Students should understand the interference and diffraction of waves, so they can:
Apply the principles of interference to coherent sources in order to:
Describe the conditions under which the waves reaching an observation point from two or more sources will all interfere constructively, or under which the waves from two sources will interfere destructively.
Determine locations of interference maxima or minima for two sources or determine the frequencies or wavelengths that can lead to constructive or destructive interference at a certain point.
Relate the amplitude produced by two or more sources that interfere constructively to the amplitude and intensity produced by a single source.
Students should understand the interference and diffraction of waves, so they can:
Apply the principles of interference and diffraction to waves that pass through a single or double slit or through a diffraction grating, so they can:
Sketch or identify the intensity pattern that results when monochromatic waves pass through a single slit and fall on a distant screen, and describe how this pattern will change if the slit width or the wavelength of the waves is changed.
Calculate, for a single-slit pattern, the angles or the positions on a distant screen where the intensity is zero.
Sketch or identify the intensity pattern that results when monochromatic waves pass through a double slit, and identify which features of the pattern result from single-slit diffraction and which from two-slit interference.
Calculate, for a two-slit interference pattern, the angles or the positions on a distant screen at which intensity maxima or minima occur.
Describe or identify the interference pattern formed by a diffraction grating, calculate the location of intensity maxima, and explain qualitatively why a multiple-slit grating is better than a two-slit grating for making accurate determinations of wavelength.
Students should understand the interference and diffraction of waves, so they can:
Apply the principles of interference to light reflected by thin films, so they can:
State under what conditions a phase reversal occurs when light is reflected from the interface between two media of different indices of refraction.
Determine whether rays of monochromatic light reflected perpendicularly from two such interfaces will interfere constructively or destructively, and thereby account for Newton’s rings and similar phenomena, and explain how glass may be coated to minimize reflection of visible light.
Compact Discs, Digital Video Discs, and the Use of Interference (AP?)
X-Ray Diffraction (AP?)
Chapter 27: Interference and the Wave Nature of Light Section 1: The Principle of Linear Superposition
The Principle of Linear Superposition When two or more light waves pass through a given point, their electric fields combine according to the principle of superposition. The waves emitted by the sources start out in phase and arrive at point P in phase, leading to constructive interference.
The Principle of Linear Superposition The waves emitted by the sources start out in phase and arrive at point P out of phase, leading to destructive interference.
The Principle of Linear Superposition If constructive or destructive interference is to continue occurring at a point, the sources of the waves must be coherent sources . Two sources are coherent if the waves they emit maintain a constant phase relation.
27.1.1. In a shallow pool of water, there are two needle-like dippers that move up and down at the same constant frequency. The water waves move outward from each source as shown in the drawing. The wave crests, represented by solid lines, have an amplitude of 0.4 cm. What is the displacement of the water, relative to the undisturbed water level, at the point labeled P? a) +0.8 cm b) +0.4 cm c) zero cm d) 0.4 cm e) 0.8 cm
27.1.2. In a shallow pool of water, there are two needle-like dippers that move up and down at the same constant frequency. The water waves move outward from each source as shown in the drawing. The wave crests, represented by solid lines, have an amplitude of 0.4 cm. What is the displacement of the water, relative to the undisturbed water level, at the point labeled P? a) +0.8 cm b) +0.4 cm c) zero cm d) 0.4 cm e) 0.8 cm
27.1.3. Two wave pulses are sent down a stretched out rope. Pulse A is traveling toward the right with an amplitude of +2 mm. Pulse B is traveling toward pulse A from the right to the left with an amplitude of 4 mm. When the two pulses meet and completely overlap, what will be the maximum displacement of the rope relative to its undisurbed position? a) zero mm b) + 6 mm c) + 2 mm d) 2 mm e) 4 mm
27.1.4. Complete the following sentence: In order for light to be considered completely coherent, a) the phase difference of light at any two points must be constant. b) it must originate from the same source. c) its intensity at every point must be constant. d) it must follow the same path. e) it must be traveling at its vacuum speed.
27.1.5. Why is no interference pattern observed when light from two sources of differing wavelength interfere? a) The intensities of the two waves will be necessarily different. b) The light from the two different sources is not likely to be coherent. c) If the two light sources are close enough to each other, they will produce an interference pattern.
Chapter 27: Interference and the Wave Nature of Light Section 2: Young’s Double-Slit Experiment
Young’s Double Slit Experiment In Young’s experiment, two slits acts as coherent sources of light. Light waves from these slits interfere constructively and destructively on the screen.
Young’s Double Slit Experiment The waves coming from the slits interfere constructively or destructively, depending on the difference in distances between the slits and the screen.
Young’s Double Slit Experiment Bright fringes of a double-slit Dark fringes of a double-slit
Example 1 Young’s Double-Slit Experiment Red light (664 nm) is used in Young’s experiment with slits separated by 0.000120 m. The screen is located a distance 2.75 m from the slits. Find the distance on the screen between the central bright fringe and the third-order bright fringe.
Conceptual Example 2 White Light and Young’s Experiment The figure shows a photograph that illustrates the kind of interference fringes that can result when white light is used in Young’s experiment. Why does Young’s experiment separate white light into its constituent colors? In any group of colored fringes, such as the two singled out, why is red farther out from the central fringe than green is? Why is the central fringe white?
27.2.1. You are sitting in a closed room with no windows. The only light in the room originates from two identical bare, incandescent light bulbs. One is located on the wall to your left; and the other is located on the wall to your right. Bored, you look up at the ceiling and realize there is no interference pattern. Why is there no interference pattern? a) The two light sources are not polarized. b) The two light sources are not coherent. c) The two light sources are in phase. d) The interference pattern is too small to observe with the naked eye. e) Interference of light is never observed, but the diffraction of light can easily be observed.
27.2.2. In a Young’s double slit experiment, green light is incident of the two slits; and the resulting interference pattern is observed a screen. Which one of the following changes would cause the fringes to be spaced further apart? a) Move the screen closer to the slits. b) Move the light source closer to the slits. c) Increase the distance between the slits. d) Use orange light instead of green light. e) Use blue light instead of green light.
27.2.3. What happens to the locations of the maxima for double slit interference when the size of the slits is reduced? a) Reducing the size of the slits has no effect on the locations of the maxima. b) The distances between the maxima increase as the widths are reduced. c) The distances between the maxima decrease as the widths are reduced. d) Reducing the slit size only increases the number of maxima, but the locations of the initial maxima are not changed. e) Reducing the slit size only decreases the number of maxima, but the locations of the initial maxima are not changed.
27.2.4. What happens to the width of the maxima for double slit interference when the size of the slits is reduced? a) Reducing the size of the slits has no effect on the size of the maxima. b) The widths of the maxima increase as the slit size is reduced. c) The widths of the maxima decrease as the slit size is reduced. d) Reducing the slit size only increases the number of maxima, but the widths of the initial maxima are not changed. e) Reducing the slit size only decreases the number of maxima, but the widths of the initial maxima are not changed.
27.2.5. Without changing the slit width in the double slit experiment, what effect on the interference pattern does reducing the height of the slit have? Assume that the height always remains somewhat larger than the wavelength of light incident on the slits. a) There is no effect on the pattern. b) The distances between the maxima will increase. c) The widths of the maxima will increase. d) The number of maxima will increase. e) The height of the maxima will decrease, but there is otherwise no effect.
Chapter 27: Interference and the Wave Nature of Light Section 3: Thin-Film Interference
Thin Film Interference Because of reflection and refraction, two light waves enter the eye when light shines on a thin film of gasoline floating on a thick layer of water. Because of the extra distance traveled, there can be interference between the two waves.
Example 3 A Colored Thin Film of Gasoline A thin film of gasoline floats on a puddle of water. Sunlight falls perpendicularly on the film and reflects into your eyes. The film has a yellow hue because destructive interference eliminates the color of blue (469 nm) from the reflected light. The refractive indices of the blue light in gasoline and water are 1.40 and 1.33. Determine the minimum non-zero thickness of the film.
Conceptual Example 4 Multicolored Thin Films Under natural conditions, thin films, like gasoline on water or like the soap bubble in the figure, have a multicolored appearance that often changes while you are watching them. Why are such films multicolored and why do they change with time?
27.3.1. A special system is set up in a lab that lets its user select any wavelength between 400 nm and 700 nm with constant intensity. This light is directed at a thin glass film ( n = 1.53) with a thickness of 350 nm and that is surrounded by air. As one scans through these possible wavelengths, which wavelength of light reflected from the glass film will appear to be the brightest, if any?
a) 428 nm
b) 535 nm
c) 657 nm
d) 700 nm
Since the intensity of the light is constant,
all wavelengths of light reflected from the
glass will appear to be the same.
27.3.2. Blue light ( = 512 nm) is illuminating a thin film of plastic ( n P = 1.60) that is on top of a glass sheet ( n G = 1.45). Which of the following statements best describes the light that an observer sees coming from the thin film, if it has a uniform thickness of 0.200 m? a) You would see alternating bright and dark bands. b) You would see the spectrum of colors. c) You would see the film as uniformly bright due to constructive interference. d) You would see the film as uniformly dark due to destructive interference. e) You cannot see any effects because the film is too thin.
27.3.3. Blue light ( = 512 nm) is illuminating a thin film of plastic ( n P = 1.45) that is on top of a glass sheet ( n G = 1.53). Which of the following statements best describes the light that an observer sees coming from the thin film, if it has a uniform thickness of 0.265 m? a) You would see alternating bright and dark bands. b) You would see the spectrum of colors. c) You would see the film as uniformly bright due to constructive interference. d) You would see the film as uniformly dark due to destructive interference. e) You cannot see any effects because the film is too thin.
27.3.4. Which one of the following choices does not affect interference of light when the thickness of a thin film is much less than the wavelength of light? a) path length difference b) phase shifts upon reflection c) the angle of incidence
27.3.5. Which one of the following statements provides the most convincing evidence that visible light is a form of electromagnetic radiation? a) Two light sources can be coherent. b) Light can be reflected from a surface. c) Light can form a double-slit interference pattern. d) Light can be diffracted through an aperture. e) Light travels through vacuum at the same speed as X-rays.
Chapter 27: Interference and the Wave Nature of Light Section 4: The Michelson Interferometer (AP?)
Michelson Interferometer A schematic drawing of a Michelson interferometer.
27.4.1. The drawing shows the Michelson interferometer with the addition of a closed cylinder, the ends of which are transparent. The sources provides a beam of monochromatic light of wavelength . Initially, the cylinder, which has a length L , is evacuated. This apparatus may be used to determine the index of refraction for the gas in the cylinder. As gas is very slowly allowed to enter the cylinder, a student counts the number of fringes N that pass by a fixed point in the viewing telescope. Which of the following is the correct expression that determines the index of refraction for the gas? a) b) c) d) e)
Chapter 27: Interference and the Wave Nature of Light Section 5: Diffraction
27.5.1. A human hair is placed directly in front of the opening of a laser pointer. The light has a wavelength of 532 nm. On a screen 2.0 m in front of the laser, a diffraction pattern is observed with minima spaced 0.0164 m apart. Determine the approximate thickness of the hair. a) 40 m b) 48 m c) 53 m d) 65 m e) 79 m
27.5.2. A laser uniformly illuminates two narrow, identical slits and an interference pattern is observed on a screen. Now, imagine that one of the two slits is completely covered so that no light can pass through it. Which of the following statements best describes what is subsequently observable on the screen, if anything? a) The width of the maxima and their spacing looks the same as before. b) The maxima are spaced farther apart, but their width remains the same. c) The maxima are spaced closer together and their width is smaller. d) The maxima are spaced farther apart and their width increases. e) Only a narrow band of light is observed on the screen.
27.5.3. Which of the following must be satisfied if interference is to occur for light passing though a single slit? a) The light source must be a point source. b) The light must be traveling with an angle of incidence of 0 toward the slit. c) The distance from the slit to the observation screen must be greater than the width of the slit. d) The width of the slit must be comparable to the wavelength of light. e) The light must be comprised of a single wavelength.
27.5.4. In a dark room, Jennifer is conducting an experiment. The two sides of a slit-like opening are initially 5.0 cm apart. Jennifer shines green laser light on the opening. She then continually brings the two sides closer together, narrowing the slit-like opening. At what point will Jennifer observe an interference (diffraction) pattern on a screen behind the opening? a) She’ll she interference as soon as the slit width becomes similar to the path difference between the Huygen’s wavelets. b) She’ll be unable to observe interference unless the waves undergo edge effects at the opening. c) Once the opening becomes comparable to an integer multiple of the wavelength, she’ll be able to see the interference. d) She’ll be unable to observe interference by using green light. She should use white light.
27.5.5. Which one of the following statements best explains why the diffraction of sound is more apparent than the diffraction of light under most circumstances? a) Sound waves are longitudinal, and light waves are transverse. b) Sound requires a physical medium for propagation. c) Light waves can be represented by rays while sound waves cannot. d) The speed of sound in air is six orders of magnitude smaller than that of light. e) The wavelength of light is considerably smaller than the wavelength of sound.
27.5.6. In a single slit experiment, what effect on the diffraction pattern would result as the slit width is decreased? a) The width of the central band would increase. b) The width of the central band would decrease. c) The width of the central band would not change.
27.5.7. In a single slit experiment, what effect on the first two minima in the diffraction pattern would result as the slit width is decreased? a) The width of the two minima would increase. b) The width of the two minima would decrease. c) The width of the two minima would not change.
27.5.8. In a single slit experiment, what effect on the central minimum in the diffraction pattern would result as the wavelength of the light is decreased? a) The width of the central maximum would increase. b) The width of the central maximum would decrease. c) The width of the central maximum would not change.
27.5.9. Light of wavelength 600 nm is incident upon a single slit with width 4 × 10 4 m. The figure shows the pattern observed on a screen positioned 2 m from the slits. Determine the distance s . a) 0.002 m b) 0.003 m c) 0.004 m d) 0.006 m e) 0.008 m
27.5.10. Light of 600.0 nm is incident upon a single slit. The resulting diffraction pattern is observed on a screen that is 0.50 m from the slit. The distance between the first and third minima of the diffraction pattern is 0.80 mm. Which range of values listed below contains the width of the slit? a) 0.1 mm to 0.4 mm b) 0.4 mm to 0.8 mm c) 0.8 mm to 1.2 mm d) 1.2 mm to 1.6 mm e) 1.6 mm to 2.0 mm
27.5.11. Visible light of wavelength 589 nm is incident on a diffraction grating that has 3500 lines/cm. At what angle with respect to the central maximum is the fifth order maximum observed? a) 17.9 b) 23.8 c) 35.7 d) 71.3 e) A fifth order maximum cannot be observed with this grating.
Chapter 27: Interference and the Wave Nature of Light Section 6: Resolving Power (AP?)
Resolving Power Three photographs of an automobile’s headlights, taken at progressively greater distances.
Effect of Diffraction of Resolving Power First minimum of a circular diffraction pattern diameter of hole
Rayleigh criterion Two point objects are just resolved when the first dark fringe in the diffraction pattern of one falls directly on the central bright fringe in the diffraction patter of the other.
Conceptual Example 8 What You See is Not What You Get The French postimpressionist artist Georges Seurat developed a technique of painting in which dots of color are placed close together on the canvas. From sufficiently far away the individual dots are not distinguishable, and the images in the picture take on a more normal appearance. Why does the camera resolve the dots, while his eyes do not?
27.6.1. A spy satellite is at an altitude 650 km above the Earth’s surface. How large must the satellite’s camera lens be so that its resolution is 25 cm? Assume the average wavelength of light of 480 nm. a) 1.8 m b) 2.7 m c) 0.55 m d) 1.5 m e) 0.85 m
27.6.2. A special microscope has been set up that allows the user to view a specimen using light from among the colors listed below. Which of these would you choose to use for the best resolution? a) yellow b) red c) violet d) blue e) green
27.6.3. The Hubble Space Telescope in orbit above the Earth has a 2.4 m circular aperture. The telescope has equipment for detecting ultraviolet light. What is the minimum angular separation between two objects that the Hubble Space Telescope can resolve in ultraviolet light of wavelength 95 nm? a) 4.8 × 10 8 rad b) 7.0 × 10 8 rad c) 1.9 × 10 7 rad d) 1.5 × 10 7 rad e) 3.3 × 10 9 rad
27.6.4. A spy satellite is in orbit at a distance of 1.0 × 10 6 m above the ground. It carries a telescope that can resolve the two rails of a railroad track that are 1.4 m apart using light of wavelength 600 nm. Which one of the following statements best describes the diameter of the lens in the telescope? a) It is less than 0.14 m. b) It is greater than 0.14 m and less than 0.23 m. c) It is greater than 0.23 m and less than 0.35 m. d) It is greater than 0.35 m and less than 0.52 m. e) It is greater than 0.52 m.
27.6.5. The headlights of a car are 1.6 m apart and produce light of wavelength 575 nm in vacuum. The pupil of the eye of the observer has a diameter of 4.0 mm and a refractive index of 1.4. What is the maximum distance from the observer that the two headlights can be distinguished? a) 8.0 km b) 9.1 km c) 11 km d) 13 km e) 16 km
Chapter 27: Interference and the Wave Nature of Light Section 7: The Diffraction Grating
Diffraction Grating An arrangement consisting of a large number of closely spaced, parallel slits is called a diffraction grating.
Explanation of Diffraction The conditions shown here lead to the first- and second-order intensity maxima in the diffraction pattern.
Grating vs Double Slit The bright fringes produced by a diffraction grating are much narrower than those produced by a double slit. Principal maxima of a diffraction grating distance between slits
Example 9 Separating Colors With a Diffraction Grating A mixture of violet (410 nm) light and red (660 nm) light falls onto a grating that contains 1.0x10 4 lines/cm. For each wavelength, find the angle that locates the first-order maximum.
27.7.1. Two monochromatic beams of light, one red and one blue, are directed at the same spot on a diffraction grating. The resultant diffraction pattern is observed on a screen located a short distance behind the grating. Which of the following best describes the observed pattern? a) The central maximum appears to be purple. The maxima on each side would alternate, first red, then blue. b) The central maximum appears to be purple. The maxima on each side would alternate, first blue, then red. c) The central maximum is red. The maxima on each side would alternate, first blue, then red. d) The central maximum is blue. The maxima on each side would alternate, first red, then blue. e) The central maximum is blue. The maxima on each side would alternate, first blue, then red.
27.7.2. Three monochromatic light beams are directed at a diffraction grating. The resulting pattern, shown in grayscale below, is observed on a screen 2 m from the grating. What is the correct order from top to bottom of the three light beams used? a) green, red, blue b) red, green, blue c) blue, green, red d) green, blue, red e) blue, red, green
27.7.3. Diffraction occurs when light passes through a single slit. Rank the following three choices in decreasing order, according to the extent of the diffraction that occurs (largest diffraction first): A - blue light, narrow slit, B - red light, narrow slit, C - blue light, wide slit Note : The blue light referred to in choices A and C is the same wavelength. Also, the narrow slit referred to in choices A and B is the same width. a) A , B , C b) B , A , C c) C , A , B d) A , C , B e) B , C , A
Chapter 27: Interference and the Wave Nature of Light Section 8: Compact Discs, Digital Video Discs, and the Use of Interference (AP?)