University Physics with Modern Physics

Hugh D. Young

Chapter 36

Diffraction - all with Video Answers

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Chapter Questions

Problem 1

Monochromatic light from a distant source is incident on a slit 0.750 $\mathrm{mm}$ wide. On a screen 2.00 $\mathrm{m}$ away, the distance from the central maximum of the diffraction pattern to the first minimum is measured to be 1.35 $\mathrm{mm}$ . Calculate the wavelength of the light.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 2

Parallel rays of green mercury light with a wavelength of 546 $\mathrm{nm}$ pass through a slit covering a lens with a focal length of 60.0 $\mathrm{cm} .$ In the focal plane of the lens the distance from the central maximum to the first minimum is 10.2 $\mathrm{mm} .$ What is the width of the slit?

Donald Albin

Numerade Educator

Problem 3

Light of wavelength 585 nm falls on a slit 0.0666 $\mathrm{mm}$ wide. (a) On a very large and distant screen, how many totally dark fringes (indicating complete cancellation) will there be, including
both sides of the central bright spot? Solve this problem without calculating all the angles! (Hint: What is the largest that sin $\theta$ can be? What does this tell you is the largest that $m$ can be? (b) At
what angle will the dark fringe that is most distant from the central bright fringe occur?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 4

Light of wavelength 633 $\mathrm{nm}$ from a distant source is incident on a slit 0.750 $\mathrm{mm}$ wide, and the resulting diffraction pattern is observed on a screen 3.50 $\mathrm{m}$ away. What is the distance between the two dark fringes on either side of the central bright fringe?

Haoran Sun

Kent State University

Problem 5

Diffraction occurs for all types of waves, including sound waves. High-frequency sound from a distant source with wave- length 9.00 $\mathrm{cm}$ passes through a slit 12.0 $\mathrm{cm}$ wide. A microphone- is placed 8.00 $\mathrm{m}$ directly in front of the center of the slit, corresponding to point $O$ in Fig. 36.5 $\mathrm{a}$ . The microphone is then moved in a direction perpendicular to the line from the center of the slit to point $O .$ At what distances from $O$ will the intensity detected by
the microphone be zero?

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Numerade Educator

Problem 6

On December $26,2004,$ a violent earthquake of magnitude 9.1 occurred off the coast of Sumatra. This quake triggered a huge tsunami (similar to a tidal wave) that killed more than $150,000$ people. Scientists observing the wave on the open ocean measured the time between crests to be 1.0 $\mathrm{h}$ and the speed of the wave to be 800 $\mathrm{km} / \mathrm{h}$ . Computer models of the evolution of this enormous wave showed that it bent around the continents and spread to all the oceans of the earth. When the wave reached the gaps between continents, it diffracted between them as through a slit. (a) What was the wavelength of this tsunami? (b) The distance between the southern tip of Africa and northern Antarctica is about $4500 \mathrm{km},$ while the distance between the southern end of Australia and Antarctica is about 3700 $\mathrm{km} .$ As an approximation, we can model this wave's behavior by using Fraunhofer diffraction. Find the smallest angle away from the central maximum for which the waves would cancel after going through each of these continental gaps.

Donald Albin

Numerade Educator

Problem 7

A series of parallel linear water wave fronts are traveling directly toward the shore at 15.0 $\mathrm{cm} / \mathrm{s}$ on an otherwise placid lake. A long concrete barrier that runs parallel to the shore at a distance of 3.20 $\mathrm{m}$ away has a hole in it. You count the wave crests and observe that 75.0 of them pass by each minute, and you also observe that no waves reach the shore at $\pm 61.3 \mathrm{cm}$ from the point directly opposite the hole, but waves do reach the shore every- where within this distance. (a) How wide is the hole in the barrier? (b) At what other angles do you find no waves hitting the shore?

Donald Albin

Numerade Educator

Problem 8

Monochromatic electromagnetic radiation with wavelength $\lambda$ from a distant source passes through a slit. The diffraction pattern is observed on a screen 2.50 $\mathrm{m}$ from the slit. If the width of the central maximum is 6.00 $\mathrm{mm}$ , what is the slit width $a$ if the wavelength
is (a) 500 $\mathrm{nm}$ (visible light); (b) 50.0$\mu \mathrm{m}$ (infrared radiation); (c) 0.500 $\mathrm{nm}(\mathrm{x}$ rays)?

Haoran Sun

Kent State University

Problem 9

Doorway Diffraction. Sound of frequency 1250 $\mathrm{Hz}$ leaves a room through a $1.00-\mathrm{m}$ -wide doorway (see Exercise 36.5$)$ . At which angles relative to the centerline perpendicular to the doorway will someone outside the room hear no sound? Use 344 $\mathrm{m} / \mathrm{s}$ for the speed of sound in air and assume that the source and listener are both far enough from the doorway for Fraunhofer diffraction to apply. You can ignore effects of reflections.

Donald Albin

Numerade Educator

Problem 10

Light waves, for which the electric field is given by $E_{y}(x, t)=E_{\max } \sin \left[\left(1.20 \times 10^{7} \mathrm{m}^{-1}\right) x-\omega t\right],$ pass through a slit and produce the first dark bands at $\pm 28.6^{\circ}$ from the center of the diffraction pattern. (a) What is the frequency of this light? (b) How wide is the slit? (c) At which angles will other dark bands occur?

Donald Albin

Numerade Educator

Problem 11

Parallel rays of light with wavelength 620 $\mathrm{nm}$ pass through a slit covering a lens with a focal length of 40.0 $\mathrm{cm} .$ The diffraction pattern is observed in the focal plane of the lens, and
the distance from the center of the central maximum to the first minimum is 36.5 $\mathrm{cm} .$ What is the width of the slit? (Note: The angle that locates the first minimum is not small.)

Donald Albin

Numerade Educator

Problem 12

Red light of wavelength 633 nm from a helium-neon laser passes through a slit 0.350 $\mathrm{mm}$ wide. The diffraction pattern is observed on a screen 3.00 $\mathrm{m}$ away. Define the width of a bright fringe as the distance between the minima on either side. (a) What is the width of the central bright fringe? (b) What is the width of the first bright fringe on either side of the central one?

Bret Rosen

Numerade Educator

Problem 13

Monochromatic light of wavelength 580 nm passes through a single slit and the diffraction pattern is observed on a screen. Both the source and screen are far enough from the slit for Fraunhofer diffraction to apply.(a) If the first diffraction minima are at $\pm 90.0^{\circ},$ so the central maximum completely fills the screen, what is the width of the slit? (b) For the width of the slit as calculated in part (a), what is the ratio of the intensity at $\theta=45.0^{\circ}$ to the intensity at $\theta=0 ?$

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 14

Monochromatic light of wavelength $\lambda=620 \mathrm{nm}$ from a distant source passes through a slit 0.450 $\mathrm{mm}$ wide. The diffraction pattern is observed on a screen 3.00 $\mathrm{m}$ from the slit. In terms of the intensity $I_{0}$ at the peak of the central maximum, what is the intensity of the light at the screen the following distances from the center of the central maximum: (a) $1.00 \mathrm{mm} ;$ (b) 3.00 $\mathrm{mm}$ ; (c) 5.00 $\mathrm{mm}$ ?

Haoran Sun

Kent State University

Problem 15

A slit 0.240 $\mathrm{mm}$ wide is illuminated by parallel light rays of wavelength 540 $\mathrm{nm}$ . The diffraction pattern is observed on a screen that is 3.00 $\mathrm{m}$ from the slit. The intensity at the center of the central maximum $\left(\theta=0^{\circ}\right)$ is $6.00 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2} .$ (a) What is the distance on the screen from the center of the central maximum to the first minimum? (b) What is the intensity at a point on the screen midway between the center of the central maximum and the first minimum?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

Monochromatic light of wavelength 486 $\mathrm{nm}$ from a distant source passes through a slit that is 0.0290 $\mathrm{mm}$ wide. In the resulting diffraction pattern, the intensity at the center of the central maximum $\left(\theta=0^{\circ}\right)$ is $4.00 \times 10^{-5} \mathrm{W} / \mathrm{m}^{2} .$ What is the intensity at a point on the screen that corresponds to $\theta=1.20^{\circ} .$

Donald Albin

Numerade Educator

Problem 17

A single-slit diffraction pattern is formed by monochromatic electromagnetic radiation from a distant source passing through a slit 0.105 $\mathrm{mm}$ wide. At the point in the pattern $3.25^{\circ}$ .
from the center of the central maximum, the total phase difference between wavelets from the top and bottom of the slit is 56.0 rad. (a) What is the wavelength of the radiation? (b) What is the intensity at this point, if the intensity at the center of the central maxi-mum is $I_{0} ?$

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 18

Consider a single-slit diffraction experiment in which the amplitude of the wave at point $O$ in Fig. 36.5$a$ is $E_{0} .$ For each of the following cases, draw a phasor diagram like that in Fig. 36.8 $\mathrm{c}$ and determine graphically the amplitude of the wave at the point in question. (Hint: Use Eq. $(36.6)$ to determine the value of $\beta$ for each case.) Compute the intensity and compare to Eq. (36.5). (a) $\sin \theta=\lambda / 2 a ;(b) \sin \theta=\lambda / a ;(c) \sin \theta=3 \lambda / 2 a$

Donald Albin

Numerade Educator

Problem 19

Public Radio station KXPR-FM in Sacramento broad-casts at 88.9 $\mathrm{MHz}$ . The radio waves pass between two tall skyscrapers that are 15.0 $\mathrm{m}$ apart along their closest walls. (a) At what horizontal angles, relative to the original direction of the waves, will a distant antenna not receive any signal from this station? (b) If the maximum intensity is 3.50 $\mathrm{W} / \mathrm{m}^{2}$ at the antenna, what is the intensity at $\pm 5.00^{\circ}$ from the center of the central maximum at the distant antenna?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 20

Diffraction and Interference Combined. Consider the interference pattern produced by two parallel slits of width $a$ and separation $d,$ in which $d=3 a .$ The slits are illuminated by normally incident light of wavelength $\lambda$ . (a) First we ignore diffraction effects due to the slit width. At what angles $\theta$ from the central-maximum will the next four maxima in the two-slit interference pattern occur? Your answer will be in terms of $d$ and $\lambda .$ (b) Now we include the effects of diffraction. If the intensity at $\theta=0$ is $I_{0}$ , what is the intensity at each of the angles in part (a)? (c) Which double-slit interference maxima are missing in the pattern? (d) Compare your results to those illustrated in Fig. 36.12 $\mathrm{c}$ . In what ways is your result different?

Donald Albin

Numerade Educator

Problem 21

Number of Fringes in a Diffraction Maximum. In Fig. 36.12 $\mathrm{c}$ the central diffraction maximum contains exactly seven interference fringes, and in this case $d / a=4 .$ (a) What must the
ratio $d / a$ be if the central maximum contains exactly five fringes? (b) In the case considered in part (a), how many fringes are contained within the first diffraction maximum on one side of the central maximum?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

An interference pattern is produced by eight parallel and equally spaced, narrow slits. There is an interference minimum when the phase difference $\phi$ between light from adjacent slits is
$\pi / 4 .$ The phasor diagram is given in Fig. 36.14 $\mathrm{b} .$ For which pairs of slits is there totally destructive interference?

Donald Albin

Numerade Educator

Problem 23

An interference pattern is produced by light of wave- length 580 $\mathrm{nm}$ from a distant source incident on two identical parallel slits separated by a distance (between centers) of 0.530 $\mathrm{mm}$ . (a) If the slits are very narrow, what would be the angular positions of the first-order and second-order, two-slit, interference maxima? (b) Let the slits have width 0.320 $\mathrm{mm}$ . In terms of the intensity $I_{0}$ at the center of the central maximum, what is the intensity at each
of the angular positions in part (a)?

Donald Albin

Numerade Educator

Problem 24

Parallel rays of monochromatic light with wavelength 568 nm illuminate two identical slits and produce an interference pattern on a screen that is 75.0 $\mathrm{cm}$ from the slits. The centers of the
slits are 0.640 $\mathrm{mm}$ apart and the width of each slit is 0.434 $\mathrm{mm}$ . If the intensity at the center of the central maximum is $5.00 \times$ $10^{-4} \mathrm{W} / \mathrm{m}^{2},$ what is the intensity at a point on the screen that is 0.900 $\mathrm{mm}$ from the center of the central maximum?

Haoran Sun

Kent State University

Problem 25

An interference pattern is produced by four parallel and equally spaced, narrow slits. By drawing appropriate phasor diagrams, show that there is an interference minimum when the phase difference $\phi$ from adjacent slits is (a) $\pi / 2 ;$ (b) $\pi ;(\mathrm{c}) 3 \pi / 2$ . In each case, for which pairs of slits is there totally destructive interference?

Donald Albin

Numerade Educator

Problem 26

A diffraction experiment involving two thin parallel slits yields the pattern of closely spaced bright and dark fringes shown in Fig. E36.26. Only the central portion of the pattern is shown in the figure. The bright spots are equally spaced at 1.53 $\mathrm{mm}$ center to center (except for the missing spots) on a screen 2.50 $\mathrm{m}$ from the slits. The light source was a He-Ne laser producing a wavelength of 632.8 nm. (a) How far apart are the two slits? (b) How wide is each one?

SB

Sarah Brandsen

Numerade Educator

Problem 27

Laser light of wavelength 500.0 nm illuminates two identical slits, producing an interference pattern on a screen 90.0 $\mathrm{cm}$ from the slits. The bright bands are 1.00 $\mathrm{cm}$ apart, and the third bright bands on either side of the central maximum are missing in the pattern. Find the width and the separation of the two slits.

Shoukat Ali

Other Schools

Problem 28

Monochromatic light is at normal incidence on a plane transmission grating. The first-order maximum in the interference pattern is at an angle of $8.94^{\circ} .$ What is the angular position of the fourth-order maximum?

Yaqub Khan

Numerade Educator

Problem 29

If a diffraction grating produces its third-order bright band at an angle of $78.4^{\circ}$ for light of wavelength $681 \mathrm{nm},$ find (a) the number of slits per centimeter for the grating and (b) the angular location of the first-order and second-order bright bands. (c) Will there be a fourth-order bright band? Explain.

Nishant Kumar

Numerade Educator

Problem 30

If a diffraction grating produces a third-order bright spot for red light (of wavelength 700 $\mathrm{nm}$ ) at $65.0^{\circ}$ from the central maximum, at what angle will the second-order bright spot be for violet light (of wavelength 400 $\mathrm{nm} ) ?$

Haoran Sun

Kent State University

Problem 31

Visible light passes through a diffraction grating that has 900 slits/cm, and the interference pattern is observed on a screen that is 2.50 $\mathrm{m}$ from the grating. (a) Is the angular position of the
first-order spectrum small enough for $\sin \theta \approx \theta$ to be a good approximation? (b) In the first-order spectrum, the maxima for two different wavelengths are separated on the screen by 3.00 $\mathrm{mm}$ . What is the difference in these wavelengths?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 32

The wavelength range of the visible spectrum is approximately $380-750 \mathrm{nm} .$ White light falls at normal incidence on a diffraction grating that has 350 slits/mm. Find the angular width of
the visible spectrum in (a) the first order and (b) the third order. (Note: An advantage of working in higher orders is the greater angular spread and better resolution. A disadvantage is the overlapping of different orders, as shown in Example $36.4 . )$

Haoran Sun

Kent State University

Problem 33

When laser light of wavelength 632.8 nm passes through a diffraction grating, the first bright spots occur at $\pm 17.8^{\circ}$ from the central maximum. (a) What is the line density (in lines/cm) of this grating? (b) How many additional bright spots are there beyond the first bright spots, and at what angles do they occur?

Ryan Hood

Numerade Educator

Problem 34

(a) What is the wavelength of light that is deviated in the first order through an angle of $13.5^{\circ}$ by a transmission grating having 5000 slits/cm? (b) What is the second order deviation of this wavelength? Assume normal incidence.

Narendra Kumar

Numerade Educator

Problem 35

$\cdot$ Plane monochromatic waves with wavelength 520 $\mathrm{nm}$ are incident normally on a plane transmission grating having 350 slits/mm. Find the angles of deviation in the first, second, and third orders.

Donald Albin

Numerade Educator

Problem 36

Identifying Isotones hy Snectra Different isotones of the same element emit light at slightly different wavelengths. A wavelength in the emission spectrum of a hydrogen atom is 656.45 nm; for deuterium, the corresponding wavelength is 656.27 $\mathrm{nm} .$ (a) What minimum number of slits is required to resolve these two wavelengths in second order? (b) If the grating has 500.00 slits/mm, find the angles and angular separation of these two wavelengths in the second order.

Donald Albin

Numerade Educator

Problem 37

A typical laboratory diffraction grating has $5.00 \times$ $10^{3}$ lines $/ \mathrm{cm},$ and these lines are contained in a $3.50-\mathrm{cm}$ width of grating. (a) What is the chromatic resolving power of such a grating in the first order? (b) Could this grating resolve the lines of the sodium doublet (see Section 36.5 ) in the first order? (c) While doing $10^{3}$ lines $/ \mathrm{cm},$ and these lines are contained in a $3.50-\mathrm{cm}$ width of grating. (a) What is the chromatic resolving power of such a grating in the first order? (b) Could this grating resolve the lines of the
sodium doublet (see Section 36.5 ) in the first order? (c) While doing spectral analysis of a star, you are using this grating in the second order to resolve spectral lines that are very close to the 587.8002 -nm spectral line of iron. (i) For wavelengths longer than the iron line, what is the shortest wavelength you could distinguish from the iron line? (ii) For wavelengths shorter than the iron line, what is the longest wavelength you could distinguish from the iron line? (iii) What is the range of wavelengths you could not distinguish from the iron line?

Donald Albin

Numerade Educator

Problem 38

The light from an iron arc includes many different wavelengths. Two of these are at $\lambda=587.9782 \mathrm{nm}$ and $\lambda=$ 587.8002 $\mathrm{nm} .$ You wish to resolve these spectral lines in first order using a grating 1.20 $\mathrm{cm}$ in length. What minimum number of slits per centimeter must the grating have?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 39

$\mathrm{X}$ rays of wavelength 0.0850 $\mathrm{nm}$ are scattered from the atoms of a crystal. The second-order maximum in the Bragg reflection occurs when the angle $\theta$ in Fig. 36.22 is $21.5^{\circ} .$ What is the spacing between adjacent atomic planes in the crystal?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 40

If the planes of a crystal are 3.50$\hat{\mathrm{A}}$ (1 $\hat{\mathrm{A}}=10^{-10} \mathrm{m}=$ Angstrom unit) apart, (a) what wavelength of electromagnetic waves is needed so that the first strong interference maximum in the Bragg reflection occurs, when the waves strike the planes at an angle of $15.0^{\circ},$ and in what part of the electromagnetic spectrum do these waves lie? (See Fig. $32.4 . )$ (b) At what other angles will strong interference maxima occur?

Donald Albin

Numerade Educator

Problem 41

Monochromatic $\mathrm{x}$ rays are incident on a crystal for which the spacing of the atomic planes is 0.440 $\mathrm{nm} .$ The first-order maximum in the Bragg reflection occurs when the incident and reflected $x$ rays make an angle of $39.4^{\circ}$ with the crystal planes. What is the wavelength of the $\mathrm{x}$ rays?

Haoran Sun

Kent State University

Problem 42

If you can read the bottom row of your doctor's eye chart, your eye has a resolving power of 1 arcminute, equal to $\frac{1}{60}$ degree. If this resolving power is diffraction limited, to what effective diameter of your eye's optical system does this correspond? Use Rayleigh's criterion and assume $\lambda=550 \mathrm{nm} .$

Donald Albin

Numerade Educator

Problem 43

Two satellites at an altitude of 1200 $\mathrm{km}$ are separated by 28 km. If they broadcast $3.6-\mathrm{cm}$ microwaves, what minimum receiving-dish diameter is needed to resolve (by Rayleigh's criterion) the two transmissions?

Kelley Commeford

Kelley Commeford

Numerade Educator

Problem 44

The VLBA (Very Long Baseline Array) uses a number of individual radio telescopes to make one unit having an equivalent diameter of about 8000 $\mathrm{km} .$ When this radio telescope is
focusing radio waves of wavelength $2.0 \mathrm{cm},$ what would have to be the diameter of the mirror of a visible-light telescope focusing light of wavelength 550 $\mathrm{nm}$ so that the visible-light telescope has the same resolution as the radio telescope?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 45

Monochromatic light with wavelength 620 nm passes through a circular aperture with diameter 7.4$\mu \mathrm{m} .$ The resulting diffraction pattern is observed on a screen that is 4.5 $\mathrm{m}$ from the aperture. What is the diameter of the Airy disk on the screen?

Vishal Gupta

Numerade Educator

Problem 46

Photography. A wildlife photographer uses a moderate telephoto lens of focal length 135 $\mathrm{mm}$ and maximum aperture $f / 4.00$ to photograph a bear that is 11.5 $\mathrm{m}$ away. Assume the wavelength is 550 $\mathrm{nm}$ (a) What is the width of the smallest feature on
the bear that this lens can resolve if it is opened to its maximum aperture? (b) If, to gain depth of field, the photographer stops the lens down to $f / 22.0,$ what would be the width of the smallest resolvable feature on the bear?

Donald Albin

Numerade Educator

Problem 47

Observing Jupiter. You are asked to design a space telescope for earth orbit. When Jupiter is $5.93 \times 10^{8} \mathrm{km}$ away (its closest approach to the earth), the telescope is to resolve, by Rayleigh's criterion, features on Jupiter that are 250 $\mathrm{km}$ apart. What minimum- diameter mirror is required? Assume a wavelength of 500 $\mathrm{nm} .$

Mayukh Banik

Numerade Educator

Problem 48

A converging lens 7.20 $\mathrm{cm}$ in diameter has a focal length of 300 $\mathrm{mm} .$ If the resolution is diffraction limited, how far away can an object be if points on it 4.00 $\mathrm{mm}$ apart are to be resolved (according to Rayleigh's criterion)? Use $\lambda=550 \mathrm{nm} .$

Mayukh Banik

Numerade Educator

Problem 49

Hubble Versus Arecibo. The Hubble Space Telescope has an aperture of 2.4 $\mathrm{m}$ and focuses visible light $(380-750 \mathrm{nm}) .$ The Arecibo radio telescope in Puerto Rico is 305 $\mathrm{m}(1000 \mathrm{ft})$ in diameter (it is built in a mountain valley) and focuses radio waves of wavelength 75 $\mathrm{cm} .$ (a) Under optimal viewing conditions, what is the smallest crater that each of these telescopes could resolve on our moon? (b) If the Hubble Space Telescope were to be converted to surveillance use, what is the highest orbit above the surface of the earth it could have and still be able to resolve the license plate (not the letters, just the plate) of a car on the ground? Assume optimal viewing conditions, so that the resolution is diffraction limited.

Donald Albin

Numerade Educator

Problem 50

Searching for Starspots. The Hale Telescope on Palomar Mountain in California has a mirror 200 in. $(5.08 \mathrm{m})$ in diameter and it focuses visible light. Given that a large sunspot is about
$10,000$ mi in diameter, what is the most distant star on which this telescope could resolve a sunspot to see whether other stars have them? (Assume optimal viewing conditions, so that the resolution is diffraction limited.) Are there any stars this close to us, besides our sun?

Donald Albin

Numerade Educator

Problem 51

Thickness of Human Hair. Although we have discussed single-slit diffraction only for a slit, a similar result holds when light bends around a straight, thin object, such as a strand of hair. In that case, $a$ is the width of the strand. From actual laboratory measurements on a human hair, it was found that when a beam of light of wavelength 632.8 $\mathrm{nm}$ was shone on a single strand of hair, and the diffracted light was viewed on a screen 1.25 $\mathrm{m}$ away, the first dark fringes on either side of the central bright spot were 5.22 $\mathrm{cm}$ apart. How thick was this strand of hair?

Donald Albin

Numerade Educator

Problem 52

Suppose the entire apparatus (slit, screen, and space in between) in Exercise 36.4 is immersed in water $(n=1.333)$ Then what is the distance between the two dark fringes?

Donald Albin

Numerade Educator

Problem 53

Laser light of wavelength 632.8 nm falls normally on a slit that is 0.0250 $\mathrm{mm}$ wide. The transmitted light is viewed on a distant screen where the intensity at the center of the central bright
fringe is 8.50 $\mathrm{W} / \mathrm{m}^{2} .$ (a) Find the maximum number of totally dark
fringes on the screen, assuming the screen is large enough to show them all. (b) At what angle does the dark fringe that is most distant from the center occur? (c) What is the maximum intensity of the
bright fringe that occurs immediately before the dark fringe in part (b)? Approximate the angle at which this fringe occurs by assuming it is midway between the angles to the dark fringes on either side of it.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 54

A loudspeaker having a diaphragm that vibrates at 1250 $\mathrm{Hz}$ is traveling at 80.0 $\mathrm{m} / \mathrm{s}$ directly toward a pair of holes in a very large wall in a region for which the speed of sound is 344 $\mathrm{m} / \mathrm{s} .$ You observe that the sound comind through the openings first cancels at $\pm 11.4^{\circ}$ with respect to the original direction of the speaker when observed far from the wall. (a) How far apart are the two openings? (b) At what angles would the sound first cancel if the source stopped moving?

Donald Albin

Numerade Educator

Problem 55

Measuring Refractive Index. A thin slit illuminated by light of frequency $f$ produces its first dark band at $\pm 38.2^{\circ}$ in air. When the entire apparatus (slit, screen, and space in between) is immersed in an unknown transparent liquid, the slit's first dark bands occur instead at $\pm 21.6^{\circ} .$ Find the refractive index of the liquid.

Donald Albin

Numerade Educator

Problem 56

Grating Design. Your boss asks you to design a diffraction grating that will disperse the first-order visible spectrum through an angular range of $21.0^{\circ}$ (see Example 36.4 in Section
36.5$)$ . (a) What must the number of slits per centimeter be for this grating? (b) At what angles will the first-order visible spectrum begin and end?

Donald Albin

Numerade Educator

Problem 57

A slit 0.360 $\mathrm{mm}$ wide is illuminated by parallel rays of light that have a wavelength of 540 $\mathrm{nm}$ . The diffraction pattern is observed on a screen that is 1.20 $\mathrm{m}$ from the slit. The intensity at the center of the central maximum $\left(\theta=0^{\circ}\right)$ is $I_{0}$ . (a) What is the distance on the screen from the center of the central maximum to the first minimum? (b) What is the distance on the screen from the center of the central maximum to the point where the intensity has
fallen to $I_{0} / 2 ?$

Donald Albin

Numerade Educator

Problem 58

The intensity of light in the Fraunhofer diffraction pattern of a single slit is $$I=I_{0}\left(\frac{\sin \gamma}{\gamma}\right)^{2}$$ where $$\gamma=\frac{\pi a \sin \theta}{\lambda}$$
(a) Show that the equation for the values of $\gamma$ at which $I$ is a maximum is $\tan \gamma=\gamma .($ b) Determine the three smallest positive values of $\gamma$ that are solutions of this equation. (Hint: You can use a trial-and-error procedure. Guess a value of $\gamma$ and adjust your guess to bring $\tan \gamma$ closer to $\gamma .$ A graphical solution of the equation is very helpful in locating the solutions approximately, to get good initial guesses.)

Donald Albin

Numerade Educator

Problem 59

Angular Width of a Principal Maximum. Consider $N$ evenly spaced, narrow slits. Use the small-angle approximation $\sin \theta=\theta($ for $\theta$ in radians) to prove the following: For an intensity
maximum that occurs at an angle $\theta$ , the intensity minima immediately adjacent to this maximum are at angles $\theta+\lambda / N d$ and $\theta-\lambda / N d,$ so that the angular width of the principal maximum is 2$\lambda / N d .$ This is proportional to $1 / N,$ as we concluded in Section 36.4 on the basis of energy conservation.

Mayukh Banik

Numerade Educator

Problem 60

In a large vacuum chamber, monochromatic laser light passes through a narrow slit in a thin aluminum plate and forms a diffraction pattern on a screen that is 0.620 $\mathrm{m}$ from the slit. When the aluminum plate has a temperature of $20.0^{\circ} \mathrm{C},$ the width of the central maximum in the diffraction pattern is 2.75 $\mathrm{mm}$ . What is the change in the width of the central maximum when the temperature of the plate is raised to $520.0^{\circ} \mathrm{C}$ ? Does the width of the central diffraction maximum increase or decrease when the temperature is increased?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 61

Phasor Diagram for Eight Slits. An interference pat-tern is produced by eight equally spaced, narrow slits. Figure 36.14 shows phasor diagrams for the cases in which the phase difference $\phi$ between light from adjacent slits is $\phi=\pi, \phi=\pi / 4,$ and $\phi=\pi / 2 .$ Each of these cases gives an intensity minimum. The caption for Fig. 36.14 also claims that minima occur for $\phi=3 \pi / 4, \phi=5 \pi / 4, \phi=3 \pi / 2,$ and $\phi=7 \pi / 4 .$ (a) Draw the phasor diagram for each of these four cases, and explain why each diagram proves that there is in fact a minimum. (Note: You may find it helpful to use a different colored pencil for each slit') (b) For each of the four cases $\phi=3 \pi / 4, \phi=5 \pi / 4, \phi=3 \pi / 2$ and $\phi=7 \pi / 4,$ for which pairs of slits is there totally destructive interference?

Donald Albin

Numerade Educator

Problem 62

In a laboratory, light from a particular spectrum line of helium passes through a diffraction grating and the second-order maximum is at $18.9^{\circ}$ from the center of the central bright fringe. The same grating is then used for light from a distant galaxy that is moving away from the earth with a speed of $2.65 \times 10^{7} \mathrm{m} / \mathrm{s} .$ For the light from the galaxy, what is the angular location of the second-order maximum for the same spectral line as was observed in the lab? (See Section $16.8 . )$

Donald Albin

Numerade Educator

Problem 63

What is the longest wavelength that can be observed in the third order for a transmission grating having 9200 slits/cm? Assume normal incidence.

Mayukh Banik

Numerade Educator

Problem 64

(a) Figure 36.16 shows plane waves of light incident nomally on a diffraction grating. If instead the light strikes the grating at an angle of incidence $\theta^{\prime}$ (measured from the normal), show that the condition for an intensity maximum is not Eq. $(36.13),$ but rather $$d\left(\sin \theta+\sin \theta^{\prime}\right)=m \lambda(m=0, \pm 1, \pm 2, \pm 3, \ldots)$$(b) For the grating described in Example 36.4 (Section 36.5), with 600 slits/mm, find the angles of the maxima corresponding to
$m=0,1,$ and $-1$ with red light $(\lambda=650 \mathrm{nm})$ for the cases $\theta^{\prime}=0$ (normal incidence) and $\theta^{\prime}=20.0^{\circ} .$

Mayukh Banik

Numerade Educator

Problem 65

A diffraction grating has 650 slits/mm. What is the highest order that contains the entire visible spectrum? (The wavelength range of the visible spectrum is approximately $380-750 \mathrm{nm.}$ )

Donald Albin

Numerade Educator

Problem 66

Quasars, an abbreviation for quasi-stellar radio sources, are distant objects that look like stars through a telescope but that emit far more electromagnetic radiation than an entire normal
galaxy of stars. An example is the bright object below and to the left of center in Fig. P36.66; the other elongated objects in this image are normal galaxies. The leading model for the structure of a
quasar is a galaxy with a supermassive black hole at its center. In this model, the radiation is emitted by interstellar gas and dust within the galaxy as this material falls toward the black hole. The
radiation is thought to emanate from a region just a few light-years radiation is thought to emanate from a region just a few light-years in diameter. (The diffuse glow surrounding the bright quasar shown in Fig. $P 36.66$ is thought to be this quasar's host galaxy.) To investigate this model of quasars and to study other exotic astronomical objects, the Russian Space Agency plans to place a radio telescope in an orbit that extends to $77,000 \mathrm{km}$ from the earth. When the signals from this telescope are combined with signals from the ground-based telescopes of the VLBA, the resolution will be that of a single radio telescope $77,000 \mathrm{km}$ in diameter. What is the size of the smallest detail that this arrangement could resolve ir quasar $3 \mathrm{C} 405,$ which is $7.2 \times 10^{8}$ light-years from earth, using radio waves at a frequency of 1665 $\mathrm{MHz}$ ? (Hint: Use Rayleigh's criterion.) Give your answer in light-years and in kilometers.

Donald Albin

Numerade Educator

Problem 67

Phased-Array Radar. In one common type of radar installation, a rotating antenna sweeps a radio beam around the sky. But in a phased-array radar system, the antennas remain stationary and the beam is swept electronically. To see how this is done, consider an array of $N$ antennas that are arranged along the horizontal $x$ -axis at $x=0, \pm d, \pm 2 d, \ldots, \pm(N-1) d / 2 .$ (The
number $N$ is odd.) Each antenna emits radiation uniformly in all directions in the horizontal $x y$ -plane. The antennas all emit radiation coherently, with the same amplitude $E_{0}$ and the save-
length $\lambda$ . The relative phase $\delta$ of the emission from adjacent antennas
can be varied, however. If the antenna at $x=0$ emits a signal that is given by $E_{0} \cos \omega t,$ as measured at a point next to the antenna, the antenna at $x=d$ emits a signal given by $E_{0} \cos (\omega t+\delta),$ as measured at a point next to that antenna. The corresponding quantity for the antenna at $x=-d$ is $E_{0} \cos (\omega t-\delta) ;$ for the antennas at $x=\pm 2 d,$ it is $E_{0} \cos (\omega t \pm 2 \delta) ;$ and so on. (a) If $\delta=0,$ the interference pattern at a distance from the antennas is large compared to $d$ and has a principal maximum at $\theta=0$ (that is, in the $+y$ -direction, perpendicular to the line of the antennas. Show that if $d<\lambda,$ this is the only principal interference maximum in the angular range $-90^{\circ}<\theta<90^{\circ}$ . Hence this principal maximum describes a beam emitted in the direction $\theta=0 .$ As described in
Section 36.4 , if $N$ is large, the beam will have a large intensity and be quite narrow. (b) If $\delta \neq 0,$ show that the principal intensity maximum described in part (a) is located at
$$\theta=\arcsin \left(\frac{\delta \lambda}{2 \pi d}\right)$$ where $\delta$ is measured in radians. Thus, by varying $\delta$ from positive to negative values and back again, which can easily be done electronically, the beam can be made to sweep back and forth around $\theta=0 .(\mathrm{c})$ A weather radar unit to be installed on an airplane emits radio waves at 8800 $\mathrm{MHz}$ . The unit uses 15 antennas in an array 28.0 $\mathrm{cm}$ long (from the antenna at one end of the array to the antenna at the other end. What must the maximum and minimum values of $\delta$ be (that is, the most positive and most negative values) if the radar beam is to sweep $45^{\circ}$ to the left or right of the airplane's direction of flight? Give your answer in radians.

Mayukh Banik

Numerade Educator

Problem 68

Underwater Photography. An underwater camera has a lens of focal length 35.0 $\mathrm{mm}$ and a maximum aperture of $f / 2.80 .$ The film it uses has an emulsion that is sensitive to light of frequency $6.00 \times 10^{14} \mathrm{Hz}$ . If the photographer takes a picture of an object 2.75 $\mathrm{m}$ in front of the camera with the lens wide open, what is the width of the smallest resolvable detail on the subject if the object is (a) a fish underwater with the camera in the water and (b) a person on the beach with the camera out of the water?

Donald Albin

Numerade Educator

Problem 69

An astronaut in the space shuttle can just resolve two point sources on earth that are 65.0 $\mathrm{m}$ apart. Assume that the resolution is diffraction limited and use Rayleigh's criterion. What is the astronaut's altitude above the earth? Treat his eye as a circular aperture with a diameter of 4.00 $\mathrm{mm}$ (the diameter of his pupil), and take the wavelength of the light to be 550 $\mathrm{nm} .$ Ignore the effect of fluid in the eye.

Donald Albin

Numerade Educator

Problem 70

Resolution of the Eye. The maximum resolution of the eye depends on the diameter of the opening of the pupil (a diffraction effect) and the size of the retinal cells. The size of the retinal cells (about 5.0$\mu \mathrm{m}$ in diameter) limits the size of an object at the near point $(25 \mathrm{cm})$ of the eye to a height of about 50$\mu \mathrm{m} .$ To get a reasonable estimate without having to go through complicated calculations, we shall ignore the effect of the fluid in the eye.) (a) Given that the diameter of the human pupil is about 2.0 mm, does the Rayleigh criterion allow us to resolve a 50 -\mum-tall object at 25 $\mathrm{cm}$ from the eye with light of wavelength 550 $\mathrm{nm}$ ? (b) According to the Rayleigh criterion, what is the shortest object we could resolve at the $25-\mathrm{cm}$ near point with light of wavelength 550 $\mathrm{nm} ?(\mathrm{c})$ What angle would the object in part (b) subtend at the eye? Express your answer in minutes $\left(60 \min =1^{\circ}\right),$ and compare it with the experimental value of about 1 min. (d) Which effect is
more important in limiting the resolution of our eyes: diffraction or the size of the retinal cells?

Donald Albin

Numerade Educator

Problem 71

A glass sheet is covered by a very thin opaque coating. In the middle of this sheet there is a thin scratch 0.00125 $\mathrm{mm}$ thick. The sheet is totally immersed beneath the surface of a liquid. Parallel rays of monochromatic coherent light with wavelength 612 $\mathrm{nm}$ in air strike the sheet perpendicular to its surface and pass through the scratch. A screen is placed in the liquid a distance of 30.0 $\mathrm{cm}$ away from the sheet and parallel to it. You observe that the first dark fringes on either side of the central bright fringe on the screen are 22.4 $\mathrm{cm}$ apart. What is the refractive index of the liquid?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 72

NASA is considering a project called Planet Imager that would give astronomers the ability to see details on planets orbiting other stars. Using the same principle as the Very Large Array (see Section 36.7 ), Planet Imager will use an array of infrared telescopes spread over thousands of kilometers of space. (Visible light would give even better resolution. Unfortunately, at visible wavelengths, stars are so bright that a planet would be lost in the glare. This is less of a problem at infrared wavelengths.) (a) If Planet Imager has an effective diameter of $6000 \mathrm{~km}$ and observes infrared radiation at
a wavelength of $10 \mu \mathrm{m},$ what is the greatest distance at which it would be able to observe details as small as $250 \mathrm{~km}$ across (about the size of the greater Los Angeles area) on a planet? Give your answer in light-years (see Appendix E). (Hint: Use Rayleigh's criterion.)
(b) For comparison, consider the resolution of a single infrared telescope in space that has a diameter of $1.0 \mathrm{~m}$ and that observes $10-\mu \mathrm{m}$ radiation. What is the size of the smallest details that such a telescope could resolve at the distance of the nearest star to the sun, Proxima Centauri, which is 4.22 light-years distant? How does this compare to the diameter of the earth $\left(1.27 \times 10^{4} \mathrm{~km}\right) ?$ To the average distance from the earth to the sun $\left(1.50 \times 10^{8} \mathrm{~km}\right) ?$ Would a single telescope of this kind be able to detect the presence of a planet like the earth, in an orbit the size of the earth's orbit, around any other star? Explain. (c) Suppose Planet Imager is used to observe a planet orbiting the star 70 Virginis, which is 59 lightyears from our solar system. A planet (though not an earthlike one) has in fact been detected orbiting this star, not by imaging it directly but by observing the slight "wobble" of the star as both it and the planet orbit their common center of mass. What is the size of the smallest details that Planet Imager could hope to resolve on the planet of 70 Virginis? How does this compare to the diameter of the planet, assumed to be comparable to that of Jupiter $\left(1.38 \times 10^{5} \mathrm{~km}\right) ?$ (Although the planet of 70 Virginis is thought to be at least 6.6 times more massive than Jupiter, its radius is probably not too different from that of Jupiter. The reason is that such large planets are thought to be composed primarily of gases, not rocky material, and hence can be greatly compressed by the mutual gravitational attraction of different parts of the planet.)

Donald Albin

Numerade Educator

Problem 73

It is possible to calculate the intensity in the single-slit Fraunhofer diffraction pattern without using the phasor method of Section $36.3 .$ Let $y^{\prime}$ represent the position of a point
within the slit of width $a$ in Fig. 36.5 $\mathrm{a}$ , with $y^{\prime}=0$ at the center of
the slit so that the slit extends from $y^{\prime}=-a / 2$ to $y^{\prime}=a / 2 .$ We imagine dividing the slit up into infinitesimal strips of width $d y^{\prime}$ ,
each of which acts as a source of secondary wavelets. (a) The
amplitude of the total wave at the point $O$ on the distant screen in
Fig. 36.5 $\mathrm{a}$ is $E_{0 .}$ Explain why the amplitude of the wavelet from each infinitesimal strip within the slit is $E_{0}\left(d y^{\prime} / a\right),$ so that the elec-
tric field of the wavelet a distance $x$ from the infinitesimal strip is
$d E=E_{0}\left(d y^{\prime} / a\right) \sin (k x-\omega t) \cdot$ (b) Explain why the wavelet from
each strip as detected at point $P$ in Fig. 36.5$a$ can be expressed as
$$d E=E_{0} \frac{d y^{\prime}}{a} \sin \left[k\left(D-y^{\prime} \sin \theta\right)-\omega t\right]$$ where $D$ is the distance from the center of the slit to point $P$ and
$k=2 \pi / \lambda .(c)$ By integrating the contributions $d E$ from all parts of
the slit, show that the total wave detected at point $$P$ is $E=E_{0} \sin (k D-\omega t) \frac{\sin [k a(\sin \theta) / 2]}{k a(\sin \theta) / 2}$$
$$=E_{0} \sin (k D-\omega t) \frac{\sin [\pi a(\sin \theta) / \lambda]}{\pi a(\sin \theta) / \lambda}$$
(The trigonometric identities in Appendix B will be useful.) Show
that at $\theta=0,$ corresponding to point $O$ in Fig. 36.5 $\mathrm{a}$ , the wave is
$E=E_{0} \sin (k D-\omega t)$ and has amplitude $E_{0},$ as stated in part (a)
(d) Use the result of part (c) to show that if the intensity at point $O$ is $I_{0},$ then the intensity at a point $P$ is given by Eq. $(36.7)$

Mayukh Banik

Numerade Educator

Problem 74

Intensity Pattern of $N$ Slits. (a) Consider an arrangement of $N$ slits with a distance $d$ between adjacent slits.
The slits emit coherently and in phase at wavelength $\lambda .$ Show that
at a time $t,$ the electric field at a distant point $P$ is $$\begin{aligned} E_{P}(t)=& E_{0} \cos (k R-\omega t)+E_{0} \cos (k R-\omega t+\phi) \\ &+E_{0} \cos (k R-\omega t+2 \phi)+\cdots \\ &+E_{0} \cos (k R-\omega t+(N-1) \phi) \end{aligned}$$
where $E_{0}$ is the amplitude at $P$ of the electric field due to an indi-
vidual slit, $\phi=(2 \pi d \sin \theta) / \lambda, \theta$ is the angle of the rays reaching
$P$ (as measured from the perpendicular bisector of the slit arrange-
ment), and $R$ is the distance from $P$ to the most distant slit. In this problem, assume that $R$ is much larger than $d$ . (b) To carry out the
sum in part (a), it is convenient to use the complex-number relationship
$$e^{i z}=\cos z+i \sin z$$
where $i=\sqrt{-1}$ . In this expression, cos $z$ is the real part of the
complex number $e^{i z},$ and $\sin z$ is its imaginary part. Show that the
electric field $E_{P}(t)$ is equal to the real part of the complex quantity
$$\sum_{n=0}^{N-1} E_{0} e^{i(k R-\omega t+n \phi)}$$
(c) Using the properties of the exponential function that
$e^{A} e^{B}=e^{(A+B)}$ and $\left(e^{A}\right)^{n}=e^{n A},$ show that the sum in part (b) can
be written as
$$E_{0}\left(\frac{e^{i N \phi}-1}{e^{i \phi}-1}\right) e^{i(k R-\omega t)}$
$\quad=E_{0}\left(\frac{e^{i N \phi / 2}-e^{-i N \phi / 2}}{e^{i \phi / 2}-e^{-i \phi / 2}}\right) e^{i[k R-\omega t+(N-1) \phi}$$
Then, using the relationship $e^{i z}=\cos z+i \sin z,$ show that the
(real) electric field at point $P$ is
$$I=I_{0}\left[\frac{\sin (N \phi / 2)}{\sin (\phi / 2)}\right]^{2}$$
where $I_{0}$ is the maximum intensity for an individual slit. (e) Check
the result in part (d) for the case $N=2 .$ It will help to recall that
$\sin 2 A=2 \sin A \cos A .$ Explain why your result differs from Eq. $(35.10),$ the expression for the intensity in two-source interfer-
ence, by a factor of $4 .$ Hint: Is $I_{0}$ defined in the same way in both
expressions?

Lainey Roebuck

Numerade Educator

Problem 75

Intensity Pattern of $N$ Slits, Continued. Part (d) of Challenge Problem 36.74 gives an expression for the intensity in the interference pattern of $N$ identical slits. Use this result toverify the following statements. (a) The maximum intensity in the pattern is $N^{2} I_{0}$ . ( b ) The principal maximum at the center of the pattern extends from $\phi=-2 \pi / N$ to $\phi=2 \pi / N,$ so its width is inversely proportional to 1$/ N .(\mathrm{c})$ A minimum occurs whenever $\phi$ is an integral multiple of $2 \pi / N,$ except when $\phi$ is an integral multiple of 2$\pi$ (which gives a principal maximum). (d) There are $(N-1)$ minima between each pair of principal maxima. (e) Halfway between two principal maxima, the intensity can be no greater than $I_{0} ;$ that is, it can be no greater than 1$/ N^{2}$ times the intensity at a principal maximum.

Mayukh Banik

Numerade Educator