.

24. A narrow beam of ultrasonic waves reflects off the livertumor in Figure P22.24. If the speed of the wave is 10.0%less in the liver than in the surrounding medium, deter-mine the depth of the tumor.

50.0°

12.0 cm

Liver

Tumor

Figure P22.24

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17.

9.


21.

Mirror Mirror

1.00 m

1.00 m

Incident beam5.00°

Figure P22.17



2.00 cm

d

30.0°


Linseed oil

Water

20.0°

N "

N

Air

u"

u

Figure P22.21




50.0°

12.0 cm

Liver

Tumor

Figure P22.24

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17.

9.


21.

Mirror Mirror

1.00 m

1.00 m

Incident beam5.00°

Figure P22.17



2.00 cm

d

30.0°


Linseed oil

Water

20.0°

N "

N

Air

u"

u

Figure P22.21




50.0°

12.0 cm

Liver

Tumor

Figure P22.24

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17.

9.


21.

Mirror Mirror

1.00 m

1.00 m

Incident beam5.00°

Figure P22.17



2.00 cm

d

30.0°


Linseed oil

Water

20.0°

N "

N

Air

u"

u

Figure P22.21




50.0°

12.0 cm

Liver

Tumor

Figure P22.24

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17.

9.


21.

Mirror Mirror

1.00 m

1.00 m

Incident beam5.00°

Figure P22.17



2.00 cm

d

30.0°


Linseed oil

Water

20.0°

N "

N

Air

u"

u

Figure P22.21




50.0°

12.0 cm

Liver

Tumor

Figure P22.24

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17.

9.


21.

Mirror Mirror

1.00 m

1.00 m

Incident beam5.00°

Figure P22.17



2.00 cm

d

30.0°


Linseed oil

Water

20.0°

N "

N

Air

u"

u

Figure P22.21




50.0°

12.0 cm

Liver

Tumor

Figure P22.24

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Problems 749

A beam of light both reflects and refracts at the surfacebetween air and glass, as shown in Figure P22.25. If theindex of refraction of the glass is ng , find the angle of inci-dence, !1, in the air that would result in the reflected rayand the refracted ray being perpendicular to each other.[Hint : Remember the identity sin(90° " !) # cos !.]

25. 32. The index of refraction for violet light in silica flint glassis 1.66, and that for red light is 1.62. What is the angulardispersion of visible light passing through an equilateralprism of apex angle 60.0° if the angle of incidence is50.0°? (See Fig. P22.32.)

n g

u1

Figure P22.25

26. Three sheets of plastic have unknown indices of refrac-tion. Sheet 1 is placed on top of sheet 2, and a laser beamis directed onto the sheets from above so that it strikes theinterface at an angle of 26.5° with the normal. Therefracted beam in sheet 2 makes an angle of 31.7° withthe normal. The experiment is repeated with sheet 3 ontop of sheet 2, and with the same angle of incidence, therefracted beam makes an angle of 36.7° with the normal.If the experiment is repeated again with sheet 1 on top ofsheet 3, what is the expected angle of refraction in sheet3? Assume the same angle of incidence.

27. An opaque cylindrical tank with an open top has a diame-ter of 3.00 m and is completely filled with water. When theafternoon Sun reaches an angle of 28.0° above the horizon,sunlight ceases to illuminate the bottom of the tank. Howdeep is the tank?

28. A cylindrical cistern, constructed below ground level, is3.0 m in diameter and 2.0 m deep and is filled to the brimwith a liquid whose index of refraction is 1.5. A smallobject rests on the bottom of the cistern at its center. Howfar from the edge of the cistern can a girl whose eyes are1.2 m from the ground stand and still see the object?

Section 22.4 Dispersion and Prisms29. The index of refraction for red light in water is 1.331, and

that for blue light is 1.340. If a ray of white light enters thewater at an angle of incidence of 83.00°, what are the un-derwater angles of refraction for the blue and red compo-nents of the light?

30. A certain kind of glass has an index of refraction of 1.650for blue light of wavelength 430 nm and an index of 1.615for red light of wavelength 680 nm. If a beam containingthese two colors is incident at an angle of 30.00° on apiece of this glass, what is the angle between the twobeams inside the glass?

A ray of light strikes the midpoint ofone face of an equiangular (60°–60°–60°) glass prism(n # 1.5) at an angle of incidence of 30°. (a) Trace thepath of the light ray through the glass, and find the anglesof incidence and refraction at each surface. (b) If a smallfraction of light is also reflected at each surface, find theangles of reflection at the surfaces.

31.

Visible light

Angulardispersion

Deviation ofyellow light

Screen

RO

YG

B

V

Figure P22.32

Section 22.7 Total Internal Reflection33. Calculate the critical angles for the following materials

when surrounded by air: (a) zircon, (b) fluorite, and(c) ice. Assume that $ # 589 nm.

34. For light of wavelength 589 nm, calculate the criticalangle for the following materials surrounded by air: (a) diamond and (b) flint glass.

35. Repeat Problem 34, but this time suppose that the materi-als are surrounded by water.

A beam of light is incident from air onthe surface of a liquid. If the angle of incidence is 30.0°and the angle of refraction is 22.0°, find the critical anglefor the liquid when surrounded by air.

37. A plastic light pipe has an index of refraction of 1.53. Fortotal internal reflection, what is the minimum angle ofincidence if the pipe is in (a) air? (b) water?

38. Determine the maximum angle ! for which the light raysincident on the end of the light pipe in Figure P22.38 aresubject to total internal reflection along the walls of thepipe. Assume that the light pipe has an index of refrac-tion of 1.36 and that the outside medium is air.

36.

2.00 mm

u

Figure P22.328

39. Consider a common mirage formed by superheated airjust above a roadway. A truck driver whose eyes are 2.00 mabove the road, where n # 1.000 3, looks forward. She hasthe illusion of seeing a patch of water ahead on the road,where her line of sight makes an angle of 1.20° below thehorizontal. Find the index of refraction of the air justabove the road surface. [Hint : Treat this as a problem oneinvolving total internal reflection.]

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Problems 749

A beam of light both reflects and refracts at the surfacebetween air and glass, as shown in Figure P22.25. If theindex of refraction of the glass is ng , find the angle of inci-dence, !1, in the air that would result in the reflected rayand the refracted ray being perpendicular to each other.[Hint : Remember the identity sin(90° " !) # cos !.]

25. 32. The index of refraction for violet light in silica flint glassis 1.66, and that for red light is 1.62. What is the angulardispersion of visible light passing through an equilateralprism of apex angle 60.0° if the angle of incidence is50.0°? (See Fig. P22.32.)

n g

u1

Figure P22.25

26. Three sheets of plastic have unknown indices of refrac-tion. Sheet 1 is placed on top of sheet 2, and a laser beamis directed onto the sheets from above so that it strikes theinterface at an angle of 26.5° with the normal. Therefracted beam in sheet 2 makes an angle of 31.7° withthe normal. The experiment is repeated with sheet 3 ontop of sheet 2, and with the same angle of incidence, therefracted beam makes an angle of 36.7° with the normal.If the experiment is repeated again with sheet 1 on top ofsheet 3, what is the expected angle of refraction in sheet3? Assume the same angle of incidence.

27. An opaque cylindrical tank with an open top has a diame-ter of 3.00 m and is completely filled with water. When theafternoon Sun reaches an angle of 28.0° above the horizon,sunlight ceases to illuminate the bottom of the tank. Howdeep is the tank?

28. A cylindrical cistern, constructed below ground level, is3.0 m in diameter and 2.0 m deep and is filled to the brimwith a liquid whose index of refraction is 1.5. A smallobject rests on the bottom of the cistern at its center. Howfar from the edge of the cistern can a girl whose eyes are1.2 m from the ground stand and still see the object?

Section 22.4 Dispersion and Prisms29. The index of refraction for red light in water is 1.331, and

that for blue light is 1.340. If a ray of white light enters thewater at an angle of incidence of 83.00°, what are the un-derwater angles of refraction for the blue and red compo-nents of the light?

30. A certain kind of glass has an index of refraction of 1.650for blue light of wavelength 430 nm and an index of 1.615for red light of wavelength 680 nm. If a beam containingthese two colors is incident at an angle of 30.00° on apiece of this glass, what is the angle between the twobeams inside the glass?

A ray of light strikes the midpoint ofone face of an equiangular (60°–60°–60°) glass prism(n # 1.5) at an angle of incidence of 30°. (a) Trace thepath of the light ray through the glass, and find the anglesof incidence and refraction at each surface. (b) If a smallfraction of light is also reflected at each surface, find theangles of reflection at the surfaces.

31.

Visible light

Angulardispersion

Deviation ofyellow light

Screen

RO

YG

B

V

Figure P22.32

Section 22.7 Total Internal Reflection33. Calculate the critical angles for the following materials

when surrounded by air: (a) zircon, (b) fluorite, and(c) ice. Assume that $ # 589 nm.

34. For light of wavelength 589 nm, calculate the criticalangle for the following materials surrounded by air: (a) diamond and (b) flint glass.

35. Repeat Problem 34, but this time suppose that the materi-als are surrounded by water.

A beam of light is incident from air onthe surface of a liquid. If the angle of incidence is 30.0°and the angle of refraction is 22.0°, find the critical anglefor the liquid when surrounded by air.

37. A plastic light pipe has an index of refraction of 1.53. Fortotal internal reflection, what is the minimum angle ofincidence if the pipe is in (a) air? (b) water?

38. Determine the maximum angle ! for which the light raysincident on the end of the light pipe in Figure P22.38 aresubject to total internal reflection along the walls of thepipe. Assume that the light pipe has an index of refrac-tion of 1.36 and that the outside medium is air.

36.

2.00 mm

u

Figure P22.328

39. Consider a common mirage formed by superheated airjust above a roadway. A truck driver whose eyes are 2.00 mabove the road, where n # 1.000 3, looks forward. She hasthe illusion of seeing a patch of water ahead on the road,where her line of sight makes an angle of 1.20° below thehorizontal. Find the index of refraction of the air justabove the road surface. [Hint : Treat this as a problem oneinvolving total internal reflection.]

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40. A jewel thief hides a diamond by placing it on the bottomof a public swimming pool. He places a circular raft onthe surface of the water directly above and centered overthe diamond, as shown in Figure P22.40. If the surfaceof the water is calm and the pool is 2.00 m deep, find theminimum diameter of the raft that would prevent the dia-mond from being seen.

10.0° intervals from 0° to 90.0°. (d) Do the same for lightrays traveling up to the interface through the glass.

45. A layer of ice having parallel sides floats on water. If lightis incident on the upper surface of the ice at an angle ofincidence of 30.0°, what is the angle of refraction in thewater?

46. A light ray of wavelength 589 nm is incident at an angle !on the top surface of a block of polystyrene surroundedby air, as shown in Figure P22.46. (a) Find the maximumvalue of ! for which the refracted ray will undergo totalinternal reflection at the left vertical face of the block.(b) Repeat the calculation for the case in which the poly-styrene block is immersed in water. (c) What happens ifthe block is immersed in carbon disulfide?

d

2.00 m

Raft

Diamond

Figure P22.40

A room contains air in which the speed of sound is343 m/s. The walls of the room are made of concrete, inwhich the speed of sound is 1 850 m/s. (a) Find thecritical angle for total internal reflection of sound atthe concrete–air boundary. (b) In which medium mustthe sound be traveling in order to undergo total internalreflection? (c) “A bare concrete wall is a highly efficientmirror for sound.” Give evidence for or against this state-ment.

42. Three adjacent faces (that all share a corner) of a plasticcube of index of refraction n are painted black. A clearspot at the painted corner serves as a source of divergingrays when light comes through it. Show that a ray fromthis corner to the center of a clear face is totally reflectedif

43. The light beam in Figure P22.43 strikes surface 2 at thecritical angle. Determine the angle of incidence, !i .

n " !3.

41.

Surface 1

Surf

ace

242.0°

60.0°

42.0°

ui

Figure P22.43

ADDITIONAL PROBLEMS44. (a) Consider a horizontal interface between air above and

glass with an index of 1.55 below. Draw a light rayincident from the air at an angle of incidence of 30.0°.Determine the angles of the reflected and refracted rays,and show them on the diagram. (b) Suppose instead thatthe light ray is incident from the glass at an angle ofincidence of 30.0°. Determine the angles of the reflectedand refracted rays, and show all three rays on a newdiagram. (c) For rays incident from the air onto theair–glass surface, determine and tabulate the angles ofreflection and refraction for all the angles of incidence at

u

Figure P22.46

Figure P22.47 shows the path of a beam of light throughseveral layers with different indices of refraction. (a) If !1 #30.0°, what is the angle !2 of the emerging beam?(b) What must the incident angle !1 be in order to havetotal internal reflection at the surface between themedium with n # 1.20 and the medium with n # 1.00?

47.

n = 1.60

n = 1.40

n = 1.20

n = 1.00

u1

u2

Figure P22.47

48. The walls of a prison cell are perpendicular to the fourcardinal compass directions. On the first day of spring,light from the rising Sun enters a rectangular window inthe eastern wall. The light traverses 2.37 m horizontally toshine perpendicularly on the wall opposite the window. Aprisoner observes the patch of light moving across thiswestern wall and for the first time forms his own under-standing of the rotation of the Earth. (a) With what speeddoes the illuminated rectangle move? (b) The prisonerholds a small square mirror flat against the wall at onecorner of the rectangle of light. The mirror reflects lightback to a spot on the eastern wall close beside the window.How fast does the smaller square of light move across thatwall? (c) Seen from a latitude of 40.0° north, the risingSun moves through the sky along a line making a 50.0°angle with the southeastern horizon. In what directiondoes the rectangular patch of light on the western wall of

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d

2.00 m

Raft

Diamond

Figure P22.40




n " !3.

41.

Surface 1Su

rfac

e 2

42.0°

60.0°

42.0°

ui

Figure P22.43



u

Figure P22.46


47.

n = 1.60

n = 1.40

n = 1.20

n = 1.00

u1

u2

Figure P22.47


44920_22_p726-753 1/12/05 8:52 AM Page 750






d

2.00 m

Raft

Diamond

Figure P22.40




n " !3.

41.

Surface 1

Surf

ace

2

42.0°

60.0°

42.0°

ui

Figure P22.43



u

Figure P22.46


47.

n = 1.60

n = 1.40

n = 1.20

n = 1.00

u1

u2

Figure P22.47


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Show that n is given by the expression

where A is the apex angle of the prism.58. A hiker stands on a mountain peak near sunset and

observes a rainbow caused by water droplets in the airabout 8.00 km away. The valley is 2.00 km below the moun-tain peak and entirely flat. What fraction of the completecircular arc of the rainbow is visible to the hiker?A light ray incident on a prism is refracted at the first sur-face, as shown in Figure P22.59. Let ! represent the apexangle of the prism and n its index of refraction. Find, interms of n and !, the smallest allowed value of the angle ofincidence at the first surface for which the refracted ray willnot undergo total internal reflection at the second surface.

59.

n "

sin! 12

(A # $min)"sin # A

2 $

the plastic. (b) If the light ray enters the plastic at a pointL " 50.0 cm from the bottom edge, how long does it takethe light ray to travel through the plastic?

62. A. H. Pfund’s method for measuring the index of refrac-tion of glass is illustrated in Figure P22.62. One face of aslab of thickness t is painted white, and a small holescraped clear at point P serves as a source of divergingrays when the slab is illuminated from below. Ray PBB%strikes the clear surface at the critical angle and is totallyreflected, as are rays such as PCC ’. Rays such as PAA%emerge from the clear surface. On the painted surfacethere appears a dark circle of diameter d, surrounded byan illuminated region, or halo. (a) Derive an equation forn in terms of the measured quantities d and t. (b) What isthe diameter of the dark circle if n " 1.52 for a slab0.600 cm thick? (c) If white light is used, the critical angledepends on color caused by dispersion. Is the inner edgeof the white halo tinged with red light or violet light?Explain.

uf

Figure P22.59

60. Students allow a narrow beam of laser light to strike awater surface. They arrange to measure the angle ofrefraction for selected angles of incidence and record thedata shown in the following table:

Angle of Incidence Angle of Refraction(degrees) (degrees)

10.0 7.520.0 15.130.0 22.340.0 28.750.0 35.260.0 40.370.0 45.380.0 47.7

Use the data to verify Snell’s law of refraction by plottingthe sine of the angle of incidence versus the sine of theangle of refraction. From the resulting plot, deduce theindex of refraction of water.

61. A light ray enters a rectangular block of plastic at an angle&1 " 45.0° and emerges at an angle &2 " 76.0°, as shownin Figure P22.61. (a) Determine the index of refraction of

n

2

L

1!

!

Figure P22.61

A"

B "C "

t

C B A

Pd

Clearsurface

Paintedsurface

Figure P22.62

ACTIVITIES1. Tape a coin to the bottom of a large opaque bowl, as

shown in Figure A22.1a. Stand over the bowl so that youare looking at the coin, and then move backwards awayfrom the bowl until you can no longer see the coin overthe bowl’s rim. Remain at that position, and have a friendfill the bowl with water, as shown in Figure A22.1b. Youcan now see the coin again because the light is refractedat the water–air interface.

(a)

(b)

Water

Coin

Figure A22.1

2. Tape a piece of black paper to the end of a flashlight andcut a narrow slit in the middle of the paper, as shown inFigure A22.2. Lean a flat mirror against one end of a traypartially filled with water. Shine your flashlight on that

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Chapter 23

Problems 781

PROBLEMS1, 2, 3 = straightforward, intermediate, challenging ! = full solution available in Student Solutions Manual/Study Guide

= coached solution with hints available at www.cp7e.com = biomedical application

Section 23.1 Flat Mirrors1. Does your bathroom mirror show you older or younger

than your actual age? Compute an order-of-magnitudeestimate for the age difference, based on data that youspecify.

2. Use Active Figure 23.2 to give a geometric proof that thevirtual image formed by a plane mirror is the same dis-tance behind the mirror as the object is in front of it.A person walks into a room that has, on opposite walls,two plane mirrors producing multiple images. Find thedistances from the person to the first three images seen inthe left-hand mirror when the person is 5.00 ft from themirror on the left wall and 10.0 ft from the mirror on theright wall.

4. In a church choir loft, two parallel walls are 5.30 m apart.The singers stand against the north wall. The organistfaces the south wall, sitting 0.800 m away from it. So thatshe can see the choir, a flat mirror 0.600 m wide ismounted on the south wall, straight in front of the organ-ist. What width of the north wall can she see? [Hint: Drawa top-view diagram to justify your answer.]

Section 23.2 Images Formed by Spherical MirrorsSection 23.3 Convex Mirrors and Sign Conventions

In the following problems, algebraic signs are not given. We leaveit to you to determine the correct sign to use with each quantity,based on an analysis of the problem and the sign conventions inTable 23.1.

At an intersection of hospital hallways, aconvex mirror is mounted high on a wall to help peopleavoid collisions. The mirror has a radius of curvature of0.550 m. Locate and describe the image of a patient 10.0 mfrom the mirror. Determine the magnification of the image.

6. To fit a contact lens to a patient’s eye, a keratometer can beused to measure the curvature of the cornea— the frontsurface of the eye. This instrument places an illuminatedobject of known size at a known distance p from thecornea, which then reflects some light from the object,forming an image of it. The magnification M of the imageis measured by using a small viewing telescope that allowsa comparison of the image formed by the cornea with asecond calibrated image projected into the field of viewby a prism arrangement. Determine the radius of curva-ture of the cornea when p ! 30.0 cm and M ! 0.013 0.

7. A concave spherical mirror has a radius of curvature of 20.0 cm. Locate the images for object distances of(a) 40.0 cm, (b) 20.0 cm, and (c) 10.0 cm. In each case,state whether the image is real or virtual and upright orinverted, and find the magnification.

8. A dentist uses a mirror to examine a tooth that is 1.00 cmin front of the mirror. The image of the tooth is formed10.0 cm behind the mirror. Determine (a) the mirror’sradius of curvature and (b) the magnification of the image.

9. A large church has a niche in one wall. On the floor planit appears as a semicircular indentation of radius 2.50 m.A worshiper stands on the centerline of the niche, 2.00 mout from its deepest point, and whispers a prayer. Where

5.

3.

is the sound concentrated after it reflects from the backwall of the niche?

10. While looking at her image in a cosmetic mirror, Dinanotes that her face is highly magnified when she is closeto the mirror, but as she backs away from the mirror, herimage first becomes blurry, then disappears when she isabout 30 cm from the mirror, and then inverts when sheis beyond 30 cm. Based on these observations, what canshe conclude about the properties of the mirror?A 2.00-cm-high object is placed 3.00 cm in front of a con-cave mirror. If the image is 5.00 cm high and virtual, whatis the focal length of the mirror?

12. A dedicated sports car enthusiast polishes the inside andoutside surfaces of a hubcap that is a section of a sphere.When he looks into one side of the hubcap, he sees animage of his face 30.0 cm in back of it. He then turns thehubcap over, keeping it the same distance from his face.He now sees an image of his face 10.0 cm in back ofthe hubcap. (a) How far is his face from the hubcap?(b) What is the radius of curvature of the hubcap?

13. A concave makeup mirror is designed so that a person25 cm in front of it sees an upright image magnified by afactor of two. What is the radius of curvature of the mirror?

14. A certain Christmas tree ornament is a silver sphere hav-ing a diameter of 8.50 cm. Determine an object locationfor which the size of the reflected image is three-fourthsthe size of the object. Use a principal-ray diagram toarrive at a description of the image.A man standing 1.52 m in front of a shaving mirror pro-duces an inverted image 18.0 cm in front of it. How close tothe mirror should he stand if he wants to form an uprightimage of his chin that is twice the chin’s actual size?

16. A convex spherical mirror with a radius of curvature of10.0 cm produces a virtual image one-third the size of thereal object. Where is the object?

17. A child holds a candy bar 10.0 cm in front of a convexmirror and notices that the image is only one-half the sizeof the candy bar. What is the radius of curvature of themirror?

18. It is observed that the size of a real image formed by a con-cave mirror is four times the size of the object when theobject is 30.0 cm in front of the mirror. What is the radiusof curvature of this mirror?

19. A spherical mirror is to be used to form an image, fivetimes as tall as an object, on a screen positioned 5.0 mfrom the mirror. (a) Describe the type of mirror required.(b) Where should the mirror be positioned relative to theobject?

20. A ball is dropped from rest 3.00 m directly above the ver-tex of a concave mirror having a radius of 1.00 m andlying in a horizontal plane. (a) Describe the motion of theball’s image in the mirror. (b) At what time do the balland its image coincide?

Section 23.4 Images Formed by Refraction21. A cubical block of ice 50.0 cm on an edge is placed on a

level floor over a speck of dust. Locate the image of the

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782 Chapter 23 Mirrors and Lenses

speck, when viewed from directly above, if the index ofrefraction of ice is 1.309.

22. The top of a swimming pool is at ground level. If the poolis 2 m deep, how far below ground level does the bottomof the pool appear to be located when (a) the pool iscompletely filled with water? (b) the pool is filled halfwaywith water?

23. A paperweight is made of a solid glass hemisphere withindex of refraction 1.50. The radius of the circular crosssection is 4.0 cm. The hemisphere is placed on its flat sur-face, with the center directly over a 2.5-mm-long linedrawn on a sheet of paper. What length of line is seen bysomeone looking vertically down on the hemisphere?

24. A flint glass plate (n ! 1.66) rests on the bottom of anaquarium tank. The plate is 8.00 cm thick (vertical dimen-sion) and covered with water (n ! 1.33) to a depth of12.0 cm. Calculate the apparent thickness of the plate asviewed from above the water. (Assume nearly normal inci-dence of light rays.)A transparent sphere of unknown composition is observedto form an image of the Sun on its surface opposite theSun. What is the refractive index of the sphere material?

26. A goldfish is swimming at 2.00 cm/s toward the front wallof a rectangular aquarium. What is the apparent speed ofthe fish as measured by an observer looking in fromoutside the front wall of the tank? The index of refractionof water is 1.333.

Section 23.6 Thin Lenses27. A contact lens is made of plastic with an index of refrac-

tion of 1.50. The lens has an outer radius of curvature of" 2.00 cm and an inner radius of curvature of " 2.50 cm.What is the focal length of the lens?

28. The left face of a biconvex lens has a radius of curvatureof 12.0 cm, and the right face has a radius of curvature of18.0 cm. The index of refraction of the glass is 1.44.(a) Calculate the focal length of the lens. (b) Calculatethe focal length if the radii of curvature of the two facesare interchanged.

A converging lens has a focal lengthof 20.0 cm. Locate the images for object distances of(a) 40.0 cm, (b) 20.0 cm, and (c) 10.0 cm. For each case,state whether the image is real or virtual and upright orinverted, and find the magnification.

30. Where must an object be placed to have unit magnifica-tion (!M ! ! 1.00) (a) for a converging lens of focallength 12.0 cm? (b) for a diverging lens of focal length12.0 cm?

31. A diverging lens has a focal length of 20.0 cm. Locate theimages for object distances of (a) 40.0 cm, (b) 20.0 cm,and (c) 10.0 cm. For each case, state whether the imageis real or virtual and upright or inverted, and find themagnification.

32. The use of a lens in a certain situation is described by theequation

Determine (a) the object distance and (b) the imagedistance. (c) Use a ray diagram to obtain a description ofthe image. (d) Identify a practical device described by the

1p

"1

# 3.50p!

17.50 cm

29.

25.

given equation, and write the statement of a problemhaving a solution that contains the equation.A transparent photographic slide is placed in front of aconverging lens with a focal length of 2.44 cm. The lensforms an image of the slide 12.9 cm from it. How far is thelens from the slide if the image is (a) real? (b) virtual?

34. The nickel’s image in Figure P23.34 has twice the diame-ter of the nickel when the lens is 2.84 cm from the nickel.Determine the focal length of the lens.

33.

Figure P23.34

35. A certain LCD projector contains a single thin lens. Anobject 24.0 mm high is to be projected so that its imagefills a screen 1.80 m high. The object-to-screen distance is3.00 m. (a) Determine the focal length of the projectionlens. (b) How far from the object should the lens of theprojector be placed in order to form the image on thescreen?

36. A person uses a converging lens that has a focal lengthof 12.5 cm to inspect a gem. The lens forms a virtual im-age 30.0 cm away. Determine the magnification. Is the im-age upright or inverted?

37. A diverging lens is to be used to produce a virtual imageone-third as tall as the object. Where should the object beplaced?

38. An object is 5.00 m to the left of a flat screen. A converg-ing lens for which the focal length is f ! 0.800 m is placedbetween object and screen. (a) Show that there are twolens positions that form an image on the screen, anddetermine how far these positions are from the object.(b) How do the two images differ from each other?A converging lens is placed 30.0 cm to the right of adiverging lens of focal length 10.0 cm. A beam of parallellight enters the diverging lens from the left, and the beamis again parallel when it emerges from the converginglens. Calculate the focal length of the converging lens.

40. An object is placed 20.0 cm to the left of a converginglens of focal length 25.0 cm. A diverging lens of focallength 10.0 cm is 25.0 cm to the right of the converginglens. Find the position and magnification of the final image.

41. Two converging lenses, each of focal length 15.0 cm, areplaced 40.0 cm apart, and an object is placed 30.0 cm infront of the first lens. Where is the final image formed,and what is the magnification of the system?

42. Object O1 is 15.0 cm to the left of a converging lens with a10.0-cm focal length. A second lens is positioned 10.0 cmto the right of the first lens and is observed to form a finalimage at the position of the original object O1. (a) What isthe focal length of the second lens? (b) What is the overallmagnification of this system? (c) What is the nature (i.e.,real or virtual, upright or inverted) of the final image?

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1p

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17.50 cm

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Figure P23.34








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# 3.50p!

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Problems 783

A 1.00-cm-high object is placed 4.00 cmto the left of a converging lens of focal length 8.00 cm. Adiverging lens of focal length !16.00 cm is 6.00 cm to theright of the converging lens. Find the position and heightof the final image. Is the image inverted or upright? Realor virtual?

44. Two converging lenses having focal lengths of 10.0 cmand 20.0 cm are placed 50.0 cm apart, as shown in FigureP23.44. The final image is to be located between thelenses, at the position indicated. (a) How far to the left ofthe first lens should the object be positioned? (b) What isthe overall magnification of the system? (c) Is the finalimage upright or inverted?

43. the thicknesses of the lenses can be ignored in compari-son to the other distances involved.

ADDITIONAL PROBLEMS47. An object placed 10.0 cm from a concave spherical mirror

produces a real image 8.00 cm from the mirror. If theobject is moved to a new position 20.0 cm from themirror, what is the position of the image? Is the finalimage real or virtual?

48. An object is placed 12 cm to the left of a diverging lens offocal length !6.0 cm. A converging lens of focal length12 cm is placed a distance d to the right of the diverginglens. Find the distance d that places the final image atinfinity.

49. A convergent lens with a 50.0-mm focal length is used tofocus an image of a very distant scene onto a flat screen35.0 mm wide. What is the angular width " of the sceneincluded in the image on the screen?The object in Figure P23.50 is midway between the lensand the mirror. The mirror’s radius of curvature is20.0 cm, and the lens has a focal length of !16.7 cm.Considering only the light that leaves the object and trav-els first towards the mirror, locate the final image formedby this system. Is the image real or virtual? Is it upright orinverted? What is the overall magnification of the image?

50.

f2 (20.0 cm)f1 (10.0 cm)

Final imageObject

p 31.0 cm50.0 cm

Figure P23.44

45. Lens L1 in Figure P23.45 has a focal length of 15.0 cmand is located a fixed distance in front of the film plane ofa camera. Lens L2 has a focal length of 13.0 cm, and itsdistance d from the film plane can be varied from 5.00 cmto 10.0 cm. Determine the range of distances for whichobjects can be focused on the film.

L1 L2Film

d

12.0 cm

Figure P23.45

46. Consider two thin lenses, one of focal length f1 and theother of focal length f2, placed in contact with each otheras shown in Figure P23.46. Apply the thin-lens equation toeach of these lenses and combine the results to show thatthis combination of lenses behaves like a thin lens havinga focal length f given by 1/f # 1/f1 $ 1/f2. Assume that

f2f1

Figure P23.46

Lens ObjectMirror

25.0 cm

51. The lens and mirror in Figure P23.51 are separatedby 1.00 m and have focal lengths of $80.0 cm and!50.0 cm, respectively. If an object is placed 1.00 m to theleft of the lens, locate the final image. State whether theimage is upright or inverted, and determine the overallmagnification.

Figure P23.50

MirrorLens

1.00 m1.00 m

Object

Figure P23.51

52. A diverging lens (n # 1.50) is shaped like that in ActiveFigure 23.25c. The radius of the first surface is 15.0 cm,and that of the second surface is 10.0 cm. (a) Find thefocal length of the lens. Determine the positions ofthe images for object distances of (b) infinity, (c) 3 ! f !,(d) ! f !, and (e) ! f !/2.

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Chapter 24

Problems 813



Section 24.2 Young’s Double-Slit Experiment1. A laser beam (! " 632.8 nm) is incident on two slits

0.200 mm apart. How far apart are the bright interfer-ence fringes on a screen 5.00 m away from the doubleslits?

2. In a Young’s double-slit experiment, a set of parallel slitswith a separation of 0.100 mm is illuminated by light hav-ing a wavelength of 589 nm, and the interference patternis observed on a screen 4.00 m from the slits. (a) What isthe difference in path lengths from each of the slits to thelocation of a third-order bright fringe on the screen?(b) What is the difference in path lengths from the twoslits to the location of the third dark fringe on the screen,away from the center of the pattern?

3. A pair of narrow, parallel slits separated by 0.250 mm isilluminated by the green component from a mercuryvapor lamp (! " 546.1 nm). The interference pattern isobserved on a screen 1.20 m from the plane of the paral-lel slits. Calculate the distance (a) from the central maxi-mum to the first bright region on either side of thecentral maximum and (b) between the first and seconddark bands in the interference pattern.

4. Light of wavelength 460 nm falls on two slits spaced0.300 mm apart. What is the required distance from theslit to a screen if the spacing between the first and seconddark fringes is to be 4.00 mm?

In a location where the speed of soundis 354 m/s, a 2 000-Hz sound wave impinges on two slits30.0 cm apart. (a) At what angle is the first maximumlocated? (b) If the sound wave is replaced by 3.00-cmmicrowaves, what slit separation gives the same angle forthe first maximum? (c) If the slit separation is 1.00 #m,what frequency of light gives the same first maximumangle?

6. White light spans the wavelength range between about400 nm and 700 nm. If white light passes through two slits0.30 mm apart and falls on a screen 1.5 m from the slits,find the distance between the first-order violet and thefirst-order red fringes.

7. Two radio antennas separated by 300 m, as shown in FigureP24.7, simultaneously transmit identical signals of thesame wavelength. A radio in a car traveling due northreceives the signals. (a) If the car is at the position of thesecond maximum, what is the wavelength of the signals?(b) How much farther must the car travel to encounterthe next minimum in reception? [Hint: Determine thepath difference between the two signals at the two loca-tions of the car.]

5.

8. If the distance between two slits is 0.050 mm and the dis-tance to a screen is 2.50 m, find the spacing between thefirst- and second-order bright fringes for yellow light of600-nm wavelength.

9. Waves from a radio station have a wavelength of 300 m.They travel by two paths to a home receiver 20.0 km fromthe transmitter. One path is a direct path, and the secondis by reflection from a mountain directly behind thehome receiver. What is the minimum distance from themountain to the receiver that produces destructive inter-ference at the receiver? (Assume that no phase changeoccurs on reflection from the mountain.)

10. A pair of slits, separated by 0.150 mm, is illuminated bylight having a wavelength of ! " 643 nm. An interferencepattern is observed on a screen 140 cm from the slits.Consider a point on the screen located at y " 1.80 cmfrom the central maximum of this pattern. (a) What is thepath difference $ for the two slits at the location y ?(b) Express this path difference in terms of the wave-length. (c) Will the interference correspond to a maxi-mum, a minimum, or an intermediate condition?

11. A riverside warehouse has two open doors, as in FigureP24.11. Its interior is lined with a sound-absorbing mate-rial. A boat on the river sounds its horn. To person A, the

21. Light in air that is reflected from a water surface is foundto be completely polarized at an angle %. If the light isinstead reflected from a glass coffee table, will the newangle for complete polarization be larger or smaller?

22. In one experiment, blue light passes through a diffrac-tion grating and forms an interference pattern on a

screen. In a second experiment, red light passesthrough the same diffraction grating and forms anotherinterference pattern. How do the separations betweenbright lines in the two experiments compare with eachother?

400 m

1000 m300 m

Figure P24.7

20.0 m

150 m

A

B

Figure P24.11

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814 Chapter 24 Wave Optics

sound is loud and clear. To person B, the sound is barelyaudible. The principal wavelength of the sound waves is3.00 m. Assuming person B is at the position of the firstminimum, determine the distance between the doors,center to center.

12. The waves from a radio station can reach a home receiverby two different paths. One is a straight-line path from thetransmitter to the home, a distance of 30.0 km. The sec-ond path is by reflection from a storm cloud. Assume thatthis reflection takes place at a point midway betweenreceiver and transmitter. If the wavelength broadcast bythe radio station is 400 m, find the minimum height ofthe storm cloud that will produce destructive interferencebetween the direct and reflected beams. (Assume nophase changes on reflection.)Radio waves from a star, of wavelength 250 m, reach aradio telescope by two separate paths, as shown in FigureP24.13. One is a direct path to the receiver, which is situ-ated on the edge of a cliff by the ocean. The second is byreflection off the water. The first minimum of destructiveinterference occurs when the star is 25.0° above the hori-zon. Find the height of the cliff. (Assume no phasechange on reflection.)

13.

light (wavelength 600 nm in air). Assuming the maximumoccurs in the first order, determine the thickness of theoil slick.

A possible means for making an airplane invisible to radaris to coat the plane with an antireflective polymer. If radarwaves have a wavelength of 3.00 cm and the index ofrefraction of the polymer is n ! 1.50, how thick wouldyou make the coating?

20. A beam of light of wavelength 580 nm passes through twoclosely spaced glass plates, as shown in Figure P24.20. Forwhat minimum non-zero value of the plate separation dwill the transmitted light be bright? This arrangement isoften used to measure the wavelength of light and iscalled a Fabry–Perot interferometer.

19.

Directpath

Reflectedpath

Figure P24.13

Section 24.3 Change of Phase Due to ReflectionSection 24.4 Interference in Thin Films14. Determine the minimum thickness of a soap film (n !

1.330) that will result in constructive interference of(a) the red H" line (# ! 656.3 nm); (b) the blue H$ line(# ! 434.0 nm).

15. Suppose the film shown in Figure 24.7 has an index ofrefraction of 1.36 and is surrounded by air on both sides.Find the minimum thickness that will produce construc-tive interference in the reflected light when the film isilluminated by light of wavelength 500 nm.

16. A thin film of glass (n ! 1.50) floats on a liquid of n ! 1.35and is illuminated by light of # ! 580 nm incident fromair above it. Find the minimum thickness of the glass,other than zero, that will produce destructive interferencein the reflected light.

17. A coating is applied to a lens to minimize reflections. The in-dex of refraction of the coating is 1.55, and that of the lens is1.48. If the coating is 177.4 nm thick, what wavelengthis minimally reflected for normal incidence in the lowestorder?

18. A transparent oil with index of refraction 1.29 spills onthe surface of water (index of refraction 1.33), producinga maximum of reflection with normally incident orange

d

Figure P24.20

21. Astronomers observe the chromosphere of the sun with afilter that passes the red hydrogen spectral line of wave-length 656.3 nm, called the H" line. The filter consists ofa transparent dielectric of thickness d held between twopartially aluminized glass plates. The filter is kept at a con-stant temperature. (a) Find the minimum value of d thatwill produce maximum transmission of perpendicular H"

light if the dielectric has an index of refraction of 1.378.(b) If the temperature of the filter increases above thenormal value increasing its thickness, what happens to thetransmitted wavelength? (c) The dielectric will also passwhat near-visible wavelength? One of the glass plates iscolored red to absorb this light.

22. Two rectangular optically flat plates (n ! 1.52) are in con-tact along one end and are separated along the other endby a 2.00-%m-thick spacer (Fig. P24.22). The top plate isilluminated by monochromatic light of wavelength546.1 nm. Calculate the number of dark parallel bandscrossing the top plate (including the dark band at zerothickness along the edge of contact between the plates).


An air wedge is formed between two glass plates separatedat one edge by a very fine wire, as in Figure P24.22. Whenthe wedge is illuminated from above by 600-nm light, 30dark fringes are observed. Calculate the radius of thewire.

24. A planoconvex lens with radius of curvature R ! 3.0 m isin contact with a flat plate of glass. A light source and theobserver’s eye are both close to the normal, as shown in

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Problems 815

Figure 24.8a. The radius of the 50th bright Newton’s ringis found to be 9.8 mm. What is the wavelength of the lightproduced by the source?

25. A planoconvex lens rests with its curved side on a flat glasssurface and is illuminated from above by light of wave-length 500 nm. (See Fig. 24.8.) A dark spot is observed atthe center, surrounded by 19 concentric dark rings (withbright rings in between). How much thicker is the airwedge at the position of the 19th dark ring than at thecenter?

26. Nonreflective coatings on camera lenses reduce the loss oflight at the surfaces of multilens systems and prevent inter-nal reflections that might mar the image. Find the mini-mum thickness of a layer of magnesium fluoride (n ! 1.38)on flint glass (n ! 1.66) that will cause destructive inter-ference of reflected light of wavelength 550 nm near themiddle of the visible spectrum.

A thin film of MgF2 (n ! 1.38) withthickness 1.00 " 10#5 cm is used to coat a camera lens.Are any wavelengths in the visible spectrum intensified inthe reflected light?

28. A flat piece of glass is supported horizontally above theflat end of a 10.0-cm-long metal rod that has its lower endrigidly fixed. The thin film of air between the rod and theglass is observed to be bright when illuminated by light ofwavelength 500 nm. As the temperature is slowly increasedby 25.0°C, the film changes from bright to dark and backto bright 200 times. What is the coefficient of linearexpansion of the metal?

Section 24.7 Single-Slit Diffraction29. Helium–neon laser light ($ ! 632.8 nm) is sent through

a 0.300-mm-wide single slit. What is the width of thecentral maximum on a screen 1.00 m from the slit?

30. Light of wavelength 600 nm falls on a 0.40-mm-wide slitand forms a diffraction pattern on a screen 1.5 m away.(a) Find the position of the first dark band on each sideof the central maximum. (b) Find the width of the centralmaximum.Light of wavelength 587.5 nm illuminates a slit of width 0.75 mm. (a) At what distance from the slit should ascreen be placed if the first minimum in the diffractionpattern is to be 0.85 mm from the central maximum?(b) Calculate the width of the central maximum.

32. Microwaves of wavelength 5.00 cm enter a long, narrowwindow in a building that is otherwise essentially opaqueto the incoming waves. If the window is 36.0 cm wide, whatis the distance from the central maximum to the first-order minimum along a wall 6.50 m from the window?

33. A slit of width 0.50 mm is illuminated with light of wave-length 500 nm, and a screen is placed 120 cm in front ofthe slit. Find the widths of the first and second maxima oneach side of the central maximum.

34. A screen is placed 50.0 cm from a single slit, which is illu-minated with light of wavelength 680 nm. If the distancebetween the first and third minima in the diffraction pat-tern is 3.00 mm, what is the width of the slit?

Section 24.8 The Diffraction Grating35. Three discrete spectral lines occur at angles of 10.1°,

13.7°, and 14.8°, respectively, in the first-order spectrum

31.

27.

of a diffraction-grating spectrometer. (a) If the gratinghas 3 660 slits/cm, what are the wavelengths of the light?(b) At what angles are these lines found in the second-order spectra?

36. Intense white light is incident on a diffraction grating thathas 600 lines/mm. (a) What is the highest order in whichthe complete visible spectrum can be seen with this grat-ing? (b) What is the angular separation between the violetedge (400 nm) and the red edge (700 nm) of the first-order spectrum produced by the grating?The hydrogen spectrum has a red line at 656 nm and aviolet line at 434 nm. What angular separation betweenthese two spectral lines obtained with a diffraction gratingthat has 4 500 lines/cm?

38. A grating with 1 500 slits per centimeter is illuminatedwith light of wavelength 500 nm. (a) What is the highest-order number that can be observed with this grating?(b) Repeat for a grating of 15 000 slits per centimeter.

39. A light source emits two major spectral lines: an orangeline of wavelength 610 nm and a blue-green line of wave-length 480 nm. If the spectrum is resolved by a diffractiongrating having 5 000 lines/cm and viewed on a screen2.00 m from the grating, what is the distance (in centime-ters) between the two spectral lines in the second-orderspectrum?

40. White light is spread out into its spectral components by adiffraction grating. If the grating has 2 000 lines percentimeter, at what angle does red light of wavelength640 nm appear in the first-order spectrum?

41. Sunlight is incident on a diffraction grating that has2 750 lines/cm. The second-order spectrum over the visi-ble range (400–700 nm) is to be limited to 1.75 cm alonga screen that is a distance L from the grating. What is therequired value of L?

42. Light containing two different wavelengths passesthrough a diffraction grating with 1 200 slits/cm. On ascreen 15.0 cm from the grating, the third-order maxi-mum of the shorter wavelength falls midway betweenthe central maximum and the first side maximum for thelonger wavelength. If the neighboring maxima of thelonger wavelength are 8.44 mm apart on the screen, whatare the wavelengths in the light? [Hint: Use the small-angle approximation.]

A beam of 541-nm light is incident on adiffraction grating that has 400 lines/mm. (a) Determinethe angle of the second-order ray. (b) If the entire appara-tus is immersed in water, determine the new second-orderangle of diffraction. (c) Show that the two diffractedrays of parts (a) and (b) are related through the law ofrefraction.

44. Light from a helium–neon laser ($ ! 632.8 nm) is inci-dent on a single slit. What is the maximum width forwhich no diffraction minima are observed? [Hint: Valuesof sin % & 1 are not possible.]

Section 24.9 Polarization of Light Waves45. The angle of incidence of a light beam in air onto a

reflecting surface is continuously variable. The reflectedray is found to be completely polarized when the angle ofincidence is 48.0°. (a) What is the index of refraction ofthe reflecting material? (b) If some of the incident light

43.

37.

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Problems 815














31.

27.












43.

37.

!!"#$%#!%&'()*(+(,,+-+.-$/,,'0+",12,,3456,(+/

Problems 815














31.

27.












43.

37.

!!"#$%#!%&'()*(+(,,+-+.-$/,,'0+",12,,3456,(+/

Problems 815














31.

27.












43.

37.

!!"#$%#!%&'()*(+(,,+-+.-$/,,'0+",12,,3456,(+/

Problems 815














31.

27.












43.

37.

!!"#$%#!%&'()*(+(,,+-+.-$/,,'0+",12,,3456,(+/

Problems 815














31.

27.












43.

37.

!!"#$%#!%&'()*(+(,,+-+.-$/,,'0+",12,,3456,(+/

ch.22-24 h/w questions - physicsatthebay.comphysicsatthebay.com/physics/chapter_13,14_files/ch.22-24...

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