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

Educators

Chapter Questions

Problem 1

The Apollo 11 astronauts set up a panel of efficient corner-cube retroreflectors on the Moon's surface (Fig.
35.8 $\mathrm{a}$ ). 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. What is the measured speed of light? Take the center-to-center distance from the Earth to the Moon to be $3.84 \times 10^{8} \mathrm{m}$ . Explain whether it is necessary to consider the sizes of the Earth and the Moon in your calculation.

Aatish Gupta

Aatish Gupta

Numerade Educator

Problem 2

As a result of his observations, 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 $1.50 \times 10^{8} \mathrm{km}$ as the average radius of the Earth's orbit around the Sun, calculate the speed of light from these data.

Narayan Hari

Numerade Educator

Problem 3

In an experiment to measure the speed of light using the apparatus of Fizeau (see Fig. 35.2$)$ , 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 $2.998 \times 10^{8} \mathrm{m} / \mathrm{s}$ . Calculate the minimum angular speed of the wheel for this experiment.

Zulfiqar Ali

Numerade Educator

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

Numerade Educator

Problem 5

The two mirrors illustrated in Figure $\mathrm{P} 35.5$ 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 ?

Zulfiqar Ali

Numerade Educator

Problem 6

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. In a clothing store, such a pair of mirrors shows you an image of yourself as others see you, with no apparent right-left reversal. (b) What If? Now assume the paper is replaced with a third flat mirror, touching edges with the other two and perpendicular to both. The set of three mirrors is called a corner-cube reflector. 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 oppo- site to its original direction. The Aporeflectors on the 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.

Aatish Gupta

Numerade Educator

Problem 7

The distance of a lightbulb from a large plane mirror is twice the distance of a person from the plane mirror. Light from the lightbulb reaches the person by two paths. It travels to the mirror at an angle of incidence $\theta$ and reflects from the mirror to the person. It also travels directly to the person without reflecting off the mirror. The total distance traveled by the light in the first case is twice the distance traveled by the light in the second case. Find the value of the angle $\theta .$

Aatish Gupta

Numerade Educator

Problem 8

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

Aatish Gupta

Numerade Educator

Problem 9

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 $35.0^{\circ} .$ Determine the angle of refraction and the wavelength of the light in water.

Aatish Gupta

Numerade Educator

Problem 10

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 $3.50^{\circ} .$ Determine the angle of refraction for the sound wave and the wavelength of the sound in water. Compare and contrast the behavior of the sound in this problem with the behavior of the light in Problem $9 .$

Zulfiqar Ali

Numerade Educator

Problem 11

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 12

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

Aatish Gupta

Numerade Educator

Problem 13

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.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

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 15

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

Zulfiqar Ali

Numerade Educator

Problem 16

A narrow beam of ultrasonic waves reflects off the liver tumor illustrated in Figure $\mathrm{P} 35.16$ . 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 17

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 18

An opaque cylindrical tank with an open top has a diameter of 3.00 $\mathrm{m}$ 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 19

When the light illustrated in Figure $\mathrm{P} 35.19$ passes through the glass block, it is shifted laterally by the distance $d .$ Taking $n=1.50$ , find the value of $d .$

Aatish Gupta

Numerade Educator

Problem 20

Find the time interval required for the light to pass through the glass block described in Problem 19 .

Vipender Yadav

Numerade Educator

Problem 21

The light beam shown in Figure $\mathrm{P} 35.21$ 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 index of refraction of linseed oil is $1.48 . )$

Aatish Gupta

Numerade Educator

Problem 22

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 again with sheet 1 on top of sheet $3,$ what is the expected angle of refraction in sheet 3$?$ Assume the same angle of incidence.

Aatish Gupta

Numerade Educator

Problem 23

Light passes from air into flint glass. (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 24

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

Problem 25

A prism that has an apex angle of $50.0^{\circ}$ is made of cubic zirconia, with $n=2.20 .$ What is its angle of minimum deviation?

Aatish Gupta

Numerade Educator

Problem 26

Light of wavelength 700 $\mathrm{nm}$ is incident on the face of a fused quartz prism at an angle of $75.0^{\circ}$ (with respect to the normal to the surface). The apex angle of the prism is $60.0^{\circ} .$ Use the value of $n$ from Figure 35.21 and 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 ravs.

Aatish Gupta

Numerade Educator

Problem 27

A triangular glass prism with apex angle $\Phi=60.0^{\circ}$ has an index of refraction $n=1.50$ (Fig. P35.27). What is the smallest angle of incidence $\theta_{1}$ for which a light ray can emerge from the other side?

Narayan Hari

Numerade Educator

Problem 28

A triangular glass prism with apex angle $\Phi$ has index of refraction $n$ . (See Fig. P35. 27 .) 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 29

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 30

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. Sketch a map of an ocean beach on the eastern side of a landmass. Show contour lines of constant depth under water, assuming reasonably uniform slope. (a) 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. (b) Sketch a map of a coastline with alternating bays and headlands as suggested in Figure $\mathrm{P} 35.30 .$ Again make a reasonable guess about the shape of contour lines of constant depth. 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 31

A 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 $\mathrm{P} 35.31 .$

Aatish Gupta

Numerade Educator

Problem 32

A narrow, white light beam is incident on a block of fused quartz at an angle of $30.0^{\circ} .$ Find the angular spread of the light beam inside the quartz due to dispersion.

Aatish Gupta

Numerade Educator

Problem 33

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

Aatish Gupta

Numerade Educator

Problem 34

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

Sheh Lit Chang

University of Washington

Problem 35

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 $n=1.0003,$ looks forward. She perceives the illusion of a patch of water ahead on the road, where 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. Suggestion: Treat this problem as one about total internal reflection.

Aatish Gupta

Numerade Educator

Problem 36

Determine the maximum angle $\theta$ for which the light rays incident on the end of the pipe in Figure $\mathrm{P} 35.36$ are subject to total internal reflection along the walls of the pipe. Assume the pipe has an index of refraction of 1.36 and the outside medium is air. Your answer defines the size of the cone of acceptance for the light pipe.

Aatish Gupta

Numerade Educator

Problem 37

An optical fiber has index of refraction $n$ and diameter $d .$ It is surrounded by air. Light is sent into the fiber along its axis as shown in Figure $\mathrm{P} 35.37$ . ( a) Find the smallest outside radius $R$ permitted for a bend in the fiber if no light is to escape. (b) What If? Does the result for part (a) predict reasonable behavior as $d$ approaches zero? As $n$ increases? As $n$ approaches 1 ? (c) Evaluate $R$ assuming the fiber diameter is 100$\mu \mathrm{m}$ and its index of refraction is $1.40 .$

Eduard Sanchez

Numerade Educator

Problem 38

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 traveling if it is undergo total internal reflection? (c) "A bare concrete wall is a highly efficient mirror for sound." Give evidence for or against this statement.

Aatish Gupta

Numerade Educator

Problem 39

Around 1965 , engineers at the Toro Company invented a gasoline gauge for small engines diagrammed in Figure P35.39. 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. Explain how the gauge works. Explain the design requirements, if any, for the index of refraction of the plastic.

Zulfiqar Ali

Numerade Educator

Problem 40

A digital videodisc 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. P35.40a) that scatter light from a laser beam sharply focused on them. The laser shines in through transparent plastic of thickness $t=1.20 \mathrm{mm}$ and index of refraction 1.55 (Fig. P35. 40 b) 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 from below 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 videodisc. 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. Particles on the plastic surface would have to be as large as 0.7 $\mathrm{mm}$ to obscure the beam.

Mayukh Banik

Numerade Educator

Problem 41

Figure $\mathrm{P} 35.41 \mathrm{a}$ shows a desk ornament globe containing a photograph. The flat photograph is in air, inside a vertical slot located behind a water-filled compartment having the shape of one half of a cylinder. Suppose you are looking at the center of the photograph and then rotate the globe about a vertical axis. You find that the center of the photograph disappears when you rotate the globe beyond a certain maximum angle (Fig. P35.41b). Account for this phenomenon and calculate the maximum angle. Describe what you see when you turn the globe beyond this angle.

Mayukh Banik

Numerade Educator

Problem 42

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 1.000 , 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) State how this travel time compares with the time interval required in the absence of an atmosphere.

Oliver Mcneely

Numerade Educator

Problem 43

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 44

(a) Consider a horizontal interface between air above and glass of index 1.55 below. 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 angle of incidence $30.0^{\circ} .$ 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

Numerade Educator

Problem 45

A small light fixture on the bottom of a swimming pool is 1.00 $\mathrm{m}$ 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

The walls of a prison cell 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 young prisoner observes the patch of light moving across this western wall and for the first time forms his own understanding of the rotation of the Earth. (a) With what speed does the illuminated rectangle move? (b) The prisoner holds a small, square mirror flat against the 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 prisoner's cell move? (d) In what direction does the smaller square of light on the eastern wall move?

Keshav Singh

Numerade Educator

Problem 47

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 $\mathrm{km}$ along her line of sight. The valley is 2.00 $\mathrm{km}$ below the mountain peak and entirely flat. What fraction of the complete circular arc of the rainbow is visible to the hiker? (See Fig. $35.24 .$ )

Mayukh Banik

Numerade Educator

Problem 48

Figure $\mathrm{P} 35.48$ 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 49

A laser beam strikes one end of a slab of material as shown in Figure $\mathrm{P} 35.49$ . 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.

Rashmi Sinha

Numerade Educator

Problem 50

A $4.00-\mathrm{m}$ -long pole stands vertically in a 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. Take the index of refraction for water to be 1.33 .

Oliver Mcneely

Numerade Educator

Problem 51

The light beam in Figure $\mathrm{P} 35.51$ strikes surface 2 at the critical angle. Determine the angle of incidence $\theta_{1}$ .

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 52

Builders use a leveling instrument in which the beam from a fixed helium-neon laser reflects in a horizontal plane from a small, flat mirror mounted on a vertical rotating shaft. The light is sufficiently bright and the rotation rate is sufficiently high that the reflected light appears as a horizontal line, wherever it falls on a wall. (a) Assume the mirror is at the center of a circular grain elevator of radius 3.00 $\mathrm{m}$ . The mirror spins with constant angular velocity 35.0 $\mathrm{rad} / \mathrm{s}$ . Find the speed of the spot of laser light on the curved wall. (b) Now assume the spinning mirror is at a perpendicular distance of 3.00 $\mathrm{m}$ from point $O$ on a long, flat, vertical wall. When the spot of laser light on the wall is at distance $x$ from point $O$ , what is its speed? (c) What is the minimum value for the speed? What value of $x$ corresponds to it? How does the minimum speed compare with the speed you found in part (a)? (d) What is the maximum speed of the spot on the flat wall? (e) In what time interval does the spot change from its minimum to its maximum speed?

Mayukh Banik

Numerade Educator

Problem 53

A $Q$ 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 $\mathrm{P} 35.53$ . (a) Find the maximum value of $\theta$ for which the refracted ray undergoes total internal reflection 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. You will need to explain your answers.

Aatish Gupta

Numerade Educator

Problem 54

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 when a mathematically straight line between an observer and the top of the Sun just clears the horizon and the instant at which 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

Problem 55

A shallow glass dish is 4.00 $\mathrm{cm}$ wide at the bottom as shown in Figure $\mathrm{P} 35.55$ . When an observer's eye is located as shown, the observer sees the edge of the bottom of the empty dish. When this dish is filled with water, the observer sees the center of the bottom of the dish. Find the height of the dish.

Oliver Mcneely

Numerade Educator

Problem 56

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 57

A material having an index of refraction $n$ is surrounded by a vacuum and is in the shape of a quarter circle of radius $R$ (Fig. P35.5 7). 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 $\theta .$

Mayukh Banik

Numerade Educator

Problem 58

Fermat's principle. 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. 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 P35.58, 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 at which 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

Numerade Educator

Problem 59

Refer to Problem 58 for the statement of Fermat's principle of least time. Derive the law of reflection (Eq. 35.2 ) from Fermat's principle.

Victor Salazar

Numerade Educator

Problem 60

A transparent cylinder of radius $R=2.00 \mathrm{m}$ has a mirrored surface on its right half as shown in Figure P35.60. 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.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 61

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

Mayukh Banik

Numerade Educator

Problem 62

A. H. Pfund's method for measuring the index of refraction of glass is illustrated in Figure P35.62. 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? $(\mathrm{c})$ 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

Numerade Educator

Problem 63

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 $\mathrm{P} 35.63$ . (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 64

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. 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. Explain what the shape of the graph demonstrates. Use the resulting plot to deduce the index of refraction of water, explaining how you do so.
(TABLE NOT COPY)

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 65

Review problem. 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, say, 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

Numerade Educator