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Fundamentals of Physics, Volume 2

David Halliday & Robert Resnick & Jearl Walker

Chapter 36

Diffraction - all with Video Answers

Educators

Chapter Questions

02:26

Problem 1

The distance between the first and fifth minima of a single-slit diffraction pattern is $0.35 \mathrm{~mm}$ with the screen $40 \mathrm{~cm}$ away from the slit, when light of wavelength $550 \mathrm{~nm}$ is used. (a) Find the slit width. (b) Calculate the angle $\theta$ of the first diffraction minimum.

Paul A.
Paul A.
California State Polytechnic University, Pomona
00:57

Problem 2

What must be the ratio of the slit width to the wavelength for a single slit to have the first diffraction minimum at $\theta=45.0^{\circ}$ ?

Ben Nicholson
Ben Nicholson
Numerade Educator
00:06

Problem 3

A plane wave of wavelength $590 \mathrm{~nm}$ is incident on a slit with a width of $a=0.40 \mathrm{~mm}$. A thin converging lens of focal length $+70 \mathrm{~cm}$ is placed between the slit and a viewing screen and focuses the light on the screen. (a) How far is the screen from the lens? (b) What is the distance on the screen from the center of the diffraction pattern to the first minimum?

Paul A.
Paul A.
California State Polytechnic University, Pomona
02:15

Problem 4

In conventional television, signals are broadcast from towers to home receivers. Even when a receiver is not in direct view of a tower because of a hill or building, it can still intercept a signal if the signal diffracts enough around the obstacle, into the obstacle's "shadow region." Previously, television signals had a wavelength of about $50 \mathrm{~cm}$, but digital television signals that are transmitted from towers have a wavelength of about $10 \mathrm{~mm}$. (a) Did this change in wavelength increase or decrease the diffraction of the signals into the shadow regions of obstacles? Assume that a signal passes through an opening of $5.0 \mathrm{~m}$ width between two adjacent buildings. What is the angular spread of the central diffraction maximum (out to the first minima) for wavelengths of (b) $50 \mathrm{~cm}$ and (c) $10 \mathrm{~mm}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
02:37

Problem 5

A single slit is illuminated by light of wavelengths $\lambda_a$ and $\lambda_b$, chosen so that the first diffraction minimum of the $\lambda_a$ component coincides with the second minimum of the $\lambda_b$ component. (a) If $\lambda_b=350 \mathrm{~nm}$, what is $\lambda_a$ ? For what order number $m_b$ (if any) does a minimum of the $\lambda_b$ component coincide with the minimum of the $\lambda_a$ component in the order number (b) $m_a=2$ and (c) $m_a=3$ ?

Keshav Singh
Keshav Singh
Numerade Educator
02:27

Problem 6

Monochromatic light of wavelength $441 \mathrm{~nm}$ is incident on a narrow slit. On a screen $2.00 \mathrm{~m}$ away, the distance between the second diffraction minimum and the central maximum is $1.50 \mathrm{~cm}$. (a) Calculate the angle of diffraction $\theta$ of the second minimum. (b) Find the width of the slit.

Ben Nicholson
Ben Nicholson
Numerade Educator
03:28

Problem 7

Light of wavelength $633 \mathrm{~nm}$ is incident on a narrow slit. The angle between the first diffraction minimum on one side of the central maximum and the first minimum on the other side is $1.20^{\circ}$. What is the width of the slit?

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:47

Problem 8

Sound waves with frequency $3000 \mathrm{~Hz}$ and speed $343 \mathrm{~m} / \mathrm{s}$ diffract through the rectangular opening of a speaker cabinet and into a large auditorium of length $d=100 \mathrm{~m}$. The opening, which has a horizontal width of $30.0 \mathrm{~cm}$, faces a wall $100 \mathrm{~m}$ away (Fig. 36.7). Along that wall, how far from the central axis will a listener be at the first diffraction minimum and thus have difficulty hearing the sound? (Neglect reflections.)
(Figure Cant Copy)
Figure 36.7 Problem 8.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:57

Problem 9

A slit $1.00 \mathrm{~mm}$ wide is illuminated by light of wavelength $589 \mathrm{~nm}$. We see a diffraction pattern on a screen $3.00 \mathrm{~m}$ away. What is the distance between the first two diffraction minima on the same side of the central diffraction maximum?

Paul A.
Paul A.
California State Polytechnic University, Pomona
03:07

Problem 10

Manufacturers of wire (and other objects of small dimension) sometimes use a laser to continually monitor the thickness of the product. The wire intercepts the laser beam, producing a diffraction pattern like that of a single slit of the same width as the wire diameter (Fig. 36.8). Suppose a helium-neon laser, of wavelength $632.8 \mathrm{~nm}$, illuminates a wire, and the diffraction pattern appears on a screen at distance $L=2.60 \mathrm{~m}$. If the desired wire diameter is $1.37 \mathrm{~mm}$, what is the observed distance between the two tenth-order minima (one on each side of the central maximum)?
(Figure Cant Copy)
Figure 36.8 Problem 10.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:09

Problem 11

A $0.10-\mathrm{mm}$-wide slit is illuminated by light of wavelength $589 \mathrm{~nm}$. Consider a point $P$ on a viewing screen on which the diffraction pattern of the slit is viewed; the point is at $30^{\circ}$ from the central axis of the slit. What is the phase difference between the Huygens wavelets arriving at point $P$ from the top and midpoint of the slit?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:38

Problem 12

Figure 36.9 gives $\alpha$ versus the sine of the angle $\theta$ in a single-slit diffraction experiment using light of wavelength $610 \mathrm{~nm}$. The vertical axis scale is set by $\alpha_s=12 \mathrm{rad}$. What are (a) the slit width, (b) the total number of diffraction minima in the pattern (count them on both sides of the center of the diffraction pattern), (c) the least angle for a minimum, and (d) the greatest angle for a minimum?
(Figure Cant Copy)
Figure 36.9 Problem 12.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
09:09

Problem 13

Monochromatic light with wavelength $538 \mathrm{~nm}$ is incident on a slit with width $0.025 \mathrm{~mm}$. The distance from the slit to a screen is $3.5 \mathrm{~m}$. Consider a point on the screen $1.1 \mathrm{~cm}$ from the central maximum. Calculate (a) $\theta$ for that point, (b) $\alpha$, and (c) the ratio of the intensity at that point to the intensity at the central maximum.

Paul A.
Paul A.
California State Polytechnic University, Pomona
03:25

Problem 14

In the single-slit diffraction experiment of Fig. 36.1.4, let the wavelength of the light be $500 \mathrm{~nm}$, the slit width be $6.00 \mu \mathrm{m}$, and the viewing screen be at distance $D=3.00 \mathrm{~m}$. L.et a $y$ axis extend upward along the viewing screen, with its origin at the center of the diffraction pattern. Also let $I_P$ represent the intensity of the diffracted light at point $P$ at $y=15.0 \mathrm{~cm}$. (a) What is the ratio of $I_p$ to the intensity $I_m$ at the center of the pattern? (b) Determine where point $P$ is in the diffraction pattern by giving the maximum and minimum between which it lies, or the two minima between which it lies.

Keshav Singh
Keshav Singh
Numerade Educator
03:02

Problem 15

The full width at half-maximum (FWHM) of a central diffraction maximum is defined as the angle between the two points in the pattern where the intensity is one-half that at the center of the pattern. (See Fig. 36.2.2b.) (a) Show that the intensity drops to one-half the maximum value when $\sin ^2 \alpha=\alpha^2 / 2$. (b) Verify that $\alpha=1.39 \mathrm{rad}$ (about $80^{\circ}$ ) is a solution to the transcendental equation of (a). (c) Show that the FWHM is $\Delta \theta=2 \sin ^{-1}(0.442 \lambda / a)$, where $a$ is the slit width. Calculate the FWHM of the central maximum for slit width (d) $1.00 \lambda$, (e) $5.00 \%$, and (f) $10.0 \lambda$.

Neelesh Sharma
Neelesh Sharma
Numerade Educator
01:22

Problem 16

A monochromatic beam of parallel light is incident on a "collimating" hole of diameter $x>\lambda$. Point $P$ lies in the geometrical shadow region on a distant screen (Fig. 36.10a). Two diffracting objects, shown in Fig. 36.10b, are placed in turn over the collimating hole. Object $A$ is an opaque circle with a hole in it, and $B$ is the "photographic negative" of $A$. Using superposition concepts, show that the intensity at $P$ is identical for the two diffracting objects $A$ and $B$.
(Figure Cant Copy)
Figure 36.10 Problem 16.

Keshav Singh
Keshav Singh
Numerade Educator
08:04

Problem 17

(a) Show that the values of $\alpha$ at which intensity maxima for single-slit diffraction occur can be found exactly by differentiating Eq. 36.2.2 with respect to $\alpha$ and equating the result to zero, obtaining the condition $\tan \alpha=\alpha$. To find values of $\alpha$ satisfying this relation, plot the curve $y=\tan \alpha$ and the straight line $y=\alpha$ and then find their intersections, or use a calculator to find an appropriate value of $\alpha$ by trial and error. Next, from $\alpha=\left(m+\frac{1}{2}\right) \pi$, determine the values of $m$ associated with the maxima in the single-slit pattern. (These $m$ values are not integers because secondary maxima do not lie exactly halfway between minima.) What are the (b) smallest $\alpha$ and (c) associated $m$, the (d) second smallest $\alpha$ and (e) associated $m$, and the (f) third smallest $\alpha$ and $(\mathrm{g})$ associated $m$ ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:03

Problem 18

The wall of a large room is covered with acoustic tile in which small holes are drilled $5.0 \mathrm{~mm}$ from center to center. How far can a person be from such a tile and still distinguish the individual holes, assuming ideal conditions, the pupil diameter of the observer's eye to be $4.0 \mathrm{~mm}$, and the wavelength of the room light to be $550 \mathrm{~nm}$ ?

Ben Nicholson
Ben Nicholson
Numerade Educator
11:40

Problem 19

(a) How far from grains of red sand must you be to position yourself just at the limit of resolving the grains if your pupil diameter is $1.5 \mathrm{~mm}$, the grains are spherical with radius $50 \mu \mathrm{m}$, and the light from the grains has wavelength $650 \mathrm{~nm}$ ? (b) If the grains were blue and the light from them had wavelength $400 \mathrm{~nm}$, would the answer to (a) be larger or smaller?

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:28

Problem 20

The radar system of a navy cruiser transmits at a wavelength of $1.6 \mathrm{~cm}$, from a circular antenna with a diameter of $2.3 \mathrm{~m}$. At a range of $6.2 \mathrm{~km}$, what is the smallest distance that two speedboats can be from each other and still be resolved as two separate objects by the radar system?

Ben Nicholson
Ben Nicholson
Numerade Educator
09:43

Problem 21

Estimate the linear separation of two objects on Mars that can just be resolved under ideal conditions by an observer on Earth (a) using the naked eye and (b) using the $200 \mathrm{in}$. $(=5.1 \mathrm{~m})$ Mount Palomar telescope. Use the following data: distance to Mars $=8.0 \times 10^7 \mathrm{~km}$, diameter of pupil $=5.0 \mathrm{~mm}$, wavelength of light $=550 \mathrm{~nm}$.

Paul A.
Paul A.
California State Polytechnic University, Pomona
03:50

Problem 22

Assume that Rayleigh's criterion gives the limit of resolution of an astronaut's eye looking down on Earth's surface from a typical space shuttle altitude of $400 \mathrm{~km}$. (a) Under that idealized assumption, estimate the smallest linear width on Earth's surface that the astronaut can resolve. Take the astronaut's pupil diameter to be $5 \mathrm{~mm}$ and the wavelength of visible light to be $550 \mathrm{~nm}$. (b) Can the astronaut resolve the Great Wall of China (Fig. 36.11), which is more than $3000 \mathrm{~km}$ long, 5 to $10 \mathrm{~m}$ thick at its base, $4 \mathrm{~m}$ thick at its top, and $8 \mathrm{~m}$ in height? (c) Would the astronaut be able to resolve any unmistakable sign of intelligent life on Earth's surface?
(Figure Cant Copy)
Figure 36.11 Problem 22. The Great Wall of China.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:29

Problem 23

The two headlights of an approaching automobile are $1.4 \mathrm{~m}$ apart. At what (a) angular separation and (b) maximum distance will the eye resolve them? Assume that the pupil diameter is $5.0 \mathrm{~mm}$, and use a wavelength of $550 \mathrm{~nm}$ for the light. Also assume that diffraction effects alone limit the resolution so that Rayleigh's criterion can be applied.

Keshav Singh
Keshav Singh
Numerade Educator
03:16

Problem 24

f someone looks at a bright outdoor lamp in otherwise dark surroundings, the lamp appears to be surrounded by bright and dark rings (hence halos) that are actually a circular diffraction pattern as in Fig. 36.3.1, with the central maximum overlapping the direct light from the lamp. The diffraction is produced by structures within the cornea or lens of the eye (hence entoptic). If the lamp is monochromatic at wavelength $550 \mathrm{~nm}$ and the first dark ring subtends angular diameter $2.5^{\circ}$ in the observer's view, what is the (linear) diameter of the structure producing the diffraction?

Ben Nicholson
Ben Nicholson
Numerade Educator
04:19

Problem 25

Find the separation of two points on the Moon's surface that can just be resolved by the $200 \mathrm{in} .(=5.1 \mathrm{~m})$ telescope at Mount Palomar, assuming that this separation is determined by diffraction effects. The distance from Earth to the Moon is $3.8 \times 10^5 \mathrm{~km}$. Assume a wavelength of $550 \mathrm{~nm}$ for the light.

Paul A.
Paul A.
California State Polytechnic University, Pomona
04:51

Problem 26

The telescopes on some commercial surveillance satellites can resolve objects on the ground as small as $85 \mathrm{~cm}$ across (see Google Earth), and the telescopes on military surveillance satellites reportedly can resolve objects as small as $10 \mathrm{~cm}$ across. Assume first that object resolution is determined entirely by Rayleigh's criterion and is not degraded by turbulence in the atmosphere. Also assume that the satellites are at a typical altitude of $400 \mathrm{~km}$ and that the wavelength of visible light is $550 \mathrm{~nm}$. What would be the required diameter of the telescope aperture for (a) $85 \mathrm{~cm}$ resolution and (b) $10 \mathrm{~cm}$ resolution? (c) Now, considering that turbulence is certain to degrade resolution and that the aperture diameter of the Hubble Space Telescope is $2.4 \mathrm{~m}$, what can you say about the answer to (b) and about how the military surveillance resolutions are accomplished?

Ben Nicholson
Ben Nicholson
Numerade Educator
06:46

Problem 27

If Superman really had $x$-ray vision at $0.10 \mathrm{~nm}$ wavelength and a $4.0 \mathrm{~mm}$ pupil diameter, at what maximum altitude could he distinguish villains from heroes, assuming that he needs to resolve points separated by $5.0 \mathrm{~cm}$ to do this?

Paul A.
Paul A.
California State Polytechnic University, Pomona
03:38

Problem 28

The wings of tiger beetles (Fig. 36.12) are colored by interference due to thin cuticle-like layers. In addition, these layers are arranged in patches that are $60 \mu \mathrm{m}$ across and produce different colors. The color you see is a pointillistic mixture of thin-film interference colors that varies with perspective. Approximately what viewing distance from a wing puts you at the limit of resolving the different colored patches according to Rayleigh's criterion? Use $550 \mathrm{~nm}$ as the wavelength of light and $3.00 \mathrm{~mm}$ as the diameter of your pupil.
(Figure Cant Copy)
Figure 36.12 Problem 28. Tiger beetles are colored by pointillistic mixtures of thin-film interference colors.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
13:38

Problem 29

(a) What is the angular separation of two stars if their images are barely resolved by the Thaw refracting telescope at the Allegheny Observatory in Pittsburgh? The lens diameter is $76 \mathrm{~cm}$ and its focal length is $14 \mathrm{~m}$. Assume $\lambda=550 \mathrm{~nm}$. (b) Find the distance between these barely resolved stars if each of them is 10 light-years distant from Earth. (c) For the image of a single star in this telescope, find the diameter of the first dark ring in the diffraction pattern, as measured on a photographic plate placed at the focal plane of the telescope lens. Assume that the structure of the image is associated entirely with diffraction at the lens aperture and not with lens "errors."

Paul A.
Paul A.
California State Polytechnic University, Pomona
04:56

Problem 30

The floaters you see when viewing a bright, featureless background are diffraction patterns of defects in the vitreous humor that fills most of your eye. Sighting through a pinhole sharpens the diffraction pattern. If you also view a small circular dot, you can approximate the defect's size. Assume that the defect diffracts light as a circular aperture does. Adjust the dot's distance $L$ from your eye (or eye lens) until the dot and the circle of the first minimum in the diffraction pattern appear to have the same size in your view. That is, until they have the same diameter $D^{\prime}$ on the retina at distance $L^{\prime}=2.0 \mathrm{~cm}$ from the front of the eye, as suggested in Fig. $36.13 a$, where the angles on the two sides of the eye lens are equal. Assume that the wavelength of visible light is $\lambda=550 \mathrm{~nm}$. If the dot has diameter $D=2.0 \mathrm{~mm}$ and is distance $L=45.0 \mathrm{~cm}$ from the eye and the defect is $x=6.0 \mathrm{~mm}$ in front of the retina (Fig. $36.13 b$ ), what is the diameter of the defect?
(Figure Cant Copy)
Figure 36.13 Problem 30.

Ben Nicholson
Ben Nicholson
Numerade Educator
03:47

Problem 31

Millimeter-wave radar generates a narrower beam than conventional microwave radar, making it less vulnerable to antiradar missiles than conventional radar. (a) Calculate the angular width $2 \theta$ of the central maximum, from first minimum to first minimum, produced by a $220 \mathrm{GHz}$ radar beam emitted by a $55.0-\mathrm{cm}$-diameter circular antenna. (The frequency is chosen to coincide with a low-absorption atmospheric "window.") (b) What is $2 \theta$ for a more conventional circular antenna that has a diameter of $2.3 \mathrm{~m}$ and emits at wavelength $1.6 \mathrm{~cm}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
02:58

Problem 32

(a) A circular diaphragm $60 \mathrm{~cm}$ in diameter oscillates at a frequency of $25 \mathrm{kHz}$ as an underwater source of sound used for submarine detection. Far from the source, the sound intensity is distributed as the diffraction pattern of a circular hole whose diameter equals that of the diaphragm. Take the speed of sound in water to be $1450 \mathrm{~m} / \mathrm{s}$ and find the angle between the normal to the diaphragm and a line from the diaphragm to the first minimum. (b) Is there such a minimum for a source having an (audible) frequency of $1.0 \mathrm{kHz}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
11:39

Problem 33

Nuclear-pumped x-ray lasers are seen as a possible weapon to destroy ICBM booster rockets at ranges up to $2000 \mathrm{~km}$. One limitation on such a device is the spreading of the beam due to diffraction, with resulting dilution of beam intensity. Consider such a laser operating at a wavelength of $1.40 \mathrm{~nm}$. The element that emits light is the end of a wire with diameter $0.200 \mathrm{~mm}$. (a) Calculate the diameter of the central beam at a target $2000 \mathrm{~km}$ away from the beam source. (b) What is the ratio of the beam intensity at the target to that at the end of the wire? (The laser is fired from space, so neglect any atmospheric absorption.)

Paul A.
Paul A.
California State Polytechnic University, Pomona
04:16

Problem 34

A circular obstacle produces the same diffraction pattern as a circular hole of the same diameter (except very near $\theta=0$ ). Airborne water drops are examples of such obstacles. When you see the Moon through suspended water drops, such as in a fog, you intercept the diffraction pattern from many drops. The composite of the central diffraction maxima of those drops forms a white region that surrounds the Moon and may obscure it. Figure 36.14 is a photograph in which the Moon is obscured. There are two faint, colored rings around the Moon (the larger one may be too faint to be seen in your copy of the photograph). The smaller ring is on the outer edge of the central maxima from the drops; the somewhat larger ring is on the outer edge of the smallest of the secondary maxima from the drops (see Fig. 36.3.1). The color is visible because the rings are adjacent to the diffraction minima (dark rings) in the patterns. (Colors in other parts of the pattern overlap too much to be visible.)
(a) What is the color of these rings on the outer edges of the diffraction maxima? (b) The colored ring around the central maxima in Fig. 36.14 has an angular diameter that is 1.35 times the angular diameter of the Moon, which is $0.50^{\circ}$. Assume that the drops all have about the same diameter. Approximately what is that diameter?
(Figure Cant Copy)
Figure 36.14 Problem 34. The corona around the Moon is a composite of the diffraction patterns of airborne water drops.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
16:42

Problem 35

Suppose that the central diffraction envelope of a doubleslit diffraction pattern contains 11 bright fringes and the first diffraction minima eliminate (are coincident with) bright fringes. How many bright fringes lie between the first and second minima of the diffraction envelope?

Paul A.
Paul A.
California State Polytechnic University, Pomona
02:13

Problem 36

A beam of light of a single wavelength is incident perpendicularly on a double-slit arrangement, as in Fig. 35.2.5. The slit widths are each $46 \mu \mathrm{m}$ and the slit separation is $0.30 \mathrm{~mm}$. How many complete bright fringes appear between the two firstorder minima of the diffraction pattern?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
15:58

Problem 37

In a double-slit experiment, the slit separation $d$ is 2.00 times the slit width $w$. How many bright interference fringes are in the central diffraction envelope?

Paul A.
Paul A.
California State Polytechnic University, Pomona
02:07

Problem 38

In a certain two-slit interference pattern, 10 bright fringes lie within the second side peak of the diffraction envelope and diffraction minima coincide with two-slit interference maxima. What is the ratio of the slit separation to the slit width?

Ben Nicholson
Ben Nicholson
Numerade Educator
03:43

Problem 39

Light of wavelength $440 \mathrm{~nm}$ passes through a double slit, yielding a diffraction pattern whose graph of intensity $I$ versus angular position $\theta$ is shown in Fig. 36.15. Calculate (a) the slit width and (b) the slit separation. (c) Verify the displayed intensities of the $m=1$ and $m=2$ interference fringes.
(Figure Cant Copy)
Figure 36.15 Problem 39.

Keshav Singh
Keshav Singh
Numerade Educator
05:25

Problem 40

Figure 36.16 gives the parameter $\beta$ of Eq. 36.4 .2 versus the sine of the angle $\theta$ in a two-slit interference experiment using light of wavelength $435 \mathrm{~nm}$. The vertical axis scale is set by $\beta_s=80.0 \mathrm{rad}$. What are (a) the slit separation, (b) the total number of interference maxima (count them on both sides of the pattern's center), (c) the smallest angle for a maxima, and (d) the greatest angle for a minimum? Assume that none of the interference maxima are completely eliminated by a diffraction minimum.
(Figure Cant Copy)
Figure 36.16 Problem 40.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
15:18

Problem 41

In the two-slit interference experiment of Fig. 35.2.5, the slit widths are each $12.0 \mu \mathrm{m}$, their separation is $24.0 \mu \mathrm{m}$, the wavelength is $600 \mathrm{~nm}$, and the viewing screen is at a distance of $4.00 \mathrm{~m}$. Let $I_P$ represent the intensity at point $P$ on the screen, at height $y=70.0 \mathrm{~cm}$. (a) What is the ratio of $I_P$, to the intensity $I_m$ at the center of the pattern? (b) Determine where $P$ is in the two-slit interference pattern by giving the maximum or minimum on which it lies or the maximum and minimum between which it lies. (c) In the same way, for the diffraction that occurs, determine where point $P$ is in the diffraction pattern.

Paul A.
Paul A.
California State Polytechnic University, Pomona
03:05

Problem 42

(a) In a double-slit experiment, what largest ratio of $d$ to $a$ causes diffraction to eliminate the fourth bright side fringe? (b) What other bright fringes are also eliminated? (c) How many other ratios of $d$ to $a$ cause the diffraction to (exactly) eliminate that bright fringe?

Ben Nicholson
Ben Nicholson
Numerade Educator
05:10

Problem 43

(a) How many bright fringes appear between the first diffraction-envelope minima to either side of the central maximum in a double-slit pattern if $\lambda=550 \mathrm{~nm}, d=0.150 \mathrm{~mm}$, and $a=30.0 \mu \mathrm{m}$ ? (b) What is the ratio of the intensity of the third bright fringe to the intensity of the central fringe?
Module 36.5 Diffraction Gratings

Prabhu Ramji
Prabhu Ramji
Numerade Educator
01:47

Problem 44

Perhaps to confuse a predator, some tropical gyrinid beetles (whirligig beetles) are colored by optical interference that is due to scales whose alignment forms a diffraction grating (which scatters light instead of transmitting it). When the incident light rays are perpendicular to the grating, the angle between the first-order maxima (on opposite sides of the zeroth-order maximum) is about $26^{\circ}$ in light with a wavelength of $550 \mathrm{~nm}$. What is the grating spacing of the beetle?

Ben Nicholson
Ben Nicholson
Numerade Educator
10:12

Problem 45

A diffraction grating $20.0 \mathrm{~mm}$ wide has 6000 rulings. Light of wavelength $589 \mathrm{~nm}$ is incident perpendicularly on the grating. What are the (a) largest, (b) second largest, and (c) third largest values of $\theta$ at which maxima appear on a distant viewing screen?

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:49

Problem 46

Eisible light is incident perpendicularly on a grating with 315 rulings $/ \mathrm{mm}$. What is the longest wavelength that can be seen in the fifth-order diffraction?

Ben Nicholson
Ben Nicholson
Numerade Educator
08:50

Problem 47

A grating has 400 lines $/ \mathrm{mm}$. How many orders of the entire visible spectrum $(400-700 \mathrm{~nm})$ can it produce in a diffraction experiment, in addition to the $m=0$ order?

Paul A.
Paul A.
California State Polytechnic University, Pomona
03:58

Problem 48

A diffraction grating is made up of slits of width $300 \mathrm{~nm}$ with separation $900 \mathrm{~nm}$. The grating is illuminated by monochromatic plane waves of wavelength $\lambda=600 \mathrm{~nm}$ at normal incidence. (a) How many maxima are there in the full diffraction pattern? (b) What is the angular width of a spectral line observed in the first order if the grating has 1000 slits?

Ben Nicholson
Ben Nicholson
Numerade Educator
14:48

Problem 49

Light of wavelength $600 \mathrm{~nm}$ is incident normally on a diffraction grating. Two adjacent maxima occur at angles given by $\sin \theta=0.2$ and $\sin \theta=0.3$. The fourth-order maxima are missing. (a) What is the separation between adjacent slits? (b) What is the smallest slit width this grating can have? For that slit width, what are the (c) largest, (d) second largest, and (e) third largest values of the order number $m$ of the maxima produced by the grating?

Paul A.
Paul A.
California State Polytechnic University, Pomona
02:05

Problem 50

With light from a gaseous discharge tube incident normally on a grating with slit separation $1.73 \mu \mathrm{m}$, sharp maxima of green light are experimentally found at angles $\theta= \pm 17.6^{\circ}, 37.3^{\circ}$, $-37.1^{\circ}, 65.2^{\circ}$, and $-65.0^{\circ}$. Compute the wavelength of the green light that best fits these data.

Keshav Singh
Keshav Singh
Numerade Educator
04:25

Problem 51

A diffraction grating having 180 lines $/ \mathrm{mm}$ is illuminated with a light signal containing only two wavelengths, $\lambda_1=400 \mathrm{~nm}$ and $\lambda_2=500 \mathrm{~nm}$. The signal is incident perpendicularly on the grating. (a) What is the angular separation between the second-order maxima of these two wavelengths? (b) What is the smallest angle at which two of the resulting maxima are superimposed? (c) What is the highest order for which maxima for both wavelengths are present in the diffraction pattern?

Keshav Singh
Keshav Singh
Numerade Educator
05:47

Problem 52

A beam of light consisting of wavelengths from $460.0 \mathrm{~nm}$ to $640.0 \mathrm{~nm}$ is directed perpendicularly onto a diffraction grating with 160 lines $/ \mathrm{mm}$. (a) What is the lowest order that is overlapped by another order? (b) What is the highest order for which the complete wavelength range of the beam is present? In that highest order, at what angle does the light at wavelength (c) $460.0 \mathrm{~nm}$ and (d) $640.0 \mathrm{~nm}$ appear? (e) What is the greatest angle at which the light at wavelength $460.0 \mathrm{~nm}$ appears?

Ben Nicholson
Ben Nicholson
Numerade Educator
12:32

Problem 53

A grating has 350 rulings $/ \mathrm{mm}$ and is illuminated at normal incidence by white light. A spectrum is formed on a screen $30.0 \mathrm{~cm}$ from the grating. If a hole $10.0 \mathrm{~mm}$ square is cut in the screen, its inner edge being $50.0 \mathrm{~mm}$ from the central maximum and parallel to it, what are the (a) shortest and (b) longest wavelengths of the light that passes through the hole?

Paul A.
Paul A.
California State Polytechnic University, Pomona
03:40

Problem 54

Derive this expression for the intensity pattern for a threeslit "grating":
$$
I=\frac{1}{9} I_m\left(1+4 \cos \phi+4 \cos ^2 \phi\right),
$$
where $\phi=(2 \pi d \sin \theta) / \lambda$ and $a \ll \lambda$.

Keshav Singh
Keshav Singh
Numerade Educator
06:07

Problem 55

A source containing a mixture of hydrogen and deuterium atoms emits red light at two wavelengths whose mean is $656.3 \mathrm{~nm}$ and whose separation is $0.180 \mathrm{~nm}$. Find the minimum number of lines needed in a diffraction grating that can resolve these lines in the first order.

Paul A.
Paul A.
California State Polytechnic University, Pomona
02:21

Problem 56

(a) How many rulings must a $4.00-\mathrm{cm}$-wide diffraction grating have to resolve the wavelengths 415.496 and $415.487 \mathrm{~nm}$ in the second order? (b) At what angle are the second-order maxima found?

Ben Nicholson
Ben Nicholson
Numerade Educator
06:51

Problem 57

Light at wavelength $589 \mathrm{~nm}$ from a sodium lamp is incident perpendicularly on a grating with 40000 rulings over width $76 \mathrm{~mm}$. What are the first-order (a) dispersion $D$ and (b) resolving power $R$, the second-order (c) $D$ and (d) $R$, and the third-order (e) $D$ and (f) $R$ ?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:08

Problem 58

A grating has $600 \mathrm{rulings} / \mathrm{mm}$ and is $5.0 \mathrm{~mm}$ wide. (a) What is the smallest wavelength interval it can resolve in the third order at $\lambda=500 \mathrm{~nm}$ ? (b) How many higher orders of maxima can be seen?

Ben Nicholson
Ben Nicholson
Numerade Educator
04:43

Problem 59

A diffraction grating with a width of $2.0 \mathrm{~cm}$ contains 1000 lines $/ \mathrm{cm}$ across that width. For an incident wavelength of $600 \mathrm{~nm}$, what is the smallest wavelength difference this grating can resolve in the second order?

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:31

Problem 60

The $D$ line in the spectrum of sodium is a doublet with wavelengths 589.0 and $589.6 \mathrm{~nm}$. Calculate the minimum number of lines needed in a grating that will resolve this doublet in the second-order spectrum.

Keshav Singh
Keshav Singh
Numerade Educator
07:26

Problem 61

With a particular grating the sodium doublet $(589.00 \mathrm{~nm}$ and $589.59 \mathrm{~nm}$ ) is viewed in the third order at $10^{\circ}$ to the normal and is barely resolved. Find (a) the grating spacing and (b) the total width of the rulings.

Paul A.
Paul A.
California State Polytechnic University, Pomona
04:08

Problem 62

A diffraction grating illuminated by monochromatic light normal to the grating produces a certain line at angle $\theta$. (a) What is the product of that line's half-width and the grating's resolving power? (b) Evaluate that product for the first order of a grating of slit separation $900 \mathrm{~nm}$ in light of wavelength $600 \mathrm{~nm}$.

Ben Nicholson
Ben Nicholson
Numerade Educator
14:42

Problem 63

Assume that the limits of the visible spectrum are arbitrarily chosen as 430 and $680 \mathrm{~nm}$. Calculate the number of rulings per millimeter of a grating that will spread the first-order spectrum through an angle of $20.0^{\circ}$.

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:00

Problem 64

What is the smallest Bragg angle for $\mathrm{X}$ rays of wavelength $30 \mathrm{pm}$ to reflect from reflecting planes spaced $0.30 \mathrm{~nm}$ apart in a calcite crystal?

Ben Nicholson
Ben Nicholson
Numerade Educator
08:13

Problem 65

An x-ray beam of wavelength $A$ undergoes first-order reflection (Bragg law diffraction) from a crystal when its angle of incidence to a crystal face is $23^{\circ}$, and an x-ray beam of wavelength $97 \mathrm{pm}$ undergoes third-order reflection when its angle of incidence to that face is $60^{\circ}$. Assuming that the two beams reflect from the same family of reflecting planes, find (a) the interplanar spacing and (b) the wavelength $A$.

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:05

Problem 66

An x-ray beam of a certain wavelength is incident on an $\mathrm{NaCl}$ crystal, at $30.0^{\circ}$ to a certain family of reflecting planes of spacing $39.8 \mathrm{pm}$. If the reflection from those planes is of the first order, what is the wavelength of the $\mathrm{x}$ rays?

Keshav Singh
Keshav Singh
Numerade Educator
06:46

Problem 67

Figure 36.17 is a graph of intensity versus angular position $\theta$ for the diffraction of an x-ray beam by a crystal. The horizontal scale is set by $\theta_s=2.00^{\circ}$. The beam consists of two wavelengths, and the spacing between the reflecting planes is $0.94 \mathrm{~nm}$. What are the (a) shorter and (b) longer wavelengths in the beam?
(Figure Cant Copy)
Figure 36.17 Problem 67.

Paul A.
Paul A.
California State Polytechnic University, Pomona
02:23

Problem 68

If first-order reflection occurs in a crystal at Bragg angle $3.4^{\circ}$, at what Bragg angle does second-order reflection occur from the same family of reflecting planes?

Ben Nicholson
Ben Nicholson
Numerade Educator
05:13

Problem 69

X rays of wavelength $0.12 \mathrm{~nm}$ are found to undergo second-order reflection at a Bragg angle of $28^{\circ}$ from a lithium fluoride crystal. What is the interplanar spacing of the reflecting planes in the crystal?

Paul A.
Paul A.
California State Polytechnic University, Pomona
02:18

Problem 70

In Fig. 36.18, first-order reflection from the reflection planes shown occurs when an $\mathrm{x}$-ray beam of wavelength $0.260 \mathrm{~nm}$ makes an angle $\theta=63.8^{\circ}$ with the top face of the crystal. What is the unit cell size $a_0$ ?
(Figure Cant Copy)
Figure 36.18 Problem 70.

Ben Nicholson
Ben Nicholson
Numerade Educator
04:55

Problem 71

In Fig. 36.19, let a beam of $x$ rays of wavelength $0.125 \mathrm{~nm}$ be incident on an $\mathrm{NaCl}$ crystal at angle $\theta=45.0^{\circ}$ to the top face of the crystal and a family of reflecting planes. Let the reflecting planes have separation $d=0.252 \mathrm{~nm}$. The crystal is turned through angle $\phi$ around an axis perpendicular to the plane of the page until these reflecting planes give diffraction maxima. What are the (a) smaller and (b) larger value of $\phi$ if the crystal is turned clockwise and the (c) smaller and (d) larger value of $\phi$ if it is turned counterclockwise?(Figure Cant Copy)
Figure 36.19 Problems 71 and 72 .

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:30

Problem 72

In Fig. 36.19, an x-ray beam of wavelengths from 95.0 to $140 \mathrm{pm}$ is incident at $\theta=45.0^{\circ}$ to a family of reflecting planes with spacing $d=275 \mathrm{pm}$. What are the (a) longest wavelength $\lambda$ and (b) associated order number $m$ and the (c) shortest $\lambda$ and (d) associated $m$ of the intensity maxima in the diffraction of the beam?
(Figure Cant Copy)
Figure 36.19 Problems 71 and 72 .

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
16:00

Problem 73

Consider a two-dimensional square crystal structure, such as one side of the structure shown in Fig. 36.7.2a. The largest interplanar spacing of reflecting planes is the unit cell size $a_0$. Calculate and sketch the (a) second largest, (b) third largest, (c) fourth largest, (d) fifth largest, and (e) sixth largest interplanar spacing. (f) Show that your results in (a) through (e) are consistent with the general formula
$$
d=\frac{a_0}{\sqrt{h^2+k^2}}
$$
where $h$ and $k$ are relatively prime integers (they have no common factor other than unity).

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:36

Problem 74

An astronaut in a space shuttle claims she can just barely resolve two point sources on Earth's surface, $160 \mathrm{~km}$ below. Calculate their (a) angular and (b) linear separation, assuming ideal conditions. Take $\lambda=540 \mathrm{~nm}$ and the pupil diameter of the astronaut's eye to be $5.0 \mathrm{~mm}$.

Ben Nicholson
Ben Nicholson
Numerade Educator
07:15

Problem 75

Visible light is incident perpendicularly on a diffraction grating of 200 rulings $/ \mathrm{mm}$. What are the (a) longest, (b) second longest, and (c) third longest wavelengths that can be associated with an intensity maximum at $\theta=30.0^{\circ}$ ?

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:10

Problem 76

A beam of light consists of two wavelengths, $590.159 \mathrm{~nm}$ and $590.220 \mathrm{~nm}$, that are to be resolved with a diffraction grating. If the grating has lines across a width of $3.80 \mathrm{~cm}$, what is the minimum number of lines required for the two wavelengths to be resolved in the second order?

Ben Nicholson
Ben Nicholson
Numerade Educator
10:36

Problem 77

In a single-slit diffraction experiment, there is a minimum of intensity for orange light $(\lambda=600 \mathrm{~nm})$ and a minimum of intensity for blue-green light $(\lambda=500 \mathrm{~nm})$ at the same angle of 1.00 mrad. For what minimum slit width is this possible?

Paul A.
Paul A.
California State Polytechnic University, Pomona
04:15

Problem 78

A double-slit system with individual slit widths of $0.030 \mathrm{~mm}$ and a slit separation of $0.18 \mathrm{~mm}$ is illuminated with $500 \mathrm{~nm}$ light directed perpendicular to the plane of the slits. What is the total number of complete bright fringes appearing between the two first-order minima of the diffraction pattern? (Do not count the fringes that coincide with the minima of the diffraction pattern.)

Narayan Hari
Narayan Hari
Numerade Educator
08:18

Problem 79

Consider a diffraction grating that has resolving power $R=\lambda_{\text {avg }} / \Delta \lambda=N m$. (a) Show that the corresponding frequency range $\Delta f$ that can just be resolved is given by $\Delta f=c / N m \lambda$. (b) From Fig. 36.5.5, show that the times required for light to travel along the ray at the bottom of the figure and the ray at the top differ by $\Delta t=(N d / c) \sin \theta$. (c) Show that $(\Delta f)(\Delta t)=1$, this relation being independent of the various grating parameters. Assume $N=1$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:34

Problem 80

The pupil of a person's eye has a diameter of $5.00 \mathrm{~mm}$. According to Rayleigh's criterion, what distance apart must two small objects be if their images are just barely resolved when they are $250 \mathrm{~mm}$ from the eye? Assume they are illuminated with light of wavelength $500 \mathrm{~nm}$.

Ben Nicholson
Ben Nicholson
Numerade Educator
09:13

Problem 81

Light is incident on a grating at an angle $\psi$ as shown in Fig. 36.20. Show that bright fringes occur at angles $\theta$ that satisfy the equation
$$
d(\sin \psi+\sin \theta)=m \lambda, \text { for } m=0,1,2, \ldots .
$$
(Compare this equation with Eq. 36.5.1.) Only the special case $\psi=0$ has been treated in this chapter.
(Figure Cant Copy)
Figure 36.20 Problem 81.

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:35

Problem 82

A grating with $d=1.50 \mu \mathrm{m}$ is illuminated at various angles of incidence by light of wavelength $600 \mathrm{~nm}$. Plot, as a function of the angle of incidence $\left(0\right.$ to $\left.90^{\circ}\right)$, the angular deviation of the first-order maximum from the incident direction. (See Problem 81.$)$

Ben Nicholson
Ben Nicholson
Numerade Educator
09:13

Problem 83

In two-slit interference, if the slit separation is $14 \mu \mathrm{m}$ and the slit widths are each $2.0 \mu \mathrm{m}$, (a) how many two-slit maxima are in the central peak of the diffraction envelope and (b) how many are in either of the first side peak of the diffraction envelope?

Paul A.
Paul A.
California State Polytechnic University, Pomona
04:39

Problem 84

In a two-slit interference pattern, what is the ratio of slit separation to slit width if there are 17 bright fringes within the central diffraction envelope and the diffraction minima coincide with two-slit interference maxima?

Ben Nicholson
Ben Nicholson
Numerade Educator
08:17

Problem 85

A beam of light with a narrow wavelength range centered on $450 \mathrm{~nm}$ is incident perpendicularly on a diffraction grating with a width of $1.80 \mathrm{~cm}$ and a line density of 1400 lines $/ \mathrm{cm}$ across that width. For this light, what is the smallest wavelength difference this grating can resolve in the third order?

Paul A.
Paul A.
California State Polytechnic University, Pomona
00:42

Problem 86

If you look at something $40 \mathrm{~m}$ from you, what is the smallest length (perpendicular to your line of sight) that you can resolve, according to Rayleigh's criterion? Assume the pupil of your eye has a diameter of $4.00 \mathrm{~mm}$, and use $500 \mathrm{~nm}$ as the wavelength of the light reaching you.

Ben Nicholson
Ben Nicholson
Numerade Educator
08:36

Problem 87

Two yellow flowers are separated by $60 \mathrm{~cm}$ along a line perpendicular to your line of sight to the flowers. How far are you from the flowers when they are at the limit of resolution according to the Rayleigh criterion? Assume the light from the flowers has a single wavelength of $550 \mathrm{~nm}$ and that your pupil has a diameter of $5.5 \mathrm{~mm}$.

Paul A.
Paul A.
California State Polytechnic University, Pomona
00:52

Problem 88

In a single-slit diffraction experiment, what must be the ratio of the slit width to the wavelength if the second diffraction minima are to occur at an angle of $37.0^{\circ}$ from the center of the diffraction pattern on a viewing screen?

Ben Nicholson
Ben Nicholson
Numerade Educator
05:50

Problem 89

A diffraction grating $3.00 \mathrm{~cm}$ wide produces the second order at $33.0^{\circ}$ with light of wavelength $600 \mathrm{~nm}$. What is the total number of lines on the grating?

Paul A.
Paul A.
California State Polytechnic University, Pomona
01:36

Problem 90

A single-slit diffraction experiment is set up with light of wavelength $420 \mathrm{~nm}$, incident perpendicularly on a slit of width $5.10 \mu \mathrm{m}$. The viewing screen is $3.20 \mathrm{~m}$ distant. On the screen, what is the distance between the center of the diffraction pattern and the second diffraction minimum?

Ben Nicholson
Ben Nicholson
Numerade Educator
03:57

Problem 91

A diffraction grating has 8900 slits across $1.20 \mathrm{~cm}$. If light with a wavelength of $500 \mathrm{~nm}$ is sent through it, how many orders (maxima) lie to one side of the central maximum?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:57

Problem 92

In an experiment to monitor the Moon's surface with a light beam, pulsed radiation from a ruby laser $(\lambda=0.69 \mu \mathrm{m})$ was directed to the Moon through a reflecting telescope with a mirror radius of $1.3 \mathrm{~m}$. A reflector on the Moon behaved like a circular flat mirror with radius $10 \mathrm{~cm}$, reflecting the light directly back toward the telescope on Earth. The reflected light was then detected after being brought to a focus by this telescope. Approximately what fraction of the original light energy was picked up by the detector? Assume that for each direction of travel all the energy is in the central diffraction peak.

Ben Nicholson
Ben Nicholson
Numerade Educator
02:51

Problem 93

In June 1985, a laser beam was sent out from the Air Force Optical Station on Maui, Hawaii, and reflected back from the shuttle Discovery as it sped by $354 \mathrm{~km}$ overhead. The diameter of the central maximum of the beam at the shuttle position was said to be $9.1 \mathrm{~m}$, and the beam wavelength was $500 \mathrm{~nm}$. What is the effective diameter of the laser aperture at the Maui ground station? (Hint: A laser beam spreads only because of diffraction; assume a circular exit aperture.)

Prabhu Ramji
Prabhu Ramji
Numerade Educator
01:16

Problem 94

A diffraction grating $1.00 \mathrm{~cm}$ wide has 10000 parallel slits. Monochromatic light that is incident normally is diffracted through $30^{\circ}$ in the first order. What is the wavelength of the light?

Ben Nicholson
Ben Nicholson
Numerade Educator
11:41

Problem 95

If you double the width of a single slit, the intensity of the central maximum of the diffraction pattern increases by a factor of 4 , even though the energy passing through the slit only doubles. Explain this quantitatively.

Paul A.
Paul A.
California State Polytechnic University, Pomona
00:53

Problem 96

When monochromatic light is incident on a slit $22.0 \mu \mathrm{m}$ wide, the first diffraction minimum lies at $1.80^{\circ}$ from the direction of the incident light. What is the wavelength?

Ben Nicholson
Ben Nicholson
Numerade Educator
01:51

Problem 97

A spy satellite orbiting at $160 \mathrm{~km}$ above Earth's surface has a lens with a focal length of $3.6 \mathrm{~m}$ and can resolve objects on the ground as small as $30 \mathrm{~cm}$. For example, it can easily measure the size of an aircraft's air intake port. What is the effective diameter of the lens as determined by diffraction consideration alone? Assume $\lambda=550 \mathrm{~nm}$.

Keshav Singh
Keshav Singh
Numerade Educator
01:02

Problem 98

Epidural with fiber Bragg grating. A fiber Bragg grating is an optical fiber that has had its core treated with ultraviolet light so that it has a periodic variation in its index of refraction, with a certain spacing $d$. Along a few millimeters, there are "lines" with a greater index than the rest of the core (Fig. 36.21a). When light over a broad wavelength range is sent into the fiber, one wavelength, called the Bragg wavelength $\lambda_B$, is reflected and the rest is transmitted. The value of $\lambda_B$ depends on $d$. If a force $F$ decreases the length of the grating, decreasing $d$, then $\lambda_{\mathrm{B}}$ decreases. Thus, the grating acts as a strain gauge. Figure $36.21 b$ gives the change $\Delta \lambda_{\mathrm{B}}$ in the Bragg wavelength versus applied force $F$.

Recent research suggests that a fiber Bragg grating could be used in robotic assisted surgery in an epidural procedure in which a needle is inserted into the epidural space of the spinal column to release an anesthetic fluid. The surgeon first inserts the needle into the back and then manually monitors the force magnitude required to advance the needle. This tricky procedure requires much practice so that the surgeon knows when the needle has reached the epidural space and not overshot it, an error that could result in serious complications. Figure $36.21 c$ is a graph of the force magnitude $F$ versus displacement $x$ of the needle tip in a typical epidural procedure. (The line segments have been straightened somewhat from the original data.) (1) As $x$ increases from 0 , the skin resists the needle, but at $x=$ $8.0 \mathrm{~mm}$ the force is finally great enough to pierce the skin, and then the required force decreases. (2) Next, the needle finally pierces the interspinous ligament at $x=18 \mathrm{~mm}$ and (3) the relatively tough ligamentum flavum at $x=30 \mathrm{~mm}$. (4) As the needle then enters the epidural space, the force drops sharply. A new surgeon must learn this pattern of force magnitude versus displacement to recognize when to stop pushing on the needle.
If a fiber Bragg grating could be incorporated into an epidural needle, an automated system could monitor $\Delta \hat{\lambda}_B$ to determine when the needle is properly placed. For the plot of Fig. $36.21 b$, what is $\Delta \lambda_{\mathrm{B}}$ for the peak force at (a) $x=8.0 \mathrm{~mm}$, (b) $x=18 \mathrm{~mm}$, and (c) $x=30 \mathrm{~mm}$ ?
(Figure Cant Copy)
Figure 36.21 Problem 98.

Mayukh Banik
Mayukh Banik
Numerade Educator