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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 22

Reflection and refraction of Light

Educators


Problem 1

During the Apollo XI Moon landing, a retroreflecting panel was erected on the Moon’s surface. The speed of light can be found by measuring the time it takes a laser beam to travel from Earth, reflect from the panel, and return to Earth. If this interval is found to be 2.51 s, what is the measured speed of light? Take the center-to-center distance from Earth to the Moon to be $3.84 \times 10^{8} \mathrm{m} .$ Assume the Moon is directly overhead and do not neglect the sizes of Earth and the Moon.

Suzanne W.
Numerade Educator

Problem 2

(a) What is the energy in joules of an x-ray photon with wavelength $1.00 \times 10^{-10} \mathrm{m}$ ? (b) Convert the energy to electron volts. (c) If more penetrating $\mathrm{x}$ -rays are desired, should the wavelength be increased or decreased? (d) Should the frequency be increased or decreased?

Zachary W.
Numerade Educator

Problem 3

Find the energy of (a) a photon having a frequency of $5.00 \times 10^{17} \mathrm{Hz}$ and (b) a photon having a wavelength of $3.00 \times 10^{2} \mathrm{nm}$ . Express your answers in units of electron volts, noting that $1 \mathrm{eV}=1.60 \times 10^{-19} \mathrm{J} .$

Suzanne W.
Numerade Educator

Problem 4

(a) Find a symbolic expression for the wavelength $\lambda$ of a photon in terms of its energy $E,$ Planck's constant $h,$ and the speed of light $c,$ (b) What does the equation say about the wavelengths of higher-energy photons?

Zachary W.
Numerade Educator

Problem 5

(a) How many minutes does it take a photon to travel from the Sun to the Earth? (b) What is the energy in eV of a photon with a wavelength of 558 nm? (c) What is the wavelength of a photon with an energy of 1.00 eV?

Suzanne W.
Numerade Educator

Problem 6

Find the speed of light in (a) water, (b) crown glass, and (c) diamond.

Zachary W.
Numerade Educator

Problem 7

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 $P 22.7 .$ Find the angle of refraction $\theta_{2}$ if the second medium is (a) fused quartz, (b) carbon disulfide, and (c) water.

Suzanne W.
Numerade Educator

Problem 8

The two mirrors in Figure P22.8 meet at a right angle. The beam of light in the vertical plane $P$ 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$?$

Zachary W.
Numerade Educator

Problem 9

An underwater scuba diver sees the Sun at an apparent angle of $45.0^{\circ}$ from the vertical. What is the actual direction of the Sun?

Suzanne W.
Numerade Educator

Problem 10

Two plane mirrors are at right angles to each other as shown by the side view in Figure P22.10. A light ray is incident on mirror 1 at an angle $\theta$ with the vertical. Using the law of reflection and geometry, show that after the ray is reflected off of both mirrors, the outgoing reflected ray is parallel to the incident ray.

Zachary W.
Numerade Educator

Problem 11

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

Suzanne W.
Numerade Educator

Problem 12

Light containing wavelengths of 400. nm, 500. nm, and 650. nm is incident from air on a block of crown glass at an angle of $25.0^{\circ} .$ (a) Are all colors refracted alike, or is one color bent more than the others? (b) Calculate the angle of refraction in each case to verify your answer.

Zachary W.
Numerade Educator

Problem 13

A ray of light is incident on the surface of a block of clear ice at an angle of $40.0^{\circ}$ with the normal. Part of the light is reflected, and part is refracted. Find the angle between the reflected and refracted light.

Suzanne W.
Numerade Educator

Problem 14

Two plane mirrors are at an angle of $\theta_{1}=50.0^{\circ}$ with each other as in the side view shown in Figure $P 22.14$ . If a horizontal ray is incident on mirror 1 , at what angle $\theta_{2}$ does the outgoing reflected ray make with the surface of mirror 2$?$

Zachary W.
Numerade Educator

Problem 15

The light emitted by a helium–neon laser has a wavelength of 632.8 nm in air. As the light travels from air into zircon, find its (a) speed, (b) wavelength, and (c) frequency, all in the zircon.

Suzanne W.
Numerade Educator

Problem 16

Figure P22.16 shows a light ray traveling in a slab of crown glass surrounded by air. The ray is incident on the right surface at an angle of $55^{\circ}$ with the normal and then reflects from points $A, B,$ and $C$ (a) At which of these points does part of the ray enter the air? (b) If the glass slab is surrounded by carbon disulfide, at which point does part of the ray enter the carbon disulfide?

Zachary W.
Numerade Educator

Problem 17

How many times will the incident beam shown in Figure P22.17 be reflected by each of the parallel mirrors?

Salamat A.
Numerade Educator

Problem 18

A ray of light strikes a flat, 2.00-cm-thick block of glass $(n=1.50)$ at an angle of $30.0^{\circ}$ with respect to the normal (Fig. P 22.18 ). (a) Find the angle of refraction at the top surface. (b) Find the angle of incidence at the bottom surface and the refracted angle. (c) Find the lateral distance $d$ by which the light beam is shifted. (d) Calculate the speed of light in the glass and (e) the time required for the light to pass through the glass block. (f) Is the travel time through the block affected by the angle of incidence? Explain.

Zachary W.
Numerade Educator

Problem 19

The light beam shown in Figure P22.19 makes an angle of $20.0^{\circ}$ with the normal line $N N^{\prime}$ in the linseed oil. Determine the angles $\theta$ and $\theta^{\prime} .$ (The refractive index for linseed oil is 1.48 . )

Keshav S.
Numerade Educator

Problem 20

A laser beam is incident on a $45^{\circ}-45^{\circ}-90^{\circ}$ prism perpendicular to one of its faces, as shown in Figure P 22.20 . The transmitted beam that exits the hypotenuse of the prism makes an angle of $\theta=15.0^{\circ}$ with the direction of the incident beam. Find the index of refraction of the prism.

Zachary W.
Numerade Educator

Problem 21

A man shines a flashlight from a boat into the water, illuminating a rock as in Figure P 22.21 . What is the angle of incidence $\theta_{1} ?$

Suzanne W.
Numerade Educator

Problem 22

A narrow beam of ultrasonic waves reflects off the liver tumor in Figure P22.22. If the speed of the wave is 10.0% less in the liver than in the surrounding medium, determine the depth of the tumor.

Zachary W.
Numerade Educator

Problem 23

A person looking into an empty container is able to see the far edge of the container’s bottom, as shown in Figure P22.23a. 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 P 22.23b . (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 cm and is filled with water, use the expression above to find the height of the container.

Suzanne W.
Numerade Educator

Problem 24

Photons with a wavelength of 589 $\mathrm{nm}$ in air enter a plate of crown glass with index of refraction $n=1.52 .$ Find the (a) speed, (b) wavelength, and (c) energy of a photon in the glass.

Zachary W.
Numerade Educator

Problem 25

A beam of light both reflects and refracts at the surface between air and glass, as shown in Figure P22.25. If the index of refraction of the glass is $n_{g},$ 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. Hint: Remember the identity sin $\left(90^{\circ}-\theta\right)=\cos \theta$

Suzanne W.
Numerade Educator

Problem 26

Figure P22.26 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.

Zachary W.
Numerade Educator

Problem 27

An opaque cylindrical tank with an open top has a diameter of 3.00 m and is completely filled with water. When the afternoon Sun reaches an angle of $28.0^{\circ}$ above the horizon, sunlight ceases to illuminate the bottom of the tank. How deep is the tank?

Suzanne W.
Numerade Educator

Problem 28

A certain kind of glass has an index of refraction of 1.650 for blue light of wavelength 430 nm and an index of 1.615 for red light of wavelength 680 nm. If a beam containing these two colors is incident at an angle of $30.00^{\circ}$ on a piece of this glass, what is the angle between the two beams inside the glass?

Zachary W.
Numerade Educator

Problem 29

The index of refraction for red light in water is 1.331 and that for blue light is 1.340 . If a ray of white light enters the water at an angle of incidence of $83.00^{\circ},$ what are the underwater angles of refraction for the (a) blue and (b) red components of the light?

Suzanne W.
Numerade Educator

Problem 30

The index of refraction for crown glass is 1.512 at a wavelength of 660 nm (red), whereas its index of refraction is 1.530 at a wavelength of 410 nm (violet). If both wavelengths are incident on a slab of crown glass at the same angle of incidence, $60.0^{\circ},$ what is the angle of refraction for each wavelength?

Zachary W.
Numerade Educator

Problem 31

A light beam containing red and violet wavelengths is incident on a slab of quartz at an angle of incidence of $50.00^{\circ} .$ The index of refraction of quartz is 1.455 at 660 $\mathrm{nm}$ (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.

Suzanne W.
Numerade Educator

Problem 32

The prism in Figure $P 22.32$ is made of glass with an index of refraction of 1.64 for blue light and 1.60 for red light. Find (a) $\delta_{R},$ the angle of deviation for red light, and (b) $\delta_{B},$ the angle of deviation for blue light, if white light is incident on the prism at an angle of $30.0^{\circ} .$

Zachary W.
Numerade Educator

Problem 33

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 $30.0^{\circ} .$ (a) Trace the path of the light ray through the glass and find the angles of incidence and refracted 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?

Salamat A.
Numerade Educator

Problem 34

For light of wavelength 589 nm, calculate the critical angles for the following substances when surrounded by air: (a) fused quartz, (b) polystyrene, and (c) sodium chloride.

Zachary W.
Numerade Educator

Problem 35

Repeat Problem 34, but this time assume the quartz, polystyrene, and sodium chloride are surrounded by water.

Suzanne W.
Numerade Educator

Problem 36

A beam of light is incident from air on the surface of a liquid. If the angle of incidence is $30.0^{\circ}$ and the angle of refraction is $22.0^{\circ},$ find the critical angle for the liquid when surrounded by air.

Zachary W.
Numerade Educator

Problem 37

A plastic light pipe has an index of refraction of 1.53. For total internal reflection, what is the minimum angle of incidence if the pipe is in (a) air and (b) water?

Suzanne W.
Numerade Educator

Problem 38

Determine the maximum angle $\theta$ for which the light rays incident on the end of the light pipe in Figure P22.38 are subject to total internal reflection along the walls of the pipe. Assume the light pipe has an index of refraction of 1.36 and the outside medium is air.

Zachary W.
Numerade Educator

Problem 39

A light ray is incident normally to the long face (the hypotenuse) of a $45^{\circ}-45^{\circ}-90^{\circ}$ prism surrounded by air, as shown in Figure 22.26b. Calculate the minimum index of refraction of the prism for which the ray will totally internally reflect at each of the two sides making the right angle.

Suzanne W.
Numerade Educator

Problem 40

A beam of laser light with wavelength 612 nm is directed through a slab of glass having index of refraction 1.78. (a) For what minimum incident angle would a ray of light undergo total internal reflection? (b) If a layer of water is placed over the glass, what is the minimum angle of incidence on the glass–water interface that will result in total internal reflection at the water–air interface? (c) Does the thickness of the water layer or glass affect the result? (d) Does the index of refraction of the intervening layer affect the result?

Zachary W.
Numerade Educator

Problem 41

A room contains air in which the speed of sound is 343 m/s. The walls of the room are made of concrete, in which the speed of sound is 1 850 m/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 traveling in order to undergo total internal reflection? (c) “A bare concrete wall is a highly efficient mirror for sound.” Give evidence for or against this statement.

Suzanne W.
Numerade Educator

Problem 42

Consider a light ray traveling between air and a diamond cut in the shape shown in Figure P22.42. (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 P22.42. Show that the light traveling toward point $P$ in the diamond is totally reflected. (c) If the diamond is immersed in water, find 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 P22.42 undergo total internal reflection at $P$ ? Explain. (e) If the light ray entering the diamond remains vertical as shown in Figure P22.42, 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 $P$ ?

Zachary W.
Numerade Educator

Problem 43

The light beam in Figure P22.43 strikes surface 2 at the critical angle. Determine the angle of incidence,$\theta_{1}$.

Suzanne W.
Numerade Educator

Problem 44

A boy floating on a pond watches a fish swim away from him as in Figure P22.44. If the fish is 2.25 m beneath the surface, for what maximum distance $d$ will he be able to see the fish? Neglect the height of the boy’s eyes above the water.

Zachary W.
Numerade Educator

Problem 45

A layer of ice having parallel sides floats on water. If light is incident on the upper surface of the ice at an angle of incidence of $30.0^{\circ},$ what is the angle of refraction in the water?

Suzanne W.
Numerade Educator

Problem 46

A ray of light is incident at an angle $30.0^{\circ}$ on a plane slab of flint glass surrounded by water. (a) Find the refraction angle. (b) Suppose the index of refraction of the surrounding medium can be adjusted, but the incident angle of the light remains the same. As the index of refraction of the medium approaches that of the glass, what happens to the refraction angle? (c) What happens to the refraction angle when the medium’s index of refraction exceeds that of the glass?

Zachary W.
Numerade Educator

Problem 47

When a man stands near the edge of an empty drainage ditch of depth 2.80 m, he can barely see the boundary between the opposite wall and bottom of the ditch as in Figure P22.47a. The distance from his eyes to the ground is 1.85 m. (a) What is the horizontal distance $d$ from the man to the edge of the drainage ditch? (b) After the drainage ditch is filled with water as in Figure P22.47b, what is the maximum distance $x$ the man can stand from the edge and still see the same boundary?

Keshav S.
Numerade Educator

Problem 48

A light ray of wavelength 589 nm is incident at an angle $\theta$ on the top surface of a block of polystyrene surrounded by air, as shown in Figure P22.48. (a) Find the maximum value of $\theta$ for which the refracted ray will undergo total internal reflection at the left vertical face of the block. (b) Repeat the calculation for the case in which the polystyrene block is immersed in water. (c) What happens if the block is immersed in carbon disulfide?

Zachary W.
Numerade Educator

Problem 49

Refraction causes objects submerged in water to appear less deep than they actually are. The fish in Figure P22.49 has an apparent depth of 1.25 m. Calculate its actual depth.

Suzanne W.
Numerade Educator

Problem 50

A narrow beam of light is incident from air onto a glass surface with index of refraction 1.56. Find the angle of incidence for which the corresponding angle of refraction is one-half the angle of incidence. Hint: You might want to use the trigonometric identity $\sin 2 \theta=2 \sin \theta \cos \theta$.

Zachary W.
Numerade Educator

Problem 51

One technique for measuring the angle of a prism is shown in Figure P22.51. A parallel beam of light is
directed onto the apex of the prism so that the beam reflects from opposite faces of the prism. Show that the angular separation of the two reflected beams is given by $B=2 A$.

Salamat A.
Numerade Educator

Problem 52

Endoscopes are medical instruments used to examine the gastrointestinal tract and other cavities inside the body. The light required for examination is conducted from an outside source along a long, flexible bundle of optical fibers to the tip, where it exits and illuminates the internal cavity. A lens on the tip collects an image of the lighted cavity and another fiber bundle conducts the image back along the endoscope to an eyepiece for viewing (Fig. P22.52). If each fiber in the bundle has diameter $d=1.00 \times 10^{-4} \mathrm{m}$ and refractive index $n=1.40,$ find the smallest outside radius $R$ permitted for a bend in the fiber if no light is to escape.

Zachary W.
Numerade Educator

Problem 53

A piece of wire is bent through an angle $\theta .$ The bent wire is partially submerged in benzene (index of refraction =1.50 ) so that, to a person looking along the dry part, the wire appears to be straight and makes an angle of $30.0^{\circ}$ with the horizontal. Determine the value of $\theta .$

Suzanne W.
Numerade Educator

Problem 54

A light ray traveling in air is incident on one face of a right- angle prism with index of refraction $n=1.50,$ as shown in Figure P 22.54 , and the ray follows the path shown in the figure. Assuming $\theta=60.0^{\circ}$ and the base of the prism is mirrored, determine the angle $\phi$ made by the outgoing ray with the normal to the right face of the prism.

Zachary W.
Numerade Educator

Problem 55

A transparent cylinder of radius $R=2.00 \mathrm{m}$ has a mirrored surface on its right half, as shown in Figure P 22.55 . A light ray traveling in air is incident on the left side of the cylinder. The incident light ray and the exiting light ray are parallel, and $d=2.00 \mathrm{m} .$ Determine the index of refraction of the material.

Salamat A.
Numerade Educator

Problem 56

A laser beam strikes one end of a slab of material, as in Figure P22.56. The index of refraction of the slab is 1.48. Determine the number of internal reflections of the beam before it emerges from the opposite end of the slab.

Zachary W.
Numerade Educator

Problem 57

A light ray enters a rectangular block of plastic at an angle $\theta_{1}=45.0^{\circ}$ and emerges at an angle $\theta_{2}=76.0^{\circ},$ as shown in Figure P 22.57 . (a) Determine the index of refraction of the plastic. (b) If the light ray enters the plastic at a point $L=50.0 \mathrm{cm}$ from the bottom edge, how long does it take the light ray to travel through the plastic?

Suzanne W.
Numerade Educator

Problem 58

Students allow a narrow beam of laser light to strike a water surface. They arrange to measure the angle of refraction for selected angles of incidence and record the data shown in the following table:
$$\begin{array}{ll}\hline{\text { Angle of Incidence }} & {\text { Angle of Refraction }} \\ {\text { (degrees) }} & {\text { (degrees) }} \\ \hline \quad\quad {10.0} &\quad {7.5} \\ \quad\quad {20.0} & \quad {15.1} \\ \quad\quad {30.0} & \quad {22.3} \\ \quad\quad {40.0} & \quad {28.7} \\ \quad\quad {50.0} & \quad {35.2} \\ \quad\quad {60.0} & \quad {40.3} \\ \quad\quad {70.0} & \quad {45.3} \\ \quad\quad {80.0} & \quad {47.7}\\ \hline \end{array}$$
Use the data to verify Snell’s law of refraction by plotting the sine of the angle of incidence versus the sine of the angle of refraction. From the resulting plot, deduce the index of refraction of water.

Zachary W.
Numerade Educator

Problem 59

Figure P 22.59 shows the path of a beam of light through seyeral layers 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 $n=1.00 ?$

Suzanne W.
Numerade Educator

Problem 60

Three sheets of plastic have unknown indices of refraction. Sheet 1 is placed on top of sheet 2, and a laser beam is directed onto the sheets from above so that it strikes the interface at an angle of $26.5^{\circ}$ with the normal. The refracted beam in sheet 2 makes an angle of $31.7^{\circ}$ with the normal. The experiment is repeated with sheet 3 on top of sheet $2,$ and with the same angle of incidence, the refracted beam makes an angle of $36.7^{\circ}$ with the normal. If the experiment is repeated angain with sheet 1 on top of sheet 3 , what is the expected angle of refraction in sheet 3$?$ Assume the same angle of incidence.

Zachary W.
Numerade Educator

Problem 61

A person swimming underwater on a bright day and looking up at the surface will see a bright circle surrounded by relative darkness as in Figure P22.61a, a phenomenon known as Snell’s window. Use the concept of total internal reflection and the illustration in Figure P22.61b to show that $\theta=97.2^{\circ}$ for the cone containing Snell’s window.

Suzanne W.
Numerade Educator

Problem 62

The index of refraction for violet light in silica flint glass is 1.66 and that for red light is 1.62. What is the angular dispersion of visible light passing through an equilateral prism of apex angle $60.0^{\circ}$ if the angle of incidence is $50.0^{\circ}$? (See Fig. P22.62.)

Zachary W.
Numerade Educator