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Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 37

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. An interference pattern appears on the screen. If the angle from the center fringe to the first bright fringe to the side is 0.181°, what is the wavelength of the laser light?

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Problem 2

Light of wavelength 530 nm illuminates a pair of slits separated by 0.300 mm. If a screen is placed 2.00 m from the slits, determine the distance between the first and second dark fringes.

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Problem 3

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° with the horizontal. Find the separation between the slits.

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Problem 4

A Young’s interference experiment is performed with blue green argon laser light. The separation between the slits is 0.500 mm, and the screen is located 3.30 m from the slits. The first bright fringe is located 3.40 mm from the center of the interference pattern. What is the wavelength of the argon laser light?

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Problem 5

Young’s double-slit experiment is performed with 589-nm light and a distance of 2.00 m between the slits and the screen. The tenth interference minimum is observed 7.26 mm from the central maximum. Determine the spacing of the slits.

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Problem 6

Why is the following situation impossible? Two narrow slits are separated by 8.00 mm in a piece of metal. A beam of microwaves strikes the metal perpendicularly, passes through the two slits, and then proceeds toward a wall some distance away. You know that the wavelength of the radiation is 1.00 cm 65%, but you wish to measure it more precisely. Moving a microwave detector along the wall to study the interference pattern, you measure the position of the m 5 1 bright fringe, which leads to a successful measurement of the wavelength of the radiation.

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Problem 7

A pair of narrow, parallel slits separated by 0.250 mm are illuminated by green light $(\lambda=546.1 \mathrm{nm})$. The interference pattern is observed on a screen 1.20 m away from the plane of the parallel slits. Calculate the distance (a) from the central maximum to the first bright region on either side of the central maximum and (b) between the first and second dark bands in the interference pattern.

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Problem 8

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

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Problem 9

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

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Problem 10

In a location where the speed of sound is 343 m/s, a 2 000-Hz sound wave impinges on two slits 30.0 cm apart. (a) At what angle is the first maximum of sound intensity located? (b) What If? If the sound wave is replaced by 3.00-cm microwaves, what slit separation gives the same angle for the first maximum of microwave intensity?(c) What If? If the slit separation is 1.00 mm, what frequency of light gives the same angle to the first maximum of light intensity?

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Problem 11

Two radio antennas separated by $d=300 \mathrm{m}$ as shown in Figure $\mathrm{P} 37.11$ simultaneously broadcast identical signals at the same wavelength. A car travels due north along a
straight line at position $x=1000 \mathrm{m}$ from the center point between the antennas, 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 $y=$ 400 $\mathrm{m}$ 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? Note: Do not use the small-angle approximation in this problem.

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Problem 12

A riverside warehouse has several small doors facing the river. Two of these doors are open as shown in Figure P37.12. The walls of the warehouse are lined with sound-absorbing material. Two people stand at a distance $L=$ 150 $\mathrm{m}$ from the wall with the open doors. Person A stands along a line passing through the midpoint between the open doors, and person B stands a distance $y=20 \mathrm{m}$ to his absorbing material. Two people stand at a distance $L=$ 150 $\mathrm{m}$ from the wall with the open doors. Person A stands along a line passing through the midpoint between the
open doors, and person B stands a distance $y=20 \mathrm{m}$ to his

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Problem 13

A student holds a laser that emits light of wavelength 632.8 $\mathrm{nm}$ . The laser beam passes though a pair of slits separated by 0.300 $\mathrm{mm}$ , in a glass plate attached to the front of
the laser. The beam then falls perpendicularly on a screen, creating an interference pattern on it. The student begins to walk directly toward the screen at 3.00 $\mathrm{m} / \mathrm{s}$ . The central
maximum on the screen is stationary. Find the speed of the 50 th-order maxima on the screen.

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Problem 14

A student holds a laser that emits light of wavelength l. The laser beam passes though a pair of slits separated by a distance d, in a glass plate attached to the front of the laser. The beam then falls perpendicularly on a screen, creating an interference pattern on it. The student begins to walk directly toward the screen at speed v. The central maximum on the screen is stationary. Find the speed of
the mth-order maxima on the screen, where m can be very large.

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Problem 15

Radio waves of wavelength 125 $\mathrm{m}$ from a galaxy reach a radio telescope by two separate paths as shown in Figure $\mathrm{P} 37.15 .$ One is a direct path to the receiver, which is situated on the edge of a tall cliff by the ocean, and the second is by reflection off the water. As the galaxy rises in
the east over the water, the first minimum of destructive interference occurs when the galaxy is $\theta=25.0^{\circ}$ above the horizon. Find the height of the radio telescope dish above the water.

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Problem 16

In Figure $\mathrm{P} 37.16$ (not to scale), let $L=1.20 \mathrm{m}$ and $d= 0.120 $
$\mathrm{mm}$ and assume the slit system is illuminated with monochromatic 500 -nm light. Calculate the phase difference between the two wave fronts arriving at $P$ when (a) $\theta=0.500^{\circ}$ and $(\mathrm{b}) y=5.00 \mathrm{mm} .$ (c) What is the value of $\theta$ for which the phase difference is 0.333 $\mathrm{rad} ?$ (d) What is the value of $\theta$ for which the path difference is $\lambda / 4$ ?

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Problem 17

Coherent light rays of wavelength $\lambda$ strike a pair of slits separated by distance $d$ at an angle $\theta_{1}$ with respect to the normal to the plane containing the slits as shown in Figure P37.17. The rays leaving the slits make an angle $\theta_{2}$ with respect to the normal, and an interference maximum is formed by those rays on a screen that is a great distance from the slits. Show that the angle $\theta_{2}$ is given by $\begin{aligned} \theta_{2} &=\sin ^{-1}\left(\sin \theta_{1}-\frac{m \lambda}{d}\right) \\ \text { where } m \text { is an integer. } \end{aligned}$

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Problem 18

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 $\mathrm{m}$ from the slits. The first-order bright fringe is at a position $y_{\text { bright }}=$ 4.52 $\mathrm{mm}$ measured from the center of the central maximum. From this information, we wish to predict where the fringe for $n=50$ would be located. (a) Assuming the fringes are laid out linearly along the screen, find the position of the $n=50$ fringe by multiplying the position of the $n=1$ fringe by 50.0 . (b) Find the tangent of the angle the first-order bright fringe makes with respect to the line extending from the point midway between the slits to the center of the central maximum. (c) Using the result of part (b) and Equation $37.2,$ calculate the wavelength of the light. (d) Compute the angle for the 50 th-order bright
fringe from Equation $37.2 .$ (e) Find the position of the 50 th-order bright fringe on the screen from Equation $37.5 .$ (f) Comment on the agreement between the answers to parts (a) and (e).

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Problem 19

In the double-slit arrangement of Figure $\mathrm{P} 37.19$ $d=0.150 \mathrm{mm}, L=140 \mathrm{cm}, \lambda=643 \mathrm{nm},$ and $y=1.80 \mathrm{cm} .$ (a) What is the path difference $\delta$ for the rays from the two slits arriving at $P ?$ (b) Express this path difference in terms of $\lambda .(\mathrm{c})$ Does $P$ correspond to a maximum, a minimum, or an intermediate condition? Give evidence for your answer.

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Problem 20

Young's double-slit experiment underlies the instrument landing system used to guide aircraft to safe landings at some airports when the visibility is poor. Although real systems are more complicated than the example described here, they operate on the same principles. A pilot is trying to align her plane with a runway as suggested in Figure $\mathrm{P} 37.20$ .

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Problem 21

Two slits are separated by 0.180 $\mathrm{mm}$ . An interference pattern is formed on a screen 80.0 $\mathrm{cm}$ away by 656.3 -nm light. Calculate the fraction of the maximum intensity a distance
$y=0.600 \mathrm{cm}$ away from the central maximum.

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Problem 22

Show that the two waves with wave functions given by $E_{1}=6.00 \sin (100 \pi t)$ and $E_{2}=8.00 \sin (100 \pi t+\pi / 2)$ add to give a wave with the wave function $E_{R} \sin (100 \pi t+\phi) .$
Find the required values for $E_{R}$ and $\phi .$

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Problem 23

In Figure $\mathrm{P} 37.16,$ let $L=120 \mathrm{cm}$ and $d=0.250 \mathrm{cm} .$ The slits are illuminated with coherent 600 -nm light. Calculate the distance $y$ from the central maximum for which the average intensity on the screen is 75.0$\%$ of the maximum.

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Problem 24

Monochromatic coherent light of amplitude $E_{0}$ and angular frequency $\omega$ passes through three parallel slits, each separated by a distance $d$ from its neighbor. (a) Show that the time-averaged intensity as a function of the angle $\theta$ is
$$I(\theta)=I_{\max }\left[1+2 \cos \left(\frac{2 \pi d \sin \theta}{\lambda}\right)\right]^{2} $$
(b) Explain how this expression describes both the primary and the secondary maxima. (c) Determine the ratio of the intensities of the primary and secondary maxima.

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Problem 25

The intensity on the screen at a certain point in a double-slit interference pattern is 64.0$\%$ of the maximum value. (a) What minimum phase difference (in radians) between sources produces this result? (b) Express this phase difference as a path difference for 486.1 -nm light.

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Problem 26

Green light $(\lambda=546 \mathrm{nm})$ illuminates a pair of narrow, parallel slits separated by 0.250 $\mathrm{mm}$ . Make a graph of $I / I_{\max }$ as a function of $\theta$ for the interference pattern observed on a screen 1.20 $\mathrm{m}$ away from the plane of the parallel slits. Let
$\theta$ range over the interval from $-0.3^{\circ}$ to $+0.3^{\circ} .$

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Problem 27

Two narrow, parallel slits separated by 0.850 $\mathrm{mm}$ are illuminated by 600 -nm light, and the viewing screen is 2.80 $\mathrm{m}$ away from the slits. (a) What is the phase difference between the two interfering waves on a screen at a point 2.50 $\mathrm{mm}$ from the central bright fringe? (b) What is the center of a bright fringe?

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Problem 28

A soap bubble $(n=1.33)$ floating in air has the shape of a spherical shell with a wall thickness of 120 $\mathrm{nm}$ . (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 wave- length. (c) Find the two smallest film thicknesses larger than 120 $\mathrm{nm}$ that can produce strongly reflected light of
the same wavelength.

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Problem 29

A thin film of oil $(n=1.25)$ is located on smooth, wet pavement. When viewed perpendicular to the pavement, the film reflects most strongly red light at 640 $\mathrm{nm}$ and reflects no green light at 512 $\mathrm{nm}$ . How thick is the oil film?

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Problem 30

A material having an index of refraction of 1.30 is used as an antireflective coating on a piece of glass $(n=1.50) .$ What should the minimum thickness of this film be to minimize reflection of 500 -nm light?

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Problem 31

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 $\mathrm{cm}$ and the index of refraction of the polymer is $n=1.50,$ how thick would you make the coating?

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Problem 32

A film of MgF, $(n=1.38)$ having thickness $1.00 \times 10^{-5} \mathrm{cm}$ is used to coat a camera lens. (a) What are the three longest wavelengths that are intensified in the reflected light? (b) Are any of these wavelengths in the visible spectrum?

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Problem 33

A beam of 580 -nm light passes through two closely spaced glass plates at close to normal incidence as shown in Figure P37.33. For what minimum nonzero value of the plate separation $d$ is the transmitted light bright?

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Problem 34

An oil film $(n=1.45)$ floating on water is illuminated by white light at normal incidence. The film is 280 $\mathrm{nm}$ 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 spectrum most strongly transmitted. Explain your reasoning.

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Problem 35

An air wedge is formed between two glass plates separated at one edge by a very fine wire of circular cross section as shown in Figure P37.35. When the wedge is illuminated from above by 600-nm light and viewed from above, 30 dark fringes are observed. Calculate the diameter d of the wire.

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Problem 36

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 held at a constant temperature. (a) Find the minimum value of $d$ that produces maximum transmission of perpendicular $\mathrm{H}_{\alpha}$ light if the dielectric has an index of refraction of 1.378 .
(b) What If? 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.

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Problem 37

When a liquid is introduced into the air space between the lens and the plate in a Newton's-rings apparatus, the diameter of the tenth ring changes from 1.50 to $1.31 \mathrm{cm} .$ Find the index of refraction of the liquid.

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Problem 38

A lens made of glass $\left(n_{g}=1.52\right)$ is coated with a thin film of $\mathrm{MgF}_{2}\left(n_{s}=1.38\right)$ of thickness $t .$ Visible light is incident normally on the coated lens as in Figure $\mathrm{P} 37.38$ . (a) For what minimum value of $t$ will the reflected light
of wavelength 540 $\mathrm{nm}$ (in air) be missing? (b) Are there

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Problem 39

Two glass plates 10.0 $\mathrm{cm}$ long are in contact at one end and separated at the other end by a thread with a diameter $d=$ 0.0500 $\mathrm{mm}$ (Fig. P37.35). Light containing the two wave-
lengths 400 $\mathrm{nm}$ and 600 $\mathrm{nm}$ is incident perpendicularly and viewed by reflection. At what distance from the contact point is the next dark fringe?

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Problem 40

Mirror $\mathrm{M}_{1}$ in Active Figure 37.13 is moved through a displacement $\Delta L .$ During this displacement, 250 fringe reversals (formation of successive dark or bright bands) are counted. The light being used has a wavelength of $632.8 \mathrm{nm} .$ Calculate the displacement $\Delta L$ .

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Problem 41

One leg of a Michelson interferometer contains an evacuated cylinder of length $L,$ having glass plates on each end. A gas is slowly leaked into the cylinder until a pressure of 1 atm is reached. If $N$ bright fringes pass on the screen during this process when light of wavelength $\lambda$ is used, what is the index of refraction of the gas?

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Problem 42

Radio transmitter A operating at 60.0 $\mathrm{MHz}$ is 10.0 $\mathrm{m}$ from another similar transmitter $\mathrm{B}$ that is $180^{\circ}$ out of phase with A. How far must an observer move from A toward B along the line connecting the two transmitters to reach the nearest point where the two beams are in phase?

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Problem 43

In an experiment similar to that of Example $37.1,$ green light with wavelength $560 \mathrm{nm},$ sent through a pair of slits 30.0$\mu \mathrm{m}$ apart, produces bright fringes 2.24 $\mathrm{cm}$ apart on a screen 1.20 $\mathrm{m}$ away. If the apparatus is now submerged in a tank containing a sugar solution with index of refraction $1.38,$ calculate the fringe separation for this same arrangement.

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Problem 44

In the What If? section of Example 37.2 , it was claimed that overlapping fringes in a two-slit interference pattern for two different wavelengths obey the following relation- ship even for large values of the angle $\theta :$ $$ \frac{m^{\prime}}{m}=\frac{\lambda}{\lambda^{\prime}} $$ (a) Prove this assertion. (b) Using the data in Example $37.2,$ find the nonzero value of $y$ on the screen at which the fringes from the two wavelengths first coincide.

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Problem 45

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 $d$ of the fiber, the technician places it between two flat glass plates at their ends as in Figure P37.35. When the plates, of length $14.0 \mathrm{cm},$ are illuminated from above with light of wavelength 650 $\mathrm{nm}$ , she observes interference bands separated by $0.580 \mathrm{mm} .$ What is the diameter of the fiber?

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Problem 46

Raise your hand and hold it flat. Think of the space between your index finger and your middle finger as one slit and think of the space between middle finger and ring finger as a second slit. (a) Consider the interference resulting from sending coherent visible light perpendicularly through this pair of openings. Compute an order-of- magnitude estimate for the angle between adjacent zones of constructive interference. (b) To make the angles in the interference pattern easy to measure with a plastic protractor, you should use an electromagnetic wave with frequency of what order of magnitude? (c) How is this wave classified on the electromagnetic spectrum?

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Problem 47

Two coherent waves, coming from sources at different locations, move along the $x$ axis. Their wave functions are $$E_{1}=860 \sin \left[\frac{2 \pi x_{1}}{650}-924 \pi t+\frac{\pi}{6}\right]$$
and $$E_{2}=860 \sin \left[\frac{2 \pi x_{2}}{650}-924 \pi t+\frac{\pi}{8}\right] $$ where $E_{1}$ and $E_{2}$ are in volts per meter, $x_{1}$ and $x_{2}$ are in nanometers, and $t$ is in picoseconds. When the two waves are superposed, determine the relationship between $x_{1}$ and $x_{2}$ that produces constructive interference.

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Problem 48

In a Young's interference experiment, the two slits are separated by 0.150 $\mathrm{mm}$ and the incident light includes two wavelengths: $\lambda_{1}=540 \mathrm{nm}$ (green) and $\lambda_{2}=450 \mathrm{nm}$ (blue). The overlapping interference patterns are observed on a screen 1.40 $\mathrm{m}$ from the slits. Calculate the minimum distance from the center of the screen to a point where a bright fringe of the green light coincides with a bright fringe of the blue light.

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Problem 49

In a Young's double-slit experiment using light of wave- length $\lambda,$ a thin piece of Plexiglas having index of refraction $n$ covers one of the slits. If the center point on the screen is a dark spot instead of a bright spot, what is the minimum thickness of the Plexiglas?

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Problem 50

Review. A flat piece of glass is held stationary and horizontal above the highly polished, flat top end of a 10.0 -cm-long vertical metal rod that has its lower end rigidly fixed. The thin film of air between the rod and glass is observed to be bright by reflected light when it is illuminated by light of wavelength 500 $\mathrm{nm}$ . 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?

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Problem 51

A certain grade of crude oil has an index of refraction of $1.25 .$ A ship accidentally spills 1.00 $\mathrm{m}^{3}$ of this oil into the ocean, and the oil spreads into a thin, uniform slick. If the film produces a first-order maximum of light of wavelength 500 $\mathrm{nm}$ normally incident on it, how much surface area of the ocean does the oil slick cover? Assume the index of refraction of the ocean water is 1.34 .

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Problem 52

The waves from a radio station can reach a home receiver by two paths. One is a straight-line path from transmitter to home, a distance of $30.0 \mathrm{km} .$ The second is by reflection from the ionosphere (a layer of ionized air molecules high in the atmosphere). Assume this reflection takes place at a point midway between receiver and transmitter, the wavelength broadcast by the radio station is $350 \mathrm{m},$ and no phase change occurs on reflection. Find the minimum height of the ionospheric layer that could produce destructive interference between the direct and reflected beams.

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Problem 53

Interference effects are produced at point $P$ on a screen as a result of direct rays from a 500 -nm source and reflected rays from the mirror as shown in Figure P37.53. Assume the source is 100 $\mathrm{m}$ to the left of the screen and 1.00 $\mathrm{cm}$ above the mirror. Find the distance $y$ to the first dark band above the mirror.

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Problem 54

Measurements are made of the intensity distribution within the central bright fringe in a Young's interference pattern (see Fig. $37.6 ) .$ At a particular value of $y,$ it is found that $I / I_{\max }=0.810$ when 600 -nm light is used. What wavelength of light should be used to reduce the relative intensity at
the same location to 64.0$\%$ of the maximum intensity?

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Problem 55

Many cells are transparent and colorless. Structures of great interest in biology and medicine can be practically invisible to ordinary microscopy. To indicate the size and shape of cell structures, an interference microscope reveals a difference in index of refraction as a shift in interference fringes. The idea is exemplified in the following problem. An air wedge is formed between two glass plates in con-
tact along one edge and slightly separated at the opposite edge as in Figure $\mathrm{P} 37.35$ . 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.

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Problem 56

Consider the double-slit arrangement shown in Figure P37.56, where the slit separation is $d$ and the distance from the slit to the screen is $L .$ A sheet of transparent plastic having an index of refraction $n$ and thickness $t$ is placed over the upper slit. As a result, the central maximum of the interference pattern moves upward a distance $y^{\prime} .$ Find $y^{\prime} .$

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Problem 57

Figure $\mathrm{P} 37.57$ shows a radio-wave transmitter and a receiver separated by a distance $d=50.0 \mathrm{m}$ and both a distance $h=35.0 \mathrm{m}$ above the ground. The receiver can receive signals both directly from the transmitter and indirectly from signals that reflect from the ground. Assume the ground is level between the transmitter and receiver and a $180^{\circ}$ phase shift occurs upon reflection. Determine the longest wavelengths that interfere (a) constructively and (b) destructively.

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Problem 58

Figure $\mathrm{P} 37.57$ shows a radio-wave transmitter and a receiver separated by a distance $d$ and both a distance $h$ above the ground. The receiver can receive signals both directly from the transmitter and indirectly from signals that reflect from the ground. Assume the ground is level between the transmitter and receiver and a $180^{\circ}$ phase shift occurs upon reflection. Determine the longest wavelengths that interfere (a) constructively and (b) destructively.

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Problem 59

In a Newton's-rings experiment, a plano-convex glass $(n=$ 1.52 ) lens having radius $r=5.00 \mathrm{cm}$ is placed on a flat plate as shown in Figure $\mathrm{P} 37.59$ . When light of wavelength $\lambda=$ 650 $\mathrm{nm}$ is incident normally, 55 bright rings are observed, with the last one precisely on the edge of the lens. (a) What is the radius $R$ of curvature of the convex surface of the lens? (b) What is the focal length of the lens?

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Problem 60

Why is the following situation impossible? A piece of transparent material having an index of refraction $n=1.50$ is cut into the shape of a wedge as shown in Figure $\mathrm{P} 37.60 .$ Both the top and bottom surfaces of the wedge are in contact with air. Monochromatic light of wavelength $\lambda=632.8 \mathrm{nm}$ is normally incident from above, and the wedge is viewed from above. Let $h=1.00 \mathrm{mm}$ represent the height of the wedge and $\ell=0.500 \mathrm{m}$ its length. A thin-film interference pattern appears in the wedge due to reflection from the top and bottom surfaces. You have been given the task of counting the number of bright fringes that appear in the entire length $\ell$ of the wedge. You find this task tedious, and your concentration is broken by a noisy distraction after accurately counting 5000 bright fringes.

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Problem 61

A plano-concave lens having index of refraction 1.50 is placed on a flat glass plate as shown in Figure P37.61. Its curved surface, with radius of curvature 8.00 m, is on the bottom. The lens is illuminated from above with yellow sodium light of wavelength 589 nm, and a series of concentric bright and dark rings is observed by reflection. The interference pattern has a dark spot at the center that is surrounded by 50 dark rings, the largest of which is at the outer edge of the lens. (a) What is the thickness of the air layer at the center of the interference pattern? (b) Calculate the radius of the outermost dark ring. (c) Find the focal length of the lens.

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Problem 62

A plano-convex lens has index of refraction $n .$ The curved side of the lens has radius of curvature $R$ and rests on a flat glass surface of the same index of refraction, with a film of index $n_{\text { film between them, as shown in Figure 37.62 . The lens is illuminated from above by light of wavelength $\lambda$ . Show that the dark Newton's rings have radii given approximately by $r \approx \sqrt{\frac{m \lambda R}{n_{\text { film }}}}$ where $r<R$ and $m$ is an integer.

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Problem 63

Interference fringes are produced using Lloyd's mirror and a source S of wavelength $\lambda=606 \mathrm{nm}$ as shown in Figure $\mathrm{P} 37.63$ . Fringes separated by $\Delta y=1.20 \mathrm{mm}$ are formed on a screen a distance $L=2.00 \mathrm{m}$ from the source. Find the vertical distance $h$ of the source above the reflecting surface.

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Problem 64

The quantity $n t$ in Equations 37.17 and 37.18 is called the optical path length corresponding to the geometrical distance $t$ and is analogous to the quantity $\delta$ in Equation 37.1 , the path difference. The optical path length is proportional to $n$ because a larger index of refraction shortens
the wavelength, so more cycles of a wave fit into a particular geometrical distance. (a) Assume a mixture of corn syrup and water is prepared in a tank, with its index of refraction $n$ increasing uniformly from 1.33 at $y=20.0 \mathrm{cm}$ at the top to 1.90 at $y=0 .$ Write the index of refraction $n(y)$ as a function of $y$ . (b) Compute the optical path length correspond-
$$\int_{0}^{20} \mathrm{cm}$$
(c) Suppose a narrow beam of light is directed into the mixture at a nonzero angle with respect to the normal to the surface of the mixture. Qualitatively describe its path.

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Problem 65

Astronomers observe a 60.0 -MHz radio source both directly and by reflection from the sea as shown in Figure P37.15. If the receiving dish is 20.0 $\mathrm{m}$ above sea level, what is the angle of the radio source above the horizon at first maximum?

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Problem 66

Figure $\mathrm{CQ} 37.2$ shows an unbroken soap film in a circular frame. The film thickness increases from top to bottom, slowly at first and then rapidly. As a simpler model, consider a soap film $(n=1.33)$ contained within a rectangular wire frame. The frame is held vertically so that the film drains downward and forms a wedge with flat faces. The thickness of the film at the top is essentially zero. The film is viewed in reflected white light with near-normal incidence, and the first violet $(\lambda=420 \mathrm{nm})$ interference band is observed 3.00 $\mathrm{cm}$ from the top edge of the film. (a) Locate the first red $(\lambda=680 \mathrm{nm})$ interference band. (b) Determine the film thickness at the positions of the violet and red bands. (c) What is the wedge angle of the film?

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Problem 67

Our discussion of the techniques for determing constructive and destructive interference by reflection from a thin film in air has been confined to rays striking the film at nearly normal incidence. What If? Assume a ray is incident at an angle of $30.0^{\circ}$ (relative to the normal) on a film with index of refraction 1.38 surrounded by vacuum. Calculate the minimum thickness for constructive interference of sodium light with a wavelength of 590 $\mathrm{nm}$ .

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Problem 68

The condition for constructive interference by reflection from a thin film in air as developed in Section 37.5
assumes nearly normal incidence. What If? Suppose the light is incident on the film at a nonzero angle $\theta_{1}$ (relative to the normal). The index of refraction of the film is $n$ , and the film is surrounded by vacuum of the film is $n$ , and constructive interference that relates the thickness $t$ of the film, the index of refraction $n$ of the film, the wavelength $\lambda$ of the light, and the angle of incidence $\theta_{1}$ .

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Problem 69

Both sides of a uniform film that has index of refraction $n$ and thickness $d$ are in contact with air. For normal incidence of light, an intensity minimum is observed in the reflected light at $\lambda_{2}$ and an intensity maximum is observed at $\lambda_{1},$ where $\lambda_{1}>\lambda_{2}$ (a) Assuming no intensity minima are observed between $\lambda_{1}$ and $\lambda_{2},$ find an expression for the
integer $m$ in Equations 37.17 and 37.18 in terms of the wave- lengths $\lambda_{1}$ and $\lambda_{2} .$ (b) Assuming $n=1.40, \lambda_{1}=500 \mathrm{nm},$ and $\lambda_{2}=370 \mathrm{nm},$ determine the best estimate for the thickness of the film.

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Problem 70

Slit 1 of a double slit is wider than slit 2 so that the light from slit 1 has an amplitude 3.00 times that of the light from slit $2 .$ Show that Equation 37.13 is replaced by the equation $I=I_{\max }\left(1+3 \cos ^{2} \phi / 2\right)$ for this situation.

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Problem 71

Monochromatic light of wavelength 620 $\mathrm{nm}$ passes through a very narrow slit $\mathrm{S}$ and then strikes a screen in which are two parallel slits, $\mathrm{S}_{1}$ and $\mathrm{S}_{2},$ as shown in Figure $\mathrm{P} 37.71$ on page $1110 .$ Slit $\mathrm{S}_{1}$ is directly in line with $\mathrm{S}$ and at a distance of $L=1.20 \mathrm{m}$ away from $\mathrm{S}$ , whereas $\mathrm{S}_{2}$ is displaced a distance $d$ to one side. The light is detected at point $P$ on a sec-
ond screen, equidistant from $\mathrm{S}_{1}$ and $\mathrm{S}_{2} .$ When either slit $\mathrm{S}_{1}$ or $\mathrm{S}_{2}$ is open, equal light intensities are measured at point $P .$ When both slits are open, the intensity is three times larger. Find the minimum possible value for the slit
separation $d .$

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Problem 72

A plano-convex lens having a radius of curvature of $r=$ 4.00 $\mathrm{m}$ is placed on a concave glass surface whose radius of curvature is $R=12.0 \mathrm{m}$ as shown in Figure $\mathrm{P} 37.72$ . Assuming 500 -nm light is incident normal to the flat surface of the lens, determine the radius of the 100 th bright ring.

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