Review. A mirror is often silvered with aluminum. By

In each of the following situations, a wave passes through an opening in an absorbing wall. Rank the situations in order from the one in which the wave is best described by the ray approximation to the one in which the wave coming through the opening spreads out most nearly equally in all directions in the hemisphere beyond the wall. (a) The sound of a low whistle at 1 kHz passes through a doorway 1 m wide. (b) Red light passes through the pupil of your eye. (c) Blue light passes through the pupil of your eye. (d) The wave broadcast by an AM radio station passes through a doorway 1 m wide. (e) An x-ray passes through the space between bones in your elbow joint.

A source emits monochromatic light of wavelength 495 nm in air. When the light passes through a liquid, its wavelength reduces to 434 nm. What is the liquids index of refraction? (a) 1.26 (b) 1.49 (c) 1.14 (d) 1.33 (e) 2.03

Carbon disulfide (n 5 1.63) is poured into a container made of crown glass (n 5 1.52). What is the critical angle for total internal reflection of a light ray in the liquid when it is incident on the liquid-to-glass surface? (a) 89.2 (b) 68.8 (c) 21.2 (d) 1.07 (e) 43.0

A light wave moves between medium 1 and medium 2. Which of the following are correct statements relating its speed, frequency, and wavelength in the two media, the indices of refraction of the media, and the angles of incidence and refraction? More than one statement may be correct. (a) v1/sin u1 5 v2/sin u2 (b) csc u1/n1 5 csc u2/n2 (c) l1/sin u1 5 l2/sin u2 (d) f1/sin u1 5 f2/sin u2 (e) n1/cos u1 5 n2/cos u2

What happens to a light wave when it travels from air into glass? (a) Its speed remains the same. (b) Its speed increases. (c) Its wavelength increases. (d) Its wavelength remains the same. (e) Its frequency remains the same.

The index of refraction for water is about 4 3. What happens as a beam of light travels from air into water? (a) Its speed increases to 4 3c, and its frequency decreases. (b) Its speed decreases to 3 4c, and its wavelength decreases by a factor of 3 4. (c) Its speed decreases to 3 4c, and its wavelength increases by a factor of 4 3. (d) Its speed and frequency remain the same. (e) Its speed decreases to 3 4c, and its frequency increases.

Light can travel from air into water. Some possible paths for the light ray in the water are shown in Figure OQ35.7. Which path will the light most likely follow? (a) A (b) B (c)C (d) D (e) E

What is the order of magnitude of the time interval required for light to travel 10 km as in Galileos attempt to measure the speed of light? (a) several seconds (b) several milliseconds (c) several microseconds (d) several nanoseconds

A light ray containing both blue and red wavelengths is incident at an angle on a slab of glass. Which of the sketches in Figure OQ35.9 represents the most likely outcome? (a) A (b) B (c) C (d) D (e) none of them

For the following questions, choose from the following possibilities: (a) yes; water (b) no; water (c) yes; air (d) no; air. (i) Can light undergo total internal reflection at a smooth interface between air and water? If so, in which medium must it be traveling originally? (ii) Can sound undergo total internal reflection at a smooth interface between air and water? If so, in which medium must it be traveling originally?

A light ray travels from vacuum into a slab of material with index of refraction n1 at incident angle u with respect to the surface. It subsequently passes into a second slab of material with index of refraction n2 before passing back into vacuum again. The surfaces of the different materials are all parallel to one another. As the light exits the second slab, what can be said of the final angle f that the outgoing light makes with the normal? (a) f . u (b) f , u (c) f 5 u (d) The angle depends on the magnitudes of n1 and n2. (e) The angle depends on the wavelength of the light.

Suppose you find experimentally that two colors of light, A and B, originally traveling in the same direction in air, are sent through a glass prism, and A changes direction more than B. Which travels more slowly in the prism, A or B? Alternatively, is there insufficient information to determine which moves more slowly?

The core of an optical fiber transmits light with minimal loss if it is surrounded by what? (a) water (b) diamond (c) air (d) glass (e) fused quartz

Which color light refracts the most when entering crown glass from air at some incident angle u with respect to the normal? (a) violet (b) blue (c) green (d) yellow (e) red

Light traveling in a medium of index of refraction n1 is incident on another medium having an index of refraction n2. Under which of the following conditions can total internal reflection occur at the interface of the two media? (a) The indices of refraction have the relation n2 . n1. (b) The indices of refraction have the relation n1 . n2. (c) Light travels slower in the second medium than in the first. (d) The angle of incidence is less than the critical angle. (e) The angle of incidence must equal the angle of refraction.

Try this simple experiment on your own. Take two opaque cups, place a coin at the bottom of each cup near the edge, and fill one cup with water. Next, view the cups at some angle from the side so that the coin in water is just visible as shown on the left in Figure CQ35.16. Notice that the coin in air is not visible as shown on the right in Figure CQ35.16. Explain this observation.

Figure CQ35.17a shows a desk ornament globe containing a photograph. The flat photograph is in air, inside a vertical slot located behind a water-filled compart ment 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. CQ35.17b). (a) Account for this phenomenon and (b) describe what you see when you turn the globe beyond this angle.

The reflecting surfaces of two intersecting flat mirrors are at an angle u (0 , u , 90) as shown in Figure P35.18. For a light ray that strikes the horizontal mirror, show that the emerging ray will intersect the incident ray at an angle b 5 180 2 2u.

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?

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 u1. Prove that the final direction of the ray, after reflection from both mirrors, is opposite its initial direction. (b) What If? Now assume the paper is replaced with a third flat mirror, touching edges with the other two and perpendicular to both, creating a corner-cube retroreflector (Fig. 35.8a). 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 Moons orbit is increasing at the rate of 3.8cm/yr as it loses kinetic energy because of tidal friction.

The two mirrors illustrated in Figure P35.21 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?

When the light ray illustrated in Figure P35.22 passes through the glass block of index of refraction n 5 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.

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 m of ice along the way. Determine the difference in the pulses times of arrival at the detector.

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

A laser beam with vacuum wavelength 632.8 nm is incident from air onto a block of Lucite as shown in Figure 35.10b. 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.

A narrow beam of ultrasonic waves reflects off the liver tumor illustrated in Figure P35.26. The speed of the wave is 10.0% less in the liver than in the surrounding medium. Determine the depth of the tumor.

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 above the horizon, sunlight ceases to illuminate any part of the bottom of the tank. How deep is the tank?

A triangular glass prism with apex angle 60.0 has an index of refraction of 1.50. (a) Show that if its angle of incidence on the first surface is u1 5 48.6, light will pass symmetrically through the prism as shown in Figure 35.17. (b) Find the angle of deviation dmin for u1 5 48.6. (c) What If? Find the angle of deviation if the angle of incidence on the first surface is 45.6. (d) Find the angle of deviation if u1 5 51.6

Light of wavelength 700 nm is incident on the face of a fused quartz prism (n 5 1.458 at 700 nm) at an incidence angle of 75.0. The apex angle of the prism is 60.0. 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

Figure P35.30 shows a light ray incident on a series of slabs having different refractive indices, where n1 , n2 , n3 , n4. 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.

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 with the normal. The refracted beam in sheet 2 makes an angle of 31.7 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 3sheet 2 interface, the refracted beam makes an angle of 36.7 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 1sheet 3 interface, what is the expected angle of refraction in sheet 3?

A person looking into an empty container is able to see the far edge of the containers bottom as shown in Figure P35.32a. 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 containers bottom as shown in Figure P35.32b. (a) Show that the ratio h/d is given by h d 5 n2 2 1 4 2 n2 (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. (c) For what range of values of n will the center of the coin not be visible for any values of h and d?

A laser beam is incident on a 454590 prism perpendicular to one of its faces as shown in Figure P35.33. The transmitted beam that exits the hypotenuse of the prism makes an angle of u 5 15.0 with the direction of the incident beam. Find the index of refraction of the prism.

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 waterair 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.

A beam of light both reflects and refracts at the surface between air and glass as shown in Figure P35.35. If the refractive index of the glass is ng , find the angle of incidence u1 in the air that would result in the reflected ray and the refracted ray being perpendicular to each other.

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, what are the underwater angles of refraction for the (a) red and (b) blue components of the light?

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

The speed of a water wave is described by v 5 !gd , 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 northnortheast. 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 P35.38. 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

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 if the angle of incidence is 50.0? See Figure P35.39.

The index of refraction for violet light in silica flint glass is nV, and that for red light is nR. What is the angular spread of visible light passing through a prism of apex angle F if the angle of incidence is u? See Figure P35.39.

A glass optical fiber (n 5 1.50) is submerged in water (n 5 1.33). What is the critical angle for light to stay inside the fiber?

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

A triangular glass prism with apex angle F 5 60.0 has an index of refraction n 5 1.50 (Fig. P35.43). What is the smallest angle of incidence u1 for which a light ray can emerge from the other side?

A triangular glass prism with apex angle F has an index of refraction n (Fig. P35.43). What is the smallest angle of incidence u1 for which a light ray can emerge from the other side?

Assume a transparent rod of diameter d 5 2.00 mm has an index of refraction of 1.36. Determine the maximum angle u for which the light rays incident on the end of the rod in Figure P35.45 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.

Consider a light ray traveling between air and a diamond cut in the shape shown in Figure P35.46. (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 P35.46. 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 diamondwater interface? (d) When the diamond is immersed in water, does the light ray entering the top surface in Figure P35.46 undergo total internal reflection at P? Explain. (e) If the light ray entering the diamond remains vertical as shown in Figure P35.46, 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?

Consider a common mirage formed by superheated air immediately above a roadway. A truck driver whose eyes are 2.00 m above the road, where n 5 1.000 293, 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 below the horizontal. Find the index of refraction of the air immediately above the road surface.

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 concreteair 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.

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 P35.49. (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 min assuming the fiber diameter is 100 mm and its index of refraction is 1.40.

Around 1968, Richard A. Thorud, an engineer at The Toro Company, invented a gasoline gauge for small engines diagrammed in Figure P35.50. 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 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.

A beam of light is incident from air on the surface of a liquid. If the angle of incidence is 30.0 and the angle of refraction is 22.0, find the critical angle for total internal reflection for the liquid when surrounded by air

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. 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. 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 airglass surface, determine and tabulate the angles of reflection and refraction for all the angles of incidence at 10.0 intervals from 0 to 90.0. (d) Do the same for light rays coming up to the interface through the glass.

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?

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 above the horizontal.

A digital video disc (DVD) records information in a spiral track approximately 1 mm wide. The track consists of a series of pits in the information layer (Fig. P35.55a) that scatter light from a laser beam sharply focused on them. The laser shines in from below through transparent plastic of thickness t 5 1.20 mm and index of refraction 1.55 (Fig. P35.55b). Assume the width of the laser beam at the information layer must be a 5 1.00 mm 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 5 0.700 mm. A lens makes the beam converge into a cone with an apex angle 2u1 before it enters the DVD. Find the incidence angle u1 of the light at the edge of the conical beam. This design is relatively immune to small dust particles degrading the video quality.

How many times will the incident beam shown in Figure P35.56 be reflected by each of the parallel mirrors?

When light is incident normally on the interface between two transparent optical media, the intensity of the reflected light is given by the expression S 1 r 5 a n2 2 n1 n2 1 n1 b 2 S 1 In this equation, S1 represents the average magnitude of the Poynting vector in the incident light (the incident intensity), S1 9 is the reflected intensity, and n1 and n2 are the refractive indices of the two media. (a) What fraction of the incident intensity is reflected for 589-nm light normally incident on an interface between air and crown glass? (b) Does it matter in part (a) whether the light is in the air or in the glass as it strikes the interface?

Refer to Problem 57 for its description of the reflected intensity of light normally incident on an interface between two transparent media. (a) For light normally incident on an interface between vacuum and a transparent medium of index n, show that the intensity S 2 of the transmitted light is given by S2/S1 = 4n/(n 1 1)2. (b) Light travels perpendicularly through a diamond slab, surrounded by air, with parallel surfaces of entry and exit. Apply the transmission fraction in part (a) to find the approximate overall transmission through the slab of diamond, as a percentage. Ignore light reflected back and forth within the slab.

A light ray enters the atmosphere of the Earth and descends vertically to the surface a distance h 5 100 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 5 1.000 293 at the Earths 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 Earths atmosphere?

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?

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 2u 5 2 sin u cos u.

One technique for measuring the apex angle of a prism is shown in Figure P35.62. Two parallel rays of light are directed onto the apex of the prism so that the rays reflect from opposite faces of the prism. The angular separation g of the two reflected rays can be measured. Show that f 5 1 2g.

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 centered over the jewel as shown in Figure P35.63. The surface of the water is calm. The raft, of diameter d 5 4.54 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?

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

The light beam in Figure P35.65 strikes surface 2 at the critical angle. Determine the angle of incidence u1.

Why is the following situation impossible? A laser beam strikes one end of a slab of material of length L 5 42.0 cm and thickness t 5 3.10 mm as shown in Figure P35.66 (not to scale). It enters the material at the center of the left end, striking it at an angle of incidence of u 5 50.0. The index of refraction of the slab is n 5 1.48. The light makes 85 internal reflections from the top and bottom of the slab before exiting at the other end.

A 4.00-m-long pole stands vertically in a freshwater lake having a depth of 2.00 m. The Sun is 40.0 above the horizontal. Determine the length of the poles shadow on the bottom of the lake.

A light ray of wavelength 589 nm is incident at an angle u on the top surface of a block of polystyrene as shown in Figure P35.68. (a) Find the maximum value of u 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.

A light ray traveling in air is incident on one face of a right-angle prism with index of refraction n 5 1.50 as shown in Figure P35.69, and the ray follows the path shown in the figure. Assuming u 5 60.0 and the base of the prism is mirrored, determine the angle f made by the outgoing ray with the normal to the right face of the prism.

As sunlight enters the Earths 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 Earths 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 Earths equator so that the apparent path of the rising and setting Sun is perpendicular to the horizon.

A material having an index of refraction n is surrounded by vacuum and is in the shape of a quarter circle of radius R (Fig. P35.71). 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 u. Determine an expression for u in terms of n, R, and L.

A ray of light passes from air into water. For its deviation angle d 5 uu1 2 u2u to be 10.0, what must its angle of incidence be?

As shown in Figure P35.73, a light ray is incident normal to one face of a 306090 block of flint glass (a prism) that is immersed in water. (a) Determine the exit angle u3 of the ray. (b) A substance is dissolved in the water to increase the index of refraction n2. At what value of n2 does total internal reflection cease at point P ?

A transparent cylinder of radius R 5 2.00 m has a mirrored surface on its right half as shown in Figure P35.74. 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 5 2.00 m. Determine the index of refraction of the material.

Figure P35.75 shows the path of a light beam through several slabs with different indices of refraction. (a) If u1 5 30.0, what is the angle u2 of the emerging beam? (b) What must the incident angle u1 be to have total internal reflection at the surface between the medium with n 5 1.20 and the medium with n 5 1.00?

A. H. Pfunds method for measuring the index of refraction of glass is illustrated in Figure P35.76. 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 PBB9 strikes the clear surface at the critical angle and is totally reflected, as are rays such as PCC9. Rays such as PAA9 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 5 1.52 for a slab 0.600 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.

A light ray enters a rectangular block of plastic at an angle u1 5 45.0 and emerges at an angle u2 5 76.0 as shown in Figure P35.77. (a) Determine the index of refraction of the plastic. (b) If the light ray enters the plastic at a point L 5 50.0 cm from the bottom edge, what time interval is required for the light ray to travel through the plastic?

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 Snells 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. Angle of Incidence Angle of Refraction (degrees) (degrees) 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

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 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 north, the rising Sun moves through the sky along a line making a 50.0 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?

Figure P35.80 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 u 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 u for which the ray can exit after multiple reflections? If so, sketch one of the rays paths.

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?

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.

Figure P35.83 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 v 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?

Pierre de Fermat (16011665) 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 Fermats 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 Snells law of refraction from Fermats principle. Proceed as follows. In Figure P35.84, 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 5 r1 v1 1 r2 v 2 5 n1"a 2 1 x 2 c 1 n2"b 2 1 1d 2 x2 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 n1x "a 2 1 x 2 5 n2 1d 2 x2 "b 2 1 1d 2 x2 2 (c) Show that this expression in turn gives Snells law, n1 sin u1 5 n2 sin u2

Refer to Problem 84 for the statement of Fermats principle of least time. Derive the law of reflection (Eq. 35.2) from Fermats principle.

Suppose a luminous sphere of radius R1 (such as the Sun) is surrounded by a uniform atmosphere of radius R2 . R1 and index of refraction n. When the sphere is viewed from a location far away in vacuum, what is its apparent radius (a) when R2 . nR1 and (b) when R2 , nR1?

This problem builds upon the results of Problems 57 and 58. Light travels perpendicularly through a diamond slab, surrounded by air, with parallel surfaces of entry and exit. The intensity of the transmitted light is what fraction of the incident intensity? Include the effects of light reflected back and forth inside the slab.