Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 59 | Concept Questions (a) Point A is the midpoint of one of the sides of a square. On the side opposite this spot, two in-phase loudspeakers are located at adjacent corners, as in Figure 27-41. Standing at point A, you hear a loud sound because constructive interference occurs between the identical sound waves coming from the speakers. Why does constructive interference occur at point A? (b) As you walk along the side of the square toward either empty corner, the loudness diminishes gradually to nothing and then increases again until you hear a maximally loud sound at the corner. Explain these observations in terms of destructive and constructive interference. (c) The general condition that leads to constructive interference entails a number of possibilities for the variable m, which designates the order of the interference maxima. Which one of those possibilities, if any, applies at either empty corner of the square? Explain.
Problem The square referred to in the Concept Questions is 4.6 m on a side. Find the wavelength of the sound waves.

| 60 | Concept Questions (a) The screen in Figure 27-8 has a fixed width and is centered on the midpoint between the two slits. Should the screen be moved to the left or to the right to ensure that the third-order bright fringe does not lie on the screen? (b) Do fewer fringes lie on the screen when the distance L between the double-slit and the screen is smaller or larger? Account for your answers.
Problem In a setup like that in Figure 27-8, a wavelength of 625 nm is used in a Young’s double-slit experiment. The separation between the slits is . The total width of the screen is 0.20 m. In one version of the setup, the separation between the double slit and the screen is LA = 0.35 m, whereas in another version it is LB = 0.50 m. On one side of the central bright fringe, how many bright fringes lie on the screen in the two versions of the setup? Do not include the central bright fringe in your counting. Verify that your answer is consistent with your answers to the Concept Questions.

| 61 | Concept Questions (a) What, if any, phase change occurs when light, traveling in air, reflects from the interface between the air and a soap film (n = 1.33)? (b) What, if any, phase change occurs when light, traveling in a soap film, reflects from the interface between the soap film and a glass plate (n = 1.52)? (c) Is the wavelength of the light in a soap film greater than, smaller than, or equal to the wavelength in a vacuum?
Problem A soap film (n = 1.33) is 465 nm thick and lies on a glass plate (n = 1.52). Sunlight, whose wavelengths (in vacuum) extend from 380 to 750 nm, travels through the air and strikes the film perpendicularly. For which wavelength(s) in this range does destructive interference cause the film to look dark in reflected light?

| 62 | Concept Questions (a) In a single-slit diffraction pattern the width of the central bright fringe is defined by the location of the first dark fringe that lies on either side of it. For a given slit width, does the width of the central bright fringe increase, decrease, or remain the same as the wavelength of the light increases? (b) For a given wavelength, does the width of the central bright fringe increase, decrease, or remain the same as the slit width increases? (c) When both the wavelength and the slit width change, it is possible for the width of the central bright fringe to remain the same. What condition must be satisfied for this to happen? In each case, give your reasoning.
Problem A slit has a width of . When light with a wavelength of passes through this slit, the width of the central bright fringe on a flat observation screen has a certain value. With the screen kept in the same place, this slit is replaced with a second slit (width W2) and a wavelength of is used. The width of the central bright fringe on the screen is observed to be unchanged. Find W2.

| 63 | Concept Questions An inkjet color printer uses tiny dots of red, green, and blue ink to produce an image. At normal viewing distances, the eye does not resolve the individual dots, so that the image has a normal look. The angle is the minimum angle that two dots can subtend at the eye and still be resolved separately. (a) For which color does have the largest value? (b) For which color does have the smallest value? (c) Corresponding to each value of , there is a value for the separation distance s between the dots. How is related to s and the viewing distance L? (d) Assume that the dot separation on the printed page is the same for all colors and is chosen so that none of the colored dots can be seen as separate objects. Should the maximum allowable dot separation be less than sred, sgreen, or sblue? For each answer, give your reasoning.
Problem The wavelengths for red, green, and blue are , , and . The diameter of the pupil through which light enters the eye is 2.0 mm. For a viewing distance of 0.40 m, what is the value for the dot separation identified in Concept Question (d)?

| 64 | Concept Questions (a) Two diffraction gratings are located at the same distance from observation screens. Light with the same wavelength is used for each. The principal maxima of grating A are observed to be closer together on the screen than the principal maxima of grating B. Which grating diffracts the light to a greater extent? (b) Which grating has the smaller slit separation d? (c) Which grating has the greater number of lines permeter? Justify each of your answers.
Problem The separation between adjacent principal maxima for grating A is 2.7 cm, and for grating B it is 3.2 cm. Grating A has 2000 lines per meter. How many lines per meter does grating B have? The diffraction angles are small enough that sin . Be sure that your answer is consistent with your answers to the Concept Questions.

| 65 | Concept Questions A beam of light is sent directly down onto a glass plate (n = 1.5) and a plastic plate (n = 1.2) that form a thin wedge of air (see the drawing). An observer looking down through the glass plate sees the fringe pattern shown in the lower part of the drawing, with dark fringes at the ends A and B. (a) What, if any, phase change occurs when the light, traveling in the glass, reflects from the interface between the glass and the air wedge? (b) What, if any, phase change occurs when the light, traveling in the air wedge, reflects from the interface between the air and the top surface of the plastic plate? (c) Why is there a dark fringe at A? (d) How many wavelengths of light fit into the air between the glass and plastic plates at B? Account for your answers to each of the questions.

Problem The wavelength of the light is 520 nm. Using the fringe pattern shown in the drawing, determine the thickness of the air wedge at B. Be sure your answer is consistent with part (d) of the Concept Questions.

| 66 | Concept Questions Light shines on a diffraction grating, and a diffraction pattern is produced on a viewing screen that consists of a central bright fringe and higher-order bright fringes (see the drawing). (a) From trigonometry, how is the distance y from the central bright fringe to the second-order bright fringe related to the diffraction angle and the distance L between the grating and the screen? (b) From physics, how is related to the order m of the bright fringe, the wavelength of the light, and the separation d between the slits? (c) In this problem, the angle is small (less than a few degrees). When the angle is small, is approximately equal to , or . Using this approximation, obtain an expression for y in terms of L, m, , and d. (d) If the entire apparatus in the drawing is submerged in water, would you expect the distance y to increase, decrease, or remain unchanged? Why?

Problem Light of wavelength 480 nm (in vacuum) is incident on a diffraction grating that has a slit separation of 5.0 × 10−7 m. The distance between the grating and the viewing screen is 0.15 m. (a) Determine the distance y from the central bright fringe to the second-order bright fringe. (b) If the entire apparatus is submerged in water (nwater = 1.33), what is the distance y? Be sure your answer is consistent with part (d) of the Concept Questions.

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