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Physics for Scientists and Engineers with Modern Physics

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

Chapter 34

The Nature of Light and the Principles of Ray Optics - all with Video Answers

Educators

Chapter Questions

01:52

Problem 1

In an experiment to measure the speed of light using the apparatus of Armand H. L. Fizeau (see Fig. 34.2 ), the distance between light source and mirror was $11.45 \mathrm{~km}$ and the wheel had 720 notches. The experimentally determined value of $c$ was $2.998 \times 10^{8} \mathrm{~m} / \mathrm{s}$ when the outgoing light passed through one notch and then returned through the next notch. Calculate the minimum angular speed of the wheel for this experiment.

Zulfiqar Ali
Zulfiqar Ali
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Problem 2

The Apollo 11 astronauts set up a panel of efficient corner-cube retroreflectors on the Moon's surface (Fig. $34.8 \mathrm{a}$ ). The speed of light can be found by measuring the time interval required for a laser beam to travel from the Earth, reflect from the panel, and return to the Earth. Assume this interval is measured to be $2.51 \mathrm{~s}$ at a station where the Moon is at the zenith and take the center-to-center distance from the Earth to the Moon to be equal to $3.84 \times 10^{8} \mathrm{~m}$. (a) What is the measured speed of light? (b) Explain whether it is necessary to consider the sizes of the Earth and the Moon in your calculation.

Victor Salazar
Victor Salazar
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00:51

Problem 3

As a result of his observations, Ole Roemer concluded that eclipses of Io by Jupiter were delayed by 22 min during a six-month period as the Earth moved from the point in its orbit where it is closest to Jupiter to the diametrically opposite point where it is farthest from Jupiter. Using the value $1.50 \times 10^{8} \mathrm{~km}$ as the average radius of the Earth's orbit around the Sun, calculate the speed of light from these data.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
03:48

Problem 4

A dance hall is built without pillars and with a horizontal ceiling $7.20 \mathrm{~m}$ above the floor. A mirror is fastened flat against one section of the ceiling. Following an earthquake, the mirror is in place and unbroken. An engineer makes a quick check of whether the ceiling is sagging by directing a vertical beam of laser light up at the mirror and observing its reflection on the floor. (a) Show that if the mirror has rotated to make an angle $\phi$ with the horizontal, the normal to the mirror makes an angle $\phi$ with the vertical. (b) Show that the reflected laser light makes an angle $2 \phi$ with the vertical. (c) Assume the reflected laser light makes a spot on the floor $1.40 \mathrm{~cm}$ away from the point vertically below the laser. Find the angle $\phi$.

Henrique Saito
Henrique Saito
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Problem 5

You are working for an optical research company during a summer break. Part of the apparatus in one particular experiment is shown in Figure $34.7 \mathrm{~b}$. In fact, the experimenter used this textbook to set up this part of the experiment, and also used the result in the What If? section to determine the
angular change in direction of the light beam:
$$
\beta=360^{\circ}-2 \phi
$$
The experimenter is constantly grumbling that the measuring device to determine the angle $\phi$ on the inside of the mirrors is constantly getting in the way of the light beam and making his life difficult. You quickly draw Figure $\mathrm{P} 34.5$ and then say, "Then why don't you use the measuring device to measure the angle $\delta$ outside the mirror, and then your device won't get in the way of the light?"
The experimenter, who has never thought of this, tries to save face and says to you, "Well, Smarty, then tell me how angle $\beta$ depends on angle $\delta ! "$ You provide him the answer quickly.

Victor Salazar
Victor Salazar
Numerade Educator
02:19

Problem 6

The reflecting surfaces of two intersecting flat mirrors are at an angle $\theta\left(0^{\circ}<\theta<90^{\circ}\right)$ as shown in Figure $\mathrm{P} 34.6$. For a light ray that strikes the horizontal mirror, show that the emerging ray will intersect the incident ray at an angle $\beta=180^{\circ}-2 \theta$

Keshav Singh
Keshav Singh
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02:28

Problem 7

The two mirrors illustrated in Figure $\mathrm{P} 34.7$ meet at a right angle. The beam of light in the vertical plane indicated by the dashed lines strikes mirror 1 as shown.
(a) Determine the distance the reflected light beam travels before striking mirror 2 . (b) In what direction does the light beam travel after being reflected from mirror $2 ?$

Zulfiqar Ali
Zulfiqar Ali
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Problem 8

Two flat, rectangular mirrors, both perpendicular to a horizontal sheet of paper, are set edge to edge with their reflecting surfaces perpendicular to each other.
(a) A light ray in the plane of the paper strikes one of the mirrors at an arbitrary angle of incidence $\theta_{1} .$ Prove that the final direction of the ray, after reflection from both mirrors, is opposite its initial direction. (b) What If? Now assume the paper is replaced with a third flat mirror, touching edges with the other two and perpendicular to both, creating a corner-cube retroreflector (Fig. $34.8 \mathrm{a}$ ). A ray of light is incident from any direction within the octant of space bounded by the reflecting surfaces. Argue that the ray will reflect once from each mirror and that its final direction will be opposite its original direction. The Apollo 11 astronauts placed a panel of corner-cube retroreflectors on the Moon. Analysis of timing data taken with it reveals that the radius of the Moon's orbit is increasing at the rate of $3.8 \mathrm{~cm} / \mathrm{yr}$ as it loses kinetic energy because of tidal friction.

Victor Salazar
Victor Salazar
Numerade Educator
01:51

Problem 9

Find the speed of light in (a) flint glass, (b) water, and (c) cubic zirconia.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
02:55

Problem 10

A ray of light strikes a flat block of glass $(n=1.50)$ of thickness $2.00 \mathrm{~cm}$ at an angle of $30.0^{\circ}$ with the normal. Trace the light beam through the glass and find the angles of incidence and refraction at each surface.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
01:29

Problem 11

A ray of light travels from air into another medium, making an angle of $\theta_{1}=45.0^{\circ}$ with the normal as in Figure P34.11. Find the angle of refraction $\theta_{2}$ if the second medium is (a) fused quartz, (b) carbon disulfide, and (c) water.

Salamat Ali
Salamat Ali
Numerade Educator
06:42

Problem 12

A plane sound wave in air at $20^{\circ} \mathrm{C}$, with wavelength $589 \mathrm{~mm}$, is incident on a smooth surface of water at $25^{\circ} \mathrm{C}$ at an angle of incidence of $13.0^{\circ} .$ Determine (a) the angle of refraction for the sound wave and (b) the wavelength of the sound in water. A narrow beam of sodium yellow light, with wavelength $589 \mathrm{nm}$ in vacuum, is incident from air onto a smooth water surface at an angle of incidence of $13.0^{\circ} .$ Determine (c) the angle of refraction and (d) the wavelength of the light in water. (e) Compare and contrast the behavior of the sound and light waves in this problem.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
02:54

Problem 13

A laser beam is incident at an angle of $30.0^{\circ}$ from the vertical onto a solution of corn syrup in water. The beam is refracted to $19.24^{\circ}$ from the vertical. (a) What is the index of refraction of the corn syrup solution? Assume that the light is red, with vacuum wavelength $632.8 \mathrm{nm} .$ Find its (b) wavelength, (c) frequency, and (d) speed in the solution.

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
08:46

Problem 14

A ray of light strikes the midpoint of one face of an equiangular $\left(60^{\circ}-60^{\circ}-60^{\circ}\right)$ glass prism $(n=1.5)$ at an angle of incidence of $30^{\circ} .$ (a) Trace the path of the light ray through the glass and find the angles of incidence and refraction at each surface. (b) If a small fraction of light is also reflected at each surface, what are the angles of reflection at the surfaces?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:07

Problem 15

When you look through a window, by what time interval is the light you see delayed by having to go through glass instead of air? Make an order-of-magnitude estimate on the basis of data you specify. By how many wavelengths is it delayed?

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
07:12

Problem 16

Light passes from air into flint glass at a nonzero angle of incidence. (a) Is it possible for the component of its velocity perpendicular to the interface to remain constant? Explain your answer. (b) What If? Can the component of velocity parallel to the interface remain constant during refraction? Explain your answer.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
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Problem 17

You have just installed a new bathroom in your home. Your shower doors have frosted glass to provide privacy for the person using the shower. The frosted surface is on the outside of the shower door, facing the rest of the bathroom. The frosting is done by acid etching the surface so that light incident on the rough surface is scattered in all directions. Proud of your new bathroom, you take a photo of it with your smartphone. You notice in the photograph that you can see a reflection of the flash in the shower doors and the reflection is surrounded by a halo of light. Curious, you turn on a laser pointer and aim it at the shower door. Looking closely at the reflection, you again see a halo that consists of a dark area surrounding the reflection of the pointer and then an area of brightness outside this dark ring. You grab a micrometer and a ruler and measure the thickness of the glass to be $6.35 \mathrm{~mm}$ and the inner radius of the bright halo to be $10.7 \mathrm{~mm}$. From these measurements, you determine the index of refraction of the glass.

Victor Salazar
Victor Salazar
Numerade Educator
08:26

Problem 18

A triangular glass prism with apex angle $60.0^{\circ}$ has an index of refraction of $1.50 .$ (a) Show that if its angle of incidence on the first surface is $\theta_{1}=48.6^{\circ}$, light will pass symmetrically through the prism as shown in Figure 34.16 . (b) Find the angle of deviation $\delta_{\min }$ for $\theta_{1}=48.6^{\circ}$. (c) What If? Find the angle of deviation if the angle of incidence on the first surface is $45.6^{\circ}$
(d) Find the angle of deviation if $\theta_{1}=51.6^{\circ}$.

Aatish Gupta
Aatish Gupta
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Problem 19

You are working at your university swimming center. The athletic department decides that it would like to install a flag pole of height $10.0 \mathrm{~m}$ at the south end of one of the outdoor pools, which lies along a north-south axis. The pool is $3.00 \mathrm{~m}$ deep and the flag pole is to be installed $4.00 \mathrm{~m}$ from the south edge of the pool, midway along the length of the south edge.
(a) Your supervisor knows of your expertise in physics and asks you to determine the distance of the shadow of the tip of the flag pole on the bottom of the pool from the south wall of the pool on a summer day when the Sun appears directly south and at an angle of $65.0^{\circ}$ above the horizon.
(b) Your supervisor also asks if there is any time during the year that the flag pole will not cast a shadow along the bottom of the pool when the Sun is due south. The highest the Sun reaches in the sky at this location is $68.5^{\circ}$ at the summer solstice.

Victor Salazar
Victor Salazar
Numerade Educator
02:56

Problem 20

A person looking into an empty container is able to see the far edge of the container's bottom as shown in Figure $\mathrm{P} 34.20 \mathrm{a}$. The height of the container is $h,$ and its width is $d .$ When the container is completely filled with a fluid of index of refraction $n$ and viewed from the same angle, the person can see the center of a coin at the middle of the container's bottom as shown in Figure $\mathrm{P} 34.20 \mathrm{~b}$. (a) Show that the ratio $h / d$ is given by
$$
\frac{h}{d}=\sqrt{\frac{n^{2}-1}{4-n^{2}}}
$$
(b) Assuming the container has a width of $8.00 \mathrm{~cm}$ and is filled with water, use the expression above to find the height of the container. (c) For what range of values of $n$ will the center of the coin not be visible for any values of $h$ and $d ?$

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
01:06

Problem 21

Figure $\quad \mathrm{P} 34.21$ shows a light ray incident on a series of slabs having different refractive indices, where $n_{1}<n_{2}<$ $n_{3}<n_{4} .$ Notice that the path of the ray steadily bends toward the normal. If the variation in $n$ were continuous, the path would form a smooth curve. Use this idea and a ray diagram to explain why you can see the Sun at sunset after it has fallen below the horizon.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
08:38

Problem 22

A submarine is $300 \mathrm{~m}$ horizontally from the shore of a freshwater lake and $100 \mathrm{~m}$ beneath the surface of the water. A laser beam is sent from the submarine so that the beam strikes the surface of the water $210 \mathrm{~m}$ from the shore. A building stands on the shore, and the laser beam hits a target at the top of the building. The goal is to find the height of the target above sea level. (a) Draw a diagram of the situation, identifying the two triangles that are important in finding the solution. (b) Find the angle of incidence of the beam striking the water-air interface. (c) Find the angle of refraction. (d) What angle does the refracted beam make with the horizontal? (e) Find the height of the target above sea level.

Bethany Campbell
Bethany Campbell
Numerade Educator
04:49

Problem 23

A beam of light both reflects and refracts at the surface between air and glass as shown in Figure P34.23. If the refractive index of the glass is $n_{g^{\prime}}$ find the angle of incidence $\theta_{1}$ in the air that would result in the reflected ray and the refracted ray being perpendicular to each other.

Mark J
Mark J
Numerade Educator
02:09

Problem 24

A light beam containing red and violet wavelengths is incident on a slab of quartz at an angle of incidence of $50.0^{\circ}$. The index of refraction of quartz is 1.455 at $600 \mathrm{nm}$ (red light), and its index of refraction is 1.468 at $410 \mathrm{nm}$ (violet light). Find the dispersion of the slab, which is defined as the difference in the angles of refraction for the two wavelengths.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
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Problem 25

The index of refraction for violet light in silica flint glass is $n_{V},$ and that for red light is $n_{R^{*}}$ What is the angular spread of visible light passing through a prism of apex angle $\Phi$ if the angle of incidence is $\theta$ ? See Figure $\mathrm{P} 34.25$.

Victor Salazar
Victor Salazar
Numerade Educator
01:15

Problem 26

The speed of a water wave is described by $v=\sqrt{g d}$, where $d$ is the water depth, assumed to be small compared to the wavelength. Because their speed changes, water waves refract when moving into a region of different depth. (a) Sketch a map of an ocean beach on the eastern side of a landmass. Show contour lines of constant depth under water, assuming a reasonably uniform slope. (b) Suppose waves approach the coast from a storm far away to the north-northeast. Demonstrate that the waves move nearly perpendicular to the shoreline when they reach the beach. (c) Sketch a map of a coastline with alternating bays and headlands as suggested in Figure $\mathrm{P} 34.26 .$ Again make a reasonable guess about the shape of contour lines of constant depth. (d) Suppose waves approach the coast, carrying energy with uniform density along originally straight wave fronts. Show that the energy reaching the coast is concentrated at the headlands and has lower intensity in the bays.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:39

Problem 27

For 589 -nm light, calculate the critical angle for the following materials surrounded by air: (a) cubic zirconia, (b) flint glass, and (c) ice.

Rodger Claar
Rodger Claar
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Problem 28

Consider a light ray traveling between air and a diamond cut in the shape shown in Figure $\mathrm{P} 34.28$. (a) Find the critical angle for total internal reflection for light in the diamond incident on the interface between the diamond and the outside air. (b) Consider the light ray incident normally on the top surface of the diamond as shown in Figure P34.28. Show that the light traveling toward point $P$ in the diamond is totally reflected. What If? Suppose the diamond is immersed in water. (c) What is the critical angle at the diamond-water interface? (d) When the diamond is immersed in water, does the light ray entering the top surface in Figure $\mathrm{P} 34.28$ undergo total internal reflection at $P ?$ Explain. (e) If the light ray entering the diamond remains vertical as shown in Figure $\mathrm{P} 34.28$, which way should the diamond in the water be rotated about an axis perpendicular to the page through $O$ so that light will exit the diamond at $P ?$ (f) At what angle of rotation in part (e) will light first exit the diamond at point $P ?$

Victor Salazar
Victor Salazar
Numerade Educator
04:03

Problem 29

A room contains air in which the speed of sound is $343 \mathrm{~m} / \mathrm{s}$ The walls of the room are made of concrete in which the speed of sound is $1850 \mathrm{~m} / \mathrm{s}$. (a) Find the critical angle for total internal reflection of sound at the concrete-air boundary. (b) In which medium must the sound be initially traveling if it is to undergo total internal reflection? (c) "A bare concrete wall is a highly efficient mirror for sound." Give evidence for or against this statement.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
10:36

Problem 30

Around 1968, Richard A. Thorud, an engineer at The Toro Company, invented a gasoline gauge for small $$ $$ engines diagrammed in Figure $\mathrm{P} 34.30$. The gauge has no moving parts. It consists of a flat slab of transparent plastic fitting vertically into a slot in the cap on the gas tank. None of the plastic has a reflective coating. The plastic projects from the horizontal top down nearly to the bottom of the opaque tank. Its lower edge is cut with facets making angles of $45^{\circ}$ with the horizontal. A lawn mower operator looks down from above and sees a boundary between bright and dark on the gauge. The location of the boundary, across the width of the plastic, indicates the quantity of gasoline in the tank. (a) Explain how the gauge works. (b) Explain the design requirements, if any, for the index of refraction of the plastic.

Mark J
Mark J
Numerade Educator
01:30

Problem 31

An optical fiber has an index of refraction $n$ and diameter $d$. It is surrounded by vacuum. Light is sent into the fiber along its axis as shown in Figure $\mathrm{P} 34.31 .$ (a) Find the smallest outside radius $R_{\min }$ permitted for a bend in the fiber if no light is to escape. (b) What If? What result does part (a) predict as $d$ approaches zero? Is this behavior reasonable? Explain. ( $\mathrm{r}$ ) As $n$ increases? (d) As $n$ approaches 1? (e) Evaluate $R_{\min }$ assuming the fiber diameter is $100 \mu \mathrm{m}$ and its index of refraction is 1.40 .

Dominador Tan
Dominador Tan
Numerade Educator
02:41

Problem 32

Consider a horizontal interface between air above and glass of index of refraction 1.55 below. (a) Draw a light ray incident from the air at angle of incidence $30.0^{\circ} .$ Determine the angles of the reflected and refracted rays and show them on the diagram. (b) What If? Now suppose the light ray is incident from the glass at an angle of $30.0^{\circ}$. Determine the angles of the reflected and refracted rays and show all three rays on a new diagram. (c) For rays incident from the air onto the air-glass surface, determine and tabulate the angles of reflection and refraction for all the angles of incidence at $10.0^{\circ}$ intervals from $0^{\circ}$ to $90.0^{\circ}$.
(d) Do the same for light rays coming up to the interface through the glass.

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
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Problem 33

How many times will the incident beam in Figure $\mathrm{P} 34.33$ (page 922 ) be reflected by each of the parallel mirrors?

Victor Salazar
Victor Salazar
Numerade Educator
04:09

Problem 34

Consider a beam of light from the left entering a prism of apex angle $\Phi$ as shown in Figure $\mathrm{P} 34.34 .$ Two angles of incidence, $\theta_{1}$ and $\theta_{3},$ are shown as well as two angles of refraction, $\theta_{2}$ and $\theta_{4} .$ Show that $\Phi=\theta_{2}+\theta_{3}$

Vishal Gupta
Vishal Gupta
Numerade Educator
02:35

Problem 35

Why is the following situation impossible? While at the bottom of a calm freshwater lake, a scuba diver sees the Sun at an apparent angle of $38.0^{\circ}$ above the horizontal.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
02:16

Problem 36

Why is the following situation impossible? A laser beam strikes one end of a slab of material of length $L=42.0 \mathrm{~cm}$ and thickness $t=3.10 \mathrm{~mm}$ as shown in Figure $\mathrm{P} 34.36$ (not to scale). It enters the material at the center of the left end, striking it at an angle of incidence of $\theta=50.0^{\circ} .$ The index of refraction of the slab is $n=1.48 .$ The light makes 85 internal reflections from the top and bottom of the slab before exiting at the other end.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
01:51

Problem 37

When light is incident normally on the interface between two transparent optical media, the intensity of the reflected light is given by the expression
$$
S_{1}^{\prime}=\left(\frac{n_{2}-n_{1}}{n_{2}+n_{1}}\right)^{2} S_{1}
$$
In this equation, $S_{1}$ represents the average magnitude of the Poynting vector in the incident light (the incident intensity), $S_{1}^{\prime}$ is the reflected intensity, and $n_{1}$ and $n_{2}$ are the refractive indices of the two media. (a) What fraction of the incident intensity is reflected for 589 -nm light normally incident on an interface between air and crown glass? (b) Does it matter in part (a) whether the light is in the air or in the glass as it strikes the interface?

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
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Problem 38

Refer to Problem 37 for its description of the reflected intensity of light normally incident on an interface between two transparent media. (a) For light normally incident on an interface between vacuum and a transparent medium of index $n,$ show that the intensity $S_{2}$ of the transmitted light is given by $S_{2} / S_{1}=4 n /(n+1)^{2}$ (b) Light travels perpendicularly through a diamond slab, surrounded by air, with parallel surfaces of entry and exit. Apply the transmission fraction in part (a) to find the approximate overall transmission through the slab of diamond, as a percentage. Ignore light reflected back and forth within the slab.

Victor Salazar
Victor Salazar
Numerade Educator
07:15

Problem 39

A light ray enters the atmosphere of the Earth and descends vertically to the surface a distance $h=100 \mathrm{~km}$ below. The index of refraction where the light enters the atmosphere is 1.00 , and it increases linearly with distance to have the value $n=1.000293$ at the Earth's surface.
(a) Over what time interval does the light traverse this path?
(b) By what percentage is the time interval larger than that required in the absence of the Earth's atmosphere?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
04:28

Problem 40

A light ray enters the atmosphere of a planet and descends vertically to the surface a distance $h$ below. The index of refraction where the light enters the atmosphere is 1.00 , and it increases linearly with distance to have the value $n$ at the planet surface. (a) Over what time interval does the light traverse this path? (b) By what fraction is the time interval larger than that required in the absence of an atmosphere?

Prashant Bana
Prashant Bana
Numerade Educator
08:30

Problem 41

A light ray of wavelength $589 \mathrm{nm}$ is incident at an angle $\theta$ on the top surface of a block of polystyrene as shown in Figure $\mathrm{P} 34.41 .$ (a) Find the maximum value of $\theta$ for which
the refracted ray undergoes total internal reflection at the point $P$ located at the left vertical face of the block. What If? Repeat the calculation for the case in which the polystyrene block is immersed in (b) water and (c) carbon disulfide. Explain your answers.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
03:00

Problem 42

One technique for measuring the apex angle of a prism is shown in Figure P34.42. Two parallel rays of light are directed onto the apex of the prism so that the rays reflect from opposite faces of the prism. The angular separation $\gamma$ of the two reflected rays can be measured. Show that $\Phi=\frac{1}{2} \gamma$

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
03:05

Problem 43

A material having an index of refraction $n$ is surrounded by vacuum and is in the shape of a quarter circle of radius $R$ (Fig. $\mathrm{P} 34.43$ ). A light ray parallel to the base of the material is incident from the left at a distance $L$ above the base and emerges from the material at the angle $\theta .$ Determine an expression for $\theta$ in terms of $n, R,$ and $L$

Mayukh Banik
Mayukh Banik
Numerade Educator
02:18

Problem 44

A mirror is often "silvered" with aluminum. By adjusting the thickness of the metallic film, one can make a sheet of glass into a mirror that reflects anything between $3 \%$ and $98 \%$ of the incident light, transmitting the rest. Prove that it is impossible to construct a "one-way mirror" that would reflect $90 \%$ of the electromagnetic waves incident from one side and reflect $10 \%$ of those incident from the other side. Suggestion: Use Clausius's statement of the second law of thermodynamics.

Dominador Tan
Dominador Tan
Numerade Educator
01:47

Problem 45

Figure $\mathrm{P} 34.45$ shows the path of a light beam through several slabs with different indices of refraction. (a) If $\theta_{1}=$ $30.0^{\circ},$ what is the angle $\theta_{2}$ of the emerging beam? (b) What must the incident angle $\theta_{1}$ be to have total internal reflection at the surface between the medium with $n=1.20$ and the medium with $n=1.00 ?$

Mayukh Banik
Mayukh Banik
Numerade Educator
03:54

Problem 46

As sunlight enters the Earth's atmosphere, it changes direction due to the small difference between the speeds of light in vacuum and in air. The duration of an optical day is defined as the time interval between the instant when the top of the rising Sun is just visible above the horizon and the instant when the top of the Sun just disappears below the horizontal plane. The duration of the geometric day is defined as the time interval between the instant a mathematically straight line between an observer and the top of the Sun just clears the horizon and the instant this line just dips below the horizon. (a) Explain which is longer, an optical day or a geometric day. (b) Find the difference between these two time intervals. Model the Earth's atmosphere as uniform, with index of refraction 1.000293 , a sharply defined upper surface, and depth $8614 \mathrm{~m}$. Assume the observer is at the Earth's equator so that the apparent path of the rising and setting Sun is perpendicular to the horizon.

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
03:04

Problem 47

A ray of light passes from air into water. For its deviation angle $\delta=\left|\theta_{1}-\theta_{2}\right|$ to be $10.0^{\circ},$ what must its angle of incidence be?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
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Problem 48

In your work for an optical research company, you are asked to consider the triangular shaped prism shown in Figure $\mathrm{P} 34.48 .$ Light enters the left slanted side of the prism from air at normal incidence, reflects from the top surface by total internal reflection, and then refracts out of the right slanted surface. (a) Your supervisor asks you to determine the range of angles over which visible light exits the right slanted surface due to dispersion in the material. (b) An actual physical prism of the shape in Figure $\mathrm{P} 34.48$ is then made from cubic zirconia with $\alpha=60^{\circ}$ and $\gamma=30^{\circ}$ and it doesn't work as planned. Explain to your supervisor why not.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 49

A. H. Pfund's method for measuring the index of refraction of glass is illustrated in Figure $\mathrm{P} 34.49 .$ One face of a slab of thickness $t$ is painted white, and a small hole scraped clear at point $P$ serves as a source of diverging rays when the slab is illuminated from below. Ray $P B B^{\prime}$ strikes the clear surface at the critical angle and is totally reflected, as are rays such as $P C C^{\prime} .$ Rays such as $P A A^{\prime}$ emerge from the clear surface. On the painted surface, there appears a dark circle of diameter $d$ surrounded by an illuminated region, or halo. (a) Derive an equation for $n$ in terms of the measured quantities $d$ and $t$. (b) What is the diameter of the dark circle if $n=1.52$ for a slab $0.600 \mathrm{~cm}$ thick? (c) If white light is used, dispersion causes the critical angle to depend on color. Is the inner edge of the white halo tinged with red light or with violet light? Explain.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 50

Figure $\mathrm{P} 34.50$ shows a top view of a square enclosure. The inner surfaces are plane mirrors. A ray of light enters a small hole in the center of one mirror. (a) At what angle $\theta$ must the ray enter if it exits through the hole after being reflected once by each of the other three mirrors? (b) What If? Are there other values of $\theta$ for which the ray can exit after multiple reflections? If so, sketch one of the ray's paths.

Victor Salazar
Victor Salazar
Numerade Educator
04:29

Problem 51

The walls of an ancient shrine are perpendicular to the four cardinal compass directions. On the first day of spring, light from the rising Sun enters a rectangular window in the eastern wall. The light traverses $2.37 \mathrm{~m}$ horizontally to shine perpendicularly on the wall opposite the window. A tourist observes the patch of light moving across this western wall.
(a) With what speed does the illuminated rectangle move?
(b) The tourist holds a small, square mirror flat against the western wall at one corner of the rectangle of light. The mirror reflects light back to a spot on the eastern wall close beside the window. With what speed does the smaller square of light move across that wall?
(c) Seen from a latitude of $40.0^{\circ}$ north, the rising Sun moves through the sky along a line making a $50.0^{\circ}$ angle with the southeastern horizon. In what direction does the rectangular patch of light on the western wall of the shrine move?
(d) In what direction does the smaller square of light on the eastern wall move?

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
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Problem 52

The perpendicular distance of a lightbulb from a large plane mirror is twice the perpendicular distance of a person from the mirror. Light from the lightbulb reaches the person by two paths: (1) it travels to the mirror and reflects from the mirror to the person, and (2) it travels directly to the person without reflecting off the mirror. The total distance traveled by the light in the first case is 3.10 times the distance traveled by the light in the second case.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 53

$\begin{array}{llll}\text { Figure } & \text { P34.53 } & \text { shows } & \text { an }\end{array}$ overhead view of a room of square floor area and side L. At the center of the room is a mirror set in a vertical plane and rotating on a vertical shaft at angular speed $\omega$ about an axis coming out of the page. A bright red laser beam enters from the center point on one wall of the room and strikes the mirror. As the mirror rotates, the reflected laser beam creates a red spot sweeping across the walls of the room. (a) When the spot of light on the wall is at distance $x$ from point $O$, what is its speed? (b) What value of $x$ corresponds to the minimum value for the speed? (c) What is the minimum value for the speed? (d) What is the maximum speed of the spot on the wall? (e) In what time interval does the spot change from its minimum to its maximum speed?

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 54

Pierre de Fermat $(1601-1665)$ showed that whenever light travels from one point to another, its actual path is the path that requires the smallest time interval. This statement is known as Fermat's principle. The simplest example is for light propagating in a homogeneous medium. It moves in a straight line because a straight line is the shortest distance between two points. Derive Snell's law of refraction from Fermat's principle. Proceed as follows. In Figure $\mathrm{P} 34.54$, a light ray travels from point $P$ in medium 1 to point $Q$ in medium $2 .$ The two points are, respectively, at perpendicular distances $a$ and $b$ from the interface. The displacement from $P$ to $Q$ has the component $d$ parallel to the interface, and we let $x$ represent the coordinate of the point where the ray enters the second medium. Let $t=0$ be the instant the light starts from $P$. (a) Show that the time at which the light arrives at $Q$ is $$
t=\frac{r_{1}}{v_{1}}+\frac{r_{2}}{v_{2}}=\frac{n_{1} \sqrt{a^{2}+x^{2}}}{c}+\frac{n_{2} \sqrt{b^{2}+(d-x)^{2}}}{c}
$$
(b) To obtain the value of $x$ for which $t$ has its minimum value, differentiate $t$ with respect to $x$ and set the derivative equal to zero. Show that the result implies $$
\frac{n_{1} x}{\sqrt{a^{2}+x^{2}}}=\frac{n_{2}(d-x)}{\sqrt{b^{2}+(d-x)^{2}}}
$$
(c) Show that this expression in turn gives Snell's law,
$$
n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}
$$

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 55

Refer to Problem 54 for the statement of Fermat's principle of least time. Derive the law of reflection (Eq. 34.1 ) from Fermat's principle.

Victor Salazar
Victor Salazar
Numerade Educator
02:42

Problem 56

Suppose a luminous sphere of radius $R_{1}$ (such as the Sun) is surrounded by a uniform atmosphere of radius $R_{2}>R_{1}$ and index of refraction $n$. When the sphere is viewed from a location far away in vacuum, what is its apparent radius (a) when $R_{2}>n R_{1}$ and (b) when $R_{2}<n R_{1} ?$

Mayukh Banik
Mayukh Banik
Numerade Educator
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Problem 57

This problem builds upon the results of Problems 37 and 38. Light travels perpendicularly through a diamond slab, surrounded by air, with parallel surfaces of entry and exit. The intensity of the transmitted light is what fraction of the incident intensity? Include the effects of light reflected back and forth inside the slab.

Victor Salazar
Victor Salazar
Numerade Educator