Interference, Single and Double Slit Diffraction, and Thin Film Interference
Practice AND Review Problems
I. Interference—Two Stationary Sources
1. *Two in-phase speakers, A and B, are separated by 3.2 m. A listener is stationed at point C, 2.4 m directly in front of speaker B. ABC is a right triangle. Both speakers are playing identical 214 Hz tones and the speed of sound is 343 m/s. Does the listener hear a loud sound or no sound? Mathematically prove your answer. (loud)
2. What if the speakers in the above problem were vibrating out of phase? (no sound)
3. If speaker A is moved further to the left (while ABC remains a right triangle and the same 214 Hz frequency is emitted), what is the new separation between the two speakers when constructive interference occurs again at C? (5.06 m)
4. Suppose that the distance between speaker A and B is 5 m and the speakers are vibrating in phase. They are playing identical 125 Hz tones and the speed of sound is 343 m/s. What is the distance between speaker B and the observer at C such that he observes the first point of destructive interference? (8.4 m, and it’s a lot of algebra)
II. Single Slit Diffraction for Sound and Light
Sound (which have larger wavelengths and diffraction angles than light)
5. *A 1500 Hz sound and a 8500 Hz sound emerge from a loudspeaker through a 0.3m opening. Find the diffraction angle for each sound. (49.76 degrees and 7.73 degrees)
6. Sound emerges from a doorway that is 77 cm wide. Find the diffraction angle when the frequency of the sound is
a. 5 kHz. (5.11 degrees)
b. 5 E2 Hz (63 degrees)
7. A speaker has a single slit opening of 0.3 m.
a. Find the diffraction angle for a 2 kHz tone. (34.87 deg)
b. What should the slit length be in order to generate a 6 kHz tone whose diffraction angle is as wide as that for the 2 kHz tone? (0.1 m)
8. Sound exits a diffraction horn loudspeaker through an opening that is 0.06 m. A person sitting at some angle off to the side of the horn doesn’t hear a sound wave that has a frequency of 8100 Hz. When she is sitting at an angle that is ½ the first angle, there is a different frequency that she doesn’t hear. What is it? (1.5 E 4 Hz)
9. A row of seats is parallel to a stage at a distance of 8.7 m from it. At the center front of the stage, there is a diffraction horn loudspeaker with a 7.5 cm opening. The speaker is playing a 1 E 4 Hz tone. What is the separation between the two seats located on opposite sides of the row at which the tone can’t be heard? (8.9 m)
Light (which have smaller wavelengths and smaller diffraction angles than sound)
10. *Light passes through a slit and shines on a flat screen that is located 0.4 m away. The width of the slit is 4 E -6 m. The distance between the middle of the central (zero order) bright fringe and the beginning of the first order dark fringe is “x”. Determine the width of the central bright fringe (2x) when the wavelength of the light in a vacuum is
a. 690 nm. (0.14 m)
b. 410 nm. (0.082 m)
11. When a monochromatic light source shines through a 0.2mm wide slit onto a screen 3.5m away, the first dark band in the pattern appears 9.1mm from the center of the bright band. What is the wavelength of the light? (5.2 x 10-7m)
12. Light shines through a single slit whose width is 5.6 x 10-4 m. A diffraction pattern is formed on a flat screen located 4 m away. The distance between the middle of the central bright spot and the first order dark bend is 3.4 mm. What is the wavelength of the light used? (4.76 x 10-7 m)
13. *The distance between the two minima on either side of the central fringe is measured to be 2.2 cm on a screen located 2.5 m from a slit when light of wavelength 550 nm is used.
a. What is the distance of the opening? (1.25 E-4 m)
b. What is the distance between the first and the second minima? (0.011 m)
c. Between the second and the third? (0.011 m)
d. Between the first and the third? (0.022 m)
14. A monochromatic beam of light with a wavelength of 680 nm is directed at a single 3.5 mm wide opening. The resulting diffraction pattern is measured along a wall 8.0 m from the opening. What is the distance between the first- and second-order dark fringes? (~1.55 E-3 m)
15. When waves with a wavelength of 425 nm uniformly illuminates a single slit, the central bright fringe, observed on a screen located 0.630 m from the slit, has a width of 0.0166 m. What is the width of the slit? (3.23 E-5 m)
III. Double Slit Diffraction
16. *The first order bright line appears 0.25 cm from the center bright line when a double slit grating is used. The distance between the slits is 0.5mm and the screen is 2.7m from the grating.
a. Find the wavelength. (4.63x10-7m)
b. Where will the 2nd order bright fringe occur? (5 E –3 m)
c. Where will the 3rd order bright fringe occur? (7.5 E –3 m)
d. Where will the 1st order dark fringe occur? (1.25 E –3 m)
e. Where will the 2nd order dark fringe occur? (3.75 E –3 m)
17. *In a double slit experiment, 600 nm light is used to form the first order maximum at an angle of 3°.
a. What is the slit separation? (1.15 x 10-5 m)
b. What angle will the third order line occur? (9°)
18. Two slits separated by 0.80 mm create a first order bright line on a screen 50 cm away. The first order fringe is 0.304 mm from the central bright spot. What wavelength of light is seen? (486.4 nm or 4.86 E –7 m)
19. A Young’s double slit experiment is performed using a slit separation of 0.2 cm and a slit-to-screen distance of 100 cm.
a. How far from the central bright spot will the 1st order bright line occur when 500 nm light is used? (2.5 E -4 m)
b. What about the 2nd order bright fringe? (5 E –4 m)
c. The 3rd? (7.5 E –4 m)
20. Relative to the central fringe, what is the position on a screen located 2 m away of the
a. third order bright fringe in a double slit arrangement (d = 0.058 mm) when the slits are illuminated with a wavelength of 480 nm light? (4.97 E –2 m)
b. What about the 3rd order dark fringe? (4.14 E-2 m)
21. Two parallel slits are illuminated by light composed of two different wavelengths, one of which is 486 nm. On viewing the screen, the light whose wavelength is known produces its 3rd dark fringe at the same place where the light whose wavelength is unknown produces its fourth-order-bright-fringe. The fringes are counted relative to the central zero-order-bright-fringe. What is the unknown wavelength? (304 nm)
Summary of SS and DS Diffraction:
SS:
· The central maximum is 2x wide
· All other fringes are only x wide
· x = the distance from the center of the central max to the next (1st order) minimum
· AND x = the distance between each adjacent minimum (OR each adjacent non-zero order maximum)
· The equations enable you to find the dark fringes (minimums)
DS:
· The center bright spot (the zero order bright fringe) is only 1x wide
· All other fringes are x wide, too
· x = the distance from the center of the center bright spot to the center of the next (1st order) bright spot
· AND x = the distance between each adjacent maximum (or each adjacent minimum)
· ½x = the distance between each bright spot and the next minimum (including the distance between the center bright spot and the 1st order dark fringe)
· The equations enable you to find the bright fringes (maximums)
IV. Thin Film Interference
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22. *What is
a. the speed of light in glass (n=1.43)? (2.098 x 108 m/s)
b. What is the wavelength of 589 nm light in the glass? (411.89 nm)
23. *The surface of a glass plate (n=1.50) is coated with a transparent thin film (n=1.25). A beam of 600 nm light traveling in air is incident on the film. The beam is partially transmitted and partially reflected.
a. Calculate the frequency of the light (remember, frequency never changes once a wave has been created). (5 E 14 Hz)
b. Calculate the wavelength of the light in the film. (480 nm)
c. Calculate the minimum non-zero thickness of film such that the resulting intensity of light reflected back into the air is a minimum. (1.2 E –7 m or 120 nm)
d. Calculate the minimum non-zero thickness of film such that the resultant intensity of reflected light is a maximum. (240 nm)
24. *A certain region of a soap bubble (n = 1.36) floating on water (n = 1.33) appears red (λ = 630 nm) due to reflection and interference. What are the three smallest possible thicknesses of this region of the bubble? (115 nm, 350 nm, 580 nm)
25. *A soap film (n= 1.33) is 375 nm thick and is surrounded on both sides by air. Sunlight, whose wavelengths of light (in air and/or a vacuum as they are really the same value) extend from 380 nm to 750 nm strike the film perpendicularly. What wavelengths (in this range) does constructive interference cause the film to look bright in reflected light? (665 nm and 399 nm)
26. Repeat the above and determine what wavelengths of light will exhibit destructive interference and be minimized. (499 nm; 333 nm is close to the visible spectrum but you still can’t see it)
27. MgF2 (n=1.38) coats a lens to reduce reflection from a glass (n=1.50) surface. How thick a coating (minimum thickness) is needed to produce minimum reflection for 550 nm light (measured in air or a vacuum)? (99.64 nm)
28. Orange light (λv= 611 nm) shines on a soap film (n = 1.33) that has air (n = 1) on either side of it. What is the minimum thickness of the film for which constructive interference causes it to look bright in reflected light? (115 nm)
29. A thin gas (n=1.40) film floats on water (n-1.33). It appears yellow because blue light (469 nm in a vacuum) has been destructively interfered from the reflected light.
a. Determine the minimum thickness of the film for this to occur. (167.5 nm)
b. Repeat assuming that the gas film is floating on glass (n=1.52) rather than water. (83.75 nm)
30. A soap bubble appears green (540 nm) at the point on its front (air-side) surface nearest the viewer. What is its minimum thickness? Assume the soap bubble is resting on water (n=1.33, n of soap is 1.35). (100 nm)
31. A lens appears greenish yellow (570 nm) when white light reflects from it. What minimum thickness of coating (n=1.25) is used on such a glass lens (n=1.52)? (228 nm)
32. A thin film of gasoline floats on a puddle of water. Sunlight falls on the film and reflects into your eyes. The film has a yellow hue due to destructive interference eliminating the color blue (λv = 469 nm) from the reflected light. The refractive indices of the blue light in gasoline and water are 1.4 and 1.33, respectively. Determine the minimum non-zero thickness f the film. (168 nm)
33. A layer of transparent plastic (n = 1.61) on glass (n = 1.52) looks dark when viewed in reflected light whose wavelength is 589 nm in a vacuum or air. Find the two smallest possible non-zero values for the thickness in the layer. (183 nm and 366 nm).
34. Repeat the above problem but for constructively interfered reflected light that gives a bright appearance. (91.5 nm and 274 nm)
35. A mixture of yellow light (580 nm in a vacuum) and violet light (410 nm in a vacuum) falls on a film of gasoline (n = 1.40) floating on water (n = 1.33). What is the minimum non-zero thickness of the film in a spot that looks
a. Yellow due to destructive interference of violet? (146 nm)
b. Violet due to destructive interference of yellow? (207 nm)
36. Under natural conditions, thin films, like gas on water or a soap bubble on water, have a multicolored appearance that often changes while you are watching it. Why are the films multicolored and why do the colors change over time?
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Interference and Thin-Film Practice and Review Problems