Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program such as WileyPLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details. ssm Solution is in the Student Solutions Manual.
This icon represents a biomedical application.
Section 27.1 The Principle of Linear Superposition,
Section 27.2 Young’s Double-Slit Experiment

| 1 | ssm In a Young’s double-slit experiment, the angle that locates the second-order bright fringe is 2.0°. The slit separation is . What is the wavelength of the light?

| 2 | In a Young’s double-slit experiment, the wavelength of the light used is 520 nm (in vacuum), and the separation between the slits is . Determine the angle that locates (a) the dark fringe for which m = 0, (b) the bright fringe for which m = 1, (c) the dark fringe for which m = 1, and (d) the bright fringe for which m = 2.

| 3 | Two in-phase sources of waves are separated by a distance of 4.00 m. These sources produce identical waves that have a wave length of 5.00 m. On the line between them, there are two places at which the same type of interference occurs. (a) Is it constructive or destructive interference, and (b) where are the places located?

| 4 | The dark fringe for m = 0 in a Young’s double-slit experiment is located at an angle of . What is the angle that locates the dark fringe for m = 1?

| 5 | ssm A flat observation screen is placed at a distance of 4.5 m from a pair of slits. The separation on the screen between the central bright fringe and the first-order bright fringe is 0.037 m. The light illuminating the slits has a wavelength of 490 nm. Determine the slit separation.

* | 6 | At most, how many bright fringes can be formed on either side of the central bright fringe when light of wavelength 625 nm falls on a double slit whose slit separation is ?

* | 7 | Refer to Interactive Solution 27.7 at for help in solving this problem. In a Young’s double-slit experiment the separation y between the second-order bright fringe and the central bright fringe on a flat screen is 0.0180 m when the light has a wavelength of 425 nm. Assume that the angles that locate the fringes on the screen are small enough so that sin . Find the separation y when the light has a wavelength of 585 nm.

** | 8 | In Young’s experiment a mixture of orange light (611 nm) and blue light (471 nm) shines on the double slit. The centers of the first-order bright blue fringes lie at the outer edges of a screen that is located 0.500 m away from the slits. However, the first-order bright orange fringes fall off the screen. By how much and in which direction (toward or away from the slits) should the screen be moved, so that the centers of the first-order bright orange fringes just appear on the screen? It may be assumed that is small, so that .

** | 9 | ssm www A sheet that is made of plastic (n = 1.60) covers one slit of a double slit (see the drawing). When the double slit is illuminated by monochromatic light (), the center of the screen appears dark rather than bright. What is the minimum thickness of the plastic?

Problem 9

Section 27.3 Thin-Film Interference

| 10 | A mix of red light and green light is directed perpendicularly onto a soap film (n = 1.33) that has air on either side. What is the minimum nonzero thickness of the film, so that destructive interference causes it to look red in reflected light?

| 11 | ssm Light of wavelength 691 nm (in vacuum) is incident perpendicularly on a soap film (n = 1.33) suspended in air. What are the two smallest nonzero film thicknesses (in nm) for which the reflected light undergoes constructive interference?

| 12 | A tank of gasoline (n = 1.40) is open to the air (n = 1.00). A thin film of liquid floats on the gasoline and has a refractive index that is between 1.00 and 1.40. Light that has a wavelength of 625 nm (in vacuum) shines perpendicularly down through the air onto this film, and in this light the film looks bright due to constructive interference. The thickness of the film is 242 nm and is the minimum nonzero thickness for which constructive interference can occur. What is the refractive index of the film?

| 13 | Interactive LearningWare 27.2 at provides some pertinent background for this problem. A transparent film (n = 1.43) is deposited on a glass plate (n = 1.52) to form a nonreflecting coating. The film has a thickness that is . What is the longest possible wavelength (in vacuum) of light for which this film has been designed?

| 14 | Review Conceptual Example 4 before beginning this problem. A soap film with different thicknesses at different places has an unknown refractive index n and air on both sides. In reflected light it looks multicolored. One region looks yellow because destructive interference has removed blue from the reflected light, while another looks magenta because destructive interference has removed green . In these regions the film has the minimum nonzero thickness t required for the destructive inteference to occur. Find the ratio tsagenta/tyellow.

* | 15 | ssm www Orange light shines on a soap film (n = 1.33) that has air on either side of it. The light strikes the film perpendicularly. What is the minimum thickness of the film for which constructive interference causes it to look bright in reflected light?

* | 16 | The drawing shows a cross section of a plano-concave lens resting on a flat glass plate. (A plano-concave lens has one surface that is a plane and the other that is concave spherical.) The thickness t is . The lens is illuminated with monochromatic light , and a series of concentric bright and dark rings is formed, much like Newton’s rings. How many bright rings are there? (Hint: The cross section shown in the drawing reveals that a kind of air wedge exists between the place where the two pieces of glass touch and the top of the curved surface where the distance t is marked.)

* | 17 | Consult Interactive Solution 27.17 at to review a model for solving this problem. A film of oil lies on wet pavement. The refractive index of the oil exceeds that of the water. The film has the minimum nonzero thickness such that it appears dark due to destructive interference when viewed in red light (wavelength = 640.0 nm in vacuum). Assuming that the visible spectrum extends from 380 to 750 nm, for which visible wavelength(s) (in vacuum) will the film appear bright due to constructive interference?

** | 18 | Interactive LearningWare 27.2 at reviews the concepts that are important in this problem. A uniform layer of water (n = 1.33) lies on a glass plate (n = 1.52). Light shines perpendicularly on the layer. Because of constructive interference, the layer looks maximally bright when the wavelength of the light is 432 nm in vacuum and also when it is 648 nm in vacuum. (a) Obtain the minimum thickness of the film. (b) Assuming that the film has the minimum thickness and that the visible spectrum extends from 380 to 750 nm, determine the visible wavelength(s) (in vacuum) for which the film appears completely dark.
Section 27.5 Diffraction

| 19 | ssm A diffraction pattern forms when light passes through a single slit. The wavelength of the light is 675 nm. Determine the angle that locates the first dark fringe when the width of the slit is (a) and (b) .

| 20 | (a) As Section 17.3 discusses, high-frequency sound waves exhibit less diffraction than low-frequency sound waves do. However, even high-frequency sound waves exhibit much more diffraction under normal circumstances than do light waves that pass through the same opening. The highest frequency that a healthy ear can typically hear is . Assume that a sound wave with this frequency travels at 343 m/s and passes through a doorway that has a width of 0.91 m. Determine the angle that locates the first minimum to either side of the central maximum in the diffraction pattern for the sound. This minimum is equivalent to the first dark fringe in a single-slit diffraction pattern for light. (b) Suppose that yellow light (wave length = 580 nm, in vacuum) passes through a doorway and that the first dark fringe in its diffraction pattern is located at the angle determined in part (a). How wide would this hypothetical doorway have to be?

| 21 | Light that has a wavelength of 668 nm passes through a slit wide and falls on a screen that is 1.85 m away. What is the distance on the screen from the center of the central bright fringe to the third dark fringe on either side?

| 22 | A flat screen is located 0.60 m away from a single slit. Light with a wavelength of 510 nm (in vacuum) shines through the slit and produces a diffraction pattern. The width of the central bright fringe on the screen is 0.050 m. What is the width of the slit?

| 23 | ssm www Light shines through a single slit whose width is . A diffraction pattern is formed on a flat screen located 4.0 m away. The distance between the middle of the central bright fringe and the first dark fringe is 3.5 mm. What is the wave length of the light?

* | 24 | How many dark fringes will be produced on either side of the central maximum if light is incident on a single slit that is wide?

* | 25 | ssm The central bright fringe in a single-slit diffraction pattern has a width that equals the distance between the screen and the slit. Find the ratio of the wavelength of the light to the width of the slit.

* | 26 | In a single-slit diffraction pattern on a flat screen, the central bright fringe is 1.2 cm wide when the slit width is . When the slit is replaced by a second slit, the wavelength of the light and the distance to the screen remaining unchanged, the central bright fringe broadens to a width of 1.9 cm. What is the width of the second slit? It may be assumed that is so small that sin .

** | 27 | In a single-slit diffraction pattern, the central fringe is 450 times as wide as the slit. The screen is 18 000 times farther from the slit than the slit is wide. What is the ratio , where is the wavelength of the light shining through the slit and W is the width of the slit? Assume that the angle that locates a dark fringe on the screen is small, so that .
Section 27.6 Resolving Power

| 28 | A hunter who is a bit of a braggart claims that, from a distance of 1.6 km, he can selectively shoot either of two squirrels who are sitting ten centimeters apart on the same branch of a tree. What’s more, he claims that he can do this without the aid of a telescopic sight on his rifle. (a) Determine the diameter of the pupils of his eyes that would be required for him to be able to resolve the squirrels as separate objects. In this calculation use a wavelength of 498 nm (in vacuum) for the light. (b) State whether his claim is reasonable, and provide a reason for your answer. In evaluating his claim, consider that the human eye automatically adjusts the diameter of its pupil over a typical range of 2 to 8 mm, the larger values coming into play as the lighting becomes darker. Note also that under dark conditions, the eye is most sensitive to a wavelength of 498 nm.

| 29 | Multiple-Concept Example 7 reviews the concepts that are important in this problem. You are looking down at the earth from inside a jetliner flying at an altitude of 8690 m. The pupil of your eye has a diameter of 2.00 mm. Determine how far apart two cars must be on the ground if you are to have any hope of distinguishing between them in (a) red light (wavelength = 665 nm in vacuum) and (b) violet light (wavelength = 405 nm in vacuum).

| 30 | Two stars are apart and are equally distant from the earth. A telescope has an objective lens with a diameter of 1.02 m and just detects these stars as separate objects. Assume that light of wavelength 550 nm is being observed. Also assume that diffraction effects, rather than atmospheric turbulence, limit the resolving power of the telescope. Find the maximum distance that these stars could be from the earth.

| 31 | ssm Multiple-Concept Example 7 illustrates how the concepts needed in this problem are applied. The largest refracting telescope in the world is at the Yerkes Observatory in Williams Bay, Wiscon sin. The objective of the telescope has a diameter of 1.02 m. Two objects are from the telescope. With light of wavelength 565 nm, how close can the objects be to each other so that they are just resolved by the telescope?

| 32 | Review Conceptual Example 8 as background for this problem. In addition to the data given there, assume that the dots in the painting are separated by 1.5 mm and that the wavelength of the light is . Find the distance at which the dots can just be resolved by (a) the eye and (b) the camera.

| 33 | Astronomers have discovered a planetary system orbiting the star Upsilon Andromedae, which is at a distance of from the earth. One planet is believed to be located at a distance of from the star. Using visible light with a vacuum wavelength of 550 nm, what is the minimum necessary aperture diameter that a telescope must have so that it can resolve the planet and the star?

* | 34 | Consult Multiple-Concept Example 7 to see a model for solving this kind of problem. Refer to Interactive Solution 27.34 at to review a method by which this problem can be solved. You are using a microscope to examine a blood sample. Recall from Section 26.12 that the sample should be placed just outside the focal point of the objective lens of the microscope, (a) If the specimen is being illuminated with light of wavelength and the diameter of the objective equals its focal length, determine the closest distance between two blood cells that can just be resolved. Express your answer in terms of . (b) Based on your answer to (a), should you use light with a longer wavelength or a shorter wavelength if you wish to resolve two blood cells that are even closer together?

* | 35 | ssm www A spotlight sends red light (wavelength = 694.3 nm) to the moon. At the surface of the moon, which is away, the light strikes a reflector left there by astronauts. The reflected light returns to the earth, where it is detected. When it leaves the spotlight, the circular beam of light has a diameter of about 0.20 m, and diffraction causes the beam to spread as the light travels to the moon. In effect, the first circular dark fringe in the diffraction pattern defines the size of the central bright spot on the moon. Determine the diameter (not the radius) of the central bright spot on the moon.

** | 36 | Two concentric circles of light emit light whose wavelength is 555 nm. The larger circle has a radius of 4.0 cm, and the smaller circle has a radius of 1.0 cm. When taking a picture of these lighted circles, a camera admits light through an aperture whose diameter is 12.5 mm. What is the maximum distance at which the camera can (a) distinguish one circle from the other and (b) reveal that the inner circle is a circle of light rather than a solid disk of light?
Section 27.7 The Diffraction Grating,
Section 27.8 Compact Discs, Digital Video Discs,
and the Use of Interference

| 37 | ssm The wavelength of the laser beam used in a compact disc player is 780 nm. Suppose that a diffraction grating produces first-order tracking beams that are 1.2 mm apart at a distance of 3.0 mm from the grating. Estimate the spacing between the slits of the grating.

| 38 | The light shining on a diffraction grating has a wavelength of 495 nm (in vacuum). The grating produces a second-order bright fringe whose position is defined by an angle of 9.34°. How many lines per centimeter does the grating have?

| 39 | For a wavelength of 420 nm, a diffraction grating produces a bright fringe at an angle of 26°. For an unknown wavelength, the same grating produces a bright fringe at an angle of 41°. In both cases the bright fringes are of the same order m. What is the unknown wavelength?

| 40 | A diffraction grating is 1.50 cm wide and contains 2400 lines. When used with light of a certain wavelength, a third-order maximum is formed at an angle of 18.0°. What is the wavelength (in nm)?

| 41 | ssm The diffraction gratings discussed in the text are transmission gratings because light passes through them. There are also gratings in which the light reflects from the grating to form a pattern of fringes. Equation 27.7 also applies to a reflection grating with straight parallel lines when the incident light shines perpendicularly on the grating. The surface of a compact disc (CD) has a multicolored appearance because it acts like a reflection grating and spreads sunlight into its colors. The arms of the spiral track on the CD are separated by . Using Equation 27.7, estimate the angle that corresponds to the first-order maximum for a wavelength of (a) 660 nm (red) and (b) 410 nm (violet).

* | 42 | A diffraction grating has 2604 lines per centimeter, and it produces a principal maximum at . The grating is used with light that contains all wavelengths between 410 and 660 nm. What is (are) the wavelength(s) of the incident light that could have produced this maximum?

* | 43 | ssm Violet light (wavelength =410 nm) and red light (wavelength = 660 nm) lie at opposite ends of the visible spectrum. (a) For each wavelength, find the angle that locates the first-order maximum produced by a grating with 3300 lines/cm. This grating converts a mixture of all colors between violet and red into a rainbow-like dispersion between the two angles. Repeat the calculation above for (b) the second-order maximum and (c) the third-order maximum, (d) From your results, decide whether there is an overlap between any of the “rainbows” and, if so, specify which orders overlap.

* | 44 | Three, and only three, bright fringes can be seen on either side of the central maximum when a grating is illuminated with light . What is the maximum number of lines/cm for the grating?

** | 45 | Two gratings A and B have slit separations dA and dB, respectively. They are used with the same light and the same observation screen. When grating A is replaced with grating B, it is observed that the first-order maximum of A is exactly replaced by the second-order maximum of B. (a) Determine the ratio dB/dA of the spacings between the slits of the gratings. (b) Find the next two principal maxima of grating A and the principal maxima of B that exactly replace them when the gratings are switched. Identify these maxima by their order numbers.

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