Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 35

Interference - all with Video Answers

Educators

Chapter Questions

Problem 1

In Fig. $35-31$, a light wave along ray $r_{1}$ reflects once from a mirror and a light wave along ray $r_{2}$ reflects twice from that same mirror and once from a tiny mirror at distance $L$ from the bigger mirror. (Neglect the slight tilt Figure 35-31 Problems 1 and 2 . of the rays) The waves have wavelength $620 \mathrm{~nm}$ and are initially in phase. (a) What is the smallest value of $L$ that puts the final light waves exactly out of phase? (b) With the tiny mirror initially at that value of $L$, how far must it be moved away from the bigger mirror to again put the final waves out of phase?

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

In Fig. $35-31$, a light wave along ray $r_{1}$ reflects once from a mirror and a light wave along ray $r_{2}$ reflects twice from that same mirror and once from a tiny mirror at distance $L$ from the bigger mirror. (Neglect the slight tilt of the rays.) The waves have wavelength $\lambda$ and are initially exactly out of phase. What are the
(a) smallest, (b) second smallest, and (c) third smallest values of $L / \lambda$ that result in the final waves being exactly in phase?

Keshav Singh

Numerade Educator

Problem 3

In Fig. $35-4$, assume that two waves of light in air, of wavelength $400 \mathrm{~nm}$, are initially in phase. One travels through a glass layer of index of refraction $n_{1}=1.60$ and thickness $L .$ The other travels through an equally thick plastic layer of index of refraction $n_{2}=1.50 .$ (a) What is the smallest value $L$ should have if the waves are to end up with a phase difference of $5.65 \mathrm{rad} ?(\mathrm{~b})$ If the waves arrive at some common point with the same amplitude, is their interference fully constructive, fully destructive, intermediate but closer to fully constructive, or intermediate but closer to fully destructive?

st

Sebastien Tawa

Numerade Educator

Problem 4

In Fig. $35-32 a$, a beam of light in material 1 is incident on a boundary at an angle of $30^{\circ}$. The extent to which the light is bent due to refraction depends, in part, on the index of refraction $n_{2}$ of material 2. Figure $35-32 b$ gives the angle of refraction $\theta_{2}$ versus $n_{2}$ for a range of possible $n_{2}$ values, from $n_{a}=1.30$ to $n_{b}=1.90 .$ What is the speed of light in material 1 ?

Ben Nicholson

Numerade Educator

Problem 5

How much faster, in meters per second, does light travel in sapphire than in diamond? See Table $33-1$.

st

Sebastien Tawa

Numerade Educator

Problem 6

The wavelength of yellow sodium light in air is $589 \mathrm{~nm}$. (a) What is its frequency? (b) What is its wavelength in glass whose index of refraction is $1.52 ?$ (c) From the results of (a) and (b), find its speed in this glass.

Ben Nicholson

Numerade Educator

Problem 7

The speed of yellow light (from a sodium lamp) in a certain liquid is measured to be $1.92 \times 10^{8} \mathrm{~m} / \mathrm{s}$. What is the index of refraction of this liquid for the light?

st

Sebastien Tawa

Numerade Educator

Problem 8

In Fig. $35-33$, two light pulses 2 are sent through layers of plastic with thicknesses of either $L$ or $2 L$ as shown and indexes of refraction
$n_{1}=1.55, n_{2}=1.70, n_{3}=1.60, n_{4}=1$
$1.45, n_{5}=1.59, n_{6}=1.65$, and $n_{7}=$
1.50. (a) Which pulse travels Figure 35-33 Problem 8. through the plastic in less time?
(b) What multiple of $L / c$ gives the difference in the traversal $\operatorname{tin}$

Keshav Singh

Numerade Educator

Problem 9

In Fig. $35-4$, assume that the two light waves, of wavelength $620 \mathrm{~nm}$ in air, are initially out of phase by $\pi$ rad. The indexes of refraction of the media are $n_{1}=1.45$ and $n_{2}=1.65 .$ What are the
(a) smallest and (b) second smallest value of $L$ that will put the waves exactly in phase once they pass through $\quad$ Air $\left|n_{1}\right| \quad n_{3}|\quad| \quad$ Air the two media?

Keshav Singh

Numerade Educator

Problem 10

In Fig. 35-34, a light ray is incident at angle $\theta_{1}=50^{\circ}$ on a series of five transparent layers with parallel boundaries. For layers 1 and $3, L_{1}=$ $20 \mu \mathrm{m}, L_{3}=25 \mu \mathrm{m}, n_{1}=$
Figure 35-34 Problem 10.
1.6, and $n_{3}=1.45$. (a) At what angle does the light emerge back into air at the right? (b) How much time does the light take to travel through layer 3 ?

Keshav Singh

Numerade Educator

Problem 11

Suppose that the two waves in Fig. $35-4$ have wavelength $\lambda=500 \mathrm{~nm}$ in air. What multiple of $\lambda$ gives their phase difference when they emerge if (a) $n_{1}=1.50, n_{2}=1.60$, and $L=8.50 \mu \mathrm{m}$;
(b) $n_{1}=1.62, n_{2}=1.72$, and $L=8.50 \mu \mathrm{m} ;$ and
(c) $n_{1}=1.59, n_{2}=$
1.79, and $L=3.25 \mu \mathrm{m}$ ? (d) Suppose that in each of these three situations the waves arrive at a common point (with the same amplitude) after emerging. Rank the situations according to the brightness the waves produce at the common point.

st

Sebastien Tawa

Numerade Educator

Problem 12

In Fig. $35-35$, two light rays go through different paths by reflecting $\mathrm{B}$ from the various flat surfaces shown. The light waves have a wavelength of $420.0 \mathrm{~nm}$ and are initially in phase. What are the (a) smallest and (b) second smallest value of distance $L$ that will put the waves ex- $\mathrm{R}$ actly out of phase as they emerge from the region?

Keshav Singh

Numerade Educator

Problem 13

Two waves of light in air, of wavelength $\lambda=600.0 \mathrm{~nm}$, are initially in phase. They then both travel through a layer of plastic as shown in Fig. $35-36$, with $L_{1}=4.00 \mu \mathrm{m}$, $L_{2}=3.50 \mu \mathrm{m}, n_{1}=1.40$, and $n_{2}=1.60$
(a) What multiple of $\lambda$ gives their phase difference after they both have emerged from the layers? (b) If the waves later arrive at some common point with the same amplitude, is their interference fully constructive, fully de- Figure 35-36 Problem structive, intermediate but closer to fully constructive, or intermediate but closer to fully destructive?

Keshav Singh

Numerade Educator

Problem 14

In a double-slit arrangement the slits are separated by a distance equal to 100 times the wavelength of the light passing through the slits.
(a) What is the angular separation in radians between the central maximum and an adjacent maximum? (b) What is the distance between these maxima on a screen $50.0 \mathrm{~cm}$ from the slits?

Ben Nicholson

Numerade Educator

Problem 15

A double-slit arrangement produces interference fringes for sodium light $(\lambda=589 \mathrm{~nm})$ that have an angular separation of $3.50 \times 10^{-3}$ rad. For what wavelength would the angular separation be $10.0 \%$ greater?

st

Sebastien Tawa

Numerade Educator

Problem 16

A double-slit arrangement produces interference fringes for sodium light $(\lambda=589 \mathrm{~nm})$ that are $0.20^{\circ}$ apart. What is the angular separation if the arrangement is immersed in water $(n=1.33)$ ?

Ben Nicholson

Numerade Educator

Problem 17

In Fig. $35-37$, two radiofrequency point sources $S_{1}$ and $S_{2}$, separated by distance $d=2.0 \mathrm{~m}$, are radiating in phase with $\lambda=0.50 \mathrm{~m} .$ A Figure 35-37 Problems 17 detector moves in a large circular path and $22 .$ around the two sources in a plane containing them. How many maxima does it detect?

Keshav Singh

Numerade Educator

Problem 18

In the two-slit experiment of Fig. $35-10$, let angle $\theta$ be $20.0^{a}$, the slit separation be $4.24 \mu \mathrm{m}$, and the wavelength be $\lambda=500 \mathrm{~nm} .$ (a) What multiple of $\lambda$ gives the phase difference between the waves of rays $r_{1}$ and $r_{2}$ when they arrive at point $P$ on the distant screen? (b) What is the phase difference in radians? (c) Determine where in the interference pattern point $P$ lies by giving the maximum or minimum on which it lies, or the maximum and minimum between which it lies.

Ben Nicholson

Numerade Educator

Problem 19

Suppose that Young's experiment is performed with blue-green light of wavelength $500 \mathrm{~nm}$. The slits are $1.20 \mathrm{~mm}$ apart, and the viewing screen is $5.40 \mathrm{~m}$ from the slits. How far apart are the bright fringes near the center of the interference pattern?

st

Sebastien Tawa

Numerade Educator

Problem 20

Monochromatic green light, of wavelength $550 \mathrm{~nm}$, illuminates two parallel narrow slits $7.70 \mu \mathrm{m}$ apart. Calculate the angular deviation ( $\theta$ in Fig. $35-10$ ) of the third-order $(m=3)$ bright fringe
(a) in radians and (b) in degrees.

Ben Nicholson

Numerade Educator

Problem 21

In a double-slit experiment, the distance between slits is $5.0 \mathrm{~mm}$ and the slits are $1.0 \mathrm{~m}$ from the screen. Two interference patterns can be seen on the screen: one due to light of wavelength $480 \mathrm{~nm}$, and the other due to light of wavelength $600 \mathrm{~nm}$. What is the separation on the screen between the third-order $(m=3)$ bright fringes of the two interference patterns?

st

Sebastien Tawa

Numerade Educator

Problem 22

In Fig. $35-37$, two isotropic point sources $S_{1}$ and $S_{2}$ emit identical light waves in phase at wavelength $\lambda$. The sources lie at separation $d$ on an $x$ axis, and a light detector is moved in a circle of large radius around the midpoint between them. It detects 30 points of zero intensity, including two on the $x$ axis, one of them to the left of the sources and the other to the right of the sources. What is the value of $d / \lambda$ ?

Ben Nicholson

Numerade Educator

Problem 23

In Fig. $35-38$, sources $A$ and $\bar{B}$ emit long-range radio waves of wavelength $400 \mathrm{~m}$, with the phase of the emission from $A$ ahead of that from source $B$ by $90^{\circ}$. The distance $r_{A}$ from $A$ to detector $D$ is greater than the corresponding distance $r_{B}$ by $100 \mathrm{~m}$. What is the phase difference of the waves at $D$ ?

Keshav Singh

Numerade Educator

Problem 24

In Fig. $35-39$, two isotropic point sources $S_{1}$ and $S_{2}$ emit light in phase at wavelength $\lambda$ and at the same amplitude. The sources are separated by distance $2 d=6.00 \lambda$. They lie on an axis that is parallel to an $x$ axis, which runs along a viewing screen at distance $D=$ $20.0 \lambda$. The origin lies on the perpendicular bisector between the sources. The Figure 35-39 Problem 24. figure shows two rays reaching point $P$ on the screen, at position $x_{P}$, (a) At what value of $x_{p}$ do the rays have the minimum possible phase difference? (b) What multiple of $\lambda$ gives that minimum phase difference? (c) At what value of $x_{P}$ do the rays have the maximum possible phase difference? What multiple of $\lambda$ gives (d) that maximum phase difference and (e) the phase difference when $x_{P}=6.00 \lambda$ ? (f) When $x_{P}=6.00 \lambda$, is the resulting intensity at point $P$ maximum, minimum, intermediate but closer to maximum, or intermediate but closer to minimum?

Ben Nicholson

Numerade Educator

Problem 25

In Fig. $35-40$, two isotropic point sources of light $\left(S_{1}\right.$ and $S_{2}$ ) are separated by distance $2.70 \mu \mathrm{m}$ along a $y$ axis and emit in phase at wavelength $900 \mathrm{~nm}$ and at the same amplitude. $\mathrm{A}$ light detector is located at point $P$ at Figure 35-40 Problems 25 coordinate $x_{p}$ on the $x$ axis. What is and 28 . the greatest value of $x_{P}$ at which the detected light is minimum due to destructive interference?

Keshav Singh

Numerade Educator

Problem 26

In a double-slit experiment, the fourth-order maximum for a wavelength of $450 \mathrm{~nm}$ occurs at an angle of $\theta=90^{\circ} .$ (a) What range of wavelengths in the visible range $(400 \mathrm{~nm}$ to $700 \mathrm{~nm})$ are not present in the third-order maxima? To eliminate all visible light in the fourth-order maximum, (b) should the slit separation be increased or decreased and (c) what least change is needed?

Ben Nicholson

Numerade Educator

Problem 27

A thin flake of mica $(n=1.58)$ is used to cover one slit of a double-slit interference arrangement. The central point on the viewing screen is now occupied by what had been the seventh bright side fringe $(m=7)$. If $\lambda=550 \mathrm{~nm}$, what is the thickness of the mica?

st

Sebastien Tawa

Numerade Educator

Problem 28

Figure $35-40$ shows two isotropic point sources of light $\left(S_{1}\right.$ and $\left.S_{2}\right)$ that emit in phase at wavelength $400 \mathrm{~nm}$ and at the same amplitude. A detection point $P$ is shown on an $x$ axis that extends through source $S_{1}$. The $\quad$ Figure 35-41 Problem 28 . phase difference $\phi$ between the light arriving at point $P$ from the two sources is to be measured as $P$ is moved along the $x$ axis from $x=0$ out to $x=+\infty$. The results out to $x_{5}=10 \times 10^{-7} \mathrm{~m}$ are given in Fig. $35-41 .$ On the way out to $+\infty$, what is the greatest value of $x$ at which the light arriving at $P$ from $S_{1}$ is exactly out of phase with the light arriving at $P$ from $S_{2} ?$

Keshav Singh

Numerade Educator

Problem 29

Two waves of the same frequency have amplitudes $1.00$ and $2.00 .$ They interfere at a point where their phase difference is $60.0^{\circ} .$ What is the resultant amplitude?

Dading Chen

Numerade Educator

Problem 30

Find the sum $y$ of the following quantities:
$$
y_{1}=10 \sin \omega t \quad \text { and } \quad y_{2}=8.0 \sin \left(\omega t+30^{\circ}\right) .
$$

Ben Nicholson

Numerade Educator

Problem 31

Add the quantities $y_{1}=10 \sin \omega t, y_{2}=15 \sin \left(\omega t+30^{\circ}\right)$, and $y_{3}=5.0 \sin \left(\omega t-45^{\circ}\right)$ using the phasor method.

st

Sebastien Tawa

Numerade Educator

Problem 32

In the double-slit experiment of Fig. $35-10$, the electric fields of the waves arriving at point $P$ are given by
$$
\begin{aligned}
&E_{1}=(2.00 \mu \mathrm{V} / \mathrm{m}) \sin \left[\left(1.26 \times 10^{15}\right) t\right] \\
&E_{2}=(2.00 \mu \mathrm{V} / \mathrm{m}) \sin \left[\left(1.26 \times 10^{15}\right) t+39.6 \mathrm{rad}\right]
\end{aligned}
$$
where time $t$ is in seconds. (a) What is the amplitude of the resultant electric field at point $P ?$ (b) What is the ratio of the intensity $I_{P}$ at point $P$ to the intensity $I_{\text {cen }}$ at the center of the interference pattern? (c) Describe where point $P$ is in the interference pattern by giving the maximum or minimum on which it lies, or the maximum and minimum between which it lies. In a phasor diagram of the electric fields, (d) at what rate would the phasors rotate around the origin and (e) what is the angle between the phasors?

Keshav Singh

Numerade Educator

Problem 33

Three electromagnetic waves travel through a certain point $P$ along an $x$ axis. They are polarized parallel to a $y$ axis, with the following variations in their amplitudes. Find their resultant at $P$.
$$
\begin{aligned}
&E_{1}=(10.0 \mu \mathrm{V} / \mathrm{m}) \sin \left[\left(2.0 \times 10^{14} \mathrm{rad} / \mathrm{s}\right) t\right] \\
&E_{2}=(5.00 \mu \mathrm{V} / \mathrm{m}) \sin \left[\left(2.0 \times 10^{14} \mathrm{rad} / \mathrm{s}\right) t+45.0^{\circ}\right] \\
&E_{3}=(5.00 \mu \mathrm{V} / \mathrm{m}) \sin \left[\left(2.0 \times 10^{14} \mathrm{rad} / \mathrm{s}\right) t-45.0^{\circ}\right]
\end{aligned}
$$

Keshav Singh

Numerade Educator

Problem 34

In the double-slit experiment of Fig. $35-10$, the viewing screen is at distance $D=4.00 \mathrm{~m}$, point $P$ lies at distance $y=20.5$ $\mathrm{cm}$ from the center of the pattern, the slit separation $d$ is $4.50 \mu \mathrm{m}$, and the wavelength $\lambda$ is $580 \mathrm{~nm}$. (a) Determine where point $P$ is in the interference pattern by giving the maximum or minimum on which it lies, or the maximum and minimum between which it lies.
(b) What is the ratio of the intensity $I_{P}$ at point $P$ to the intensity $I_{\text {cen }}$ at the center of the pattern?

Ben Nicholson

Numerade Educator

Problem 35

We wish to coat flat glass $(n=$ 1.50) with a transparent material $(n=1.25)$ so that reflection of light at wavelength 600 $\mathrm{nm}$ is eliminated by interference. What minimum thickness can the coating have to do this?

st

Sebastien Tawa

Numerade Educator

Problem 36

A 600 -nm-thick soap film $(n=1.40)$ in air is illuminated with white light in a direction perpendicular to the film. For how many different wavelengths in the 300 to $700 \mathrm{~nm}$ range is there (a) fully constructive interference and (b) fully destructive interference in the reflected light?

Ben Nicholson

Numerade Educator

Problem 37

The rhinestones in costume jewelry are glass with index of refraction $1.50 .$ To make them more reflective, they are often coated with a layer of silicon monoxide of index of refraction $2.00 .$ What is the minimum coating thickness needed to ensure that light of wavelength $560 \mathrm{~nm}$ and of perpendicular incidence will be reflected from the two surfaces of the coating with fully constructive interference?

st

Sebastien Tawa

Numerade Educator

Problem 38

White light is sent downward onto a horizontal thin film that is sandwiched between two materials. The indexes of refraction are $1.80$ for the top material, $1.70$ for the thin film, and $1.50$ for the bottom material. The film thickness is $5.00 \times 10^{-7} \mathrm{~m}$. Of the visible wavelengths $(400$ to $700 \mathrm{~nm})$ that result in fully constructive interference at an observer above the film, which is the (a) longer and (b) shorter wavelength? The materials and film are then heated so that the film thickness increases. (c) Does the light resulting in fully constructive interference shift toward longer or shorter wavelengths?

Ben Nicholson

Numerade Educator

Problem 39

Light of wavelength $624 \mathrm{~nm}$ is incident perpendicularly on a soap film $(n=1.33)$ suspended in air. What are the (a) least and (b) second least thicknesses of the film for which the reflections from the film undergo fully constructive interference?

st

Sebastien Tawa

Numerade Educator

Problem 40

A thin film of acetone $(n=1.25)$ coats a thick glass plate $(n=1.50)$. White light is incident normal to the film. In the reflections, fully destructive interference occurs at $600 \mathrm{~nm}$ and fully constructive interference at $700 \mathrm{~nm}$. Calculate the thickness of the acetone film.

Ben Nicholson

Numerade Educator

Problem 41

Reflection by thin layers. In Fig. $35-42$, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3 . (The rays are tilted only for clarity.) The waves of rays $r_{1}$ and $r_{2}$ interfere, and here we consider the type of interference to be either Figure 35-42 Problems 41
this situation, each problem in Table 35 2 refers to the indexes of refraction $n_{1}, n_{2}$, and $n_{3}$, the type of interference, the thin-layer thickness $L$ in nanometers, and the wavelength $\lambda$ in nanometers of the light as measured in air. Where $\lambda$ is missing, give the wavelength that is in the visible range. Where $L$ is missing, give the second least thickness or the third least thickness as indicated.

Keshav Singh

Numerade Educator

Problem 41

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 42

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 43

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 44

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 45

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 46

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 47

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 48

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 49

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 50

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 51

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 52

Reflection by Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 53

The reflection of perpendicularly incident white light by a soap film in air has an interference maximum at $600 \mathrm{~nm}$ and a minimum at $450 \mathrm{~nm}$, with no minimum in between. If $n=1.33$ for the film, what is the film thickness, assumed uniform?

st

Sebastien Tawa

Numerade Educator

Problem 54

A plane wave of monochromatic light is incident normally on a uniform thin film of oil that covers a glass plate. The wavelength of the source can be varied continuously. Fully destructive interference of the reflected light is observed for wavelengths of 500 and $700 \mathrm{~nm}$ and for no wavelengths in between. If the index of refraction of the oil is $1.30$ and that of the glass is $1.50$, find the thickness of the oil film.

Keshav Singh

Numerade Educator

Problem 55

A disabled tanker leaks kerosene $(n=1.20)$ into the Persian Gulf, creating a large slick on top of the water $(n=1.30) .$ (a) If you are looking straight down from an airplane, while the Sun is overhead, at a region of the slick where its thickness is $460 \mathrm{~nm}$, for which wavelength(s) of visible light is the reflection brightest because of constructive interference? (b) If you are. scuba diving directly under this same region of the slick, for which wavelength(s) of visible light is the transmitted intensity strongest?

st

Sebastien Tawa

Numerade Educator

Problem 56

A thin film, with a thickness of $272.7 \mathrm{~nm}$ and with air on both sides, is illuminated with a beam of white light. The beam is perpendicular to the film and consists of the full range of wavelengths for the visible spectrum. In the light reflected by the film, light with a wavelength of $600.0 \mathrm{~nm}$ undergoes fully constructive interference. At what wavelength does the reflected light undergo fully destructive interference? (Hint: You must make a reasonable assumption about the index of refraction.)

Keshav Singh

Numerade Educator

Problem 57

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 58

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 59

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 60

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 61

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 62

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 63

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 64

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 65

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 66

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 67

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 68

Transmission Through Thin Layers. See the setup for these problems.

Keshav Singh

Numerade Educator

Problem 69

In Fig. $35-44$, a broad beam of light of wavelength $630 \mathrm{~nm}$ is incident at $90^{\circ}$ on a thin, wedge-shaped film with index of refraction $1.50$. Transmission gives 10 bright and 9 dark fringes along the film's length. What is the left-to-right change in film thickness?

Keshav Singh

Numerade Educator

Problem 70

In Fig. $35-45$, a broad beam of light of wavelength $620 \mathrm{~nm}$ is sent directly downward through the top plate of a pair of glass plates touching at the left end. The air between the plates acts as a thin film, and an interference pattern can be seen from above the plates. Initially, a dark fringe lies at the left end, a bright fringe lies at the right end, and nine dark fringes lie between Figure 35-45 Problems $70-74$. those two end fringes. The plates are then very gradually squeezed together at a constant rate to decrease the angle between them. As a result, the fringe at the right side changes between being bright to being dark every $15.0$
s. (a) At what rate is the spacing between the plates at the right end being changed? (b) By how much has the spacing there changed when both left and right ends have a dark fringe and there are five dark fringes between them?

Keshav Singh

Numerade Educator

Problem 71

In Fig. $35-45$, two microscope slides touch at one end and are separated at the other end. When light of wavelength $500 \mathrm{~nm}$ shines vertically down on the slides, an overhead observer sees an interference pattern on the slides with the dark fringes separated by $1.2 \mathrm{~mm}$. What is the angle between the slides?

Dading Chen

Numerade Educator

Problem 72

In Fig. 35-45, a broad beam of monochromatic light is directed perpendicularly through two glass plates that are held together at one end to create a wedge of air between them. An observer intercepting light reflected from the wedge of air, which acts as a thin film, sees 4001 dark fringes along the length of the wedge. When the air $\lambda$ cant figures the index of refraction of air from these data.

Keshav Singh

Numerade Educator

Problem 73

In Fig. $35-45$, a broad beam of light of wavelength $683 \mathrm{~nm}$ is sent directly downward through the top plate of a pair of glass plates. The plates are $120 \mathrm{~mm}$ long. touch at the left end, and are separated by $48.0 \mu \mathrm{m}$ at the right end. The air between the plates acts as a thin film. How many bright fringes will be seen by an observer looking down through the top plate?

Dading Chen

Numerade Educator

Problem 74

Two rectangular glass plates $(n=$ $1.60$ ) are in contact along one edge and are separated along the opposite edge (Fig. $35-45$ ). Light with a wavelength of 600 $\mathrm{nm}$ is incident perpendicularly onto the top plate. The air between the plates acts as a thin film. Nine dark fringes and eight bright fringes are observed from above the top plate. If the distance between the two plates along the separated edges is increased by $600 \mathrm{~nm}$, how many dark fringes will there then be across the top plate?

Ben Nicholson

Numerade Educator

Problem 75

Figure $35-46 a$ shows a lens with radius of curvature $R$ lying on a flat glass plate and illuminated from above by light with wavelength $\lambda$. Figure $35-46 b$ (a photograph taken from above the lens) shows that circular interference fringes (known as Newton's rings) appear, associated with the variable thickness $\underline{d}$ of the air film between the lens and the plate. Find the radii $r$ of the interference maxima assuming $r / R<1$.
$(a)$

Keshav Singh

Numerade Educator

Problem 76

The lens in a Newton's rings experiment (see Problem 75) has diameter $20 \mathrm{~mm}$ and radius of curvature $R=5.0 \mathrm{~m}$. For $\lambda=589 \mathrm{~nm}$ in air, how many bright rings are produced with the setup (a) in air and (b) immersed in water $(n=1.33)$ ?

Ben Nicholson

Numerade Educator

Problem 77

A Newton's rings apparatus is to be used to determine the radius of curvature of a lens (see Fig. $35-46$ and Problem 75). The radii of the $n$ th and $(n+20)$ th bright rings are found to be $0.162$ and $0.368 \mathrm{~cm}$, respectively, in light of wavelength $546 \mathrm{~nm} .$ Calculate the radius of curvature of the lower surface of the lens.

Dading Chen

Numerade Educator

Problem 78

A thin film of liquid is held in a horizontal circular ring. with air on both sides of the film. A beam of light at wavelength $550 \mathrm{~nm}$ is directed perpendicularly onto the film, and the intensity $I$ of its reflection is monitored. Figure $35-47$ gives intensity $I$ as a function of time $t ;$ the horizontal scale is set by $t_{s}=20.0 \mathrm{~s}$. The intensity changes because of evaporation from the two sides of the film. Assume that the film is flat and has parallel sides, a radius of $1.80 \mathrm{~cm}$, and an index of refraction of $1.40 .$ Also assume that the film's volume decreases at a constant rate. Find that rate.

Ben Nicholson

Numerade Educator

Problem 79

If mirror $M_{2}$ in a Michelson interferometer (Fig. $35-21$ ) is moved through $0.233 \mathrm{~mm}$, a shift of 792 bright fringes occurs. What is the wavelength of the light producing the fringe pattern?

Dading Chen

Numerade Educator

Problem 80

A thin film with index of refraction $n=1.40$ is placed in one arm of a Michelson interferometer, perpendicular to the optical path. If this causes a shift of $7.0$ bright fringes of the pattern produced by light of wavelength $589 \mathrm{~nm}$, what is the film thickness?

Ben Nicholson

Numerade Educator

Problem 81

In Fig. 35-48, an airtight chamber of length $d=$ $5.0 \mathrm{~cm}$ is placed in one of the arms of a Michelson interferometer. (The glass window on each end of the chamber has negligible thickness) Light of wavelength $\quad \lambda=500 \mathrm{~nm}$ is used. Evacuating the air from the chamber causes a shift of 60 bright fringes. From these data and to six significant figures, find the index of refraction of air at atmospheric pressure.

Dading Chen

Numerade Educator

Problem 82

The element sodium can emit light at two wavelengths, $\lambda_{1}=$ $588.9950 \mathrm{~nm}$ and $\lambda_{2}=589.5924 \mathrm{~nm} . \quad$ Figure 35-48 Problem 81 . Light from sodium is being used in a Michelson interferometer (Fig. $35-21$ ). Through what distance must mirror $M_{2}$ be moved if the shift in the fringe pattern for one wavelength is to be $1.00$ fringe more than the shift in the fringe pattern for the other wavelength?

Keshav Singh

Numerade Educator

Problem 83

Two light rays, initially in phase and with a wavelength of $500 \mathrm{~nm}$, go through different paths by reflecting from the various mirrors shown in Fig. $35-49$. (Such a reflection does not itself produce a phase shift.)
(a) What least value of distance $d$ will put the rays exactly out of phase when they emerge from the region? (Ignore the slight tilt of the path for ray 2.) (b) Repeat the question assuming that the entire apparatus is immersed in a protein solution with an index of refraction of $1.38$.

Dading Chen

Numerade Educator

Problem 84

In Figure $35-50$, two isotropic point sources $S_{1}$ and $S_{2}$ emit light in phase at wavelength $\lambda$ and at the same amplitude. The sources are separated by distance $d=6.00 \lambda$ on an $x$ axis. A viewing screen is at distance $D=20.0 \lambda$ from $S_{2}$ and parallel to the $y$ axis. The figure shows two rays reaching point $P$ on Figure 35-50 Problem 84 . the screen, at height $y_{P}$. (a) At what value of $y_{P}$ do the rays have the minimum possible phase difference? (b) What multiple of $\lambda$ gives that minimum phase difference? (c) At what value of $y_{P}$ do the rays have the maximum possible phase difference? What multiple of $\lambda$ gives (d) that maximum phase difference and (e) the phase difference when $y_{P}=d ?$ (f) When $y_{P}=d$, is the resulting intensity at point $P$ maximum, minimum, intermediate but closer to maximum, or intermediate but closer to minimum?

Keshav Singh

Numerade Educator

Problem 85

A double-slit arrangement produces bright interference fringes for sodium light (a distinct yellow light at a wavelength of $\lambda=589 \mathrm{~nm}$ ). The fringes are angularly separated by $0.30^{\circ}$ near the center of the pattern. What is the angular fringe separation if the entire arrangement is immersed in water, which has an index of refraction of $1.33$ ?

Dading Chen

Numerade Educator

Problem 86

In Fig. $35-51 a$, the waves along rays 1 and 2 are initially in phase, with the same wavelength $\lambda$ in air. Ray 2 goes through a material with length $L$ and index of refraction $n .$ The rays are then reflected by mirrors to a common point $P$ on a screen. Suppose that we can vary $n$ from $n=1.0$ to $n=2.5 .$ Suppose also that, from $n=1.0$ to $n=n_{s}=1.5$, the intensity $I$ of the light at point $P$ varies with $n$ as given in Fig. $35-51 b$. At what values of $n$ greater than $1.4$ is intensity $I$ (a) maximum and (b) zero? (c) What multiple of $\lambda$ gives the phase difference between the rays at point $P$ when $n=2.0 ?$

Ben Nicholson

Numerade Educator

Problem 87

In Fig. $35-51 a$, the waves along rays 1 and 2 are initially in phase, with the same wavelength $\lambda$ in air. Ray 2 goes through a material with length $L$ and index of refraction $n$. The rays are then reflected by mirrors to a common point $P$ on a screen. Suppose that we can vary $L$ from 0 to $2400 \mathrm{~nm}$. Suppose 0 (nm) also that, from $L=0$ to $L_{s}=900 \mathrm{~nm}$, the intensity $I$ of the light at point $P$ Figure 35-52 Problem 87 . varies with $L$ as given in Fig. $35-52$. At what values of $L$ greater than $L_{s}$ is intensity $I$ (a) maximum and
(b) zero? (c) What multiple of $\lambda$ gives the phase difference between ray 1 and ray 2 at common point $P$ when $L=1200 \mathrm{~nm}$ ?

Keshav Singh

Numerade Educator

Problem 88

Light of wavelength $700.0 \mathrm{~nm}$ is sent along a route of length $2000 \mathrm{~nm}$. The route is then filled with a medium having an index of refraction of $1.400 .$ In degrees, by how much does the medium phase-shift the light? Give (a) the full shift and (b) the equivalent shift that has a value less than $360^{\circ}$.

Ben Nicholson

Numerade Educator

Problem 89

In Fig. $35-53$, a microwave transmitter at height $a$ above the water level of a wide lake transmits microwaves of wavelength $\lambda$ toward a receiver on the opposite shore, a distance $\bar{x}$ above the water level. The microwaves reflecting from the water interfere with the microwaves arriving directly from the transmitter.

Keshav Singh

Numerade Educator

Problem 90

In Fig. $35-54$, two isotropic point sources $S_{1}$ and $S_{2}$ emit light at wavelength $\lambda=400 \mathrm{~nm} .$ Source $S_{1}$ is located at $y=640$ $\mathrm{nm}$; source $S_{2}$ is located at $y=-640 \mathrm{~nm}$. At point $P_{1}$ (at $x=720 \mathrm{~nm}$ ), the wave from $S_{2}$ arrives ahead of the wave from $S_{1}$ by a phase difference of $0.600 \pi$ rad. (a) What multiple of $\lambda$ gives the phase difference between the waves from the two sources as the waves arrive at point $P_{2}$, which is located at $y=720 \mathrm{~nm} ?$ (The figure is not drawn to scale.) (b) If the waves arrive at $P_{2}$ with equal amplitudes, is the interference there fully constructive, fully destructive, intermediate but closer to fully constructive, or intermediate but closer to fully Shoreline destructive?

Keshav Singh

Numerade Educator

Problem 91

Ocean waves moving at a }}$ speed of $4.0 \mathrm{~m} / \mathrm{s}$ are approaching a beach at angle $\theta_{1}=30^{\circ}$ to the normal, as shown from above in Fig. $35-55 .$ Suppose the water depth changes abruptly at a certain distance from the beach and the wave Figure 35-55 Problem 91. speed there drops to $3.0 \mathrm{~m} / \mathrm{s}$.
(a) Close to the beach, what is the angle $\theta_{2}$ between the direction of wave motion and the normal? (Assume the same law of refraction as for light.) (b) Explain why most waves come in normal to a shore even though at large distances they approach at a variety of angles.

Keshav Singh

Numerade Educator

Problem 92

Figure $35-56 a$ shows two light rays that are initially in phase as they travel upward through a block of plastic, with wavelength $400 \mathrm{~nm}$ as measured in air. Light ray $r_{1}$ exits directly into air. However, before light ray $r_{2}$ exits into air, it travels through a liquid in a hollow cylinder within the plastic. Initially the height $L_{\text {liq }}$ of the liquid is $40.0 \mu \mathrm{m}$, but then the liquid begins to evaporate. Let $\phi$ be the phase difference between rays $r_{1}$ and $r_{2}$ once they both exit into the air. Figure $35-56 b$ shows $\phi$ versus the liquid's height $L_{\text {ic }}$ until the liquid disappears, with $\phi$ given in terms of wavelength and the horizontal scale set by $L_{x}=40.00 \mu \mathrm{m}$. What are (a) the index of refraction of the plastic and (b) the index of refraction of the liquid?

Keshav Singh

Numerade Educator

Problem 93

If the distance between the first and tenth minima of a double-slit pattern is $18.0 \mathrm{~mm}$ and the slits are separated by $0.150$ $\mathrm{mm}$ with the screen $50.0 \mathrm{~cm}$ from the slits, what is the wavelength of the light used?

Dading Chen

Numerade Educator

Problem 94

Figure $35-57$ shows an optical fiber in which a central plastic core of index of refraction $n_{1}=$ $1.58$ is surrounded by a plastic sheath of index of refraction $n_{2}=$ 1.53. Light can travel along dif- Figure 35-57 Problem 94. ferent paths within the central core, leading to different travel times through the fiber. This causes an initially short pulse of light to spread as it travels along the fiber, resulting in information loss. Consider light that travels directly along the central axis of the fiber and light that is repeatedly reflected at the critical angle along the core-sheath interface, reflecting from side to side as it travels down the central core. If the fiber length is $300 \mathrm{~m}$, what is the difference in the travel times along these two routes?

Keshav Singh

Numerade Educator

Problem 96

A camera lens with index of refraction greater than $1.30$ is coated with a thin transparent film of index of refraction $1.25$ to eliminate by interference the reflection of light at wavelength $\lambda$ that is incident perpendicularly on the lens. What multiple of $\lambda$ gives the minimum film thickness needed?

Ben Nicholson

Numerade Educator

Problem 97

Light of wavelength $\lambda$ is used in a Michelson interferometer. Let $x$ be the position of the movable mirror, with $x=0$ when the arms have equal lengths $d_{2}=d_{1} .$ Write an expression for the intensity of the observed light as a function of $x$, letting $I_{m}$ be the maximum intensity.

Dading Chen

Numerade Educator

Problem 98

In two experiments, light is to be sent along the two paths shown in Fig. $35-35$ by reflecting it from the various flat surfaces shown. In the first experiment, rays 1 and 2 are initially in phase and have a wavelength of $620.0 \mathrm{~nm} .$ In the second experiment, rays 1 and 2 are initially in phase and have a wavelength of $496.0 \mathrm{~nm}$. What least value of distance $L$ is required such that the $620.0 \mathrm{~nm}$ waves emerge from the region exactly in phase but the $496.0 \mathrm{~nm}$ waves emerge exactly out of phase?

Ben Nicholson

Numerade Educator

Problem 99

Figure $35.58$ shows the design of a Texas arcade game. Four laser pistols are pointed toward the center of an array of plastic layers where a clay armadillo is the target. The indexes of refraction of the layers are $n_{1}=1.55, n_{2}=1.70, n_{3}=1.45, n_{4}=1.60$, $n_{5}=1.45, n_{6}=1.61, n_{7}=1.59, n_{8}=1.70$, and $n_{9}=1.60 .$ The layer
thicknesses are either $2.00 \mathrm{~mm}$ or $4.00 \mathrm{~mm}$, as drawn. What is the travel time through the layers for the laser burst from (a) pistol 1 ,
(b) pistol 2, (c) pistol 3, and
(d) pistol $4 ?$ (e) If the pistols are fired simultaneously, which laser burst hits the target first?

Dading Chen

Numerade Educator

Problem 100

A thin film suspended in air is $0.410 \mu \mathrm{m}$ thick and is illuminated with white light incident perpendicularly on its surface. The index of refraction of the film is $1.50$. At what wavelength will visible light that is reflected from the two surfaces of the film undergo fully constructive interference?

Ben Nicholson

Numerade Educator

Problem 101

Find the slit separation of a double-slit arrangement that will produce interference fringes $0.018$ rad apart on a distant screen when the light has wavelength $\lambda=589 \mathrm{~nm}$.

Dading Chen

Numerade Educator

Problem 102

In a phasor diagram for any point on the viewing screen for the two-slit experiment in Fig. $35-10$, the resultant-wave phasor rotates $60.0^{\circ}$ in $2.50 \times 10^{-16} \mathrm{~s}$. What is the wavelength?

Ben Nicholson

Numerade Educator

Problem 103

In Fig. $35-59$, an oil drop $(n=1.20)$ floats on the surface of water $(n=1.33)$ and is viewed from overhead when illuminated by sunlight shining vertically downward and reflected vertically upward. (a) Are the outer (thinnest) regions of the drop bright or dark? The oil film displays several spectra of colors. (b) Move from the rim inward to the third blue band and, using a wavelength of $475 \mathrm{~nm}$ for blue light, determine the film thickness there. (c) If the oil thickness increases, why do the colors gradually fade and then disappear?

Dading Chen

Numerade Educator

Problem 104

Lloyd's Mirror. In Fig, $35-60$, monochromatic light of wavelength $\lambda$ diffracts through a narrow slit $S$ in an otherwise opaque screen. On the other side, a plane mirror is perpendicular to the screen and a distance $h$ from the slit. A viewing screen $A$ is a distance much greater than $h$. (Because it sits in a plane through the focal point of the lens,
screen $A$ is effectively very distant. The lens plays no other role in the experiment and can otherwise be neglected.) Light that travels from the slit directly to $A$ interferes with light from the slit that reflects from the mirror to $A$. The reflection causes a halfwavelength phase shift. (a) Is the fringe that corresponds to a zero path length difference bright or dark? Find expressions (like Eqs $35-14$ and $35-16$ ) that locate
(b) the bright fringes and (c) the dark fringes in the interference pattern. (Hint: Consider the image of $S$ produced by the mirror as seen from a point on the viewing screen, and then consider Young's two-slit interference.)

Keshav Singh

Numerade Educator

Problem 105

The two point sources in Fig. $35-61$ emit coherent waves. Show that all curves (such as the one shown), over which the phase difference for rays $r_{1}$ and $r_{2}$ is a constant, are hyperbolas. (Hint: A constant phase difference implies a constant difference in length between $r_{1}$ and $r_{2}$.)

Dading Chen

Numerade Educator