Albert A. Michelson very carefully measured the speed of

During the Apollo XI Moon landing, a retroreecting panel was erected on the Moons surface. The speed of light can be found by measuring the time it takes a laser beam to travel from Earth, reect from the panel, and return to Earth. If this interval is found to be 2.51 s, what is the measured speed of light? Take the center-to-center distance from Earth to Moon to be 3.84  108 m. Assume that the Moon is directly overhead and do not neglect the sizes of the Earth and Moon.

Figure P22.2 shows the apparatus used by Armand H. L. Fizeau (18191896) to measure the speed of light. The basic idea is to measure the total time it takes light to travel from some point to a distant mirror and back. If d is the distance between the light source and the mirror, and if the transit time for one round-trip is t, then the speed of light is c  2d/t. To measure the transit time, Fizeau used a rotating toothed wheel, which converts an otherwise continuous beam of light to a series of light pulses. The rotation of the wheel controls what an observer at the light source sees. For example, assume that the toothed wheel of the Fizeau experiment has 360 teeth and is rotating at a speed of 27.5 rev/s when the light from the source is extinguishedthat is, when a burst of light passing through opening A in Figure P22.2 is blocked by tooth B on return. If the distance to the mirror is 7 500 m, nd the speed of light.

In an experiment designed to measure the speed of light using the apparatus of Fizeau described in the preceding problem, 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  108 m/s. Calculate the minimum angular speed of the wheel for this experiment.

Albert A. Michelson very carefully measured the speed of light using an alternative version of the technique developed by Fizeau. (See Problem 22.2.) Figure P22.4 shows the approach Michelson used. Light was reected from one face of a rotating eight-sided mirror towards a stationary mirror 35.0 km away. At certain rates of rotation, the returning beam of light was directed toward the eye of an observer as shown. (a) What minimum angular speed must the rotating mirror have in order that side A will have rotated to position B, causing the light to be reected to the eye? (b) What is the next-higher angular velocity that will enable the source of light to be seen?

Figure P22.5 shows an apparatus used to measure the distribution of the speeds of gas molecules. The device consists of two slotted rotating disks separated by a distance d, with the slots displaced by the angle  . Suppose the speed of light is measured by sending a light beam toward the left disk of this apparatus. (a) Show that a light beam will be seen in the detector (that is, will make it through both slots) only if its speed is given by c   d/  , where  is the angular speed of the disks and  is measured in radians. (b) What is the measured speed of light if the distance between the two slotted rotating disks is 2.500 m, the slot in the second disk is displaced 1/60 of 1 degree from the slot in the rst disk, and the disks are rotating at 5 555 rev/s?

The two mirrors in Figure P22.6 meet at a right angle. The beam of light in the vertical plane P strikes mirror 1 as shown. (a) Determine the distance the reected light beam travels before striking mirror 2. (b) In what direction does the light beam travel after being reected from mirror 2?

An underwater scuba diver sees the Sun at an apparent angle of 45.0 from the vertical. What is actual direction of the Sun?

Light is incident normal to a 1.00-cm layer of water that lies on top of a at Lucite plate with a thickness of 0.500 cm. How much more time is required for light to pass through this double layer than is required to traverse the same distance in air (nLucite  1.59)?

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

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

Light of wavelength  0 in a vacuum has a wavelength of 438 nm in water and a wavelength of 390 nm in benzene. (a) What is the wavelength  0? (b) Using only the given wavelengths, determine the ratio of the index of refraction of benzene to that of water.

Light of wavelength 436 nm in air enters a shbowl lled with water, then exits through the crown-glass wall of the container. Find the wavelengths of the light (a) in the water and (b) in the glass.

A ray of light is incident on the surface of a block of clear ice at an angle of 40.0 with the normal. Part of the light is reected and part is refracted. Find the angle between the reected and refracted light.

The laws of refraction and reection are the same for sound as for light. The speed of sound is 340 m/s in air and 1 510 m/s in water. If a sound wave traveling in air approaches a plane water surface at an angle of incidence of 12.0, what is the angle of refraction?

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

A ashlight on the bottom of a 4.00-m-deep swimming pool sends a ray upward and at an angle so that the ray strikes the surface of the water 2.00 m from the point directly above the ashlight. What angle (in air) does the emerging ray make with the waters surface?

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

A ray of light strikes a at 2.00-cm-thick block of glass (n  1.50) at an angle of 30.0 with the normal (Fig. P22.18). Trace the light beam through the glass, and nd the angles of incidence and refraction at each surface.

When the light ray in Problem 18 passes through the glass block, it is shifted laterally by a distance d (Fig. P22.18). Find the value of d.

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

The light beam shown in Figure P22.21 makes an angle of 20.0 with the normal line NN in the linseed oil. Determine the angles  and  . (The refractive index for linseed oil is 1.48.)

A submarine is 300 m horizontally out from the shore and 100 m beneath the surface of the water. A laser beam is sent from the sub so that it strikes the surface of the water at a point 210 m from the shore. If the beam just strikes the top of a building standing directly at the waters edge, nd the height of the building.

Two light pulses are emitted simultaneously from a source. The pulses take parallel paths to a detector 6.20 m away, but one moves through air and the other through a block of ice. Determine the difference in the pulses times of arrival at the detector.

A narrow beam of ultrasonic waves reects off the liver tumor in Figure P22.24. If the speed of the wave is 10.0% less in the liver than in the surrounding medium, determine the depth of the tumor.

A beam of light both reects and refracts at the surface between air and glass, as shown in Figure P22.25. If the index of refraction of the glass is ng, nd the angle of incidence,  1, in the air that would result in the reected ray and the refracted ray being perpendicular to each other. [Hint: Remember the identity sin(90   )  cos  .]

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 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, 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, what is the expected angle of refraction in sheet 3? Assume the same angle of incidence.

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

A cylindrical cistern, constructed below ground level, is 3.0 m in diameter and 2.0 m deep and is lled to the brim with a liquid whose index of refraction is 1.5. A small object rests on the bottom of the cistern at its center. How far from the edge of the cistern can a girl whose eyes are 1.2 m from the ground stand and still see the object?

The index of refraction for red light in water is 1.331, and that for blue light is 1.340. If a ray of white light enters the water at an angle of incidence of 83.00, what are the underwater angles of refraction for the blue and red components of the light?

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

A ray of light strikes the midpoint of one face of an equiangular (606060) glass prism (n  1.5) at an angle of incidence of 30. (a) Trace the path of the light ray through the glass, and nd the angles of incidence and refraction at each surface. (b) If a small fraction of light is also reected at each surface, nd the angles of reection at the surfaces.

The index of refraction for violet light in silica int glass is 1.66, and that for red light is 1.62. What is the angular dispersion of visible light passing through an equilateral prism of apex angle 60.0 if the angle of incidence is 50.0? (See Fig. P22.32.)

Calculate the critical angles for the following materials when surrounded by air: (a) zircon, (b) uorite, and (c) ice. Assume that

For light of wavelength 589 nm, calculate the critical angle for the following materials surrounded by air: (a) diamond and (b) int glass.

Repeat Problem 34, but this time suppose that the materials are surrounded by water.

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, nd the critical angle for the liquid when surrounded by air.

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

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

Consider a common mirage formed by superheated air just above a roadway. A truck driver whose eyes are 2.00 m above the road, where n  1.000 3, looks forward. She has the illusion of seeing a patch of water ahead on the road, where her line of sight makes an angle of 1.20 below the horizontal. Find the index of refraction of the air just above the road surface. [Hint: Treat this as a problem one involving total internal reection.]

A jewel thief hides a diamond 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 diamond, as shown in Figure P22.40. If the surface of the water is calm and the pool is 2.00 m deep, nd the minimum diameter of the raft that would prevent the diamond from being seen.

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 reection of sound at the concreteair boundary. (b) In which medium must the sound be traveling in order to undergo total internal reection? (c) A bare concrete wall is a highly efcient mirror for sound. Give evidence for or against this statement.

Three adjacent faces (that all share a corner) of a plastic cube of index of refraction n are painted black. A clear spot at the painted corner serves as a source of diverging rays when light comes through it. Show that a ray from this corner to the center of a clear face is totally reected if

The light beam in Figure P22.43 strikes surface 2 at the critical angle. Determine the angle of incidence,  i.

(a) Consider a horizontal interface between air above and glass with an index of 1.55 below. Draw a light ray incident from the air at an angle of incidence of 30.0. Determine the angles of the reected and refracted rays, and show them on the diagram. (b) Suppose instead that the light ray is incident from the glass at an angle of incidence of 30.0. Determine the angles of the reected 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 reection and refraction for all the angles of incidence at 10.0 intervals from 0 to 90.0. (d) Do the same for light rays traveling up to the interface through the glass.

A layer of ice having parallel sides oats on water. If light is incident on the upper surface of the ice at an angle of incidence of 30.0, what is the angle of refraction in the water?

A light ray of wavelength 589 nm is incident at an angle  on the top surface of a block of polystyrene surrounded by air, as shown in Figure P22.46. (a) Find the maximum value of  for which the refracted ray will undergo total internal reection at the left vertical face of the block. (b) Repeat the calculation for the case in which the polystyrene block is immersed in water. (c) What happens if the block is immersed in carbon disulde?

Figure P22.47 shows the path of a beam of light through several layers with different indices of refraction. (a) If  1  30.0, what is the angle  2 of the emerging beam? (b) What must the incident angle  1 be in order to have total internal reection at the surface between the medium with n  1.20 and the medium with n  1.00?

The walls of a prison cell are perpendicular to the four cardinal compass directions. On the rst 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 prisoner observes the patch of light moving across this western wall and for the rst 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 at against the wall at one corner of the rectangle of light. The mirror reects light back to a spot on the eastern wall close beside the window. How fast 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 prisoners cell move? (d) In what direction does the smaller square of light on the eastern wall move?

As shown in Figure P22.49, a light ray is incident normal to on one face of a 306090 block of dense int glass (a prism) that is immersed in water. (a) Determine the exit angle  4 of the ray. (b) A substance is dissolved in the water to increase the index of refraction. At what value of n2 does total internal reection cease at point P?

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

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

An optical ber with index of refraction n and diameter d is surrounded by air. Light is sent into the ber along its axis, as shown in Figure P22.52. (a) Find the smallest outside radius R permitted for a bend in the ber if no light is to escape. (b) Does the result for part (a) predict reasonable behavior as d approaches zero? As n increases? As n approaches unity? (c) Evaluate R, assuming that the diameter of the ber is 100  m and its index of refraction is 1.40.

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

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 P22.54, and the ray follows the path shown in the gure. Assuming that   60.0 and the base of the prism is mirrored, determine the angle made by the outgoing ray with the normal to the right face of the prism.

A transparent cylinder of radius R  2.00 m has a mirrored surface on its right half, as shown in Figure P22.55. A light ray traveling in air is incident on the left side of the cylinder. The incident light ray and the exiting light ray are parallel, and d  2.00 m. Determine the index of refraction of the material.

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

For this problem, refer to Figure 22.15. For various angles of incidence, it can be shown that the deviation angle  is a minimum when the ray passes through the glass so that the interior ray is parallel to the base of the prism. A measurement of this minimum angle of deviation enables one to nd the index of refraction of the prism material. Show that n is given by the expression where A is the apex angle of the prism.

A hiker stands on a mountain peak near sunset and observes a rainbow caused by water droplets in the air about 8.00 km away. The valley is 2.00 km below the mountain peak and entirely at. What fraction of the complete circular arc of the rainbow is visible to the hiker?

A light ray incident on a prism is refracted at the rst surface, as shown in Figure P22.59. Let represent the apex angle of the prism and n its index of refraction. Find, in terms of n and , the smallest allowed value of the angle of incidence at the rst surface for which the refracted ray will not undergo total internal reection at the second surface.

Students allow a narrow beam of laser light to strike a water surface. They arrange to measure the angle of refraction for selected angles of incidence and record the data shown in the following table: 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 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. From the resulting plot, deduce the index of refraction of water.

A light ray enters a rectangular block of plastic at an angle  1  45.0 and emerges at an angle  2  76.0, as shown in Figure P22.61. (a) Determine the index of refraction of the plastic. (b) If the light ray enters the plastic at a point L  50.0 cm from the bottom edge, how long does it take the light ray to travel through the plastic?

A. H. Pfunds method for measuring the index of refraction of glass is illustrated in Figure P22.62. 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 PBB strikes the clear surface at the critical angle and is totally reected, as are rays such as PCC. Rays such as PAA 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 cm thick? (c) If white light is used, the critical angle depends on color caused by dispersion. Is the inner edge of the white halo tinged with red light or violet light? Explain.

Tape a coin to the bottom of a large opaque bowl, as shown in Figure A22.1a. Stand over the bowl so that you are looking at the coin, and then move backwards away from the bowl until you can no longer see the coin over the bowls rim. Remain at that position, and have a friend ll the bowl with water, as shown in Figure A22.1b. You can now see the coin again because the light is refracted at the waterair interface.

Tape a piece of black paper to the end of a ashlight and cut a narrow slit in the middle of the paper, as shown in Figure A22.2. Lean a at mirror against one end of a tray partially lled with water. Shine your ashlight on that part of the mirror which is under water, and hold a sheet of white paper such that the reected light shines on the paper. You should observe a spectrum of colors on the paper as the light is dispersed when it travels from air into water and then from water into air. According to your observations, which color is bent the most? Which is bent the least?

Create an articial rainbow by standing with your back to the sun and spraying water into the air with a hose. You should cover the end of the hose slightly with a nger or use a nozzle so that the water is broken up into tiny droplets. Which color is on the outside of the arc? The inside? If the droplets are close to you in the experiment above, you may be able to see two rainbows, one for each eye. Close one eye, and only one bow is seen. With the Sun high in the sky, stand on a ladder and spray the hose toward the ground. In this case, you should be able to form a complete circle rainbow.

By observing the shadows formed by large and small light sources, you can demonstrate that light rays travel in straight-line paths. Figure A22.4 shows the shadows formed on a screen by a baseball when the light from a lightbulb (a large source) falls on it. Use the gure to explain why the shadow is dark at location A and less dark at B. Replace the light source with a small source, such as a high-intensity lamp with a small, clear bulb. (A substitute for a high-intensity lamp is an ordinary lightbulb with a piece of cardboard close in front of it with a hole about the size of a penny punched in the cardboard. The hole serves as the small source of light.) What kind of shadow is formed in this case? Why?