Section 1 Interference: Practice Problems

Transcription

1 Section 1 Interference: Practice Problems 1. Violet light falls on two slits separated by m. A first-order bright band appears 13.2 mm from the central bright band on a screen m from the slits. What is λ? 3. In a double-slit experiment, physics students use a laser with λ = nm. A student places the screen m from the slits and finds the firstorder bright band 65.5 mm from the central line. What is the slit separation? 2. Yellow-orange light from a sodium lamp of wavelength 596 nm is aimed at two slits that are separated by m. What is the distance from the central band to the first-order yellow band if the screen is m from the slits? 4. CHALLENGE Yellow-orange light with a wavelength of 596 nm passes through two slits that are separated by m and makes an interference pattern on a screen. If the distance from the central line to the first-order yellow band is m, how far is the screen from the slits? esolutions Manual - Powered by Cognero Page 1

2 5. In the situation in Example Problem 2, what would be the thinnest film that would create a reflected red (λ = 635 nm) band? 6. A glass lens has a nonreflective coating placed on it. If a film of magnesium fluoride is placed on the glass, how thick should the layer be to keep yellow-green light (λ = 555 nm) from being reflected? See the sketch in Figure 9. Because n film > n air, there is a phase inversion on the first reflection. Because n glass > n film, there is a phase inversion on the second reflection. For destructive interference to keep yellow-green from being reflected: For the thinnest film, m = 0. esolutions Manual - Powered by Cognero Page 2

3 7. You can observe thin-film interference by dipping a bubble wand into some bubble solution and holding the wand in the air. What is the thickness of the thinnest soap film at which you would see a black stripe if the light illuminating the film has a wavelength of 521 nm? Use n = 1.33 for the bubble solution. Because n film > nair, there is a phase change on the first reflection. Because n air < n film, there is no phase change on the second reflection. For destructive interference to get a black stripe 9. CHALLENGE A silicon solar cell has a nonreflective coating placed on it. If a film of silicon monoxide, n = 1.45, is placed on the silicon, n = 3.5, how thick should the layer be to keep yellow-green light (λ = 555 nm) from being reflected? Because n film > n air, there is a phase inversion on the first reflection. Because n silicon > n film, there is a phase inversion on the second reflection. For destructive interference to keep yellow-green from being reflected: For the thinnest film, m = 0. For the thinnest film, m = 1. Section 1 Interference: Review 8. What is the thinnest soap film (n = 1.33) for which light of wavelength 521 nm will constructively interfere with itself? 10. MAIN IDEA Two very narrow slits are cut close to each other in a large piece of cardboard. They are illuminated by monochromatic red light. A sheet of white paper is placed far from the slits, and a pattern of bright and dark bands is seen on the paper. Describe how a wave behaves when it encounters a slit, and explain why some regions are bright while others are dark. When a wave encounters a slit, the wave bends. Light is diffracted by the slits. Light from one slit interferes with light from the other. If interference is constructive, there is a bright band; if destructive, the region is dark. esolutions Manual - Powered by Cognero Page 3

4 11. Interference Patterns Sketch the pattern described in the previous problem. 13. Lucien is blowing bubbles and holds the bubble wand with a soap film (n = 1.33) in it vertically. a. What is the second thinnest width of the soap film at which he could see a bright stripe if the light illuminating the film has a wavelength of 575 nm? b. What other widths produce a bright stripe at 575 nm? 12. Interference Patterns Sketch what happens to the pattern in the previous two problems when the red light is replaced by blue light. a. There is one phase inversion, so constructive interference will be when. For the second thinnest thickness, m = 1. The light bands become more closely spaced than when using blue light. b. For the general case, where m can be any integer: Thus, any odd-integer multiple of 108 nm will produce a bright stripe. esolutions Manual - Powered by Cognero Page 4

5 Section 1 Interference: Review 14. Light of wavelength 542 nm falls on a double slit. Use the values from Figure 11 to determine how far apart the slits are. Section 2 Diffraction: Practice Problems 16. Monochromatic green light of wavelength 546 nm falls on a single slit with a width of mm. The slit is located 75 cm from a screen. How wide will the central bright band be? 17. Yellow light with a wavelength of 589 nm passes through a slit of width mm and makes a pattern on a screen. If the width of the central bright band is m, how far is it from the slits to the screen? 15. Critical Thinking The equation for wavelength from a double-slit experiment uses the simplification that θ is small so that sin θ tan θ. Up to what angle is this a good approximation when your data has two significant figures? Would the maximum angle for a valid approximation increase or decrease as you increase the precision of your angle measurement? sin θ = tan θ to two significant digits up to 9.9. An increase in the precision of the measurement reduces this angle to esolutions Manual - Powered by Cognero Page 5

6 18. Light from a He-Ne laser (λ = nm) falls on a slit of unknown width. A pattern is formed on a screen 1.15 m away, on which the central bright band is 15.0 mm wide. How wide is the slit? 20. CHALLENGE White light falls on a single slit that is mm wide. A screen is placed 1.00 m away. A student first puts a blue-violet filter (λ = 441 nm) over the slit, then a red filter (λ = 622 nm). The student measures the width of the central bright band. a. Which filter produced the wider band? b. Calculate the width of the central bright band for both filters. a. Red, because the width of the central bright band is proportional to wavelength. 19. Yellow light falls on a single slit mm wide. On a screen that is 60.0 cm away, the central bright band is 24.0 mm wide. What is the wavelength of the light? b. 21. White light shines through a grating onto a screen. Describe the pattern that is produced. A full spectrum of color is seen. Because of the variety of wavelengths, dark fringes of one wavelength are filled by bright fringes of another color. esolutions Manual - Powered by Cognero Page 6

7 22. If blue light of wavelength 434 nm shines on a diffraction grating and the spacing of the resulting lines on a screen that is 1.05 m away is 0.55 m, what is the spacing between the slits in the grating? 24. Blue light shines on the DVD in Example Problem 3. If the dots produced on a wall that is 0.65 m away are separated by 58.0 cm, what is the wavelength of the light? 25. Challenge Light of wavelength 632 nm passes through a diffraction grating and creates a pattern on a screen that is 0.55 m away. If the first bright band is 5.6 cm from the central bright band, how many slits per centimeter does the grating have? 23. A diffraction grating with slits separated by m is illuminated by violet light with a wavelength of 421 nm. If the screen is 80.0 cm from the grating, what is the separation of the lines in the diffraction pattern? esolutions Manual - Powered by Cognero Page 7

8 Section 2 Diffraction: Review 26. MAIN IDEA Many narrow slits are close to each other and equally spaced in a large piece of cardboard. They are illuminated by monochromatic red light. A sheet of white paper is placed far from the slits, and a pattern of bright and dark bands is visible on the paper. Sketch the pattern that would be seen on the screen. 28. Line Spacing You shine a red laser light through one diffraction grating and form a pattern of red dots on a screen. Then you substitute a second diffraction grating for the first one, forming a different pattern. The dots produced by the first grating are spread out more than those produced by the second. Which grating has more lines per millimeter?, so the greater the dot spacing (x) the narrower the slit spacing (d) and thus more lines per millimeter. Band spacing is exactly the same as in the pattern produced by the two slits, but now light bands are much thinner and separated by wider dark bands. 29. First-Order Dark Bands Monochromatic green light of wavelength 546 nm falls on a single slit of width and location from a screen shown in Figure 22. What is the separation of the first-order dark bands? 27. Rayleigh Criterion The brightest star in the winter sky in the northern hemisphere is Sirius. In reality, Sirius is a system of two stars that orbit each other. If the Hubble Space Telescope (diameter 2.4 m) is pointed at the Sirius system, which is 8.44 light-years from Earth, what is the minimum separation there would need to be between the stars in order for the telescope to be able to resolve them? Assume that the light coming from the stars has a wavelength of 550@nm. esolutions Manual - Powered by Cognero Page 8

9 30. Critical Thinking You are shown a spectrometer and are not told whether it has been constructed with a prism or a diffraction grating. If you look at a spectrum of white light passing through the spectrometer, how could you determine which device produced the spectrum? Determine if the violet or the red end of the spectrum makes the largest angle with the direction of the beam of incident white light. A prism bends the violet end of the spectrum the most, whereas a grating diffracts red wavelengths the most. Chapter Assessment Section 1 Interference: Mastering Concepts 31. Why is it important that monochromatic light was used to make the interference pattern in the doubleslit experiment? When monochromatic light is used, you get a sharp interference pattern; if you use white light, you get sets of colored bands. 32. Explain why the position of the central bright band of a double-slit interference pattern cannot be used to determine the wavelength of the light. All wavelengths produce the line in the same place. 33. Describe how you could use light of a known wavelength to find the distance between two narrow slits. Let the light fall on the double slit, and let the interference pattern fall on a sheet of paper. Measure the spacing between the bright bands, x, and use the equation. 34. Describe in your own words what happens in thinfilm interference when a colored band is produced by light shining on a soap film suspended in air. Make sure you include in your explanation how the wavelength of the light and the thickness of the film are related. When the light strikes the front of the film, some reflects off this surface and some passes through the film and reflects off the back surface of the film. When light reflects off a medium with a higher index of refraction, it undergoes a phase shift of one-half wavelength; this happens to the light that initially reflects. In order for a colored band to be produced, two light rays must be in phase. If the thickness of the film is such that the ray reflecting off the back surface goes through a multiple of half of a cycle while passing through the film, the light rays arriving at your eye will be in phase and constructively interfere. Remember that the index of refraction of the film determines whether the wave is inverted, so that the thickness of the film must equal a multiple of half a wavelength of the light, divided by the film s index of refraction. esolutions Manual - Powered by Cognero Page 9

10 Chapter Assessment Section 1 Interference: Mastering Problems 35. Light falls on a pair of slits 19.0 µm apart and 80.0 cm from a screen, as shown in Figure 23.The first-order bright band is 1.90 cm from the central bright band. What is the wavelength of the light? (Level 1) 36. Oil Slick After a short spring rain shower, Tom and Ann take their basset hound for a walk and notice a thin film of oil (n = 1.45) on a puddle of water, producing different colors. What is the minimum thickness of a place where the oil creates constructive interference for light with a wavelength equal to 545 nm? (Level 1) There is one phase inversion, so constructive interference will be when For the minimum thickness, m = 0. esolutions Manual - Powered by Cognero Page 10

11 37. Film Thickness A plastic reflecting film (n = 1.83) is placed on an auto glass window (n = 1.52). What is the thinnest film that will reflect yellow-green light (λ = 555 nm)? Unfortunately, a film this thin cannot be manufactured. What is the next-thinnest film that will produce the same effect? (Level 2) Because n film > n air, there is an inversion on the first reflection. Because n glass < n film, there is no inversion on the second reflection. For constructive interference to reflect yellowgreen light: 38. Insulation Film Winter is approaching and Alejandro is helping to cover the windows in his home with thin sheets of clear plastic (n = 1.81) to keep the drafts out. After the plastic is taped up around the windows, the plastic is heated with a hair dryer to shrink-wrap the window, which alters the thickness but not the refractive index of the plastic. Alejandro notices a place on the plastic where there is a blue stripe of color. He realizes that this is created by thin-film interference. What are three possible thicknesses of the plastic where the blue stripe is produced if the wavelength of the light is 445 nm? (Level 2) For the thinnest film, m = 0. For the next-thinnest film, m = 1. esolutions Manual - Powered by Cognero Page 11

12 39. Ranking Task Five different lasers produce double-slit interference patterns. In each case, the slit separation is mm. Rank them according to the wavelength of the lasers, from shortest to longest. Specifically indicate any ties. (Level 3) A: The screen is 0.95 m from the slits, and adjacent bright spots are separated by 12 mm. B: The screen is 0.95 m from the slits, and adjacent bright spots are separated by 16 mm. C: The screen is 1.3 m from the slits, and adjacent bright spots are separated by 20 mm. D: The screen is 2.8 m from the slits, and adjacent bright spots are separated by 40 mm. E: The screen is 2.8 m from the slits, and adjacent bright spots are separated by 50 mm. Chapter Assessment Section 2 Diffraction: Mastering Concepts 40. BIG IDEA White light shines through a diffraction grating. Are the resulting red lines spaced more closely or farther apart than the resulting violet lines? Why? The spacing is directly proportional to the wavelength, and because red light has a longer wavelength than violet, the red lines will be spaced farther apart than the violet lines. 41. Why do diffraction gratings have large numbers of slits? Why are these slits so close together? The large number of grooves in diffraction gratings increases the intensity of the diffraction patterns. The grooves are close together, producing sharper images of light. 42. Telescopes Why would a telescope with a small diameter be unable to resolve the images of two closely spaced stars? Small apertures have large diffraction patterns that limit resolution. 43. Reverse Problem Write a physics problem with real-life objects for which the following equation would be part of the solution: Answers will vary, but a correct form of the answer is, Light of wavelength 530 nm is shone through a slit of width 0.20 mm. If a diffraction pattern is made on a screen 2.00 m away, what is the distance from the center of the pattern to the first dark band? 44. Problem Posing Complete this problem so that it can be solved using the Rayleigh criterion: A telescope currently being designed is to have an aperture that is 8.0 m in diameter. Answers will vary. A possible form of the correct answer would be,.if resolution were to be diffraction-limited, what would be the smallest angle that could be resolved if the incoming light had a wavelength of 550 nm? 45. For a given diffraction grating, which color of visible light produces a bright line closest to the central bright band? violet light, the color with the smallest wavelength esolutions Manual - Powered by Cognero Page 12

13 46. When you look at an incandescent lamp through a pair of fun glasses you see thin stripes of spectral colors going out from the light in eight directions as in Figure 24. You recognize that the glasses act like diffraction gratings. What are the directions of the scratches in these glasses? Chapter Assessment Section 2 Diffraction: Mastering Problems 47. Monochromatic light passes through a single slit cm wide and falls on a screen 100 cm away, as shown in Figure 25. If the width of the central band is 1.20 cm, what is the wavelength of the light? (Level 1) Horizontal scratches produce up and down spectra; vertical scratches produce the left and right. Two other stripes are from scratches at 45 o above and below the horizontal. 48. A good diffraction grating has lines per cm. What is the distance between two lines? (Level 1) esolutions Manual - Powered by Cognero Page 13

14 49. Light with a wavelength of 455 nm passes through a single slit and falls on a screen 100 cm away. If the slit is cm wide, what is the distance from the center of the pattern to the first dark band? (Level 2) 51. Monochromatic light with a wavelength of 425 nm passes through a single slit and falls on a screen 75 cm away. If the central bright band is 0.60 cm wide, what is the width of the slit? (Level 3) 50. Kaleidoscope The mirrors have been removed from a kaleidoscope. The diameter of the eyehole at the end is 7.0 mm. If two bluish-purple specks on the other end of the kaleidoscope separated by 40 μm are barely distinguishable, what is the length of the kaleidoscope? Use λ = 650 nm and assume the resolution is diffraction limited through the eyehole. (Level 3) esolutions Manual - Powered by Cognero Page 14

15 52. Hubble Space Telescope Suppose the Hubble Space Telescope, 2.4 m in diameter, is in orbit m above Earth and is turned to view Earth, as shown in Figure 26. If you ignore the effect of the atmosphere, how large an object can the telescope resolve? Use λ = 515 nm. (Level 2) 53. Spectroscope A spectroscope uses a grating with 12,000 lines/cm. Find the angles at which red light, 632 nm, and blue light, 421 nm, have first-order bright lines. (Level 3) Chapter Assessment: Applying Concepts 54. Science Fair At a science fair, one exhibition is a very large soap film that has a fairly consistent thickness. It is illuminated by a light with a wavelength of 432 nm, and nearly the entire surface appears to be a lovely shade of purple. What would you see in the following situations? a. Film thickness was doubled. b. The film thickness was increased by half a wavelength of the illuminating light. c. The film thickness was decreased by one-quarter of a wavelength of the illuminating light. a. complete destructive interference b. complete constructive interference c. complete destructive interference esolutions Manual - Powered by Cognero Page 15

16 55. What are the differences in the characteristics of the diffraction patterns formed by diffraction gratings containing 10 4 lines/cm and 10 5 lines/cm? The lines in the diffraction pattern are narrower for the 10 5 lines/cm grating. 59. For each of the following examples, indicate whether the color is produced by thin-film interference, refraction, or the presence of pigments. a. soap bubbles b. rose petals c. oil films d. a rainbow 56. Laser-Pointer Challenge You have two laser pointers, a red one and a green one. Your friends Mark and Carlos disagree about which has the longer wavelength. Mark insists that red light has a longer wavelength, while Carlos is sure that green has the longer wavelength. You have a CD handy. Describe what demonstration you would do with this equipment and how you would explain the results to Carlos and Mark to settle their disagreement. Shine each laser pointer at the grating, reflecting onto a nearby wall. The color with the longer wavelength will produce spots with a greater spacing on the wall because the spacing is directly proportional to the wavelength. (Mark is correct; red light has a longer wavelength than green light.) a. interference b. pigments c. interference d. refraction 60. Describe the changes in a single-slit diffraction pattern as the width of the slit is decreased. The bands get wider and dimmer. Chapter Assessment: Mixed Review 57. How can you tell whether a pattern is produced by a single slit or a double slit? A double-slit interference pattern consists of equally spaced lines of almost equal brightness. A single-slit diffraction pattern has a bright, broad central band and dimmer side bands. 58. Optical Microscope Why is blue light used for illumination in an optical microscope? Less diffraction results from the short wavelength of blue light. esolutions Manual - Powered by Cognero Page 16

17 61. Record Marie uses an old 33 1/3 rpm record as a diffraction grating. She shines a laser, λ = nm, on the record, as shown in Figure 27. On a screen 4.0 m from the record, a series of red dots 21 mm apart are visible. (Level 1) 62. Camera When a camera with a 50-mm lens is set at, its aperture has an opening 6.25 mm in diameter. (Level 3) a. A CCD detector is 50.0 mm away and senses light at λ = 550 nm. What is the resolution of the lens? b. The owner of the camera knows that it has 6.3 megapixels in its CCD. The manufacturer says that each pixel is 7.6 μm on a side. Compare the size of the pixel with the resolution of the lens calculated in part a. a. a. How many grooves are there in a centimeter along the radius of the record? b. Marie checks her results by noting that the ridges represent a song that lasts 4.01 min and takes up 16 mm on the record. How many grooves should there be in a centimeter? b. a. b. esolutions Manual - Powered by Cognero Page 17

18 63. A glass lens has antireflective coating, n = 1.2, with a thickness of 125 nm. For which color(s) of light does complete destructive interference occur? (Level 2) Because n film > n air, there is a phase inversion on the first reflection. Because n lens = 1.52 > n film, there is a phase inversion on the second reflection. For destructive interference: Chapter Assessment: Thinking Critically 64. Apply Concepts Yellow light falls on a diffraction grating. On a screen behind the grating, you see three spots: one at zero degrees, where there is no diffraction, and one each at +30 and 30. You now add a blue light of equal intensity that is in the same direction as the yellow light. What pattern of spots will you now see on the screen? A green spot at 0, yellow spots at +30 and 30, and two blue spots slightly closer in. 65. Apply Concepts Blue light of wavelength λ passes through a single slit of width w. A diffraction pattern appears on a screen. If you replace the blue light with a green light of wavelength 1.5λ, what slit width will produce the original pattern? The angle of diffraction depends on the ratio of slit width to wavelength. Thus, you would increase the width to 1.5w. The light is reddish-orange. For other values of m, the wavelength is shorter than that of light. esolutions Manual - Powered by Cognero Page 18

19 66. Analyze and Conclude At night, the pupil of a human eye has an aperture diameter of 8.0 mm. The diameter is smaller in daylight. An automobile s headlights are separated by 1.8 m. How far away can the human eye distinguish the two headlights at night? Hint: Assume a wavelength of 525 nm. What besides diffraction might be limiting factors? 68. The gemstone opal has an iridescent sheen. Research and describe how these colors are produced. Descriptions will vary, but should include a description of thin-film interference. 69. Many telescopes have adaptive optics that reduce the atmospheric effects that cause stars to twinkle. Research and describe how these systems work. Student answers will vary. Answers could include the correction of distorted wavefronts with a deformable mirror to improve resolution. Diffraction doesn t limit the sensing ability of your eyes, which are limited by aperture at a few hundred meters. More probable factors are the refractive effects of the atmosphere, like those that cause stars to twinkle, or the limitations of the retina and the optic area of the brain to separate two dim sources. 70. Research and interpret the role of diffraction in medicine and astronomy. Describe at least two applications in each field. Student answers will vary. Answers could include diffraction in telescopes and microscopes, as well as spectroscopy. Chapter Assessment: Writing in Physics 67. Research and describe Thomas Young s contributions to physics. Evaluate the impact of his research on the scientific thought about light's nature. Student answers will vary. Answers should include Young s two-slit experiment that allowed him to precisely measure the wavelength of light. esolutions Manual - Powered by Cognero Page 19

20 Chapter Assessment: Cumulative Review 71. How much work must be done to push a 0.5-m 3 block of wood to the bottom of a 4-m-deep swimming pool? The density of wood is 500 kg/m A concave mirror has a 48.0-cm radius. A 2.0-cmtall object is placed 12.0 cm from the mirror. Calculate the image position and image height. The block would float, but to submerge it would require an extra force downward. 72. What are the wavelengths of microwaves in an oven if their frequency is 2.4 GHz? esolutions Manual - Powered by Cognero Page 20

21 74. The focal length of a convex lens is 21.0 cm. A cm-tall candle is 7.50 cm from the lens. Use the thinlens equation to find the image position and image height. esolutions Manual - Powered by Cognero Page 21