Physics for Scientists and Engineers with Modern Physics

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

Chapter 35

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

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Chapter Questions

Problem 1

The Apollo 11 astronauts set up a panel of efficient corner-cube retroreflectors on the Moon’s surface (Fig. 35.8a). 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 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.

Zulfiqar Ali

Zulfiqar Ali

Numerade Educator

Problem 2

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

Numerade Educator

Problem 3

In an experiment to measure the speed of light using the apparatus of Armand H. L. Fizeau (see Fig. 35.2), the distance between light source and mirror was 11.45 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

Numerade Educator

Problem 4

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

Numerade Educator

Problem 5

The wavelength of red helium-neon laser light in air is 632.8 $\mathrm{nm}$ . (a) What is its frequency? (b) What is its wave-length in glass that has an index of refraction of 1.50 ? (c) What is its speed in the glass?

Zulfiqar Ali

Numerade Educator

Problem 6

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

Zulfiqar Ali

Numerade Educator

Problem 7

A ray of light is incident on a flat surface of a block of crown glass that is surrounded by water. The angle of refraction is $19.6^{\circ} .$ Find the angle of reflection.

Zulfiqar Ali

Numerade Educator

Problem 8

Figure $\mathrm{P} 35.8$ shows a refracted light beam in linseed oil making an angle of $\alpha=20.0^{\circ}$ with the normal line $N N^{\prime} .$ The index of refraction of linseed oil is
1.48. Determine the angles (a) $\theta$ and (b) $\theta^{\prime}$ .

Zulfiqar Ali

Numerade Educator

Problem 9

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

Zulfiqar Ali

Numerade Educator

Problem 10

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 floor 1.40 $\mathrm{cm}$ away from the point vertically below the laser. Find the angle $\phi$ .

Henrique Saito

Numerade Educator

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 P35.11. Find the angle of refraction the angle of refraction is if the second medium is (a) fused quartz, (b) carbon disulfide, and (c) water.

Zulfiqar Ali

Numerade Educator

Problem 12

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

Numerade Educator

Problem 13

A prism that has an apex angle of $50.0^{\circ}$ is made of cubic zirconia. What is its minimum angle of deviation?

Zulfiqar Ali

Numerade Educator

Problem 14

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 cornercube retroreflector (Fig. $35.8 \mathrm{a} ) .$ 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.

Mayukh Banik

Numerade Educator

Problem 15

The two mirrors illustrated in Figure $\mathrm{P} 35.15$ 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

Numerade Educator

Problem 16

When the light ray illustrated in Figure P35. 16 passes through the glass block of index of refraction $n=1.50,$ it is shifted laterally by the distance $d$ . (a) Find the value of $d$ . (b) Find the time interval required for the light to pass through the glass block.

Zulfiqar Ali

Numerade Educator

Problem 17

Two light pulses are emitted simultaneously from a source. Both pulses travel through the same total length of air to a detector, but mirrors shunt one pulse along a path that carries it through an extra length of 6.20 $\mathrm{m}$ of ice along the way. Determine the difference in the pulses' times of arrival at the detector.

Zulfiqar Ali

Numerade Educator

Problem 18

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

Numerade Educator

Problem 19

A laser beam with vacuum wavelength 632.8 $\mathrm{nm}$ is incident from air onto a block of Lucite as shown in Active Figure 35.10 $\mathrm{b}$ . The line of sight of the photograph is perpendicular to the plane in which the light moves. Find (a) the speed, (b) the frequency, and (c) the wavelength of the light in the Lucite. Suggestion: Use a protractor.

Zulfiqar Ali

Numerade Educator

Problem 20

A narrow beam of ultrasonic waves reflects off the liver tumor illustrated in Figure P35.20. The
speed of the wave is 10.0$\%$ less in the liver than in the surrounding medium. Determine the depth of the tumor.

Aatish Gupta

Numerade Educator

Problem 21

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 any part of the bottom of the tank. How deep is the tank?

Zulfiqar Ali

Numerade Educator

Problem 22

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 35.17 . (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

Numerade Educator

Problem 23

Light of wavelength 700 $\mathrm{nm}$ is incident on the face of a fused quartz prism $(n=1.458 \text { at } 700 \mathrm{nm})$ at an incidence angle of $75.0^{\circ} .$ The apex angle of the prism is $60.0^{\circ} .$ Calculate the angle (a) of refraction at the first surface, (b) of incidence at the second surface, (c) of refraction at the second surface, and (d) between the incident and emerging rays.

Aatish Gupta

Numerade Educator

Problem 24

Figure $\mathrm{P} 35.24$ 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

Numerade Educator

Problem 25

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. The laser beam enters sheet 1 and then strikes the interface between sheet 1 and sheet 2 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 on the sheet 3 -sheet 2 interface, the refracted beam makes an angle of $36.7^{\circ}$ with the normal. If the experiment is repeated again with sheet 1 on top of sheet $3,$ with that same angle of incidence on the sheet 1 -sheet 3 interface, what is the expected angle of refraction in sheet 3 ?

Mayukh Banik

Numerade Educator

Problem 26

A person looking into an empty container is able to see the far edge of the container's bottom as shown in Figure P35.26a. 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 P35.26b.
(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

Numerade Educator

Problem 27

A laser beam is incident on a $45^{\circ}-45^{\circ}-90^{\circ}$ prism perpendicular to one of its faces as shown in Figure P35. 27 The transmitted beam that exits the hypotenuse of the prism makes an angle of prism makes an angle of of the incident beam. Find the index of refraction of the prism.

Vishal Gupta

Numerade Educator

Problem 28

A submarine is 300 m horizontally from the shore of a freshwater lake and 100 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 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.

Zulfiqar Ali

Numerade Educator

Problem 29

A beam of light both reflects and refracts at the surface between air and glass as shown in Figure P35.29. If the refractive index 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.

Zulfiqar Ali

Numerade Educator

Problem 30

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 $83.0^{\circ},$ what are the underwater angles of refraction for the (a) red and (b) blue components of the light?

Salamat Ali

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 $50.0^{\circ} .$ 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

Numerade Educator

Problem 32

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 $\operatorname{P35.32}$ . 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

Numerade Educator

Problem 33

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 spread of visible light passing through a prism of apex angle $60.0^{\circ}$ if the angle of incidence is $50.0^{\circ} ?$ See Figure P35.33.

Aatish Gupta

Numerade Educator

Problem 34

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 $\mathrm{P} 35.33$ .

Mayukh Banik

Numerade Educator

Problem 35

A glass optical fiber $(n=1.50)$ is submerged in water $(n=$ $1.33 ) .$ What is the critical angle for light to stay inside the fiber?

Zulfiqar Ali

Numerade Educator

Problem 36

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

Numerade Educator

Problem 37

A triangular glass prism with apex angle $\Phi=60.0^{\circ}$ has an index of refraction $n=1.50$ (Fig, P35.37). What is the smallest angle of incidence $\theta_{1}$ for which a light ray can emerge from the other side?

Mayukh Banik

Numerade Educator

Problem 38

A triangular glass prism with apex angle $\Phi$ has an index of refraction $n$ (Fig. P35.37). What is the smallest angle of incidence $\theta_{1}$ for which a light ray can emerge from the other side?

Sheh Lit Chang

University of Washington

Problem 39

Assume a transparent rod of diameter $d=2.00 \mu \mathrm{m}$ has an index of refraction of 1.36 Determine the maximum angle $\theta$ for which the light rays incident on the end of the rod in Figure P35.39 are subject to total internal reflection along the walls of the rod. Your answer defines the size of the cone of acceptance for the rod.

Zulfiqar Ali

Numerade Educator

Problem 40

Consider a light ray traveling between air and a diamond cut in the shape shown in Figure P35.40. (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 P35.40. 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 P35.40 undergo total internal reflection at P? Explain. (e) If the light ray entering the diamond remains vertical as shown in Figure P35.40, 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 $P ?$

Mayukh Banik

Numerade Educator

Problem 41

Consider a common mirage formed by superheated air immediately above a roadway. A truck driver whose eyes are 2.00 $\mathrm{m}$ above the road, where $n=1.000293,$ looks forward. She perceives the illusion of a patch of water ahead on the road. The road appears wet only beyond a point on the road at which her line of sight makes an angle of $1.20^{\circ}$ below the horizontal. Find the index of refraction of the air immediately above the road surface.

Zulfiqar Ali

Numerade Educator

Problem 42

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

Numerade Educator

Problem 43

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 $\mathrm{P} 35.43$ on page 1036 . (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. (c) As $n$ increases? (d) As $n$ approaches 1 (e) Evaluate $R_{\text { min }}$ assuming the fiber diameter is 100$\mu \mathrm{m}$ and its index of refraction is 1.40 .

Mayukh Banik

Numerade Educator

Problem 44

Around 1965, engineers at the Toro Company invented a gasoline gauge for small engines diagrammed in Figure P35.44. 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.

Zulfiqar Ali

Numerade Educator

Problem 45

A small light fixture on the bottom of a swimming pool is 1.00 m below the surface. The light emerging from the still water forms a circle on the water surface. What is the diameter of this circle?

Aatish Gupta

Numerade Educator

Problem 46

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 inci- dence 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.

Mayukh Banik

Numerade Educator

Problem 47

A digital video disc (DVD) records information in a spiral track approximately 1$\mu \mathrm{m}$ wide. The track consists of a series of pits in the information layer (Fig. P35. 47 $\mathrm{a}$ ) that scatter light from a laser beam sharply focused on them. The laser shines in from below through transparent plastic of thickness $t=1.20 \mathrm{mm}$ and index of refraction 1.55 (Fig.P35.47b). Assume the width of the laser beam at the information layer must be $a=1.00 \mu \mathrm{m}$ to read from only one track and not from its neighbors. Assume the width of the beam as it enters the transparent plastic is $w=0.700 \mathrm{mm}$ . A lens makes the beam converge into a cone with an apex angle 2$\theta_{1}$ before it enters the DVD. Find the incidence angle $\theta_{1}$ of the light at the edge of the conical beam. This design is relatively immune to small dust particles degrading the video quality.

Mayukh Banik

Numerade Educator

Problem 48

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

Numerade Educator

Problem 49

How many times will the incident beam shown in Figure P35.49 be reflected by each of the parallel mirrors?

Zulfiqar Ali

Numerade Educator

Problem 50

One technique for measuring the apex angle of a prism is shown in Figure P35.50. Two parallel rays of light are directed onto the apex of the prism so that the rays reflect the prism. The angular separation $\gamma$ of the two reflected rays can be measured. Show that $\phi=\frac{1}{2} \gamma$ .

Mayukh Banik

Numerade Educator

Problem 51

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

Numerade Educator

Problem 52

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?

Mayukh Banik

Numerade Educator

Problem 53

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

Aatish Gupta

Numerade Educator

Problem 54

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 $\mathrm{P} 35.54$ (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

Numerade Educator

Problem 55

A thief hides a precious jewel by placing it on the bottom of a public swimming pool. He places a circular raft on the surface of the water directly above and centeredover the jewel as shown in figure P35.55. The surface of the water is calm. The raft, of diameter $d=4.54 \mathrm{m}$ , prevents the jewel from being seen by any observer above the water, either on the raft or on the side of the pool. What is the maximum depth $h$ of the pool for the jewel to remain unseen?

Mayukh Banik

Numerade Educator

Problem 56

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

Numerade Educator

Problem 57

The light beam in Figure P35.57 strikes surface 2 at the critical angle. Determine the angle of incidence $\theta_{1}$.

Zulfiqar Ali

Numerade Educator

Problem 58

Figure P35.58 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.

Zulfiqar Ali

Numerade Educator

Problem 59

A 4.00 -m-long pole stands vertically in a freshwater lake having a depth of 2.00 $\mathrm{m}$ . The Sun is $40.0^{\circ}$ above the horizontal. Determine the length of the pole’s shadow on the bottom of the lake.

Mayukh Banik

Numerade Educator

Problem 60

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 P35.60. (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

Numerade Educator

Problem 61

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 $\mathrm{P} 35.61$ , 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 Warner

Numerade Educator

Problem 62

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 1.000 293, a sharply defined upper surface, and depth 8 614 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.

Mayukh Banik

Numerade Educator

Problem 63

A material having an index of refraction $n$ is surrounded by vacuum and is in the shape of a quarter circle of radius $R(\text { Fig. } \mathrm{P} 35.63) .$ 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

Numerade Educator

Problem 64

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

Numerade Educator

Problem 65

As shown in Figure $\mathrm{P} 35.65$ , a light ray is incident normal to one face of a $30^{\circ}-60^{\circ}-90^{\circ}$ block of flint glass (a prism) that is immersed in water. (a) Determine the exit angle $\theta_{3}$ of the ray. (b) A substance is dissolved in the water to increase the index of refraction $n_{2} .$ At what value of $n_{2}$ does total internal reflection cease at point $P ?$

Mayukh Banik

Numerade Educator

Problem 66

A transparent cylinder of radius $R=2.00 \mathrm{m}$ has a mirrored surface on its right half as shown in Figure $\mathrm{P} 35.66 .$ A light ray traveling in air is incident on the left side of the cylinder. The incident light ray and exiting light ray are parallel, and $d=2.00 \mathrm{m} .$ Determine the index of refraction of the material.

Vishal Gupta

Numerade Educator

Problem 67

Figure $\mathrm{P} 35.67$ 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

Numerade Educator

Problem 68

A. H. Pfund’s method for measuring the index of refraction of glass is illustrated in Figure P35.68. 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.

Mayukh Banik

Numerade Educator

Problem 69

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 P35.69. (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, what time interval is required for the light ray to travel through the plastic?

Mayukh Banik

Numerade Educator

Problem 70

Students allow a narrow beam of laser light to strike a water surface. They measure the angle of refraction for selected angles of incidence and record the data shown in the accompanying table. (a) 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. (b) Explain what the shape of the graph demonstrates. (c) Use the resulting plot to deduce the index of refraction of water, explaining how you do so.
$$\begin{array}{cc}{\text { Angle of Incidence }} & {\text { Angle of Refraction }} \\ \text { (degrees) } & {(\text { degrees) }} \\ \hline {10.0} & {7.5} \\ {20.0} & {15.1} \\ {30.0} & {22.3} \\ {40.0} & {28.7}\\ {50.0} & {35.2} \\ {60.0} & {40.3} \\ {70.0} & {45.3} \\ {80.0} & {47.7} \\ \hline\end{array}$$

Mayukh Banik

Numerade Educator

Problem 71

A hiker stands on an isolated mountain peak near sunset and observes a rainbow caused by water droplets in the air at a distance of 8.00 km along her line of sight to the most intense light from the rainbow. The valley is 2.00 km below the mountain peak and entirely flat. What fraction of the complete circular arc of the rainbow is visible to the hiker?

Mayukh Banik

Numerade Educator

Problem 72

Why is the following situation impossible? 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.

Mayukh Banik

Numerade Educator

Problem 73

Figure P35.73 shows an 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?

Mayukh Banik

Numerade Educator

Problem 74

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 P35.74, 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 5 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,
$$n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}$$

Mayukh Banik

Numerade Educator

Problem 75

Refer to Problem 74 for the statement of Fermat’s principle of least time. Derive the law of reflection (Eq. 35.2) from Fermat’s principle.

Mayukh Banik

Numerade Educator

Problem 76

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

Numerade Educator