WCh30 - Exercises and Problems 541 . l For homework...

Unformatted text preview: Exercises and Problems 541 ., ' l @ For homework assigned on A/lasteringPhys/cs, go to wwwmasteringphysics. com E BIO Biology and/or medical-related prob/ems C Computer problems For Thogan isssion w 1. Why is it usually inappropriate to consider low-frequency sound waves as traveling in rays? Why is the ray approximation more appropriate for high-frequency sound and for light? 2. Why does a spoon appear bent when it’s in a glass of water? 3. Why do a diamond and an identically shaped piece of glass sparkle differently? 4. White light goes from air through a glass slab with parallel sur- faces. Will its colors be dispersed when it emerges from the glass? 5. You send White light through two identical glass prisms, oriented as shown in Fig. 30.14. Describe the beam that emerges from the right-hand prism. White light FIGURE 30.14 ForThought and Discussion 5 6. In glass, which end of the visible spectrum has the smallest criti- cal angle for total internal reflection? 7. Why can’t you walk to the end of the rainbow? 8. Whai’s wrong with Fig. 30.15, which shows rainbows over Nia— gara Falls? (Hint: The rainbow subtends a half—angle of 42°.) FIGURE 30.15 ForThought and Discussion 8.The painting is Harry Fenn’s Niagara. 9. Why are polarizing sunglasses better than glasses that simply re— duce the total amount of light? 10. Under what conditions will the polarizing angle be smaller than 45 °? Exercises and Problems W Wmummmmmm Exercises Section 30.1 Reflection 11. Through what angle should you rotate a mirror so that a reflected ray rotates through 30°? 12. The mirrors in Fig. 30.16 make a 60° angle. A light ray enters parallel to the symmetry axis, as shown. (a) How many reflec- tions does it make? (b) Where and in what direction does it exit the mirror system? FIGURE 30.16 Exercises 12 and 14 and Problem 28 l3. 14. To what angular accuracy must two ostensibly perpendicular mirrors be aligned so that an incident ray returns within 1° of its incident direction? If a light ray enters the mirror system of Fig. 30.16 propagating in the plane of the page and parallel to one mirror, through what angle will it be turned? Section 30.2 Refraction 13. 16. 17. 18. 19. 20. 21. In which substance in Table 30.1 does the speed of light have the value 2.292X108 m/s? ,> Information in a compact disc is stored in “pits” whose depth is es— sentially one-fourth the wavelength of the laser light used to “read” the information. That wavelength‘is 780 nm in air, but the wave- length on which the pit depth is bli‘sed is measured in the n = 1.55 plastic that makes up most of the disc. Find the pit depth. Light is incident on an air—glass interface, and the refracted light in the glass makes a 40° angle with the normal to the interface. The glass has refractive index 1.52. Find'the incidence angle. A light ray propagates in a transparent material at 15° to the nor— mal to the surface. It emerges into the surrounding air at 24° to the normal. Find the material’s refractive index. Light propagating in the glass (n = 1.52) wall of an aquarium tank strikes the wall’s interior surface with incidence angle 12.4°. What’s the angle of refraction in the water? Find the polarizing angle for diamond when light is incident from air. J Find the refractive index of a material for which the polarizing angle in air is 62°. Section 30.3 Total internal Reflection 22. 23. 24. Find the critical angle for total internal reflection in (:1) ice, (b) polystyrene, and (c) rutile, when the sunrounding medium is air. A drop of water is trapped in a block of ice. What’s the critical angle for total internal reflection at the water—ice interface? What is the critical angle for light propagating in glass with n = 1.52 when the glass is immersed in (a) water, (1)) benzene, and (C) diiodomethane? Total internal reflection occurs at an interface between plastic and air at incidence angles greater than 37°. Find the plastic’s refractive index. Section 30.4 Dispersion 26. 27. Blue and red laser beams strike an air—glass interface with inci— dense angle 50°. If the glass has refractive indices of 1.680 for the blue light and 1.621 for the red, what will be the angle between the two beams in the glass? White light propagating in air is incident at 45° on the equilateral prism of Fig. 30.17. Find the angular dispersion y of the outgo— ing beam if the prism has refractive indices mm = 1.582 and 11mm : 1.633. Violet FIGURE 30.17 Exercise 27 (angles ofdispersed rays aren't accurate) Problems 28. Suppose the 60° angle in Fig. 30.16 is changed to 75°. A ray en— ters the mirror system parallel to the axis. (a) How many reflec— tions does it make? (b) Through what angle is it turned when it exits the system? 542 Chapter30 Reflection and Refraction 29. The refractive index of a human cornea is 1.40. If 550—nm light BIO strikes a cornea at incidence angle 25°, find (a) the angle of re— 30. 31. 32. 33. 34. 35. 36. fraction and (b) the wavelength in the cornea. Two plane mirrors make an angle qb. A light ray enters the sys— tem and is reflected once off each mirror. Show that the ray is turned through an angle 360° — 2gb. An unlabeled bottle of liquid has spilled, and you’re trying to find out whether it’s relatively harmless ethyl alcohol or toxic benzene. You submerge a glass block with n = 1.52 in the liq- uid, and shine a laser beam so it strikes the submerged glass with incidence angle 31.5°. You measure the angle of refraction in the glass at 279°. Which liquid is it? (See Table 30.1.) A meter stick lies on the bottom of the rectangular tank in Fig. 30.18, with its zero mark at the tank’s left edge. You look into the long dimension of the tank at a 45° angle, with your line of sight just grazing the top edge, as shown. What mark on the meter stick do you see when the tank is (a) empty, (1‘)) half full of water, and (0) full of water? FIGURE 30.18 Problem 32 You look at the center of one face of a solid glass cube of glass, on a line of sight making a 55° angle with the normal to the cube face. What minimum refractive index of the. glass will let you see through the cube’s opposite face? At the aquarium where you work, a fish has gone missing in a 10—m—deep, 11—m—diameter cylindrical tank. You shine a flashlight in from the top edge of the tank, hoping to see if the missing fish is on the bottom. What’s the smallest angle your flashlight beam can make with the horizontal if it’s to illuminate the bottom? You’re standing 2.3 m horizontally from the edge of ‘a 4.5—m—deep lake, with your eyes 1.7 m above the water’s surface. A diver hold— ing a flashlight at the lake bottom shines the light so you can see it. If the light in the water makes a 42° angle with the vertical, at what horizontal distance is the diver from the edge of the lake? You’ve dropped your car keys at night off the end of a dock into water 1.6 In deep. A flashlight held directly above the dock edge and 0.50 m above the water illuminates the keys when it’s aimed at 40° to the vertical, as shown in Fig. 30.19. What’s the horizon- tal distance x from the edge of the dock to the keys? FIGURE 30.19 Problem 36 37. Laser eye surgery uses ultraviolet light with wavelength 193 mn. Bio What’s the UV light’s wavelength within the eye’s lens, where 38. 39. 40. 41. 45. 46. 47. 48. 49. 50. 51. 52. 53. n = 1.39? The prism in Fig. 30.20 has n 2 1.52 and a : 60° and is sur- rounded by air. A light beam is incident at 61 = 37°. Find the an gle 5 through which the beam is deflected. FIGURE 30.20 Problems 38 and 39 '91 Repeat Problem 38 for the case n = 1.75, a = 40°, and 61 = 25°. Find the minimum refractive index for the prism in Fig. 30.10 if total internal reflection occurs as shown when the prism is sur— rounded by air. ‘ Where and in what direction would the main beam emerge if the prism in Fig. 30.10 were made of ice, surrounded by air? Find the speed of light in a material for which the critical angle at an interface with air is 61°. The prism of Fig. 30.10 has n : 1.52. When it’s immersed in a liquid, a beam incident as shown in the figure ceases to undergo total reflection. What’s the minimum value for the liquid’s re— fractive index? For the interface between air (refractive index 1) and a material with refractive index n, show that the critical angle and the polar~ izing angle are related by sin do = cotBP. A scuba diver sets off a camera flash at depth h in water with refractive index rt. Show that light emerges from the water’s surface through a circle of diameter 211/ v n2 — 1. Suppose the red and blue beams of Exercise 26 are now propa- gating in the same direction inside the glass. For what range of incidence angles on the glass—air interface will one beam be to— tally reflected and the other not? A compound lens is made from crown glass (n = 1.52) bonded to flint glass (n = 1.89). What’s the critical angle for light inci— dent on the flint—crown interface? Find a simple expression for the speed of light in a material in terms of C and the critical angle at an interface between the mate— rial and vacuum. Find the polarizing angle for light incident from below on the surface of a pond. A cylindrical tank 2.4 m deep is full to the brim with water. Sun— light first hits part of the tank bottom when the rising Sun makes a 22° angle with the horizon. Find the tank’s diameter. For what diameter tank in Problem 50 will sunlight strike some part of the tank bottom whenever the Sun is above the horizon? Light is incident from air on the flat wall of a polystyrene water tank. If the’incidence angle is 40°, what is the angle of refraction in the water? You’re an optometrist, mounting a projector at the back of your 310 4.2—m~long exam room, 26 m above the floor. It shines an eye- 54. 55. test pattern on the opposite wall. Patients will sit with their eyes 3.3 m from the wall and 1 .4 1n above the floor to View the pattern. At what height should you center the pattern on the wall? Find an expression for the displacement x in Fig. 30.6, in terms of 61 d , and n. Figure 30.21 shows light passing through a spherical raindrop, undergoing two refractions and total internal reflection, resulting 57. 58. 59. 60. 61. in an angle qb between the incident and outgoing rays. Show that <15 = 4sin“( sin 6/n) — 29, where 9 is the incidence angle. Incident ray \ Exiting ray FIGURE 30.21 Problem 55 (a) Differentiate the result of Problem 55 to show that the maxi- mum value of ob occurs when the incidence angle 6 is given by 00820 = an2 e l). (b) Use this result and that of Problem 55 to find the maximum cu in a raindrop with n = 1.333. This is the angle at which the rainbow appears, as shown in the Application on page 538. 7. _ Figure 30.22 shows the approximate path of a light ray that un— dergoes internal reflection twice in a spherical water drop. Re- peat Problems 55 and 56 for this case to find the angle at which the secondary rainbow occurs. f? FIGURE 30.22 Problem 57 Show that a three«dimensional corner reflector (three mutually perpendicular mirrors, or a solid cube in which total internal re— flection occurs) turns an incident light ray through 180°. (Hint: Let 5 : qxi +y 11),} + (11/2 be a vector in the propagation direc— tion. How does this vector get changed 'on reflection by a mirror in a plane defined by two of the coordinate axes?) Fermat’s principle states that a light ray’s path is such that the time to traverse that path is an extremum (a minimum or a maxi- mum) when compared with times for nearby paths. Show that Fermat’s principle implies Snell’s law by proving that a light ray going from point A in one medium to point B in a second medium will take the least time if it obeys Snell’s law. You’re an automotive engineer charged with evaluating safety glass, which is made by bonding a layer of flexible plastic between two layers of glass, thus eliminating dangerous glass fragments during accidents. A new product uses glass with refractive index n = 1.55 and plastic with n = 1.48. You’re asked to determine whether total internal reflection at the glass~plastic interface could cause problems with visibility. What do you conclude, and why? A slab of transparent material has thickness d and refractive index n that varies across the material: ? n1 + (n2 — n1)(x/d)2. where x is measured from one face of the slab. A light ray is inci— dent normally on the slab. Find an expression for the time it takes to traverse the slab. Passage Problems Mirages occur when air’s refractive index varies with position as a result of uneven heating. Under such conditions. light undergoes refraction continually and thus follows a curved path. Other exam- ples where a varying refractive index is important include the eye’s lens and Earth’s ionosphere, an electrically conductive layer in the ' Answers to Chapter Questions 543 upper atmosphere, where the refractive index for radio waves varies with altitude. 62. 64. mfiadidwave ricunrsoas Passage Problems 62—65. (amight path in a mirage. (b) Long— distance radio communication via (b) ionospheric refraction (not to scale). Figure 30.2351 depicts light’s path over a hot road, producing a mirage. From the path shown, you can conclude that the air’s refractive index a. increases from left to right. b. increases from right to left. 0. increases upward. (1. increases downward. The observer in Fig. 30.23a sees a shimmering mirage that looks like water but actually results from sky light folloWing the curved path. To the observer, the mirage appears to be at a. point A. b. point B. 0. point C. d. point D. Figure 30.231) shows how continuous refraction in the ionosphere enables long—distance radio communication. Waves launched at angles steeper than 49 don’t refract enough to return to Earth, so they propagate through the ionosphere and on to space. You can therefore conclude that a. all points between A and B receive stronger signals from A than point B receives. b. points between A and B can’t receive signals from A via the ionosphere. c. the refractive index must become infinite at the maximum altitude of the radio signal. The refractive index in the ionosphere is strongly dependent on radio-wave frequency, approaching l for high frequencies. Therefore, a. long-distance communication via the ionosphere is more ‘ likely at higher frequencies. b. higher frequencies won’t penetrate as far into the ionosphere. c. higher frequencies are more appropriate for satellite—based communication. Answers to Chapter Questions Answer to Chapter Opening Question Light from the bee propagates along the'solid, curved plastici“light pipe” Via total internal reflection. This same process carries e—mail and other computer data on the fiber—optic cables that constitute the \ physical structure of the Internet. Answers to GOT IT? Questions 30.1. 113 > n1> 112. 30.2. Most would emerge into the water from the diagonal interface; some would still be reflected as shown. The big idea here is that light can be considered to travel in straight rays when the objects with which it interacts are much larger than the wave— length. Under these conditions, light rays reflect and refract at interfaces between different materials. The angle of incidence and angle of reflection are equal: Snell’s law relates the angle of incidence and angle of refraction: B = 9 n sin0 = n sinB l l l l 2 2 Incident ray Reflected ray fiéflected ray // fl" Some reflection occurs. / Total internal reflection results when light is incident at greater than the critical angle, BC, on an interface with a medium with lower refractive index n2: ’12 a. . me A i cu‘x‘ , A Light polarized in the plane of the incident and refracted rays undergoes no reflection at an interface; thisgspecial polarizing angle, HP, is given by 712 tan GP = W Jr 111 ,v ’ . . w 1’ These For an air—glass interface, 0p 2 56°. 1 ' ‘ angles A combination of total internal reflection and dispersion in raindrops accounts for the rainbow. / ""More outgoing rays concentrate at around 42° deflection. ...
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