Principles of Physics a Calculus Based Text

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

Chapter 27

Wave Optics - all with Video Answers

Educators

Chapter Questions

Problem 1

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

Charles Carter

Charles Carter

Numerade Educator

Problem 2

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 P27.2. Two radio antennas (the black dots in the figure are positioned adjacent to the runway, separated by $d=40.0 \mathrm{m} .$ The antennas broadcast unmodulated coherent radio waves at $30.0 \mathrm{MHz}$. The red lines in Figure P27.2 represent paths along which maxima in the interference pattern of the radio waves exist. (a) Find the wavelength of the waves. The pilot "locks onto" the strong signal radiated along an interference maximum and steers the plane to keep the received signal strong. If she has found the central maximum, the plane will have precisely the correct heading to land when it reaches the runway as exhibited by plane $\mathrm{A}$. (b) What If? Suppose the plane is flying along the first side maximum instead as is the case for plane $\mathrm{B}$. How far to the side of the runway centerline will the plane be when it is $2.00 \mathrm{km}$ from the antennas, measured along its direction of travel? (c) It is possible to tell the pilot that she is on the wrong maximum by sending out two signals from each antenna and equipping the aircraft with a two-channel receiver. The ratio of the two frequencies must not be the ratio of small integers (such as $\frac{3}{4}$ ). Explain how this two-frequency system would work and why it would not necessarily work if the frequencies were related by an integer ratio.

Charles Carter

Numerade Educator

Problem 3

Two radio antennas separated by $d=$ $300 \mathrm{m}$ as shown in Figure P 27.3 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.

Farhanul Hasan

Numerade Educator

Problem 4

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

Charles Carter

Numerade Educator

Problem 5

Two slits are separated by $0.320 \mathrm{mm} .$ A beam of 500 -nm light strikes the slits, producing an interference pattern. Determine the number of maxima observed in the angular range $-30.0^{\circ} \leq \theta \leq 30.0^{\circ}$.

Charles Carter

Numerade Educator

Problem 6

In a location where the speed of sound is $343 \mathrm{m} / \mathrm{s}$, a $2000-\mathrm{Hz}$ sound wave impinges on two slits $30.0 \mathrm{cm}$ 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 $1.00 \mu \mathrm{m},$ what frequency of light gives the same angle to the first maximum of light intensity?

Charles Carter

Numerade Educator

Problem 7

In Figure P27.7 (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 (b) $y=5.00 \mathrm{mm} .$ (c) What is the value of $\theta$ for which the phase difference is 0.333 rad? (d) What is the value of $\theta$ for which the path difference is $\lambda / 4$ ?

Farhanul Hasan

Numerade Educator

Problem 8

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.

Charles Carter

Numerade Educator

Problem 9

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.

Vishal Gupta

Numerade Educator

Problem 10

In the double-slit arrangement of Figure P27.10 , $d=0.150 \mathrm{mm}, L=140 \mathrm{cm}, \lambda=643 \mathrm{nm},$ 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 $\lambda$. (c) Does $P$ correspond to a maximum, a minimum, or an intermediate condition? Give evidence for your answer.

Charles Carter

Numerade Educator

Problem 11

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.

Farhanul Hasan

Numerade Educator

Problem 12

Why is the following situation impossible? Two narrow slits are separated by $8.00 \mathrm{mm}$ 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 $1.00 \mathrm{cm} \pm 5 \%$, 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 $m=1$ bright fringe, which leads to a successful measurement of the wavelength of the radiation.

Charles Carter

Numerade Educator

Problem 13

A pair of narrow, parallel slits separated by $0.250 \mathrm{mm}$ are illuminated by green light $(\lambda=546.1 \mathrm{nm}) .$ The interference pattern is observed on a screen $1.20 \mathrm{m}$ 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.

Farhanul Hasan

Numerade Educator

Problem 14

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 P27.14. 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 $$\theta_{2}=\sin ^{-1}\left(\sin \theta_{1}-\frac{m \lambda}{d}\right)$$ where $m$ is an integer.

Farhanul Hasan

Numerade Educator

Problem 15

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 50th-order maxima on the screen.

Farhanul Hasan

Numerade Educator

Problem 16

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 $m$ th-order maxima on the screen, where $m$ can be very large.

Charles Carter

Numerade Educator

Problem 17

A riverside warehouse has several small doors facing the river. Two of these doors are open as shown in Figure $\mathrm{P} 27.17$ 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 $\mathrm{B}$ stands a distance $y=20 \mathrm{m}$ to his side. A boat on the river sounds its horn. To person $\mathrm{A}$, the sound is loud and clear. To person $\mathrm{B}$, the sound is barely audible. The principal wavelength of the sound waves is $3.00 \mathrm{m}$. Assuming person $\mathrm{B}$ is at the position of the first minimum, determine the distance $d$ between the doors, center to center.

Charles Carter

Numerade Educator

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 $50.0 .(\mathrm{b})$ 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 $27.2,$ calculate the wavelength of the light. (d) Compute the angle for the 50 th-order bright fringe from Equation 27.2 (e) Find the position of the 50 th-order bright fringe on the screen from Equation $27.5 .$ (f) Comment on the agreement between the answers to parts (a) and (e).

Charles Carter

Numerade Educator

Problem 19

The two speakers of a boom box are $35.0 \mathrm{cm}$ 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 as $340 \mathrm{m} / \mathrm{s}$.)

Charles Carter

Numerade Educator

Problem 20

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.

Farhanul Hasan

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 $3.00 \mathrm{cm}$ and the index of refraction of the polymer is $n=1.50,$ how thick would you make the coating?

Farhanul Hasan

Numerade Educator

Problem 22

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.

Farhanul Hasan

Numerade Educator

Problem 23

A beam of 580 -nm light passes through two closely spaced glass plates at close to normal incidence as shown in Figure $\mathrm{P} 27.23 .$ For what minimum nonzero value of the plate separation $d$ is the transmitted light bright?

Farhanul Hasan

Numerade Educator

Problem 24

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?

Farhanul Hasan

Numerade Educator

Problem 25

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 P27.25 . When the wedge is illuminated from above by $600-\mathrm{nm}$ light and viewed from above, 30 dark fringes are observed. Calculate the diameter $d$ of the wire.

Farhanul Hasan

Numerade Educator

Problem 26

A soap bubble $(n=1.33)$ is floating in air. If the thickness of the bubble wall is $115 \mathrm{nm}$, what is the wavelength of the light that is most strongly reflected?

Charles Carter

Numerade Educator

Problem 27

Coherent microwaves of wavelength $5.00 \mathrm{cm}$ enter a tall, narrow window in a building otherwise essentially opaque to the microwaves. 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?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 28

Helium-neon laser light $(\lambda=632.8 \mathrm{nm})$ is sent through a 0.300 -mm-wide single slit. What is the width of the central maximum on a screen $1.00 \mathrm{m}$ from the slit?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 29

A screen is placed $50.0 \mathrm{cm}$ from a single slit, which is illuminated with light of wavelength $690 \mathrm{nm}$. 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?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 30

A screen is placed a distance $L$ from a single slit of width $a,$ which is illuminated with light of wavelength $\lambda$. Assume $L \gg a .$ If the distance between the minima for $m=m_{1}$ and $m=m_{2}$ in the diffraction pattern is $\Delta y,$ what is the width of the slit?

Charles Carter

Numerade Educator

Problem 31

Sound with a frequency $650 \mathrm{Hz}$ from a distant source passes through a doorway $1.10 \mathrm{m}$ wide in a sound-absorbing wall. Find (a) the number and (b) the angular directions of the diffraction minima at listening positions along a line parallel to the wall.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 32

A horizontal laser beam of wavelength 632.8 nm has a circular cross section $2.00 \mathrm{mm}$ in diameter. A rectangular aperture is to be placed in the center of the beam so that when the light falls perpendicularly on a wall $4.50 \mathrm{m}$ away, the central maximum fills a rectangle $110 \mathrm{mm}$ wide and $6.00 \mathrm{mm}$ high. The dimensions are measured between the minima bracketing the central maximum. Find the required (a) width and (b) height of the aperture. (c) Is the longer dimension of the central bright patch in the diffraction pattern horizontal or vertical? (d) Is the longer dimension of the aperture horizontal or vertical? (e) Explain the relationship between these two rectangles, using a diagram.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 33

A beam of monochromatic green light is diffracted by a slit of width $0.550 \mathrm{mm} .$ The diffraction pattern forms on a wall $2.06 \mathrm{m}$ beyond the slit. The distance between the positions of zero intensity on both sides of the central bright fringe is $4.10 \mathrm{mm} .$ Calculate the wavelength of the light.

Charles Carter

Numerade Educator

Problem 34

The pupil of a cat's eye narrows to a vertical slit of width $0.500 \mathrm{mm}$ in daylight. Assume the average wavelength of the light is $500 \mathrm{nm}$. What is the angular resolution for horizontally separated mice?

Charles Carter

Numerade Educator

Problem 35

A helium-neon laser emits light that has a wavelength of $632.8 \mathrm{nm} .$ The circular aperture through which the beam emerges has a diameter of $0.500 \mathrm{cm} .$ Estimate the diameter of the beam $10.0 \mathrm{km}$ from the laser.

Charles Carter

Numerade Educator

Problem 36

Narrow, parallel, glowing gas-filled tubes in a variety of colors form block letters to spell out the name of a nightclub. Adjacent tubes are all $2.80 \mathrm{cm}$ apart. The tubes forming one letter are filled with neon and radiate predominantly red light with a wavelength of 640 nm. For another letter, the tubes emit predominantly blue light at $440 \mathrm{nm} .$ The pupil of a dark-adapted viewer's eye is 5.20 $\mathrm{mm}$ in diameter. (a) Which color is easier to resolve? State how you decide. (b) If she is in a certain range of distances away, the viewer can resolve the separate tubes of one color but not the other. The viewer's distance must be in what range for her to resolve the tubes of only one of these two colors?

Charles Carter

Numerade Educator

Problem 37

Impressionist painter Georges Seurat created paintings with an enormous number of dots of pure pigment, each of which was approximately $2.00 \mathrm{mm}$ in diameter. The idea was to have colors such as red and green next to each other to form a scintillating canvas, such as in his masterpiece, $A$ Sunday Afternoon on the Island of La Grande Jatte (Fig. $\mathrm{P} 27.37$ ). Assume $\lambda=500 \mathrm{nm}$ and a pupil diameter of $5.00 \mathrm{mm} .$ Beyond what distance would a viewer be unable to discern individual dots on the canvas?

Charles Carter

Numerade Educator

Problem 38

A spy satellite can consist of a large-diameter concave mirror forming an image on a digital-camera detector and sending the picture to a ground receiver by radio waves. In effect, it is an astronomical telescope in orbit, looking down instead of up. (a) Can a spy satellite read a license plate? (b) Can it read the date on a dime? Argue for your answers by making an order-of-magnitude calculation, specifying the data you estimate.

Charles Carter

Numerade Educator

Problem 39

A circular radar antenna on a Coast Guard ship has a diameter of $2.10 \mathrm{m}$ and radiates at a frequency of $15.0 \mathrm{GHz}$ Two small boats are located $9.00 \mathrm{km}$ away from the ship. How close together could the boats be and still be detected as two objects?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 40

White light is spread out into its spectral components by a diffraction grating. If the grating has 2000 grooves per centimeter, at what angle does red light of wavelength $640 \mathrm{nm}$ appear in first order?

Charles Carter

Numerade Educator

Problem 41

Light from an argon laser strikes a diffraction grating that has 5310 grooves per centimeter. The central and firstorder 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.

Charles Carter

Numerade Educator

Problem 42

The hydrogen spectrum includes a red line at $656 \mathrm{nm}$ and a blue-violet line at $434 \mathrm{nm}$. What are the angular separations between these two spectral lines for all visible orders obtained with a diffraction grating that has 4500 grooves/cm?

Charles Carter

Numerade Educator

Problem 43

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

Netra Sharma

University of Wisconsin - Milwaukee

Problem 44

A grating with 250 grooves $/ \mathrm{mm}$ is used with an incandescent light source. Assume the visible spectrum to range in wavelength from $400 \mathrm{nm}$ to $700 \mathrm{nm}$. In how many orders can one see (a) the entire visible spectrum and (b) the short-wavelength region of the visible spectrum?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 45

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 $37.2 \mathrm{kHz}$ from a distant source is incident perpendicular to the array. (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.

Charles Carter

Numerade Educator

Problem 46

Show that whenever white light is passed through a diffraction grating of any spacing size, the violet end of the spectrum in the third order on a screen always overlaps the red end of the spectrum in the second order.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 47

Three discrete spectral lines occur at angles of $10.1^{\circ}$ $13.7^{\circ},$ and $14.8^{\circ}$ in the first-order spectrum of a grating 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 spectrum?

Charles Carter

Numerade Educator

Problem 48

If the spacing between planes of atoms in a $\mathrm{NaCl}$ crystal is $0.281 \mathrm{nm},$ what is the predicted angle at which $0.140-\mathrm{nm}$ x-rays are diffracted in a first-order maximum?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 49

Potassium iodide (KI) has the same crystalline structure as $\mathrm{NaCl},$ with atomic planes separated by $0.353 \mathrm{nm} .$ A monochromatic x-ray beam shows a first-order diffraction maximum when the grazing angle is $7.60^{\circ} .$ Calculate the x-ray wavelength.

Charles Carter

Numerade Educator

Problem 50

In water of uniform depth, a wide pier is supported on pilings in several parallel rows $2.80 \mathrm{m}$ apart. Ocean waves of uniform wavelength roll in, moving in a direction that makes an angle of $80.0^{\circ}$ with the rows of pilings. Find the three longest wavelengths of waves that are strongly reflected by the pilings.

Charles Carter

Numerade Educator

Problem 51

A helium-neon laser can produce a green laser beam instead of red. Refer to Figure 24.17 , which omits some energy levels between $E_{2}$ and $E_{1}$. After a population inversion is established, neon atoms make a variety of downward transitions in falling from the state labeled $E_{3}^{*}$ down eventually to level $E_{1} .$ The atoms emit both red light with a wavelength of $632.8 \mathrm{nm}$ and green light with a wavelength of $543 \mathrm{nm}$ in a competing transition. Assume that the atoms are in a cavity between mirrors designed to reflect the green light with high efficiency but to allow the red light to leave the cavity immediately. Then stimulated emission can lead to the buildup of a collimated beam of green light between the mirrors, having a greater intensity than does the red light. A small fraction of the green light can be permitted to escape by transmission through one mirror to constitute the radiated laser beam. The mirrors forming the resonant cavity are not made of shiny metal, but of layered dielectrics, say silicon dioxide and titanium dioxide. (a) How thick a layer of silicon dioxide, between layers of titanium dioxide, would minimize reflection of the red light? (b) What should be the thickness of a similar but separate layer of silicon dioxide to maximize reflection of the green light?

Charles Carter

Numerade Educator

Problem 52

A wide beam of laser light with a wavelength of $632.8 \mathrm{nm}$ is directed through several narrow parallel slits, separated by $1.20 \mathrm{mm},$ and falls on a sheet of photographic film $1.40 \mathrm{m}$ away. The exposure time is chosen so that the film stays unexposed everywhere except at the central region of each bright fringe. (a) Find the distance between these interference maxima. The film is printed as a transparency; it is opaque everywhere except at the exposed lines. Next, the same beam of laser light is directed through the transparency and allowed to fall on a screen $1.40 \mathrm{m}$ beyond. (b) Argue that several narrow, parallel, bright regions, separated by $1.20 \mathrm{mm},$ appear on the screen as real images of the original slits. (A similar train of thought, at a soccer game, led Dennis Gabor to invent holography.)

Charles Carter

Numerade Educator

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 $\mathrm{P} 27.53 .$ 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.

Farhanul Hasan

Numerade Educator

Problem 54

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?

Charles Carter

Numerade Educator

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 contact along one edge and slightly separated at the opposite edge as in Figure $\mathrm{P} 27.25 .$ 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.

Charles Carter

Numerade Educator

Problem 56

Laser light with a wavelength of $632.8 \mathrm{nm}$ is directed through one slit or two slits and allowed to fall on a screen $2.60 \mathrm{m}$ beyond. Figure $\mathrm{P} 27.56$ shows the pattern on the screen, with a centimeter ruler below it. (a) Did the light pass through one slit or two slits? Explain how you can determine the answer. (b) If one slit, find its width. If two slits, find the distance between their centers.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 57

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?

Netra Sharma

University of Wisconsin - Milwaukee

Problem 58

The condition for constructive interference by reflection from a thin film in air as developed in Section 27.5 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. Find the condition for 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 $\theta_{1}$.

Charles Carter

Numerade Educator

Problem 59

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?

Charles Carter

Numerade Educator

Problem 60

A beam of $541-\mathrm{nm}$ light is incident on a diffraction grating that has 400 grooves $/ \mathrm{mm}$ (a) Determine the ang of the second-order ray. (b) What If? If the entire apparatus is immersed in water, what is the new second-order angle of diffraction? (c) Show that the two diffracted rays of parts (a) and (b) are related through the law of refraction.

Charles Carter

Numerade Educator

Problem 61

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.

Charles Carter

Numerade Educator

Problem 62

Why is the following situation impossible? A technician is sending laser light of wavelength $632.8 \mathrm{nm}$ through a pair of slits separated by $30.0 \mu \mathrm{m}$. Each slit is of width $2.00 \mu \mathrm{m}$. The screen on which he projects the pattern is not wide enough,so light from the $m=15$ interference maximum misses the edge of the screen and passes into the next lab station, startling a coworker.

Netra Sharma

University of Wisconsin - Milwaukee

Problem 63

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 27.13 and 27.14 in terms of the wavelengths $\lambda_{1}$ 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.

Charles Carter

Numerade Educator

Problem 64

Two wavelengths $\lambda$ and $\lambda+\Delta \lambda(\text { with } \Delta \lambda<<\lambda)$ are incident on a diffraction grating. Show that the angular separation between the spectral lines in the $m$ th-order spectrum is $$\Delta \theta=\frac{\Delta \lambda}{\sqrt{(d / m)^{2}-\lambda^{2}}}$$ where $d$ is the slit spacing and $m$ is the order number

Charles Carter

Numerade Educator

Problem 65

Light of wavelength $500 \mathrm{nm}$ is incident normally on a diffraction grating. If the third-order maximum of the diffraction pattern is observed at $32.0^{\circ},$ (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.

Charles Carter

Numerade Educator

Problem 66

Figure $\mathrm{P} 27.66$ shows a mega phone in use. Construct a theoretical description of how a megaphone works. You may assume that the sound of your voice radiates just through the opening of your mouth. Most of the information in speech is carried not in a signal at the fundamental frequency, but in noises and in harmonics, with frequencies of a few thousand hertz. Does your theory allow any prediction that is simple to test?

Charles Carter

Numerade Educator

Problem 67

A beam of bright red light of wavelength 654 nm passes through a diffraction grating Enclosing the space beyond the grating is a large semicylindrical screen centered on the grating, with its axis parallel to the slits in the grating. Fifteen bright spots appear on the screen. Find (a) the maximum and (b) the minimum possible values for the slit separation in the diffraction grating.

Charles Carter

Numerade Educator

Problem 68

Iridescent peacock feathers are shown in Figure $\mathrm{P} 27.68 \mathrm{a}$ (page 938 ). The surface of one microscopic barbule is composed of transparent keratin that supports rods of dark brown melanin in a regular lattice, represented in Figure $\mathrm{P} 27.68 \mathrm{b}$. (Your fingernails are made of keratin, and melanin is the dark pigment giving color to human skin. In a portion of the feather that can appear turquoise (blue-green), assume the melanin rods are uniformly separated by $0.25 \mu \mathrm{m},$ with air between them. (a) Explain how this structure can appear turquoise when it contains no blue or green pigment. (b) Explain how it can also appear violet if light falls on it in a different direction. (c) Explain how it can present different colors to your two eyes simultaneously, which is a characteristic of iridescence. (d) A compact disc can appear to be any color of the rainbow. Explain why the portion of the feather in Figure $\mathrm{P} 27.68 \mathrm{b}$ cannot appear yellow or red. (e) What could be different about the array of melanin rods in a portion of the feather that does appear to be red?

Charles Carter

Numerade Educator

Problem 69

The Very Large Array (VLA) is a set of 27 radio telescope dishes in Catron and Socorro counties, New Mexico (Fig. $\mathrm{P} 27.69$ ). The antennas can be moved apart on railroad tracks, and their combined signals give the resolving power of a synthetic aperture $36.0 \mathrm{km}$ in diameter. (a) If the detectors are tuned to a frequency of $1.40 \mathrm{GHz}$, what is the angular resolution of the VLA? (b) Clouds of interstellar hydrogen radiate at the frequency used in part (a). What must be the separation distance of two clouds at the center of the galaxy, 26000 light-years away, if they are to be resolved? (c) What If? As the telescope looks up, a circling hawk looks down. Assume the hawk is most sensitive to green light having a wavelength of $500 \mathrm{nm}$ and has a pupil of diameter $12.0 \mathrm{mm}$. Find the angular resolution of the hawk's eye. (d) A mouse is on the ground 30.0 $\mathrm{m}$ below. By what distance must the mouse's whiskers be separated if the hawk can resolve them?

Charles Carter

Numerade Educator

Problem 70

A pinhole camera has a small circular aperture of diameter
$D .$ Light from distant objects passes through the aperture into an otherwise dark box, falling on a screen located a distance $L$ away. If $D$ is too large, the display on the screen will be fuzzy because a bright point in the field of view will send light onto a circle of diameter slightly larger than $D$ On the other hand, if $D$ is too small, diffraction will blur the display on the screen. The screen shows a reasonably sharp image if the diameter of the central disk of the diffraction pattern, specified by Equation 27.17 , is equal to $D$ at the screen. (a) Show that for monochromatic light with plane wave fronts and $L \gg D,$ the condition for a sharp view is fulfilled if $D^{2}=2.44 \lambda L$. (b) Find the optimum pinhole diameter for 500 -nm light projected onto a screen $15.0 \mathrm{cm}$ away.

Charles Carter

Numerade Educator

Problem 71

Figure $\mathrm{CQ} 27.4$ 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?

Charles Carter

Numerade Educator