An optical fiber is a long transparent cylinder of

(I) When you look at yourself in a 60-cm-tall plane mirror, you see the same amount of your body whether you are close to the mirror or far away. (Try it and see.) Use ray diagrams to show why this should be true.

(I) Suppose that you want to take a photograph of yourself as you look at your image in a mirror 2.8 m away. For what distance should the camera lens be focused?

(II) Two plane mirrors meet at a 135 angle, Fig. 32-45. If light rays strike one mirror at 38 as shown, at what angle (f> do they leave the second mirror? FIGURE 32-45 Problem 3.

(II) A person whose eyes are 1.64 m above the floor stands 2.30 m in front of a vertical plane mirror whose bottom edge is 38 cm above the floor, Fig. 32-46. What is the horizontal distance x to the base of the wall supporting the mirror of the nearest point on the floor that can be seen reflected in the mirror? FIGURE 32-46 Problem 4.

(II) Show that if two plane mirrors meet at an angle 4>, a single ray reflected successively from both mirrors is deflected through an angle of 2<j> independent of the incident angle. Assume <f> < 90 and that only two reflections, one from each mirror, take place.

(II) Suppose you are 88 cm from a plane mirror. What area of the mirror is used to reflect the rays entering one eye from a point on the tip of your nose if your pupil diameter is 4.5 mm?

(II) Stand up two plane mirrors so they form a 90.0 angle as in Fig. 32-47. When you look into this double mirror, you see yourself as others see you, instead of reversed as in a single mirror. Make a ray diagram to show how this occurs.FIGURE 32-47 Problems 7 and 8.

Suppose a third mirror is placed beneath the two shown in Fig. 32-47, so that all three are perpendicular to each other. (a) Show that for such a corner reflector, any incident ray will return in its original direction after three reflections. (b) What happens if it makes only two reflections?

A solar cooker, really a concave mirror pointed at the Sun, focuses the Suns rays 18.8 cm in front of the mirror. What is the radius of the spherical surface from which the mirror was made?

How far from a concave mirror (radius 24.0 cm) must an object be placed if its image is to be at infinity?

When walking toward a concave mirror you notice that the image flips at a distance of 0.50 m. What is the radius of curvature of the mirror?

A small candle is 35 cm from a concave mirror having a radius of curvature of 24 cm. (a) What is the focal length of the mirror? (b) Where will the image of the candle be located? (c) Will the image be upright or inverted?

You look at yourself in a shiny 9.2-cm-diameter Christmas tree ball. If your face is 25.0 cm away from the balls front surface, where is your image? Is it real or virtual? Is it upright or inverted?

A mirror at an amusement park shows an upright image of any person who stands 1.7 m in front of it. If the image is three times the persons height, what is the radius of curvature of the mirror? (See Fig. 32-44.)

A dentist wants a small mirror that, when 2.00 cm from a tooth, will produce a 4.0X upright image. What kind of mirror must be used and what must its radius of curvature be?

Some rearview mirrors produce images of cars to your rear that are smaller than they would be if the mirror were flat. Are the mirrors concave or convex? What is a mirrors radius of curvature if cars 18.0 m away appear 0.33 their normal size?

You are standing 3.0 m from a convex security mirror in a store. You estimate the height of your image to be half of your actual height. Estimate the radius of curvature of the mirror.

An object 3.0 mm high is placed 18 cm from a convex mirror of radius of curvature 18 cm. (a) Show by ray tracing that the image is virtual, and estimate the image distance. (b) Show that the (negative) image distance can be computed from Eq. 32-2 using a focal length of -9.0 cm. (c) Compute the image size, using Eq. 32-3.

The image of a distant tree is virtual and very small when viewed in a curved mirror. The image appears to be 16.0 cm behind the mirror. What kind of mirror is it, and what is its radius of curvature?

Use two techniques, (a) a ray diagram, and (b) the mirror equation, to show that the magnitude of the magnification of a concave mirror is less than 1 if the object is beyond the center of curvature C (d0 > r), and is greater than 1 if the object is within C (d0 < r).

Show, using a ray diagram, that the magnification m of a convex mirror is m = ~d[/d0 , just as for a concave mirror. [Hint: Consider a ray from the top of the object that reflects at the center of the mirror.]

Use ray diagrams to show that the mirror equation, Eq. 32-2, is valid for a convex mirror as long as / is considered negative.

The magnification of a convex mirror is +0.55 X for objects 3.2 m from the mirror. What is the focal length of this mirror?

(a) Where should an object be placed in front of a concave mirror so that it produces an image at the same location as the object? (b) Is the image real or virtual? (c) Is the image inverted or upright? (d) What is the magnification of the image?

A 4.5-cm tall object is placed 26 cm in front of a spherical mirror. It is desired to produce a virtual image that is upright and 3.5 cm tall, (a) What type of mirror should be used? (b) Where is the image located? (c) What is the focal length of the mirror? (d) What is the radius of curvature of the mirror?

A shaving or makeup mirror is designed to magnify your face by a factor of 1.35 when your face is placed 20.0 cm in front of it. (a) What type of mirror is it? (b) Describe the type of image that it makes of your face. (c) Calculate the required radius of curvature for the mirror.

A concave mirror has focal length / . When an object is placed a distance dQ> f from this mirror, a real image with magnification m is formed, (a) Show that m = / / ( / - dQ). (ib) Sketch m vs. dQ over the range f < d0 < +oo where / = 0.45 m. (c) For what value of dQ will the real image have the same (lateral) size as the object? (d) To obtain a real image that is much larger than the object, in what general region should the object be placed relative to the mirror?

Let the focal length of a convex mirror be written as / = - |/ |. Show that the magnification m of an object a distance dQ from this mirror is given by m = \f\/(d0 + |/|). Based on this relation, explain why your nose looks bigger than the rest of your face when looking into a convex mirror.

A spherical mirror of focal length / produces an image of an object with magnification m. (a) Show that the object is a distance d0 = f ( 1 ) from the reflecting side of V m ) the mirror. (b) Use the relation in part (a) to show that, no matter where an object is placed in front of a convex mirror, its image will have a magnification in the range 0 < m < + 1 .

An object is placed a distance r in front of a wall, where r exactly equals the radius of curvature of a certain concave mirror. At what distance from the wall should this mirror be placed so that a real image of the object is formed on the wall? What is the magnification of the image

A short thin object (like a short length of wire) of length i is placed along the axis of a spherical mirror (perpendicular to the glass surface). Show that its image has length I' = m2l so the longitudinal magnification is equal to m2 where m is the normal lateral magnification, Eq. 32-3. Why the minus sign? [Hint: Find the image positions for both ends of the wire, and assume is very small.]

The speed of light in ice is 2.29 X 108 m/s. What is the index of refraction of ice?

What is the speed of light in (a) ethyl alcohol, (b) lucite, (c) crown glass?

Our nearest star (other than the Sun) is 4.2 light years away. That is, it takes 4.2 years for the light to reach Earth. How far away is it in meters?

How long does it take light to reach us from the Sun, 1.50 X 108 km away?

(II) The speed of light in a certain substance is 88% of its value in water. What is the index of refraction of that substance?

(II) Light is emitted from an ordinary lightbulb filament in wave-train bursts of about 10 8 s in duration. What is the length in space of such wave trains?

(I) A diver shines a flashlight upward from beneath the water at a 38.5 angle to the vertical. At what angle does the light leave the water?

(I) A flashlight beam strikes the surface of a pane of glass (n = 1.56) at a 63 angle to the normal. What is the angle of refraction?

(I) Rays of the Sun are seen to make a 33.0 angle to the vertical beneath the water. At what angle above the horizon is the Sun?

(I) A light beam coming from an underwater spotlight exits the water at an angle of 56.0. At what angle of incidence did it hit the air-water interface from below the surface?

(II) A beam of light in air strikes a slab of glass (n = 1.56) and is partially reflected and partially refracted. Determine the angle of incidence if the angle of reflection is twice the angle of refraction.

(II) A light beam strikes a 2.0-cm-thick piece of plastic with a refractive index of 1.62 at a 45 angle. The plastic is on top of a 3.0-cmthick piece of glass for which n = 1.47. What is the distance D in Fig. 32-48? FIGURE 32-48 Problem 43.

(II) An aquarium filled with water has flat glass sides whose index of refraction is 1.56. A beam of light from outside the aquarium strikes the glass at a 43.5 angle to the perpendicular (Fig. 32-49). What is the angle of this light ray when it enters (a) the glass, and then (b) the water? (c) What would be the refracted angle if the ray entered the water directly?FIGURE 32-49 Problem 44.

(II) In searching the bottom of a pool at night, a watchman shines a narrow beam of light from his flashlight, 1.3 m above the water level, onto the surface of the water at a point 2.5 m from his foot at the edge of the pool (Fig. 32-50). Where does the spot of light hit the bottom of the pool, measured from the bottom of the wall beneath his foot, if the pool is 2.1 m deep? FIGURE 32-50 Problem 45.

(II) The block of glass (n = 1.5) shown in cross section in Fig. 32-51 is surrounded by air. A ray of light enters the block at its left-hand face with incident angle 6\ and reemerges into the air from the right-hand face directed parallel to the blocks base. Determine 6 i. FIGURE 32-51 Problem 46.

(II) A laser beam of diameter d\ = 3.0 mm in air has an incident angle 6\ = 25 at a flat air-glass surface. If the index of refraction of the glass is n = 1.5, determine the diameter d2 of the beam after it enters the glass.

(II) Light is incident on an equilateral glass prism at a 45.0 angle to one face, Fig. 32-52. Calculate the angle at which light emerges from the opposite face. Assume that n = 1.54. FIGURE 32-52 Problems 48 and 65.

(II) A triangular prism made of crown glass (n = 1.52) with base angles of 30.0 is surrounded by air. If parallel rays are incident normally on its base as shown in Fig. 32-53, what is the angle <f> between the two emerging rays? FIGURE 32-53 Problem 49.

(II) Show in general that for a light beam incident on a uniform layer of transparent material, as in Fig. 32-24, the direction of the emerging beam is parallel to the incident beam, independent of the incident angle 0. Assume the air on the two sides of the transparent material is the same.

(Ill) A light ray is incident on a flat piece of glass with index of refraction n as in Fig. 32-24. Show that if the incident angle 0 is small, the emerging ray is displaced a distance d = td(n 1 )/n, where t is the thickness of the glass, 0 is in radians, and d is the perpendicular distance between the incident ray and the (dashed) line of the emerging ray (Fig. 32-24).

(I) By what percent is the speed of blue light (450 nm) less than the speed of red light (680 nm), in silicate flint glass (see Fig. 32-28)?

(I) A light beam strikes a piece of glass at a 60.00 incident angle. The beam contains two wavelengths, 450.0 nm and 700.0 nm, for which the index of refraction of the glass is 1.4831 and 1.4754, respectively. What is the angle between the two refracted beams?

(II) A parallel beam of light containing two wavelengths, Xl = 465 nm and A2 = 652 nm, enters the silicate flint glass of an equilateral prism as shown in Fig. 32-54. At what angle does each beam leave the prism (give angle with normal to the face)? See Fig. 32-28. FIGURE 32-54 Problem 54.

(Ill) A ray of light with wavelength A is incident from air at precisely 60 ( = 0) on a spherical water drop of radius r and index of refraction n (which depends on A). When the ray reemerges into the air from the far side of the drop, it has been deflected an angle cf> from its original direction as shown in Fig. 32-55. By how ^ much does the value \ of <f> for violet light (n = 1.341) differ from the value for red light (n = 1.330)?FIGURE 32-55 Problem 55.

(Ill) For visible light, the index of refraction n of glass is roughly 1.5, although this value varies by about 1% across the visible range. Consider a ray of white light incident from air at angle Qi onto a flat piece of glass, (a) Show that, upon entering the glass, the visible colors contained in this incident ray will be dispersed over a range A02 of refracted angles given approximately by A 02 ~ . : * \ / n 2 - sin2 6i n [Hint: For x in radians, (d/dx)(siri -1*) = i/Vi - *2.] (b) If Qi = 0, what is A02 in degrees? (c) If di = 90, what is A02 in degrees?

(I) What is the critical angle for the interface between water and diamond? To be internally reflected, the light must start in which material?

(I) The critical angle for a certain liquid-air surface is 49.6. What is the index of refraction of the liquid?

(II) A beam of light is emitted in a pool of water from a depth of 72.0 cm. Where must it strike the air-water interface, relative to the spot directly above it, in order that the light does not exit the water?

(II) A ray of light, after entering a light fiber, reflects at an angle of 14.5 with the long axis of the fiber, as in Fig. 32-56. Calculate the distance along the axis of the fiber that the light ray travels between successive reflections off the sides of the fiber. Assume that the fiber has an index of refraction of 1.55 and is 1.40 X 10_4m in diameter. FIGURE 32-56 Problem 60.

(II) A beam of light is emitted 8.0 cm beneath the surface of a liquid and strikes the surface 7.6 cm from the point directly above the source. If total internal reflection occurs, what can you say about the index of refraction of the liquid?

(II) Figure 32-57 shows a liquid-detecting prism device that might be used inside a washing machine or other liquid-containing appliance. If no liquid covers the prisms hypotenuse, total internal reflection of the beam from the light source produces a large signal in the light sensor. If liquid covers the hypotenuse, some light escapes from the prism into the liquid and the light sensors signal decreases. Thus a large signal from the light sensor indicates the absence of liquid in the reservoir. If this device is designed to detect the presence of water, determine the allowable range for the prisms index of refraction n. Will the device work properly if the prism is constructed from (inexpensive) lucite? For lucite, n = 1.5. FIGURE 32-57 Problem 62.

(II) Two rays A and B travel down a cylindrical optical fiber of diameter d = 75.0 /z,m, length = 1.0 km, and index of refraction ni = 1.465. Ray A travels a straight path down the fibers axis, whereas ray B propagates down the fiber by repeated reflections at the critical angle each time it impinges on the fibers boundary. Determine the extra time Af it takes for ray B to travel down the entire fiber in comparison with ray A (Fig. 32-58), assuming (a) the fiber is surrounded by air, (b) the fiber is surrounded by a cylindrical glass cladding with index of refraction n2 = 1.460. FIGURE 32-58 Problem 63.

(II) (a) What is the minimum index of refraction for a glass or plastic prism to be used in binoculars (Fig. 32-33) so that total internal reflection occurs at 45? (b) Will binoculars work if their prisms (assume n = 1.58) are immersed in water? (c) What minimum n is needed if the prisms are immersed in water?

(Ill) Suppose a ray strikes the left face of the prism in Fig. 32-52 at 45.0 as shown, but is totally internally reflected at the opposite side. If the apex angle (at the top) is 6 = 60.0, what can you say about the index of refraction of the prism?

(Ill) A beam of light enters the end of an optic fiber as shown in Fig. 32-59. (a) Show that we can guarantee total internal reflection at the side surface of the material (at point A), if the index of refraction is greater than about 1.42. In other words, regardless of the angle a, the light beam reflects back into the material at point A, assuming air outside. FIGURE 32-59 Problem 66.

(II) A 13.0-cm-thick plane piece of glass (n = 1.58) lies on the surface of a 12.0-cm-deep pool of water. How far below the top of the glass does the bottom of the pool seem, as viewed from directly above?

(II) A fish is swimming in water inside a thin spherical glass bowl of uniform thickness. Assuming the radius of curvature of the bowl is 28.0 cm, locate the image of the fish if the fish is located: (a) at the center of the bowl; (b) 20.0 cm from the side of the bowl between the observer and the center of the bowl. Assume the fish is small.

(Ill) In Section 32-8, we derived Eq. 32-8 for a convex spherical surface with n2 > nt. Using the same conventions and using diagrams similar to Fig. 32-37, show that Eq. 32-8 is valid also for (a) a convex spherical surface with n2 < nu (b) a concave spherical surface with n2 > nt, and (c) a concave spherical surface with n2 < nx.

(Ill) A coin lies at the bottom of a 0.75-m-deep pool. If a viewer sees it at a 45 angle, where is the image of the coin, relative to the coin? [Hint. The image is found by tracing back to the intersection of two rays.]

Two identical concave mirrors are set facing each other 1.0 m apart. A small lightbulb is placed halfway between the mirrors. A small piece of paper placed just to the left of the bulb prevents light from the bulb from directly shining on the left mirror, but light reflected from the right mirror still reaches the left mirror. A good image of the bulb appears on the left side of the piece of paper. What is the focal length of the mirrors?

A slab of thickness D, whose two faces are parallel, has index of refraction n. A ray of light incident from air onto one face of the slab at incident angle Qi splits into two rays A and B. Ray A reflects directly back into the air, while B travels a total distance ft within the slab before reemerging from the slabs face a distance d from its point of entry (Fig. 32-60). (a) Derive expressions for I and d in terms of D, n, and 6 i. (b) For normal incidence (i.e., Qi = 0) show that your expressions yield the expect values for and FIGURE 32-60 Problem 72.

Two plane mirrors are facing each other 2.2 m apart as in Fig. 32-61. You stand 1.5 m away from one of these mirrors and look into it. You will see multiple images of yourself. (a) How far away from you are the first three images of yourself in the mirror in front of you? (b) Are these first three images facing toward you or away from you? FIGURE 32-61 Problem 73.

We wish to determine the depth of a swimming pool filled with water by measuring the width (x = 5.50 m) and then noting that the bottom edge of the pool is just visible at an angle of 13.0 above the horizontal as shown in Fig. 32-62. Calculate the depth of the pool.

A 1.80-m-tall person stands 3.80 m from a convex mirror and notices that he looks precisely half as tall as he does in a plane mirror placed at the same distance. What is the radius of curvature of the convex mirror? (Assume that sin 0 ~ 0.) [Hint: The viewing angle is half.]

The critical angle of a certain piece of plastic in air is 0C = 39.3. What is the critical angle of the same plastic if it is immersed in water?

Each student in a physics lab is assigned to find the location where a bright object may be placed in order that a concave mirror, with radius of curvature r = 46 cm, will produce an image three times the size of the object. Two students complete the assignment at different times using identical equipment, but when they compare notes later, they discover that their answers for the object distance are not the same. Explain why they do not necessarily need to repeat the lab, and justify your response with a calculation.

A kaleidoscope makes symmetric patterns with two plane mirrors having a 60 angle between them as shown in Fig. 32-63. Draw the location of the images (some of them images of images) of the object placed between the mirrors. FIGURE 32-63 Problem 78.

When light passes through a prism, the angle that the refracted ray makes relative to the incident ray is called the deviation angle 5, Fig. 32-64. Show that this angle is a minimum when the ray passes through the prism symmetrically, perpendicular to the bisector of the apex angle (f>, and show that the minimum deviation angle, <5m, is related to the prisms index of refraction n by sin \ ((f> + Sm) n sin 0/2 [Hint: For 0 in radians, (d/d0)(sin~16) = 1/Vl - 02.] FIGURE 32-64 Problems 79 and 80.

If the apex angle of a prism is <f> = 72 (see Fig. 32-64), what is the minimum incident angle for a ray if it is to emerge from the opposite side (i.e., not be totally internally reflected), given n = 1.58?

Fermats principle states that light travels between two points along the path that requires the least time, as compared to other nearby paths. From Fermats principle derive (a) the law of reflection (6[ = 0r) and (b) the law of refraction (Snells law). [Hint: Choose two appropriate points so that a ray between them can undergo reflection or refraction. Draw a rough path for a ray between these points, and write down an expression of the time required for light to travel the arbitrary path chosen. Then take the derivative to find the minimum.]

Suppose Fig. 32-36 shows a cylindrical rod whose end has a radius of curvature R = 2.0 cm, and the rod is immersed in water with index of refraction of 1.33. The rod has index of refraction 1.53. Find the location and height of the image of an object 2.0 mm high located 23 cm away from the rod.

An optical fiber is a long transparent cylinder of diameter d and index of refraction n. If this fiber is bent sharply, some light hitting the side of the cylinder may escape rather than reflect back into the fiber (Fig. 32-65). What is the smallest radius r at a short bent section for which total internal reflection will be assured for light initially travelling parallel to the axis of the fiber?FIGURE 32-65 Problem 83

An object is placed 15 cm from a certain mirror. The image is half the height of the object, inverted, and real. How far is the image from the mirror, and what is the radius of curvature of the mirror?

The end faces of a cylindrical glass rod (n = 1.51) are perpendicular to the sides. Show that a light ray entering an end face at any angle will be totally internally reflected inside the rod when it strikes the sides. Assume the rod is in air. What if it were in water?

The paint used on highway signs often contains small transparent spheres which provide nighttime illumination of the signs lettering by retro-reflecting vehicle headlight beams. Consider a light ray from air incident on one such sphere of radius r and index of refraction n. Let 6 be its incident angle, and let the ray follow the path shown in Fig. 32-66, so that the ray exits the sphere in the direction exactly antiparallel to its incoming direction. Consid- \ ering only rays for which sin 0 can be approximated as 0, determine the required value for n. FIGURE 32-66 Problem 86.

(II) The index of refraction, n, of crown flint glass at different wavelengths (A) of light are given in the Table below. 1.50586 546.1 1.51978 365.0 1.54251 312.5 1.5600 Make a graph of n versus A. The variation in index of refraction with wavelength is given by the Cauchy equation n = A + B /A2. Make another graph of n versus 1 /A 2 and determine the constants A and B for the glass by fitting the data with a straight line.

(Ill) Consider a ray of sunlight incident from air on a spherical raindrop of radius r and index of refraction n. Defining 0 to be its incident angle, the ray then follows the path shown in Fig. 32-67, exiting the drop at a scattering angle <f> compared with its original incoming direction, (a) Show that (f) = 180 + 20 - 4 siiT1 (sin 0/n). (b) The parallel rays of sunlight illuminate a raindrop with rays of all possible incident angles from 0 to 90. Plot 4> vs. 0 in the range 0 < 0 < 90, in 0.5 steps, assuming n = 1.33 as is appropriate for water at visible-light wavelengths. (c) From your plot, you should find that a fairly large fraction of the incident angles have nearly the same scattering angle. Approximately what fraction of the possible incident angles is within roughly 1 of (f> = 139? [This subset of incident rays is what creates the rainbow. Wavelengthdependent variations in n cause the rainbow to form at slightly different <f> for the various visible colors.] FIGURE 32-67