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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 24

Wave Optics

Educators


Problem 1

A laser beam is incident on two slits with a separation of 0.200 mm, and a screen is placed 5.00 m from the slits. If the bright interference fringes on the screen are separated by 1.58 cm, what is the wavelength of the laser light?

Salamat A.
Numerade Educator

Problem 2

In a Young’s double - slit experiment, a set of parallel slits with a separation of 0.100 mm is illuminated by light having a wavelength of 589 nm, and the interference pattern is observed on a screen 4.00 m from the slits. (a) What is the difference in path lengths from each of the slits to the location of a third - order bright fringe on the screen? (b) What is the difference in path lengths from the two slits to the location of the third dark fringe on the screen, away from the center of the pattern?

Zachary W.
Numerade Educator

Problem 3

Light at 633 nm from a helium–neon laser shines on a pair of parallel slits separated by $1.45 \times 10^{-5} \mathrm{m}$ and an interference pattern is observed on a screen 2.00 $\mathrm{m}$ from the plane of the slits. (a) Find the angle from the central maximum to the first bright fringe. (b) At what angle from the central maximum does the second dark fringe appear? (c) Find the distance from the central maximum to the first bright fringe.

Salamat A.
Numerade Educator

Problem 4

Light of wavelength 620. nm falls on a double slit, and the first bright fringe of the interference pattern is seen at an angle of $15.0^{\circ}$ from the central maximum. Find the separation between the slits.

Zachary W.
Numerade Educator

Problem 5

In a location where the speed of sound is $354 \mathrm{m} / \mathrm{s},$ a 2.00 $\mathrm{kHz}$ sound wave impinges on two slits 30.0 $\mathrm{cm}$ apart. (a) At what angle is the first maximum located? (b) If the sound wave is replaced by $3.00-\mathrm{cm}$ microwaves, what slit separation gives the same angle for the first maximum? (c) If the slit separation is $1.00 \mu \mathrm{m},$ what frequency of light gives the same first maximum angle?

Salamat A.
Numerade Educator

Problem 6

A double slit separated by 0.058 0 mm is placed 1.50 m from a screen. (a) If yellow light of wavelength 588 nm strikes the double slit, what is the separation between the zeroth-order and first-order maxima on the screen? (b) If blue light of wavelength 412 nm strikes the double slit, what is the separation between the second - order and fourth - order maxima?

Zachary W.
Numerade Educator

Problem 7

Two radio antennas separated by $d=3.00 \times 10^{2} \mathrm{m},$ as shown in Figure P24.7, simultaneously broadcast identical signals at the same wavelength. A car travels due north along a
straight line at position $x=1.00 \times 10^{3} \mathrm{m}$ from the center point between the antemnas, and its radio receives the signals. (a) If the car is at the position of the second maximum after that at point $O$ when it has traveled a distance of $y=$ $4.00 \times 10^{2} \mathrm{m}$ northward, what is the wavelength of the signals? (b) How much farther must the car travel from this position to encounter the next minimum in reception? Hint: Do not use the small-angle approximation in this problem.

Salamat A.
Numerade Educator

Problem 8

Light of wavelength $6.0 \times 10^{2} \mathrm{nm}$ falls on a double slit, and the first bright fringe of the interference pattern is observed to make an angle of $12^{\circ}$ with the horizontal. Find the separation between the slits.

Zachary W.
Numerade Educator

Problem 9

Monochromatic light falls on a screen 1.75 m from two slits separated by 2.10 mm. The first - and second - order bright fringes are separated by 0.552 mm. What is the wavelength of the light?

Salamat A.
Numerade Educator

Problem 10

A pair of parallel slits separated by $2.00 \times 10^{-1} \mathrm{m}$ is illuminated by 633 -nm light and an interference pattern is observed on a screen 2.50 $\mathrm{m}$ from the plane of the slits. Calculate the difference in path lengths from each of the slits to the location on the screen of (a) a fourth-order bright fringe and (b) a fourth dark fringe.

Zachary W.
Numerade Educator

Problem 11

A riverside warehouse has two open doors, as in Figure P24.11. Its interior is lined with a sound-absorbing material. A boat on the river sounds its horn. To person A, the sound is loud and clear. To person B, the sound is barely audible. The principal wavelength of the sound waves is 3.00 m. Assuming person B is at the position of the first minimum, determine the distance between the doors, center to center.

Salamat A.
Numerade Educator

Problem 12

A student sets up a double - slit experiment using monochromatic light of wavelength $\lambda$ . The distance between the slits is equal to 25$\lambda$ .(a ) Find the angles at which the $ m=1,2,$ and 3 maxima occur on the viewing screen. (b) At what angles do the first three dark fringes occur? (c) Why are the answers so evenly spaced? Is the spacing even for all orders? Explain.

Zachary W.
Numerade Educator

Problem 13

Radio waves from a star, of wavelength $2.50 \times$ $10^{2} \mathrm{m},$ reach a radio telescope by two separate paths, as shown in Figure P 24.13. One is a direct path to the receiver, which is situated on the edge of a cliff by the ocean. The second is by reflection off the water. The first minimum of destructive interference occurs when the star is $\theta=25.0^{\circ}$ above the horizon. Find the height of the cliff. (Assume no phase change on reflection.)

Salamat A.
Numerade Educator

Problem 14

Monochromatic light of wavelength $\lambda$ is incident on a pair of slits separated by $2.40 \times 10^{-4} \mathrm{m},$ and forms an interference pattern on a screen placed 1.80 $\mathrm{m}$ away from the slits. The first-order bright fringe is 4.52 $\mathrm{mm}$ mm from the center of the central maximum. (a) Draw a picture, labeling the angle $\theta$ and the legs of the right triangle associated with the first-order bright fringe. (b) Compute the tangent of the angle $\theta$ associated with the first-order bright fringe. (c) Find the angle corresponding to the first-order bright fringe and compute the sine of that angle. Are the sine and tangent of the angle comparable in value? Does your answer always hold true? (d) Calculate the wavelength of the light. (e) Compute the angle of the fifth-order bright fringe. (f) Find its position on the screen.

Zachary W.
Numerade Educator

Problem 15

Waves from a radio station have a wavelength of $3.00 \times 10^{2} \mathrm{m}$ . They travel by two paths to a home receiver 20.0 $\mathrm{km}$ from the transmitter. One path is a direct path, and the second is by reflection from a mountain directly behind the home receiver. What is the minimum distance from the mountain to the receiver that produces destructive interference at the receiver? (Assume that no phase change occurs on reflection from the mountain.)

Salamat A.
Numerade Educator

Problem 16

A soap bubble $(n=1.93)$ having a wall thickness of 120 nm is floating in air. (a) What is the wavelength of the visible light that is most strongly reflected? (b) Explain how a bubble of different thickness could also strongly reflect light of this same wavelength. (c) Find the two smallest film thicknesses larger than the one given that can produce strongly reflected light of this same wavelength.

Zachary W.
Numerade Educator

Problem 17

A thin layer of liquid methylene iodide $(n=1.756)$ is sandwiched between two flat, parallel plates of glass $(n=1.50) .$ What is the minimum thickness of the liquid layer if normally incident light with $\lambda=6.00 \times 10^{2} \mathrm{nm}$ in air is to be strongly reflected?

Salamat A.
Numerade Educator

Problem 18

A thin film of oil $(n=1.25)$ is located on smooth, wet pavement. When viewed from a direction perpendicular to the pavement, the film reflects most strongly red light at $6.40 \times 10^{2} \mathrm{nm}$ and reflects no green light at 512 $\mathrm{nm}$ . (a) What is the minimum thickness of the oil film? (b) Let $m_{1}$ correspond to the order of the constructive interference and $m_{2}$ to the order of the destructive interference. Obtain a relationship between $m_{1}$ and $m_{2}$ that is consistent with the given data.

Zachary W.
Numerade Educator

Problem 19

A thin film of glass $(n=1.52)$ of thickness 0.420$\mu \mathrm{m}$ is viewed under white light at near normal incidence. What wavelength of visible light is most strongly reflected by the film when surrounded by air?

Salamat A.
Numerade Educator

Problem 20

A transparent oil with index of refraction 1.29 spills on the surface of water (index of refraction 1.33), producing a maximum of reflection with normally incident orange light (wavelength $6.00 \times 10^{2} \mathrm{nm}$ in air). Assuming the maximum occurs in the first order, determine the thickness of the oil slick.

Zachary W.
Numerade Educator

Problem 21

A possible means for making an airplane invisible to radar is to coat the plane with an antireflective polymer. If radar waves have a wavelength of 3.00 cm and the index of refraction of the polymer is $n=1.50,$ how thick would you make the coating? Assume $n_{\text { airplane }}>1.50$ .

Salamat A.
Numerade Educator

Problem 22

An oil film $(n=1.45)$ floating on water is illuminated by white light at normal incidence. The film is $2.80 \times 10^{2} \mathrm{nm}$ thick. Find (a) the wavelength and color of the light in the visible spectrum most strongly reflected and (b) the wavelength and color of the light in the visible spectrum most strongly transmitted. Explain your reasoning.

Zachary W.
Numerade Educator

Problem 23

Astronomers observe the chromosphere of the Sun with a filter that passes the red hydrogen spectral line of wavelength 656.3 $\mathrm{nm}$ , called the $\mathrm{H}_{\alpha}$ line. The filter consists of a
transparent dielectric of thickness $d$ held between two partially aluminized glass plates. The filter is kept at a constant temperature. (a) Find the minimum value of $d$ that will produce maximum transmission of perpendicular $\mathrm{H}_{\alpha}$ light if the dielectric has an index of refraction of $1.378 .$ (b) If the temperature of the filter increases above the normal value increasing its thickness, what happens to the transmitted wavelength? (c) The dielectric will also pass what near-visible wavelength? One of the glass plates is colored red to absorb this light.

Salamat A.
Numerade Educator

Problem 24

A spacer is cut from a playing card of thickness $2.90 \times 10^{-4} \mathrm{m}$ and used to separate one end of two rectangular, optically flat, $3.00-\mathrm{cm}$ long glass plates with $n=1.55$ , as in Figure P 24.24 Laser light at 594 nm shines straight down on the top plate. The plates have a length of 3.00 cm. (a) Count the number of phase reversals for the interfering waves. (b) Calculate the separation between dark interference bands observed on the top plate.

Zachary W.
Numerade Educator

Problem 25

An investigator finds a fiber at a crime scene that he wishes to use as evidence against a suspect. He gives the fiber to a technician to test the properties of the fiber. To measure the diameter of the fiber, the technician places it between two flat glass plates at their ends as in Figure P24.24. When the plates, of length $14.0 \mathrm{cm},$ are illuminated from above with light of wave- length $6.50 \times 10^{2} \mathrm{nm},$ she observes bright interference bands separated by 0.580 $\mathrm{mm}$ . What is the diameter of the fiber?

Salamat A.
Numerade Educator

Problem 26

A plano-convex lens with radius of curvature $R=3.0 \mathrm{m}$ is in contact with a flat plate of glass. A light source and the observer's eye are both close to the normal, as shown in Figure 24.10 a. The radius of the 50 th bright Newton's ring is found to be 9.8 $\mathrm{mm}$ . What is the wavelength of the light produced by the source?

Zachary W.
Numerade Educator

Problem 27

A thin film of oil $(n=1.45)$ of thickness 425 $\mathrm{nm}$ with air on both sides is illuminated with white light at normal incidence. Determine (a) the most strongly and (b) the most weakly reflected wavelengths in the range 400 $\mathrm{nm}$ to 600 $\mathrm{nm} .$

Salamat A.
Numerade Educator

Problem 28

Nonreflective coatings on camera lenses reduce the loss of light at the surfaces of multilens systems and prevent internal reflections that might mar the image. Find the minimum thickness of a layer of magnesium fluoride $(n=1.38)$ on flint glass $(n=1.66)$ that will cause destructive interference of reflected light of wavelength $5.50 \times 10^{2}$ nm near the middle of the visible spectrum.

Zachary W.
Numerade Educator

Problem 29

A thin film of glycerin $(n=1.473)$ of thickness 524 nm with air on both sides is illuminated with white light at near normal incidence. What wavelengths will be strongly reflected in the range 300 $\mathrm{nm}$ to 700 $\mathrm{nm}$ ?

Salamat A.
Numerade Educator

Problem 30

A lens made of glass $\left(n_{g}=1.52\right)$ is coated with a thin film of $\mathrm{MgF}_{2}\left(n_{s}=1.98\right)$ of thickness $t .$ Visible light is incident normally on the coated lens in Figure P 24.30 . (a) For what minimum value of $t$ will the reflected light of wavelength $5.40 \times 10^{2} \mathrm{nm}(\text { in air ) be }$ missing? (b) Are there other values of $t$ that will minimize the reflected light at this wavelength? Explain.

Zachary W.
Numerade Educator

Problem 31

Light of wavelength 5.40 $\times 10^{2}$ nm passes through a slit of width 0.200 $\mathrm{mm}$ . (a)
Find the width of the central maximum on a screen located 1.50 $\mathrm{m}$ from the slit. (b) Determine the width of the first-order bright fringe.

Salamat A.
Numerade Educator

Problem 32

A student and his lab partner create a single slit by carefully aligning two razor blades to a separation of 0.500 $\mathrm{mm}$ . When a helium-neon laser at 633 nm illuminates the slit, a diffraction pattern is observed on a screen 1.25 $\mathrm{m}$ beyond the slit. Calculate (a) the angle $\theta_{\text { dark }}$ to the first minimum in the diffraction pattern and (b) the width of the central maximum.

Zachary W.
Numerade Educator

Problem 33

Light of wavelength 587.5 nm illuminates a slit of width 0.75 mm. (a) At what distance from the slit should a screen be placed if the first minimum in the diffraction pattern is to be 0.85 mm from the central maximum? (b) Calculate the width of the central maximum.

Salamat A.
Numerade Educator

Problem 34

Microwaves of wavelength 5.00 $\mathrm{cm}$ enter a long, narrow window in a building that is otherwise essentially opaque to the incoming waves. If the window is 36.0 $\mathrm{cm}$ wide, what is the distance from the central maximum to the first-order minimum along a wall 6.50 $\mathrm{m}$ from the window?

Zachary W.
Numerade Educator

Problem 35

A beam of monochromatic light is diffracted by a slit of width 0.600 $\mathrm{mm}$ . The diffraction pattern forms on a wall 1.30 $\mathrm{m}$ beyond the slit. The width of the central maximum is 2.00 $\mathrm{mm}$ . Calculate the wavelength of the light.

Salamat A.
Numerade Educator

Problem 36

A screen is placed 50.0 $\mathrm{cm}$ from a single slit that is illuminated with light of wavelength $6.80 \times 10^{2} \mathrm{nm}$ . If the distance between the first and third minima in the diffraction pattern is $3.00 \mathrm{mm},$ what is the width of the slit?

Zachary W.
Numerade Educator

Problem 37

A slit of width 0.50 $\mathrm{mm}$ is illuminated with light of wavelength $5.00 \times 10^{2} \mathrm{nm},$ and a screen is placed $1.20 \times 10^{2} \mathrm{cm}$ in front of the slit. Find the widths of the first and second maxima on each side of the central maximum.

Salamat A.
Numerade Educator

Problem 38

The second-order dark fringe in a single-slit diffraction pattern is 1.40 $\mathrm{mm}$ from the center of the central maximum. Assuming the screen is 85.0 $\mathrm{cm}$ from a slit of width 0.800 $\mathrm{mm}$ and assuming monochromatic incident light, calculate the wavelength of the incident light.

Zachary W.
Numerade Educator

Problem 39

Three discrete spectral lines occur at angles of $10.1^{\circ}, 13.7^{\circ},$ and $14.8^{\circ},$ respectively, in the first-order spectrum of a diffractiongrating spectrometer. (a) If the grating has 3660 slits/cm, what are the wavelengths of the light? (b) At what angles are these lines found in the second-order spectra?

Salamat A.
Numerade Educator

Problem 40

Intense white light is incident on a diffraction grating that has 600 . lines/mm. (a) What is the highest order in which the complete visible spectrum can be seen with this grating? (b) What is the angular separation between the violet edge $(400 . \mathrm{nm})$ and the red edge $(700 . \mathrm{nm})$ of the first-order spectrum produced by the grating?

Zachary W.
Numerade Educator

Problem 41

The hydrogen spectrum has a red line at 656 $\mathrm{nm}$ and a violet line at 434 $\mathrm{nm}$ . What angular separations between these two spectral lines can be obtained with a diffraction grating that has $4.50 \times 10^{3}$ lines/cm?

Salamat A.
Numerade Educator

Problem 42

Consider an array of parallel wires with uniform spacing of 1.30 $\mathrm{cm}$ between centers. In air at $20.0^{\circ} \mathrm{C}$ , ultrasound with a frequency of 37.2 $\mathrm{kHz}$ a distant source is incident perpendicular to the array. (Take the speed of sound to be 343 $\mathrm{m} / \mathrm{s}$ ) (a) Find the number of directions on the other side of the array in which there is a maximum of intensity. (b) Find the angle for each of these directions relative to the direction of the incident beam.

Zachary W.
Numerade Educator

Problem 43

A helium-neon laser $(\lambda=632.8 \mathrm{nm})$ is used to calibrate a dif- fraction grating. If the first-order maximum occurs at $20.5^{\circ}$, what is the spacing between adjacent grooves in the grating?

Salamat A.
Numerade Educator

Problem 44

White light is spread out into its spectral components by a diffraction grating. If the grating has $2.00 \times 10^{3}$ lines/cm, at what angle does red light of wavelength $6.40 \times 10^{2} \mathrm{nm}$ appear in the first-order spectrum?

Zachary W.
Numerade Educator

Problem 45

Light from an argon laser strikes a diffraction grating that has 510 grooves/cm. The central and first-order principal maxima are separated by 0.488 $\mathrm{m}$ on a wall 1.72 $\mathrm{m}$ from the grating. Determine the wavelength of the laser light.

Salamat A.
Numerade Educator

Problem 46

White light is incident on a diffraction grating with 475 lines/ $\mathrm{mm}$ . (a) Calculate the angle $\theta_{r 2}$ to the second-order maximum for a wavelength of 675 $\mathrm{nm}$ . (b) Calculate the wavelength of light with a third-order maximum at the same angle $\theta_{r 2}$.

Zachary W.
Numerade Educator

Problem 47

Sunlight is incident on a diffraction grating that has 2 750 lines/cm. The second - order spectrum over the visible range (400.–700. nm) is to be limited to 1.75 cm along a screen that is a distance $L$ from the grating. What is the required value of $L$ ?

Salamat A.
Numerade Educator

Problem 48

Monochromatic light at 577 nm illuminates a diffraction grating with 325 lines/mm. Determine (a) the angle to the first-order maximum, (b) the highest order that can be observed with this grating at the given wavelength, and (c) the angle to this highest-order maximum.

Zachary W.
Numerade Educator

Problem 49

Light of wavelength $5.00 \times 10^{2} \mathrm{nm}$ is incident normally on a diffraction grating. If the third-order maximum of the diffraction pattern is observed at $32.0^{\circ},(\mathrm{a})$ what is the number of rulings per centimeter for the grating? (b) Determine the total number of primary maxima that can be observed in this situation.

Salamat A.
Numerade Educator

Problem 50

Light containing two different wavelengths passes through a diffraction grating with $1.20 \times 10^{3}$ slits/ $\mathrm{cm} .$ On a screen 15.0 $\mathrm{cm}$ from the grating, the third-order maximum of the shorter wavelength falls midway between the central maximum and the first side maximum for the longer wavelength. If the neighboring maxima of the longer wavelength are 8.44 $\mathrm{mm}$ apart on the screen, what are the wavelengths in the light? Hint: Use the small-angle approximation.

Zachary W.
Numerade Educator

Problem 51

The angle of incidence of a light beam in air onto a reflecting surface is continuously variable. The reflected ray is found to be completely polarized when the angle of incidence is $48.0^{\circ} .$ (a) What is the index of refraction of the reflecting material? (b) If some of the incident light (at an angle of $48.0^{\circ}$) passes into the material below the surface, what is the angle of refraction?

Salamat A.
Numerade Educator

Problem 52

Unpolarized light passes through two Polaroid sheets. The transmission axis of the analyzer makes an angle of $35.0^{\circ}$ with the axis of the polarizer. (a) What fraction of the original unpolarized light is transmitted through the analyzer? (b) What fraction of the original light is absorbed by the analyzer?

Zachary W.
Numerade Educator

Problem 53

The index of refraction of a glass plate is $1.52 .$ What is the Brewster's angle when the plate is (a) in air and (b) in water? (See Problem $57 . )$

Salamat A.
Numerade Educator

Problem 54

At what angle above the horizon is the Sun if light from it is completely polarized upon reflection from water?

Zachary W.
Numerade Educator

Problem 55

A light beam is incident on a piece of fused quartz $(n=1.458)$ at the Brewster's angle. Find (a) the value of Brewster's angle and (b) the angle of refraction for the transmitted ray.

Salamat A.
Numerade Educator

Problem 56

The critical angle for total internal reflection for sapphire surrounded by air is $34.4^{\circ}$ . Calculate the Brewster's angle for sapphire if the light is incident from the air.

Zachary W.
Numerade Educator

Problem 57

Equation 24.14 assumes the incident light is in air. If the light is incident from a medium of index $n_{1}$ onto a medium of index $n_{2},$ follow the procedure used to derive Equation 24.14 to show that $\tan \theta_{p}=n_{2} / n_{1}$

Salamat A.
Numerade Educator

Problem 58

Plane-polarized light is incident on a single polarizing disk, with the direction of $E_{0}$ parallel to the direction of the transmission axis. Through what angle should the disk be rotated so that the intensity in the transmitted beam is reduced by a factor of (a) 2.00, (b) 4.00 , and (c) 6.00$?$

Zachary W.
Numerade Educator

Problem 59

Three polarizing plates whose planes are parallel are centered on a common axis. The directions of the transmission axes relative to the common vertical direction are shown in Figure P24.59. A linearly polarized beam of light with plane of polarization parallel to the vertical reference direction is incident from the left onto the first disk with intensity $I_{i}=10.0$ units (arbitrary). Calculate the transmitted intensity $I_{f}$ when $\theta_{1}=20.0^{\circ}, \theta_{2}=40.0^{\circ},$ and $\theta_{3}=60.0^{\circ} .$ Hake repeated use of Malus' law.

Salamat A.
Numerade Educator

Problem 60

Light of intensity $L_{0}$ is polarized vertically and is incident on an analyzer rotated at an angle $\theta$ from the vertical. Find the angle $\theta$ if the transmitted light has intensity (a) $I=(0.750) I_{0}$ , (b) $I=(0.500) I_{0},(\mathrm{c}) I=(0.250) I_{0},$ and (d) $I=0$.

Zachary W.
Numerade Educator

Problem 61

Light with a wavelength in vacuum of 546.1 nm falls perpendicularly on a biological specimen that is 1.000 \mum thick. The light splits into two beams polarized at right angles, for which the indices of refraction are 1.320 and $1.333,$ respectively. (a) Calculate the wavelength of each component of the light while it is traversing the specimen. (b) Calculate the phase difference between the two beams when they emerge from the specimen.

Salamat A.
Numerade Educator

Problem 62

Light from a helium-neon laser $(\lambda=632.8 \mathrm{nm})$ is incident on a single slit. What is the maximum width of the slit for which no diffraction minima are observed?

Zachary W.
Numerade Educator

Problem 63

Laser light with a wavelength of 632.8 nm is directed through one slit or two slits and allowed to fall on a screen 2.60 m beyond. Figure P24.63 shows the pattern on the screen, with a centimeter ruler below it. Did the light pass through one slit or two slits? Explain how you can tell. If the answer is one slit, find its width. If the answer is two slits, find the distance between their centers.

Salamat A.
Numerade Educator

Problem 64

In a Young’s interference experiment, the two slits are separated by 0.150 mm and the incident light includes two wavelengths: $\lambda_{1}=5.40 \times 10^{2}$ nm (green) and $\lambda_{2}=4.50 \times$ $10^{2}$ nm (blue). The overlapping interference patterns are observed on a screen 1.40 $\mathrm{m}$ from the slits. (a) Find a relationship between the orders $m_{1}$ and $m_{2}$ that determines where a bright fringe of the green light coincides with a bright fringe of the blue light. (The order $m_{1}$ is associated with $\lambda_{1},$ and $m_{2}$ is associated with $\lambda_{2},$ (b) Find the minimum values of $m_{1}$ and $m_{2}$ such that the overlapping of the bright fringes will occur and find the position of the overlap on the screen.

Zachary W.
Numerade Educator

Problem 65

Light of wavelength 546 $\mathrm{nm}$ (the intense green line from a mercury source) produces a Young's interference pattern in which the second minimum from the central maximum is along a direction that makes an angle of 18.0 $\mathrm{min}$ of arc with the axis through the central maximum. What is the distance between the parallel slits?

Salamat A.
Numerade Educator

Problem 66

The two speakers are placed 35.0 $\mathrm{cm}$ apart. A single oscillator makes the speakers vibrate in phase at a frequency of 2.00 $\mathrm{kHz}$ . At what angles, measured from the perpendicular bisector of the line joining the speakers, would a distant observer hear maximum sound intensity? Minimum sound intensity? (Take the speed of sound to be $340 . \mathrm{m} / \mathrm{s} .$ )

Zachary W.
Numerade Educator

Problem 67

Interference effects are produced at point $P$ on a screen as a result of direct rays from a $5.00 \times 10^{2}-\mathrm{nm}$ source and reflected rays off a mirror, as shown in Figure P 24.67 . If the source is $L=1.00 \times 10^{2} \mathrm{m}$ source is $L=1.00 \times 10^{2} \mathrm{m}$ to the left of the screen to the left of the screen and $h=1.00 \mathrm{cm}$ above the mirror, find the distance $y$ (in millimeters) to the first dark band above the mirror.

Salamat A.
Numerade Educator

Problem 68

Many cells are transparent and colorless. Structures of great interest in biology and medicine can be practically invisible to ordinary microscopy. An interference microscope reveals a difference in refractive index as a shift in interference fringes to indicate the size and shape of cell structures. The idea is exemplified in the following problem: An air wedge is formed between two glass plates in contact along one edge and slightly separated at the opposite edge. When the plates are illuminated with monochromatic light from above, the reflected light has 85 dark fringes. Calculate the number of dark fringes that appear if water $(n=1.33)$ replaces the air between the plates.

Zachary W.
Numerade Educator

Problem 69

Figure P24.69 shows a radio - wave transmitter and a receiver, both $h=50.0 \mathrm{m}$ above the ground and $d=$ $6.00 \times 10^{2} \mathrm{m}$ apart. The receiver can receive signals directly from the transmitter and indirectly from signals that bounce off the ground. If the ground is level between the transmitter and receiver and a $\lambda / 2$ phase shift occurs upon reflection, determine the longest wave-lengths that interfere (a) constructively and (b) destructively.

Salamat A.
Numerade Educator

Problem 70

Three polarizers, centered on a common axis and with their planes parallel to one another, have transmission axes oriented at angles of $\theta_{1}, \theta_{2},$ and $\theta_{3}$ from the vertical, as shown in Figure P 24.59 . Light of intensity $I_{i},$ polarized with its plane of polarization oriented vertically, is incident from the left onto the first polarizer. What is the ratio $I_{f} / I_{i}$ of the final transmitted intensity to the incident intensity if (a) $\theta_{1}=45^{\circ}, \theta_{2}=90^{\circ},$ and $\theta_{3}=0^{\circ} ?(\mathrm{b}) \theta_{1}=0^{\circ}, \theta_{2}=45^{\circ},$ and $\theta_{3}=90^{\circ} ?$

Zachary W.
Numerade Educator

Problem 71

The transmitting antenna on a submarine is 5.00 $\mathrm{m}$ above the water when the ship surfaces. The captain wishes to transmit a message to a receiver on a 90.0 -m-tall cliff at the ocean shore. If the signal is to be completely polarized by reflection off the ocean surface, how far must the ship be from the shore?

Salamat A.
Numerade Educator

Problem 72

A plano - convex lens (flat on one side, convex on the other) with index of refraction $n$ rests with its curved side (radius of curvature $R$ ) on a flat glass surface of the same index of refraction with a film of index $n_{\text { film }}$ between them. The lens is illuminated from above by light of wavelength $\lambda$ . Show that the dark Newton rings that appear have radii of
$$r \approx \sqrt{m \lambda R / n_{\text { tilm }}}$$
where $m$ is an integer.

Zachary W.
Numerade Educator

Problem 73

A diffraction pattern is produced on a screen 1.40 $\mathrm{m}$ from a single slit, using monochromatic light of wavelength $5.00 \times$ $10^{2} \mathrm{nm}$ . The distance from the center of the central maximum to the first-order maximum is 3.00 $\mathrm{mm}$ . Calculate the slit width. Hint: Assume that the first-order maximum is halfway between the first- and second-order minima.

Salamat A.
Numerade Educator

Problem 74

A flat piece of glass is supported horizontally above the flat end of a 10.0 -cm-long metal rod that has its lower end rigidly fixed. The thin film of air between the rod and the glass is observed to be bright when illuminated by light of wave-length $5.00 \times 10^{2} \mathrm{nm}$ . As the temperature is slowly increased by $25.0^{\circ} \mathrm{C},$ the film changes from bright to dark and back to bright 200 times. What is the coefficient of linear expansion of the metal?

Zachary W.
Numerade Educator