Physics

James S. Walker

Chapter 28

Physical Optics: Interference and Diffraction - all with Video Answers

Educators

Chapter Questions

Problem 1

Two sources emit waves that are coherent, in phase, and have wavelengths of $26.0 \mathrm{m}$. Do the waves interfere constructively or destructively at an observation point $78.0 \mathrm{m}$ from one source and $143 \mathrm{m}$ from the other source?

Steven Emmel

Steven Emmel

University of California - Los Angeles

Problem 2

Repeat Problem 1 for observation points that are (a) $91.0 \mathrm{m}$ and $221 \mathrm{m}$ and (b) $44.0 \mathrm{m}$ and $135 \mathrm{m}$ from the two sources.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 3

Two sources emit waves that are in phase with cach other. What is the longest wavelength that will give constructive interference at an observation point $161 \mathrm{m}$ from one source and $295 \mathrm{m}$ from the other source?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 4

A person driving at $17 \mathrm{m} / \mathrm{s}$ crosses the line connecting two radio transmitters at right angles, as shown in Figure $28-31$. The transmitters emit identical signals in phase with each other, which the driver receives on the car radio. When the car is at point $A$, the radio picks up a maximum net signal. (a) What is the longest possible wavelength of the radio waves? (b) How long after the car passes point A does the radio experience a minimum in the net signal? Assume that the wavelength has the value found in part (a).

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 5

Two students in a dorm room listen to a pure tone produced by two loudspeakers that are in phase. Students A and B in Figure $28-32$ hear a maximum sound. What is the lowest possible frequency of the loudspeakers?

Steven Emmel

University of California - Los Angeles

Problem 6

If the loudspeakers in Problem 5 are $180^{\circ}$ out of phase, determine whether a $185-\mathrm{Hz}$ tone heard at location $\mathrm{B}$ is a maximum or a minimum.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

A microphone is located on the line connecting two speakers that are $0.845 \mathrm{m}$ apart and oscillating in phase. The microphone is $2.55 \mathrm{m}$ from the midpoint of the two speakers. What are the lowest two frequencies that produce an interference maximum at the microphone's location?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 8

A microphone is located on the line connecting two speakers that are $0.845 \mathrm{m}$ apart and oscillating $180^{\circ}$ out of phase. The microphone is $2.25 \mathrm{m}$ from the midpoint of the two speakers. What are the lowest two frequencies that produce an interference maximum at the microphone's location?

Steven Emmel

University of California - Los Angeles

Problem 9

Moe, Larry, and Curly stand in a line with a spacing of $1.00 \mathrm{m} .$ Larry is $3.00 \mathrm{m}$ in front of a pair of sterco speakers $0.800 \mathrm{m}$ apart, as shown in Figure $28-33$. The speakers produce a single-frequency tone, vibrating in phase with each other. What are the two lowest frequencies that allow Larry to hear a loud tone while Moe and Curly hear very little?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 10

In Figure $28-33$ the two speakers emit sound that is $180^{\circ}$ out of phase and of a single frequency, $f$.
(a) Does Larry hear a sound intensity that is a maximum or a minimum? Does your answer depend on the frequency of the sound? Explain. (b) Find the lowest two frequencies that produce a maximum sound intensity at the positions of Moe and Curly.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

Suppose the car radio in Problem 4 picks up a minimum net signal at point A. (a) What is the largest possible value for the wavelength of the radio waves? (b) If the radio transmitters use a wavelength that is half the value found in part (a), will the car radio pick up a net signal at point A that is a maximum or a minimum? Explain.
(c) What is the second largest wavelength that will result in a minimum signal at point $\mathrm{A} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

Consider a two-slit interference pattern, with monochromatic light of wavelength $\lambda$. What is the path difference $\Delta \ell$ for (a) the fourth bright fringe and (b) the third dark fringe above the central bright fringe? Give your answers in terms of the wavelength of the light.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 13

(a) Does the path-length difference $\Delta \ell$ increase or decrease as you move from one bright fringe of a two-slit experiment to the next bright fringe farther out? (b) What is $\Delta \ell$ in terms of the wavelength $\lambda$ of the light?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

A two-slit experiment with red light produces a set of bright fringes. (a) Will the spacing between the fringes increase, decrease, or stay the same if the color of the light is changed to blue? (b) Choose the best explanation from among the following:
I. The spacing between the fringes will increase because blue light has a greater frequency than red light.
II. The fringe spacing decreases because blue light has a shorter wavelength than red light.
III. Only the wave property of light is important in producing the fringes, not the color of the light. Therefore the spacing stays the same.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

A two-slit experiment with blue light produces a set of bright fringes. Will the spacing between the fringes increase, decrease, or stay the same if (a) the separation of the slits is decreased, or (b) the experiment is immersed in water?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

Laser light with a wavelength $\lambda=670$ nm illuminates a pair of slits at normal incidence. What slit separation will produce firstorder maxima at angles of $\pm 35^{\circ}$ from the incident direction?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

Monochromatic light passes through two slits separated by a distance of $0.0334 \mathrm{mm}$. If the angle to the third maximum above the central fringe is $3.21^{\circ},$ what is the wavelength of the light?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 18

In Young's two-slit experiment, the first dark fringe above the central bright fringe occurs at an angle of $0.31^{\circ} .$ What is the ratio of the slit separation, $d$, to the wavelength of the light, $\lambda ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 19

A two-slit experiment with slits separated by $48.0 \times 10^{-5} \mathrm{m}$ produces a second-order maximum at an angle of $0.0990^{\circ}$ (a) Find the wavelength of the light used in this experiment. (b) If the slit separation is increased but the secondorder maximum stays at the same angle, does the wavelength increase, decrease, or stay the same? Explain. (c) Calculate the wavelength for a slit separation of $68.0 \times 10^{-5} \mathrm{m}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

A two-slit pattern is viewed on a screen $1.00 \mathrm{m}$ from the slits. If the two third-order minima are $22.0 \mathrm{cm}$ apart, what is the width (in $\mathrm{cm}$ ) of the central bright fringe?

Steven Emmel

University of California - Los Angeles

Problem 21

Light from a He-Ne laser $(\lambda=632.8 \mathrm{nm})$ strikes a pair of slits at normal incidence, forming a double-slit interference pattern on a screen located $1.40 \mathrm{m}$ from the slits. Figure $28-34$ shows the interference pattern observed on the screen. What is the slit separation?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

Light with a wavelength of 546 nm passes through two slits and forms an interference pattern on a screen $8.75 \mathrm{m}$ away. If the linear distance on the screen from the central fringe to the first bright fringe above it is $5.36 \mathrm{cm},$ what is the separation of the slits?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 23

A set of parallel slits for optical interference can be made by holding two razor blades together (carefully!) and scratching a pair of lines on a glass microscope slide that has been painted black. When monochromatic light strikes these slits at normal incidence, an interference pattern is formed on a distant screen. The thickness of each razor blade used to make the slits is $0.230 \mathrm{mm}$, and the screen is $2.50 \mathrm{m}$ from the slits. If the centerto-center separation of the fringes is $7.15 \mathrm{mm},$ what is the wavelength of the light?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 24

Suppose the interference pattern shown in Figure $28-34$ is produced by monochromatic light passing through two slits, with a separation of $135 \mu \mathrm{m},$ and onto a screen $1.20 \mathrm{m}$ away.
(a) What is the wavelength of the light? (b) If the frequency of this light is increased, will the bright spots of the pattern move closer together or farther apart? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 25

A physics instructor wants to produce a double-slit interference pattern large enough for her class to see. For the size of the room, she decides that the distance between successive bright fringes on the screen should be at least $2.50 \mathrm{cm}$. If the slits have a separation $d=0.0220 \mathrm{mm},$ what is the minimum distance from the slits to the screen when $632.8-\mathrm{nm}$ light from a He-Ne laser is used?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

When green light $(\lambda=505 \mathrm{nm})$ passes through a pair of double slits, the interference pattern shown in Flgure $28-35$ (a) is observed. When light of a different color passes through the same pair of slits, the pattern shown in Figure $28-35$ (b) is observed.
(a) Is the wavelength of the second color longer or shorter than 505 nm? Explain.
(b) Find the wavelength of the second color. (Assume that the angles involved are small enough to set $\sin \theta \approx \tan \theta .)$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 27

The interference pattern shown in Figure $28-35$ (a) is produced by green light with a wavelength of $\lambda=505$ nm passing through two slits with a separation of $127 \mu \mathrm{m}$. After passing through the slits, the light forms a pattern of bright and dark spots on a screen located $1.25 \mathrm{m}$ from the slits.
(a) What is the distance between the two vertical, dashed lines in Figure $28-35$ (a)?
(b) If it is desired to produce a more tightly packed interference pattern, like the one shown in Figure $28-35$ (b), should the frequency of the light be increased or decreased? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

The oil film floating on water in the accompanying photo appears dark near the edges, where it is thinnest. Is the index of refraction of the oil greater than or less than that of the water? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

A soap bubble with walls 401 nm thick floats in air. If this bubble is illuminated perpendicularly with sunlight, what wavelength (and color) will be absent in the reflected light? Assume that the index of refraction of the soap film is 1.33 . (Refer to Example $25-3$ for the connection between wavelength and color.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 31

A soap film $(n=1.33)$ is 825 nm thick. White light strikes the film at normal incidence. What visible wavelengths will be constructively reflected if the film is surrounded by air on both sides? (Refer to Example $25-3$ for the range of visible wavelengths.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 32

White light is incident on a soap film $(n=1.30)$ in air. The reflected light looks bluish because the red light $(\lambda=670 \mathrm{nm})$ is absent in the reflection. What is the minimum thickness of the soap film?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

A 742 -nm-thick soap film $\left(n_{\text {film }}=1.33\right.$ ) rests on a glass plate $\left(n_{\text {glass }}=1.52\right) .$ White light strikes the film at normal incidence. What visible wavelengths will be constructively reflected from the film? (Refer to Example $25-3$ for the range of visible wavelengths.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 34

An oil film $(n=1.38)$ floats on a water puddle. You notice that green light $(\lambda=521 \mathrm{nm})$ is absent in the reflection. What is the minimum thickness of the oil film?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 35

A radio broadcast antenna is $36.00 \mathrm{km}$ from your house. Suppose an airplane is flying $2.230 \mathrm{km}$ above the line connecting the broadcast antenna and your radio, and that waves reflected from the airplane travel 88.00 wavelengths farther than waves that travel directly from the antenna to your house.
(a) Do you observe constructive or destructive interference between the direct and reflected waves? (Hint: Does a phase change occur when the waves are reflected?) (b) The situation just described occurs when the plane is above a point on the ground that is two-thirds of the way from the antenna to your house. What is the wavelength of the radio waves?

Steven Emmel

University of California - Los Angeles

Problem 36

Newton's Rings Monochromatic light with $\lambda=648 \mathrm{nm}$ shines down on a plano-convex lens lying on a piece of plate glass, as shown in Figure $28-37$. When vicwed from above, one sees a set of concentric dark and bright fringes, referred to as Newton's rings (See Figure $28-11$ for a photo of Newton's rings.). (a) If the radius of the twelfth dark ring from the center is measured to be $1.56 \mathrm{cm},$ what is the radius of curvature, $R,$ of the lens? (b) If light with a longer wavelength is used with this system, will the radius of the twelfth dark ring be greater than or less than $1.56 \mathrm{cm}$ ? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

Light is incident from above on two plates of glass, separated on both ends by small wires of diameter $d=0.600 \mu \mathrm{m}$ Considering only interference between light reflected from the bottom surface of the upper plate and light reflected from the upper surface of the lower plate, state whether the following wavelengths give constructive or destructive interference:
(a) $\lambda=600.0 \mathrm{nm}$
(b) $\lambda=800.0 \mathrm{nm}$
(c) $\lambda=343.0 \mathrm{nm}$

Steven Emmel

University of California - Los Angeles

Problem 38

(a) What is the minimum soap-film thickness $(n=1.33)$ in air that will produce constructive interference in reflection for $\operatorname{red}(\lambda=652 \mathrm{nm})$ light?
(b) Which visible wavelengths will destructively interfere when reflected from this film? (Refer to Example $25-3$ for the range of visible wavelengths.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 39

A thin layer of magnesium fluoride $(n=1.38)$ is used to (a) What thickness should the coat a flint-glass lens $(n=1.61)$. magnesium fluoride film have if the reflection of $565-\mathrm{nm}$ light is to be suppressed? Assume that the light is incident at right angles to the film. (b) If it is desired to suppress the reflection of light with a higher frequency, should the coating of magnesium fluoride be made thinner or thicker? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 40

White light is incident normally on a thin soap film $(n=1.33)$ suspended in air. (a) What are the two minimum thicknesses that will constructively reflect yellow $(\lambda=590 \mathrm{nm})$ light? (b) What are the two minimum thicknesses that will destructioely reflect yellow $(\lambda=590 \mathrm{nm})$ light?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 41

A thin coating $(t=340.0 \mathrm{nm}, n=1.480)$ is placed on a glass lens. Which visible $(400 \mathrm{nm}<\lambda<700 \mathrm{nm})$ wavelength(s) will be absent in the reflected beam if (a) the glass has an index of refraction $n=1.350,$ and $(b)$ the glass has an index of refraction $n=1.675 ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 42

Two glass plates are separated by fine wires with diameters $d_{1}=0.0500 \mathrm{mm}$ and $d_{2}=0.0520 \mathrm{mm},$ as indicated in Figure $28-38 .$ The wires are parallel and separated by a distance of $7.00 \mathrm{cm}$. If monochromatic light with $\lambda=589 \mathrm{nm}$ is incident from above, what is the distance (in $\mathrm{cm}$ ) between adjacent dark bands in the reflected light? (Consider interference only between light reflected from the bottom surface of the upper plate and light reflected from the upper surface of the lower plate.)

Steven Emmel

University of California - Los Angeles

Problem 43

A single-slit diffraction pattern is formed on a distant screen. Assuming the angles involved are small, by what factor will the width of the central bright spot on the screen change if
(a) the wavelength is doubled,
(b) the slit width is doubled, or
(c) the distance from the slit to the screen is doubled?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 44

What width single slit will produce first-order diffraction minima at angles of $\pm 23^{\circ}$ from the central maximum with $690-\mathrm{nm}$ light?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

Diffraction also occurs with sound waves. Consider $1300-\mathrm{Hz}$ sound waves diffracted by a door that is $84 \mathrm{cm}$ wide. What is the angle between the two first-order diffraction minima?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

Green light $(\lambda=546 \mathrm{nm})$ strikes a single slit at normal incidence. What width slit will produce a central maximum that is $2.50 \mathrm{cm}$ wide on a screen $1.60 \mathrm{m}$ from the slit?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 47

Light with a wavelength of $676 \mathrm{nm}$ passes through a slit $7.64 \mu \mathrm{m}$ wide and falls on a screen $1.85 \mathrm{m}$ away. Find the linear distance on the screen from the central bright fringe to the first bright fringe above it.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 48

Repeat Problem $47,$ only this time find the distance on the screen from the central bright fringe to the third dark fringe above it.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 49

A single slit is illuminated with 610 -nm light, and the resulting diffraction pattern is viewed on a screen $2.3 \mathrm{m}$ away. (a) If the linear distance between the first and second dark fringes of the pattern is $12 \mathrm{cm},$ what is the width of the slit? (b) If the slit is made wider, will the distance between the first and second dark fringes increase or decrease? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

How many dark fringes will be produced on either side of the central maximum if green light $(\lambda=553 \mathrm{nm})$ is incident on a slit that is $8.00 \mu \mathrm{m}$ wide?

Steven Emmel

University of California - Los Angeles

Problem 51

The diffraction pattern shown in Figure $28-39$ is produced by passing He-Ne laser light $(\lambda=632.8 \mathrm{nm})$ through a single slit and viewing the pattern on a screen $1.50 \mathrm{m}$ behind the slit.
(a) What is the width of the slit? (b) If monochromatic yellow light with a wavelength of $591 \mathrm{nm}$ is used with this slit instead, will the distance indicated in Figure $28-39$ be greater than or less than $15.2 \mathrm{cm}$ ? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 52

A screen is placed $1.00 \mathrm{m}$ behind a single slit. The central maximum in the resulting diffraction pattern on the screen is $1.60 \mathrm{cm}$ wide-that is, the two first-order diffraction minima are separated by $1.60 \mathrm{cm}$. What is the distance between the two second-order minima?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 53

Predict/Explain (a) In principle, do your eyes have greater resolution on a dark cloudy day or on a bright sunny day? (b) Choose the best explanation from among the following:
I. Your eyes have greater resolution on a cloudy day because your pupils are open wider to allow more light to enter the eye.
II. Your eyes have greater resolution on a sunny day because the bright light causes your pupil to narrow down to a a smaller opening.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 54

Is resolution greater with blue light or red light, all other factors being equal? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 55

Two point sources of light are separated by $5.5 \mathrm{cm}$. As viewed through a $12-\mu \mathrm{m}$ -diameter pinhole, what is the maximum distance from which they can be resolved
(a) if red light $(\lambda=690 \mathrm{nm})$ is used, or
(b) if violet light $(\lambda=420 \mathrm{nm})$ is used?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 56

A spy camera is said to be able to read the numbers on a car's license plate. If the numbers on the plate are $5.0 \mathrm{cm}$ apart, and the spy satellite is at an altitude of $160 \mathrm{km},$ what must be the diameter of the camera's aperture? (Assume light with a wavelength of $550 \mathrm{nm} .$ )

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

Splitting Binary Stars As seen from Earth, the red dwarfs Kruger $60 \mathrm{A}$ and $\mathrm{Kruger} 60 \mathrm{B}$ form a binary star system with an angular separation of 2.5 arc seconds. What is the smallest diameter telescope that could theoretically resolve these stars using $550-\mathrm{nm}$ light? $\left(\right.$ Note: 1 arc $\left.\sec =1 / 3600^{\circ}\right)$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 58

Find the minimum aperture diameter of a camera that can resolve detail on the ground the size of a person $(2.0 \mathrm{m})$ from an SR-71 Blackbird airplane flying at an altitude of $27 \mathrm{km}$. (Assume light with a wavelength of $450 \mathrm{nm}$.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 59

The Resolution of Hubble The Hubble Space Telescope (HST) orbits Earth at an altitude of $613 \mathrm{km}$. It has an objective mirror that is $2.4 \mathrm{m}$ in diameter. If the $\mathrm{HST}$ were to look down on Earth's surface (rather than up at the stars), what is the minimum separation of two objects that could be resolved using $550-\mathrm{nm}$ light? [Note: The HST is used only for astronomical work, but a (classified) number of similar telescopes are in orbit for spy purposes.]

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 60

A lens that is "optically perfect" is still limited by diffraction effects. Suppose a lens has a diameter of $120 \mathrm{mm}$ and a focal length of $640 \mathrm{mm}$. (a) Find the angular width (that is, the angle from the bottom to the top) of the central maximum in the diffraction pattern formed by this lens when illuminated with $540-\mathrm{nm}$ light. (b) What is the linear width (diameter) of the central maximum at the focal distance of the lens?

Steven Emmel

University of California - Los Angeles

Problem 61

The resolution of a telescope is ultimately limited by the diameter of its objective lens or mirror. A typical amateur astronomer's telescope may have a 6.0 -in.-diameter mirror. (a) What is the minimum angular separation (in arc seconds) of two stars that can be resolved with a 6.0 -in. scope? (Take $\lambda$ to be at the center of the visible spectrum, about $550 \mathrm{nm},$ and see Problem 57 for the definition of an arc second.) (b) What is the minimum distance (in $\mathrm{km}$ ) between two points on the Moon's surface that can be resolved by a 6.0 -in. scope? (Note: The average distance from Earth to the Moon is $384,400 \mathrm{km}$.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 62

Early cameras were little more than a box with a pinhole on the side opposite the film. (a) What angular resolution would you expect from a pinhole with a $0.50-\mathrm{mm}$ diameter? (b) What is the greatest distance from the camera at which two point objects $15 \mathrm{cm}$ apart can be resolved? (Assume light with a wavelength of $520 \mathrm{nm} .$ )

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 63

A grating has 787 lines per centimeter. Find the angles of the first three principal maxima above the central fringe when this grating is illuminated with $655-\mathrm{nm}$ light.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 64

Suppose you want to produce a diffraction pattern with X-rays whose wavelength is $0.030 \mathrm{nm}$. If you use a diffraction grating, what separation between lines is needed to generate a pattern with the first maximum at an angle of $14^{\circ}$ ? (For comparison, a typical atom is a few tenths of a nanometer in diameter.)

Steven Emmel

University of California - Los Angeles

Problem 65

A diffraction grating has 2200 lines/cm. What is the angle between the first-order maxima for red light $(\lambda=680 \mathrm{nm})$ and blue light $(\lambda=410 \mathrm{nm}) ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

A diffraction grating with 345 lines/mm is $1.00 \mathrm{m}$ in front of a screen. What is the wavelength of light whose first-order maxima will be $16.4 \mathrm{cm}$ from the central maximum on the screen?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 67

The yellow light from a helium discharge tube has a wavelength of $587.5 \mathrm{nm}$. When this light illuminates a certain diffraction grating it produces a first-order principal maximum at an angle of $1.250^{\circ} .$ Calculate the number of lines per centimeter on the grating.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 68

The second-order maximum produced by a diffraction grating with 560 lines per centimeter is at an angle of $3.1^{\circ}$. (a) What is the wavelength of the light that illuminates the grating? (b) If a grating with a larger number of lines per centimeter is used with this light, is the angle of the second-order maximum greater than or less than $3.1^{\circ}$ ? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 69

White light strikes a grating with 7600 lines/cm at normal incidence. How many complete visible spectra will be formed on either side of the central maximum? (Refer to Example $25-3$ for the range of visible wavelengths.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

White light strikes a diffraction grating (890 lines/mm) at normal incidence. What is the highest-order visible maximum that is formed? (Refer to Example $25-3$ for the range of visible wavelengths.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

White light strikes a diffraction grating $(760$ lines/mm) at normal incidence. What is the longest wavelength that forms a second-order maximum?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 72

A light source emits two distinct wavelengths $\left[\lambda_{1}=430 \mathrm{nm}\right.$ (violet); $\lambda_{2}=630 \mathrm{nm}$ (orange)]. The light strikes a diffraction grating with 450 lines/mm at normal incidence. Identify the colors of the first eight interference maxima on either side of the central maximum.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

A laser emits two wavelengths $\left(\lambda_{1}=420 \mathrm{nm} ; \lambda_{2}=630 \mathrm{nm}\right)$. When these two wavelengths strike a grating with 450 lines/mm, they produce maxima (in different orders) that coincide. (a) What is the order $(m)$ of each of the two overlapping lines? (b) At what angle does this overlap occur?

Steven Emmel

University of California - Los Angeles

Problem 74

When blue light with a wavelength of 465 nm illuminates a diffraction grating, it produces a first-order principal maximum but no second-order maximum. (a) Explain the absence of higher-order principal maxima. (b) What is the maximum spacing between lines on this grating?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 75

Monochromatic light strikes a diffraction grating at normal incidence before illuminating a screen $2.10 \mathrm{m}$ away. If the firstorder maxima a re separated by $1.53 \mathrm{m}$ on the screen, what is the distance between the two second-order maxima?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 76

A diffraction grating with a slit separation $d$ is illuminated by a beam of monochromatic light of wavelength $\lambda$. The diffracted beam is observed at an angle $\phi$ relative to the incident direction. If the plane of the grating bisects the angle between the incident and diffracted beams, show that the $m$ th maximum will be observed at an angle that satisfies the relation $m \lambda=2 d \sin (\phi / 2),$ with $m=0,\pm 1,\pm 2, \ldots$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 77

Monochromatic light with a wavelength $\lambda$ passes through a single slit of width $W$ and forms a diffraction pattern of alternating bright and dark fringes. (a) If the width of the slit is decreased, do the dark fringes move outward or inward? Explain. (b) What width is necessary for the first dark fringe to move outward to infinity? Give your answer in terms of $\lambda$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 78

(a) If a thin liquid film floating on water has an index of refraction less than that of water, will the film appear bright or dark in reflected light as its thickness goes to zero? (b) Choose the best explanation from among the following: I. The film will appear bright because as the thickness of the film goes to zero the phase difference for reflected rays goes to zero. II. The film will appear dark because there is a phase change at both interfaces, and this will cause destructive interference of the reflected rays.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 79

If the index of refraction of an eye could be magically reduced, would the eye's resolution increase or decrease? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 80

In order to increase the resolution of a camera, should its $f$ -number be increased or decreased? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 81

Diffraction effects often involve small angles, and we usually make the approximation $\sin \theta=\tan \theta .$ To see how accurate this approximation is, complete the following table.

Mayukh Banik

Numerade Educator

Problem 82

When reading the printout from a laser printer, you are actually looking at an array of tiny dots. If the pupil of your eye is $4.3 \mathrm{mm}$ in diameter when reading a page held $28 \mathrm{cm}$ from your eye, what is the minimum separation of adjacent dots that can be resolved? (Assume light with a wavelength of $540 \mathrm{nm},$ and use 1.36 as the index of refraction for the interior of the eye.)

Steven Emmel

University of California - Los Angeles

Problem 83

- The headlights of a pickup truck are $1.32 \mathrm{m}$ apart. What is the greatest distance at which these headlights can be resolved as separate points of light on a photograph taken with a camera whose aperture has a diameter of $12.5 \mathrm{mm} ?$ (Take $\lambda=555 \mathrm{nm} .)$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 84

Antireflection Coating A glass lens $\left(n_{\text {glass }}=1.52\right.$ ) has an antireflection coating of $\mathrm{MgF}_{2}(n=1.38)$ (a) For 517 -nm light, what minimum thickness of $\mathrm{MgF}_{2}$ will cause the reflected rays $\mathrm{R}_{2}$ and $R_{4}$ in Figure $28-40$ to interfere destructively, assuming normal incidence? (b) Interference will also occur between the forwardmoving rays $R_{1}$ and $R_{3}$ in Figure $28-40$. What minimum thickness of $\mathrm{MgF}_{2}$ will cause these two rays to interfere constructively?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 85

White light reflected at normal incidence from a soap bubble $(n=1.33)$ in air produces an interference maximum at $\lambda=575 \mathrm{nm}$ but no interference minima in the visible spectrum.
(a) Explain the absence of interference minima in the visible.
(b) What are the possible thicknesses of the soap film? (Refer to Example $25-3$ for the range of visible wavelengths.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 86

A thin film of oil $(n=1.30)$ floats on water $(n=1.33) .$ When sunlight is incident at right angles to this film, the only colors that are enhanced by reflection are blue $(458 \mathrm{nm})$ and red $(687 \mathrm{nm})$. Estimate the thickness of the oil film.

Steven Emmel

University of California - Los Angeles

Problem 87

The yellow light of sodium, with wavelengths of 588.99 nm and $589.59 \mathrm{nm},$ is normally incident on a grating with 494 lines/cm. Find the linear distance between the first-order maxima for these two wavelengths on a screen $2.55 \mathrm{m}$ from the grating.

Steven Emmel

University of California - Los Angeles

Problem 88

A thin soap film $(n=1.33)$ suspended in air has a uniform thickness. When white light strikes the film at normal incidence, violet light $\left(\lambda_{\mathrm{V}}=420 \mathrm{nm}\right)$ is constructively reflected.
(a) If we would like green light $\left(\lambda_{G}=560 \mathrm{nm}\right)$ to be constructively reflected, instead, should the film's thickness be increased or decreased?
(b) Find the new thickness of the film. (Assume the film has the minimum thickness that can produce these reflections.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 89

A thin film of oil $(n=1.40)$ floats on water $(n=1.33)$. When sunlight is incident at right angles to this film, the only colors that are absent from the reflected light are blue $(458 \mathrm{nm})$ and red $(687 \mathrm{nm})$. Estimate the thickness of the oil film.

Narayan Hari

Numerade Educator

Problem 90

Sodium light, with a wavelength of $\lambda=589 \mathrm{nm},$ shines downward onto the system shown in Figure $28-37$. When viewed from above, you see a series of concentric circles known as Newton's rings.
(a) Do you expect a bright or a dark spot at the center of the pattern? Explain. (b) If the radius of curvature of the plano-convex lens is $R=26.1 \mathrm{m},$ what is the radius of the tenth-largest dark ring? (Only rings of nonzero radius will be counted as "rings."

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 91

Figure $28-39$ shows a single-slit diffraction pattern formed by light passing through a slit of width $W=11.2 \mu \mathrm{m}$ and illuminating a screen 0.855 m behind the slit. (a) What is the wavelength of the light? (b) If the width of the slit is decreased, will the distance indicated in Figure $28-39$ be greater than or less than $15.2 \mathrm{cm}$ ? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 92

Images produced by structures within the eye (like lens fibers or cell fragments) are referred to as entoptic images. These images can sometimes take the form of "halos" around a bright light seen against a dark background. The halo in such a case is actually the bright outer rings of a circular diffraction pattern, like Figure $28-21$, with the central bright spot not visible because it overlaps the direct image of the light. Find the diameter of the eye structure that causes a circular diffraction pattern with the first dark ring at an angle of $3.7^{\circ}$ when viewed with monochromatic light of wavelength $630 \mathrm{nm} .$ (Typical eye structures of this type have diameters on the order of $10 \mu \mathrm{m}$. Also, the index of refraction of the vitreous humor is $1.336 .$ )

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 93

White light is incident on a soap film $(n=1.33$, thickness $=800.0 \mathrm{nm}$ ) suspended in air. If the incident light makes a $45^{\circ}$ angle with the normal to the film, what visible wavelength(s) will be constructively reflected? (Refer to Example $25-3$ for the range of visible wavelengths.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 94

A system like that shown in Figure $28-26$ consists of $N$ slits, each transmitting light of intensity $1_{0} .$ The light from each slit has the same phase and the same wavelength. The net in tensity 1 observed at an angle $\theta$ due to all $N$ slits is $$ I=I_{0}\left[\frac{\sin (N \phi / 2)}{\sin (\phi / 2)}\right]^{2} $$ In this expression, $\phi=(2 \pi d / \lambda) \sin \theta,$ where $\lambda$ is the wavelength of the light. (a) Show that the intensity in the limit $\theta \rightarrow 0$ is $l=N^{2} I_{0}$. This is the maximum intensity of the interference pattern. (b) Show that the first points of zero intensity on either side of $\theta=0$ occur at $\phi=2 \pi / N$ and $\phi=-2 \pi / N .$ (c) Does the central maximum $(\theta=0)$ of this pattern become narrower or broader as the number of slits is increased? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 95

Two plates of glass are separated on both ends by small wires of diameter d. Derive an expression for the condition for constructive interference when light of wavelength $\lambda$ is incident normally on the plates. Consider only interference between waves reflected from the bottom of the top plate and the top of the bottom plate.

Steven Emmel

University of California - Los Angeles

Problem 96

A curved piece of glass with a radius of curvature $R$ rests on a flat plate of glass. Light of wavelength $\lambda$ is incident normally on this system. Considering only interference between waves reflected from the curved (lower) surface of glass and the top surface of the plate, show that the radius of the $nth$ dark ring is $$
r_{n}=\sqrt{n \lambda R-n^{2} \lambda^{2} / 4} $$

Steven Emmel

University of California - Los Angeles

Problem 97

The resolution of the eye is ultimately limited by the pupil diameter. What is the smallest diameter spot the eye can produce on the retina if the pupil diameter is $4.25 \mathrm{mm}$ ? Assume light with a wavelength of $\lambda=550 \mathrm{nm} .$ (Note: The distance from the pupil to the retina is $25.4 \mathrm{mm} .$ In addition, the space between the pupil and the retina is filled with a fluid whose index of refraction is $n=1.36 .$ )

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 98

What is the minimum angle your eye can resolve, according to the Rayleigh criterion and the above assumptions?
A. $0.862 \times 10^{-4} \mathrm{rad}$
B. $1.05 \times 10^{-4}$ rad
C. $1.43 \times 10^{-4}$ rad
D. $1.95 \times 10^{-4} \mathrm{rad}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 99

What is the linear separation between horizontal lines on the screen?
A. $0.0235 \mathrm{mm}$
B. $0.145 \mathrm{mm}$
$\mathrm{C} \cdot 0.369 \mathrm{mm}$
D. $0.926 \mathrm{mm}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 100

- What is the angular separation of the horizontal lines as viewed from a distance of 12.0 feet?
A. $1.01 \times 10^{-4} \mathrm{rad}$
B. $2.53 \times 10^{-4} \mathrm{rad}$
C. $2.56 \times 10^{-4} \mathrm{rad}$
D. $12.1 \times 10^{-4} \mathrm{rad}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 101

According to the Rayleigh criterion, what is the closest you can be to the TV screen before resolving the individual horizontal lines? (In practice you can be considerably closer than this distance before resolving the lines.)
A. $3.51 \mathrm{ft}$
B. $4.53 \mathrm{ft}$
$\mathrm{C} .11 .5 \mathrm{ft}$
D. $14.0 \mathrm{ft}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 102

Referring to Example $28-2$ Suppose we change the slit separation to a value other than $8.5 \times 10^{-5} \mathrm{m},$ with the result that the linear distance to the tenth bright fringe above the central bright fringe increases from $12 \mathrm{cm}$ to $18 \mathrm{cm}$. The screen is still $2.3 \mathrm{m}$ from the slits, and the wavelength of the light is $440 \mathrm{nm}$ (a) Did we increase or decrease the slit separation? Explain. (b) Find the new slit separation.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 103

Referring to Example $28-2$ The wavelength of the light is changed to a value other than $440 \mathrm{nm},$ with the result that the linear distance to the secenth bright fringe above the central bright fringe is $12 \mathrm{cm}$. The screen is still $2.3 \mathrm{m}$ from the slits, and the slit separation is $8.5 \times 10^{-5} \mathrm{m}$. (a) Is the new wavelength longer or shorter than $440 \mathrm{nm} ?$ Explain. (b) Find the new wavelength.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 104

Referring to Example $28-5$ The light used in this experiment has a wavelength of $511 \mathrm{nm}$. (a) If the width of the slit is decreased, will the angle to the first dark fringe above the central bright fringe increase or decrease? Explain. (b) Find the angle to the first dark fringe if the reduced slit width is $1.50 \times 10^{-6} \mathrm{m}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 105

Referring to Example $28-5$ The width of the slit in this experiment is $2.20 \times 10^{-6} \mathrm{m}$. (a) If the frequency of the light is decreased, will the angle to the first dark fringe above the central bright fringe increase or decrease? Explain. (b) Find the angle to the first dark fringe if the reduced frequency is $5.22 \times 10^{17} \mathrm{Hz}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator