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Chapter 3: Interference
51 problems · 1 figureCheck Your Understanding 3.1 §3.2 source ↗
In the system used in the preceding examples, at what angles are the first and the second bright fringes formed?
Answer
, respectively
Check Your Understanding 3.2 §3.4 source ↗
Going further with Example 3.4, what are the next two thicknesses of soap bubble that would lead to (a) constructive interference, and (b) destructive interference?
Answer
a. 853 nm, 1097 nm; b. 731 nm, 975 nm
Check Your Understanding 3.3 §3.5 source ↗
Although m, the number of fringes observed, is an integer, which is often regarded as having zero uncertainty, in practical terms, it is all too easy to lose track when counting fringes. In Example 3.6, if you estimate that you might have missed as many as five fringes when you reported fringes, (a) is the value for the index of refraction worked out in Example 3.6 too large or too small? (b) By how much?
Answer
a. too small; b. up to
3-1 §3.1 source ↗
Young’s double-slit experiment breaks a single light beam into two sources. Would the same pattern be obtained for two independent sources of light, such as the headlights of a distant car? Explain.
Answer
No. Two independent light sources do not have coherent phase.
3-3 §3.1 source ↗
Why won’t two small sodium lamps, held close together, produce an interference pattern on a distant screen? What if the sodium lamps were replaced by two laser pointers held close together?
Answer
Because both the sodium lamps are not coherent pairs of light sources. Two lasers operating independently are also not coherent so no interference pattern results.
3-5 §3.2 source ↗
Why is monochromatic light used in the double slit experiment? What would happen if white light were used?
Answer
Monochromatic sources produce fringes at angles according to . With white light, each constituent wavelength will produce fringes at its own set of angles, blending into the fringes of adjacent wavelengths. This results in rainbow patterns.
3-7 §3.4 source ↗
How is the difference in paths taken by two originally in-phase light waves related to whether they interfere constructively or destructively? How can this be affected by reflection? By refraction?
Answer
Differing path lengths result in different phases at destination resulting in constructive or destructive interference accordingly. Reflection can cause a phase change, which also affects how waves interfere. Refraction into another medium changes the wavelength inside that medium such that a wave can emerge from the medium with a different phase compared to another wave that travelled the same distance in a different medium.
3-9 §3.4 source ↗
In placing a sample on a microscope slide, a glass cover is placed over a water drop on the glass slide. Light incident from above can reflect from the top and bottom of the glass cover and from the glass slide below the water drop. At which surfaces will there be a phase change in the reflected light?
Answer
Phase changes occur upon reflection at the top of glass cover and the top of glass slide only.
3-11 §3.4 source ↗
While contemplating the food value of a slice of ham, you notice a rainbow of color reflected from its moist surface. Explain its origin.
Answer
The surface of the ham being moist means there is a thin layer of fluid, resulting in thin-film interference. Because the exact thickness of the film varies across the piece of ham, which is illuminated by white light, different wavelengths produce bright fringes at different locations, resulting in rainbow colors.
3-13 §3.4 source ↗
A nonreflective coating like the one described in Example 3.3 works ideally for a single wavelength and for perpendicular incidence. What happens for other wavelengths and other incident directions? Be specific.
Example 3.3 Calculating the Thickness of a Nonreflective Lens Coating in the book ↗
Answer
Other wavelengths will not generally satisfy for the same value of t so reflections will result in completely destructive interference. For an incidence angle , the path length inside the coating will be increased by a factor so the new condition for destructive interference becomes .
3-15 §3.5 source ↗
Describe how a Michelson interferometer can be used to measure the index of refraction of a gas (including air).
Answer
In one arm, place a transparent chamber to be filled with the gas. See Example 3.6.
Example 3.6 Measuring the Refractive Index of a Gas in the book ↗
3-17 §3.2 source ↗
Calculate the angle for the third-order maximum of 580-nm wavelength yellow light falling on double slits separated by 0.100 mm.
Answer
3-19 §3.2 source ↗
Find the distance between two slits that produces the first minimum for 410-nm violet light at an angle of
Answer
3-21 §3.2 source ↗
What is the wavelength of light falling on double slits separated by if the third-order maximum is at an angle of ?
Answer
3-23 §3.2 source ↗
What is the highest-order maximum for 400-nm light falling on double slits separated by ?
Answer
62.5; since m must be an integer, the highest order is then .
3-25 §3.2 source ↗
What is the smallest separation between two slits that will produce a second-order maximum for 720-nm red light?
Answer
3-27 §3.2 source ↗
(a) If the first-order maximum for monochromatic light falling on a double slit is at an angle of , at what angle is the second-order maximum? (b) What is the angle of the first minimum? (c) What is the highest-order maximum possible here?
Answer
a. ; b. ; c. 5.76, the highest order is .
3-29 §3.2 source ↗
Using the result of the preceding problem, (a) calculate the distance between fringes for 633-nm light falling on double slits separated by 0.0800 mm, located 3.00 m from a screen. (b) What would be the distance between fringes if the entire apparatus were submersed in water, whose index of refraction is 1.33?
Answer
a. 2.37 cm; b. 1.78 cm
3-31 §3.2 source ↗
In a double-slit experiment, the fifth maximum is 2.8 cm from the central maximum on a screen that is 1.5 m away from the slits. If the slits are 0.15 mm apart, what is the wavelength of the light being used?
Answer
560 nm
3-33 §3.2 source ↗
If 500-nm and 650-nm light illuminates two slits that are separated by 0.50 mm, how far apart are the second-order maxima for these two wavelengths on a screen 2.0 m away?
Answer
1.2 mm
3-35 §3.3 source ↗
Ten narrow slits are equally spaced 0.25 mm apart and illuminated with yellow light of wavelength 580 nm. (a) What are the angular positions of the third and fourth principal maxima? (b) What is the separation of these maxima on a screen 2.0 m from the slits?
Answer
a. b.
3-37 §3.3 source ↗
For a three-slit interference pattern, find the ratio of the peak intensities of a secondary maximum to a principal maximum.
Answer
1:9
3-39 §3.4 source ↗
A soap bubble is 100 nm thick and illuminated by white light incident perpendicular to its surface. What wavelength and color of visible light is most constructively reflected, assuming the same index of refraction as water?
Answer
532 nm (green)
3-41 §3.4 source ↗
Calculate the minimum thickness of an oil slick on water that appears blue when illuminated by white light perpendicular to its surface. Take the blue wavelength to be 470 nm and the index of refraction of oil to be 1.40.
Answer
3-43 §3.4 source ↗
A film of soapy water () on top of a plastic cutting board has a thickness of 233 nm. What color is most strongly reflected if it is illuminated perpendicular to its surface?
Answer
620 nm (orange)
3-45 §3.4 source ↗
Suppose you have a lens system that is to be used primarily for 700-nm red light. What is the second thinnest coating of fluorite (magnesium fluoride) that would be nonreflective for this wavelength?
Answer
380 nm
3-47 §3.4 source ↗
To save money on making military aircraft invisible to radar, an inventor decides to coat them with a nonreflective material having an index of refraction of 1.20, which is between that of air and the surface of the plane. This, he reasons, should be much cheaper than designing Stealth bombers. (a) What thickness should the coating be to inhibit the reflection of 4.00-cm wavelength radar? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
Answer
a. Assuming n for the plane is greater than 1.20, then there are two phase changes: 0.833 cm. b. It is too thick, and the plane would be too heavy. c. It is unreasonable to think the layer of material could be any thickness when used on a real aircraft.
3-49 §3.5 source ↗
What is the distance moved by the traveling mirror of a Michelson interferometer that corresponds to 1500 fringes passing by a point of the observation screen? Assume that the interferometer is illuminated with a 606 nm spectral line of krypton-86.
Answer
3-51 §3.5 source ↗
In a Michelson interferometer, light of wavelength 632.8 nm from a He-Ne laser is used. When one of the mirrors is moved by a distance D, 8 fringes move past the field of view. What is the value of the distance D?
Answer
3-53 §3.AP source ↗
For 600-nm wavelength light and a slit separation of 0.12 mm, what are the angular positions of the first and third maxima in the double slit interference pattern?
Answer
3-55 §3.AP source ↗
Red light () illuminates double slits separated by a distance The screen and the slits are 3.00 m apart. (a) Find the distance on the screen between the central maximum and the third maximum. (b) What is the distance between the second and the fourth maxima?
Answer
a. 4.26 cm; b. 2.84 cm
3-57 §3.AP source ↗
Two slits apart are illuminated by light of wavelength 600 nm. What is the highest order fringe in the interference pattern?
Answer
6
3-59 §3.AP source ↗
The interference pattern of a He-Ne laser light passing through two slits 0.031 mm apart is projected on a screen 10.0 m away. Determine the distance between the adjacent bright fringes.
Answer
0.20 m
3-61 §3.AP source ↗
A double-slit experiment is to be set up so that the bright fringes appear 1.27 cm apart on a screen 2.13 m away from the two slits. The light source was wavelength 500 nm. What should be the separation between the two slits?
Answer
0.0839 mm
3-63 §3.AP source ↗
A hydrogen gas discharge lamp emits visible light at four wavelengths, 410, 434, 486, and 656 nm. (a) If light from this lamp falls on a N slits separated by 0.025 mm, how far from the central maximum are the third maxima when viewed on a screen 2.0 m from the slits? (b) By what distance are the second and third maxima separated for ?
Answer
a. 9.8, 10.4, 11.7, and 15.7 cm; b. 3.9 cm
3-65 §3.AP source ↗
Eight slits equally separated by 0.149 mm is uniformly illuminated by a monochromatic light at . What is the width of the central principal maximum on a screen 2.35 m away?
Answer
3-67 §3.AP source ↗
A transparent film of thickness 250 nm and index of refraction of 1.40 is surrounded by air. What wavelength in a beam of white light at near-normal incidence to the film undergoes destructive interference when reflected?
Answer
700 nm
3-69 §3.AP source ↗
A thin film with is surrounded by air. What is the minimum thickness of this film such that the reflection of normally incident light with is minimized?
Answer
189 nm
3-71 §3.AP source ↗
After a minor oil spill, a think film of oil () of thickness 450 nm floats on the water surface in a bay. (a) What predominant color is seen by a bird flying overhead? (b) What predominant color is seen by a seal swimming underwater?
Answer
a. green (504 nm); b. magenta (white minus green)
3-73 §3.AP source ↗
Suppose that the setup of the preceding problem is immersed in an unknown liquid. If 18 fringes per centimeter are now seen along the slide, what is the index of refraction of the liquid?
Answer
1.29
3-75 §3.AP source ↗
Two identical pieces of rectangular plate glass are used to measure the thickness of a hair. The glass plates are in direct contact at one edge and a single hair is placed between them hear the opposite edge. When illuminated with a sodium lamp (), the hair is seen between the 180th and 181st dark fringes. What are the lower and upper limits on the hair’s diameter?
Answer
and
3-77 §3.AP source ↗
A good quality camera “lens” is actually a system of lenses, rather than a single lens, but a side effect is that a reflection from the surface of one lens can bounce around many times within the system, creating artifacts in the photograph. To counteract this problem, one of the lenses in such a system is coated with a thin layer of material () on one side. The index of refraction of the lens glass is 1.68. What is the smallest thickness of the coating that reduces the reflection at 640 nm by destructive interference? (In other words, the coating’s effect is to be optimized for .)
Answer
125 nm
3-79 §3.AP source ↗
A soap bubble is blown outdoors. What colors (indicate by wavelengths) of the reflected sunlight are seen enhanced? The soap bubble has index of refraction 1.36 and thickness 380 nm.
Answer
413 nm and 689 nm
3-81 §3.AP source ↗
An experimenter detects 251 fringes when the movable mirror in a Michelson interferometer is displaced. The light source used is a sodium lamp, wavelength 589 nm. By what distance did the movable mirror move?
Answer
3-83 §3.AP source ↗
A 5.08-cm-long rectangular glass chamber is inserted into one arm of a Michelson interferometer using a 633-nm light source. This chamber is initially filled with air at standard atmospheric pressure but the air is gradually pumped out using a vacuum pump until a near perfect vacuum is achieved. How many fringes are observed moving by during the transition?
Answer
47
3-85 §3.AP source ↗
The thickness of an aluminum foil is measured using a Michelson interferometer that has its movable mirror mounted on a micrometer. There is a difference of 27 fringes in the observed interference pattern when the micrometer clamps down on the foil compared to when the micrometer is empty. The light source is a He-Ne laser with wavelength 632.8 nm. Calculate the thickness of the foil.
Answer
3-87 §3.AP source ↗
In a thermally stabilized lab, a Michelson interferometer is used to monitor the temperature to ensure it stays constant. The movable mirror is mounted on the end of a 1.00-m-long aluminum rod, held fixed at the other end. The light source is a He Ne laser, . The resolution of this apparatus corresponds to the temperature difference when a change of just one fringe is observed. What is this temperature difference?
Answer
3-89 §3.CP source ↗
Determine what happens to the double-slit interference pattern if one of the slits is covered with a thin, transparent film whose thickness is , where is the wavelength of the incident light and n is the index of refraction of the film.
Answer
Bright and dark fringes switch places.
3-91 §3.CP source ↗
A film of oil on water will appear dark when it is very thin, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the oil can be and appear dark at all visible wavelengths? Oil has an index of refraction of 1.40.
Answer
The path length must be less than one-fourth of the shortest visible wavelength in oil. The thickness of the oil is half the path length, so it must be less than one-eighth of the shortest visible wavelength in oil. If we take 380 nm to be the shortest visible wavelength in air, 33.9 nm.
3-93 §3.CP source ↗
Figure 3.14 shows two 7.50-cm-long glass slides illuminated by pure 589-nm wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on some debris at the other end, forming a wedge of air. How thick is the debris, if the dark bands are 1.00 mm apart?
(a) The rainbow-color bands are produced by thin-film interference in the air between the two glass slides. (b) Schematic of the paths taken by rays in the wedge of air between the slides. (c) If the air wedge is illuminated with monochromatic light, bright and dark bands are obtained rather than repeating rainbow colors.
Answer
3-95 §3.CP source ↗
An oil slick on water is 120 nm thick and illuminated by white light incident at a angle to its surface. What color does the oil appear (what is the most constructively reflected wavelength), given its index of refraction is 1.40?
Answer
for one phase change: 950 nm (infrared); for three phase changes: 317 nm (ultraviolet); Therefore, the oil film will appear black, since the reflected light is not in the visible part of the spectrum.
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Problems and answers are adapted from University Physics Volume 3 by Samuel J. Ling, Jeff Sanny and William Moebs, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/university-physics-volume-3/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.