(a) Show that if two thin lenses of focal lengths and are

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.

Suppose that you want to take a photograph of yourself as you look at your image in a mirror 3.1 m away. For what distance should the camera lens be focused?

Two plane mirrors meet at a \(135^\circ\) angle, Fig. 23–52. If light rays strike one mirror at \(34^\circ\) as shown, at what angle f do they leave the second mirror? ________________ Equation Transcription: ???? ???? Text Transcription: 135^o 34^o phi 34^o phi

A person whose eyes are 1.72 m above the floor stands 2.20 m in front of a vertical plane mirror whose bottom edge is 38 cm above the floor, Fig. 23–53. 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?

Stand up two plane mirrors so they form a \(90.0^\circ\) angle as in Fig. 23–54. 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. Equation Transcription: Text Transcription: 90.0^o

Two plane mirrors, nearly parallel, are facing each other 2.3 m apart as in Fig. 23–55. You stand 1.6 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 1.6 first three images facing toward you or away from you?

Suppose you are 94 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?

(I) A solar cooker, really a concave mirror pointed at the Sun, focuses the Sun’s 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 21.0 cm) must an object be placed if its image is to be at infinity?

A small candle is 38 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?

(II) An object 3.0 mm high is placed 16 cm from a convex mirror of radius of curvature 16 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. 23–2 using a focal length of –8.0 cm. \((c)\) Compute the image size, using Eq. 23–3.

(II) A dentist wants a small mirror that, when 2.00 cm from a tooth, will produce a \(4.0 \times\) upright image. What kind of mirror must be used and what must its radius of curvature be? Equation Transcription: Text Transcription: 4.0 x

You are standing 3.4 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.

The image of a distant tree is virtual and very small when viewed in a curved mirror. The image appears to be 19.0 cm behind the mirror. What kind of mirror is it, and what is its radius of curvature?

(II) A mirror at an amusement park shows an upright image of any person who stands 1.9 m in front of it. If the image is three times the person’s height, what is the radius of curvature of the mirror? (See Fig. 23–50.)

(II) In Example 23–4, show that if the object is moved 10.0 cm farther from the concave mirror, the object’s image size will equal the object’s actual size. Stated as a multiple of the focal length, what is the object distance for this “actual-sized image” situation?

You look at yourself in a shiny 8.8-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?

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 16.0 m away appear 0.33 their normal size

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) 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 shaving or makeup mirror is designed to magnify your face by a factor of 1.40 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.

(II) 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 \(C\ (d_0\ >\ r)\), and is greater than 1 if the object is within \(C\ (d_0\ >\ r)\).

(III) Show, using a ray diagram, that the magnification \(m\) of a convex mirror is \(m=-d_i/d_o\), just as for a concave mirror. [\(Hint\): Consider a ray from the top of the object that reflects at the center of the mirror.] Equation Transcription: Text Transcription: m=-d_i/d_o

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?

(I) The speed of light in ice is \(2.29 \times 10^8~\mathrm{m/s}\). What is the index of refraction of ice? Equation Transcription: Text Transcription: 2.29 x 10^8 m/s

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

The speed of light in a certain substance is 82% of its value in water. What is the index of refraction of that substance?

(I) A flashlight beam strikes the surface of a pane of glass \((n=1.56)\) at a \(67^\circ\) angle to the normal. What is the angle of refraction? Equation Transcription: Text Transcription: (n=1.56) 67^o

(I) A diver shines a flashlight upward from beneath the water at a \(35.2^\circ\) angle to the vertical. At what angle does the light leave the water? Equation Transcription: Text Transcription: 35.2^o

(I) A light beam coming from an underwater spotlight exits the water at an angle of \(56.0^\circ\). At what angle of incidence did it hit the air–water interface from below the surface? Equation Transcription: Text Transcription: 56.0^o

(I) Rays of the Sun are seen to make a \(36.0^\circ\) angle to the vertical beneath the water. At what angle above the horizon is the Sun? Equation Transcription: Text Transcription: 36.0^o

An aquarium filled with water has flat glass sides whose index of refraction is 1.54. A beam of light from outside the aquarium strikes the glass at a \(43.5^\circ\) angle to the perpendicular (Fig. 23–56). 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? Equation Transcription: Text Transcription: 43.5^o 43.5^o

A beam of light in air strikes a slab of glass \((n=1.51)\) and is partially reflected and partially refracted. Determine the angle of incidence if the angle of reflection is twice the angle of refraction. Equation Transcription: Text Transcription: (n=1.51)

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. 23–57). Where does the spot of light hit the bottom of the 2.1-m-deep pool? Measure from the bottom of the wall beneath his foot.

What is the critical angle for the interface between water and crown glass? To be internally reflected, the light must start in which material?

(I) The critical angle for a certain liquid–air surface is \(47.2^\circ\). What is the index of refraction of the liquid? Equation Transcription: Text Transcription: 47.2^o

A beam of light is emitted in a pool of water from a depth of 82.0 cm. Where must it strike the airwater interface, relative to the spot directly above it, in order that the light does not exit the water?

A beam of light is emitted 8.0 cm beneath the surface of a liquid and strikes the air 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?

(III) \((a)\) What is the minimum index of refraction for a glass or plastic prism to be used in binoculars (Fig. 23–28) so that total internal reflection occurs at \(45^\circ\)? \((b)\) Will binoculars work if their prisms (assume \(\)) are immersed in water? \((c)\) What minimum n is needed if the prisms are immersed in water? Equation Transcription: Text Transcription: 45^o n=1.58

A beam of light enters the end of an optic fiber as shown in Fig. 23–58. \((a)\) 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 \(\alpha\), the light beam reflects back into the material at point A, assuming air outside. \((b)\) What if the fiber were immersed in water? Equation Transcription: Text Transcription: alpha alpha beta gamma

A sharp image is located 391 mm behind a 215-mmfocal-length converging lens. Find the object distance (a) using a ray diagram, (b) by calculation.

Sunlight is observed to focus at a point 16.5 cm behind a lens. (a) What kind of lens is it? (b) What is its power in diopters?

\((a)\) What is the power of a 32.5-cm-focal-length lens? \((b)\) What is the focal length of a \(-6.75\)-D lens? Are these lenses converging or diverging?

A certain lens focuses light from an object 1.55 m away as an image 48.3 cm on the other side of the lens. What type of lens is it and what is its focal length? Is the image real or virtual?

(II) A 105-mm-focal-length lens is used to focus an image on the sensor of a camera. The maximum distance allowed between the lens and the sensor plane is 132 mm. (a) How far in front of the sensor should the lens (assumed thin) be positioned if the object to be photographed is 10.0m away? (b) 3.0 m away? (c) 1.0m away? (d) What is the closest object this lens could photograph sharply?

Use ray diagrams to show that a real image formed by a thin lens is always inverted, whereas a virtual image is always upright if the object is real.

A stamp collector uses a converging lens with focal length 28 cm to view a stamp 16 cm in front of the lens. (a)Where is the image located? (b) What is the magnification?

(II) It is desired to magnify reading material by a factor of \(3.0 \times\) when a book is placed 9.0 cm behind a lens. \((a)\) Draw a ray diagram and describe the type of image this would be. \((b)\) What type of lens is needed? \((c)\) What is the power of the lens in diopters? Equation Transcription: Text Transcription: 3.0 x

(II) A \(-7.00\)-D lens is held 12.5 cm from an ant 1.00 mm high. Describe the position, type, and height of the image.

An object is located 1.50 m from a 6.5-D lens. By how much does the image move if the object is moved (a) 0.90 m closer to the lens, and (b) 0.90 m farther from the lens?

\((a)\) How far from a 50.0-mm-focal-length lens must an object be placed if its image is to be magnified \(2.50 \times\) and be real? \((b)\) What if the image is to be virtual and magnified \(2.50 \times\)? Equation Transcription: Text Transcription: 2.50 x 2.50 x

(II) Repeat Problem 51 for a \(-50.0\)-mm-focal-length lens. [\(Hint\): Consider objects real or virtual (formed by some other piece of optics).]

How far from a converging lens with a focal length of 32 cm should an object be placed to produce a real image which is the same size as the object?

\((a)\) A 2.40-cm-high insect is 1.30 m from a 135-mm- focal-length lens. Where is the image, how high is it, and what type is it? \((b)\) What if \(f=-135~\mathrm{mm}\)? Equation Transcription: Text Transcription: f=-135 mm

A bright object and a viewing screen are separated by a distance of 86.0 cm. At what location(s) between the object and the screen should a lens of focal length 16.0 cm be placed in order to produce a sharp image on the screen? [Hint: First draw a diagram.]

(III) How far apart are an object and an image formed by an 85-cm-focal-length converging lens if the image is \(3.25\times\) larger than the object and is real?

In a film projector, the film acts as the object whose image is projected on a screen (Fig. 23–59). If a 105-mm- focal-length lens is to project an image on a screen 25.5 m away, how far from the lens should the film be? If the film is 24 mm wide, how wide will the picture be on the screen?

(II) A diverging lens with f = -36.5 cm is placed 14.0 cm behind a converging lens with f = 20.0 cm. Where will an object at infinity be focused?

Two 25.0-cm-focal-length converging lenses are placed 16.5 cm apart. An object is placed 35.0 cm in front of one lens. Where will the final image formed by the second lens be located? What is the total magnification?

A 38.0-cm-focal-length converging lens is 28.0 cm behind a diverging lens. Parallel light strikes the diverging lens. After passing through the converging lens, the light is again parallel. What is the focal length of the diverging lens? [Hint: First draw a ray diagram.]

(II) Two lenses, one converging with focal length 20.0 cm and one diverging with focal length \(-10.0~\mathrm{cm}\), are placed 25.0 cm apart. An object is placed 60.0 cm in front of the converging lens. Determine \((a)\) the position and \((b)\) the magnification of the final image formed. \((c)\) Sketch a ray diagram for this system.

A lighted candle is placed 36 cm in front of a converging lens of focal length \(f_1=13~\mathrm{cm}\), which in turn is 56 cm in front of another converging lens of focal length \(f_2=16~\mathrm{cm}\) (see Fig. 23–60). \((a)\) Draw a ray diagram and estimate the location and the relative size of the final image. \((b)\) Calculate the position and relative size of the final image. ________________ Equation Transcription: Text Transcription: f_1=13 cm f_2=16 cm

(I) A double concave lens has surface radii of 33.4 cm and 28.8 cm. What is the focal length if n = 1.52?

Both surfaces of a double convex lens have radii of 34.1 cm. If the focal length is 28.9 cm, what is the index of refraction of the lens material?

(I) A planoconvex lens (Fig. 23–31a) with n = 1.55 is to have a focal length of 16.3 cm. What is the radius of curvature of the convex surface?

A symmetric double convex lens with a focal length of 22.0 cm is to be made from glass with an index of refraction of 1.52. What should be the radius of curvature for each surface?

(II) A prescription for an eyeglass lens calls for \(3.50\) diopters. The lensmaker grinds the lens from a “blank” with \(n= 1.56\) and convex front surface of radius of curvature of 30.0 cm. What should be the radius of curvature of the other surface? Equation Transcription: Text Transcription: +3.50 n=1.56

(III) An object is placed 96.5 cm from a glass lens \((n=1.52)\) with one concave surface of radius 22.0 cm and one convex surface of radius 18.5 cm. Where is the final image? What is the magnification? Equation Transcription: Text Transcription: (n=1.52)

. Sunlight is reflected off the Moon. How long does it take that light to reach us from the Moon?

You hold a small flat mirror 0.50 m in front of you and can see your reflection twice in that mirror because there is a full-length mirror 1.0 m behind you (Fig. 23–61). Determine the distance of each image from you.

We wish to determine the depth of a swimming pool filled with water by measuring the width \((x=6.50~\mathrm{m})\) and then noting that the far bottom edge of the pool is just visible at an angle of \(13.0^\circ\) above the horizontal as shown in Fig. 23–62. Calculate the depth of the pool. ________________ Equation Transcription: Text Transcription: (x=6.50 m) 13.0^o 13.0^o

The critical angle of a certain piece of plastic in air is \(\theta_\mathrm C=37.8^\circ\). What is the critical angle of the same plastic if it is immersed in water? Equation Transcription: Text Transcription: theta_C=37.8^o

A pulse of light takes 2.63 ns (see Table 1–4) to travel 0.500 m in a certain material. Determine the material’s index of refraction, and identify this material.

When an object is placed 60.0 cm from a certain converging lens, it forms a real image. When the object is moved to 40.0 cm from the lens, the image moves 10.0 cm farther from the lens. Find the focal length of this lens

A 4.5-cm-tall object is placed 32 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?

Light is emitted from an ordinary lightbulb filament in wave-train bursts of about \(10^{-8}~ \mathrm s\) in duration. What is the length in space of such wave trains? Equation Transcription: Text Transcription: 10^-8 s

If the apex angle of a prism is \(\phi=75^\circ\) (see Fig. 23–63), 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 \(n = 1.58\)? Equation Transcription: ???? ???? Text Transcription: phi=75^o n=1.58 phi

\((a)\) A plane mirror can be considered a limiting case of a spherical mirror. Specify what this limit is. \((b)\) Determine an equation that relates the image and object distances in this limit of a plane mirror. \((c)\) Determine the magnification of a plane mirror in this same limit. \((d)\) Are your results in parts \((b)\) and \((c)\) consistent with the discussion of Section 23–2 on plane mirrors?

An object is placed 18 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?

Light is incident on an equilateral glass prism at a \(45.0^\circ\) angle to one face, Fig. 23–64. Calculate the angle at which light emerges from the opposite face. Assume that \(n=1.54\). ________________ Equation Transcription: Text Transcription: 45.0^o n=1.54 45.0^o

Suppose a ray strikes the left face of the prism in Fig. 23–64 at \(45.0^\circ\) as shown, but is totally internally reflected at the opposite side. If the apex angle (at the top) is \(\theta=65.0^\circ\), what can you say about the index of refraction of the prism? Equation Transcription: Text Transcription: 45.0^o theta=65.0^o 45.0^o

(a) An object 37.5 cm in front of a certain lens is imaged 8.20 cm in front of that lens (on the same side as the object). What type of lens is this, and what is its focal length? Is the image real or virtual? (b) If the image were located, instead, 44.5 cm in front of the lens, what type of lens would it be and what focal length would it have?

How large is the image of the Sun on a camera sensor with \((a)\) a 35-mm-focal-length lens, \((b)\) a 50-mm-focal-length lens, and \((c)\) a 105-mm-focal-length lens? The Sun has diameter \(1.4 \times 10^6~ \mathrm{km}\), and it is \(1.5 \times 10^8~ \mathrm{km}\) away. Equation Transcription: Text Transcription: 1.4 x 10^6 km 1.5 x 10^8 km

Figure 23–65 is a photograph of an eyeball with the image of a boy in a doorway. \((a)\) Is the eye here acting as a lens or as a mirror? \((b)\) Is the eye being viewed right side up or is the camera taking this photo upside down? \((c)\) Explain, based on all possible images made by a convex mirror or lens.

Which of the two lenses shown in Fig. 23–66 is converging, and which is diverging? Explain using ray diagrams and show how each image is formed.

Figure 23–67 shows a liquid-detecting prism device that might be used inside a washing machine. If no liquid covers the prism’s 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 sensor’s signal decreases. Thus a large signal from the light sensor indicates the absence of liquid in the reservoir. Determine the allowable range for the prism’s index of refraction\(n\). Equation Transcription: Text Transcription: 45^o 45^o

\((a)\) Show that if two thin lenses of focal lengths \(f_1\) and \(f_2\) are placed in contact with each other, the focal length of the combination is given by \(f_\mathrm T=f_1f_2/(f_1+f_2)\). \((b)\) Show that the power \(P\) of the combination of two lenses is the sum of their separate powers, \(P=P_1+P_2\). Equation Transcription: Text Transcription: f_1 f_2 f_T=f_1f_2/(f_1+f_2) P=P_1+P_2

Two converging lenses are placed 30.0 cm apart. The focal length of the lens on the right is 20.0 cm, and the focal length of the lens on the left is 15.0 cm. An object is placed to the left of the 15.0-cm-focal-length lens. A final image from both lenses is inverted and located halfway between the two lenses. How far to the left of the 15.0-cm-focal-length lens is the original object?

An object is placed 30.0 cm from a +5.0-D lens. A spher- ical mirror with focal length 25 cm is placed 75 cm behind the lens. Where is the final image? (Note that the mirror reflects light back through the lens.) Be sure to draw a diagram.

A small object is 25.0 cm from a diverging lens as shown in Fig. 23–68. A converging lens with a focal length of 12.0 cm is 30.0 cm to the right of the diverging lens. The two-lens system forms a real inverted image 17.0 cm to the right of the converging lens. What is the focal length of the diverging lens?

Problem 1MCQ Suppose you are standing about 3 m in front of a mirror. You can see yourself just from the top of your head to your waist, where the bottom of the mirror cuts off the rest of your image. If you walk one step closer to the mirror (a) you will not be able to see any more of your image. (b) you will be able to see more of your image, below your waist.

Problem 1COQ A 2.0-m-tall person is standing 2.0 m from a flat vertical mirror staring at her image. What minimum height must the mirror’s reflecting glass have if the person is to see her entire body, from the top of her head to her feet? (a) 0.50 m. (b) 1.0 m. (c) 1.5 m. (d) 2.0 m. (e) 2.5 m.

Problem 1P (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.

Problem 1Q Archimedes is said to have burned the whole Roman fleet in the harbor of Syracuse, Italy, by focusing the rays of the Sun with a huge spherical mirror. Is this† reasonable?

Problem 1SL (a) Describe the difference between a real image and a virtual image? (b) Can your eyes tell the difference? (c)How can you tell the difference on a ray diagram? (d) How could you tell the difference between a virtual image and a real image experimentally? (e) If you were to take a photograph of a virtual image, would you see the image in the photograph? (f) If you were to put a piece of photographic film at the location of a virtual image, would the image be captured on the film? (g) Explain any differences in your answers to parts (e) and (f).

Problem 2MCQ When the reflection of an object is seen in a flat mirror, the image is (a) real and upright. (b) real and inverted. (c) virtual and upright. (d) virtual and inverted.

The focal length of a lens is (a) the diameter of the lens. (b) the thickness of the lens. (c) the distance from the lens at which incoming parallel rays bend to intersect at a point. (d) the distance from the lens at which all real images are formed.

Problem 2P (I) Suppose that you want to take a photograph of yourself as you look at your image in a mirror 3.1 m away. For what distance should the camera lens be focused?

Problem 2Q What is the focal length of a plane mirror?What is the magnification of a plane mirror?

Problem 2SL Students in a physics lab are assigned to find the location where a bright object may be placed in order that a converging lens with f =12 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.

Problem 3MCQ You want to create a spotlight that will shine a bright beam of light with all of the light rays parallel to each other. You have a large concave spherical mirror and a small lightbulb. Where should you place the lightbulb? (a) At the focal point of the mirror. (b) At the radius of curvature of the mirror. (c) At any point, because all rays bouncing off the mirror will be parallel. (d) None of the above; you can’t make parallel rays with a concave mirror.

\(\text { (II) }\) Two plane mirrors meet at a \(135^{\circ}\) angle, Fig. 23–52. If light rays strike one mirror at \(34^{\circ}\) as shown, at what angle \(\varphi\) do they leave the second mirror? FIGURE 23–52 Problem 3. Equation Transcription: Text Transcription: (II) 135° 34° \varphi

Problem 3Q Although a plane mirror appears to reverse left and right, it doesn’t reverse up and down. Discuss why this happens, noting that front to back is also reversed. Also discuss what happens if, while standing, you look up vertically at a horizontal mirror on the ceiling.

Both a converging lens and a concave mirror can produce virtual images that are larger than the object. Concave mirrors can be used as makeup mirrors, but converging lenses cannot be. (a) Draw ray diagrams to explain why not. (b) If a concave mirror has the same focal length as a converging lens, and an object is placed first at a distance of \(\frac{1}{2} f\) from the lens and then at a distance of \(\frac{1}{2} f\) from the mirror, how will the magnification of the object compare in the two cases? Equation Transcription: Text Transcription: \frac1 2 f \frac1 2 f

Problem 4MCQ When you look at a fish in a still stream from the bank, the fish appears shallower than it really is due to refraction. From directly above, it appears (a) deeper than it really is. (b) at its actual depth. (c) shallower than its real depth. (d) It depends on your height above the water.

Problem 4Q An object is placed along the principal axis of a spherical mirror. The magnification of the object is Is the image real or virtual, inverted or upright? Is the mirror concave or convex? On which side of the mirror is the image located?

\(\text { (II) }\) A person whose eyes are \(1.72 \mathrm{~m}\) above the floor stands \(2.20 \mathrm{~m}\) in front of a vertical plane mirror whose bottom edge is \(38 \mathrm{~m}\) above the floor, Fig. 23–53. 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 23–53 Problem 4. Equation Transcription: Text Transcription: (II) 1.72 m 2.20 m 38 cm x 1.72 m 2.20 m 38 cm x

(a) Did the person we see in Fig. 23–69 shoot the picture we are looking at? We see her in three different mirrors. Describe (b) what type of mirror each is, and (c) her position relative to the focal point and center of curvature. FIGURE 23–69 Search and Learn 4.

Parallel light rays cross interfaces from medium 1 into medium 2 and then into medium 3 as shown in Fig. 23–51. What can we say about the relative sizes of the indices of refraction of these media? (a) \(n_{1}>n_{2}>n_{3}\) (b) \(n_{3}>n_{2}>n_{1}\) (c) \(n_{2}>n_{3}>n_{1}\) (d) \(n_{1}>n_{3}>n_{2} \) (e) \(n_{2}>n_{1}>n_{3}\) (f) None of the above. FIGURE 23–51 MisConceptual Question 5. Equation Transcription: Text Transcription: n1>n2>n3 n3>n2>n1 n2>n3>n1 n1>n3>n2 n2>n1>n3

Problem 5Q If a concave mirror produces a real image, is the image necessarily inverted? Explain.

Problem 5SL Justify the second part of sign convention 3, page 665, starting “Equivalently.” Use ray diagrams for all possible situations. Cite Figures already in the text and draw any others needed.

\(\text { (II) }\) Two plane mirrors, nearly parallel, are facing each other \(2.3 \mathrm{~m}\) apart as in Fig. 23–55. You stand \(1.6 \mathrm{~m}\) 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 23–55 Problem 6. Equation Transcription: Text Transcription: (II) 2.3 m 1.6 m 2.3 m 1.6 m

Problem 6MCQ To shoot a swimming fish with an intense light beam from a laser gun, you should aim (a) directly at the image. (b) slightly above the image. (c) slightly below the image

Problem 6Q How might you determine the speed of light in a solid, rectangular, transparent object?

Problem 6SL The only means to create a real image with a single lens would be to place (a) the object inside the focal length of a converging lens; (b) the object inside the focal length of a diverging lens; (c) the object outside the focal length of a converging lens; (d) the object outside the focal length of a diverging lens; (e) any of the above, given the correct distance from the focal point.

Problem 7MCQ When moonlight strikes the surface of a calm lake, what happens to this light? (a) All of it reflects from the water surface back to the air. (b) Some of it reflects back to the air; some enters the water. (c) All of it enters the water. (d) All of it disappears via absorption by water molecules

(III) Suppose you are \(94 cm\) 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 \(4.5 mm\)?

When you look at the Moon’s reflection from a ripply sea, it appears elongated (Fig. 23–47). Explain. FIGURE 23–47 Question 7.

Problem 7SL Make a table showing the sign conventions for mirrors and lenses. Include the sign convention for the mirrors and lenses themselves and for the image and object heights and distances for each.

Problem 8MCQ If you shine a light through an optical fiber, why does it come out the end but not out the sides? (a) It does come out the sides, but this effect is not obvious because the sides are so much longer than the ends. (b) The sides are mirrored, so the light reflects. (c) Total internal reflection makes the light reflect from the sides. (d) The light flows along the length of the fiber, never touching the sides.

Problem 8P (I) A solar cooker, really a concave mirror pointed at the Sun, focuses the Sun’s rays 18.8 cm in front of the mirror. What is the radius of the spherical surface from which the mirror was made?

What is the angle of refraction when a light ray is incident perpendicular to the boundary between two transparent materials?

Figure 23–70 shows a converging lens held above three equal-sized letters A. In (a) the lens is \(5 \mathrm{~cm}\) from the paper, and in (b) the lens is \(15 \mathrm{~cm}\) from the paper. Estimate the focal length of the lens. What is the image position for each case? FIGURE 23–70 Search and Learn 8. Equation Transcription: Text Transcription: 5 cm 15 cm

Problem 9MCQ A converging lens, such as a typical magnifying glass, (a) always produces a magnified image (taller than object). (b) always produces an image smaller than the object. (c) always produces an upright image. (d) always produces an inverted image (upside down). (e) None of these statements are true.

Problem 9Q When you look down into a swimming pool or a lake, are you likely to overestimate or underestimate its depth? Explain. How does the apparent depth vary with the viewing angle? (Use ray diagrams.)

Problem 10MCQ Virtual images can be formed by (a) only mirrors. (b) only lenses. (c) only plane mirrors. (d) only curved mirrors or lenses. (e) plane and curved mirrors, and lenses.

Draw a ray diagram to show why a stick or straw looks bent when part of it is under water (Fig. 23–23).

Problem 11MCQ A lens can be characterized by its power, which (a) is the same as the magnification. (b) tells how much light the lens can focus. (c) depends on where the object is located. (d) is the reciprocal of the focal length.

Problem 11P (II) An object 3.0 mm high is placed 16 cm from a convex mirror of radius of curvature 16 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. 23–2 using a focal length of -8.0 cm. (c) Compute the image size,

Problem 11Q When a wide beam of parallel light enters water at an angle, the beam broadens. Explain.

Problem 12MCQ You cover half of a lens that is forming an image on a screen. Compare what happens when you cover the top half of the lens versus the bottom half. (a) When you cover the top half of the lens, the top half of the image disappears; when you cover the bottom half of the lens, the bottom half of the image disappears. (b) When you cover the top half of the lens, the bottom half of the image disappears; when you cover the bottom half of the lens, the top half of the image disappears. (c) The image becomes half as bright in both cases. (d) Nothing happens in either case. (e) The image disappears in both cases.

Problem 12P (II) A dentist wants a small mirror that, when 2.00 cm from a tooth, will produce a 4.0 X upright image. What kind of mirror must be used and what must its radius of curvature be?

You look into an aquarium and view a fish inside. One ray of light from the fish is shown emerging from the tank in Fig. 23–48. The apparent position of the fish is also shown (dashed ray). In the drawing, indicate the approximate position of the actual fish. Briefly justify your answer. FIGURE 23–48 Question 12.

Problem 13MCQ Which of the following can form an image? (a) A plane mirror. (b) A curved mirror. (c) A lens curved on both sides. (d) A lens curved on only one side. (e) All of the above.

Problem 13P (II) You are standing 3.4 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.

Problem 14MCQ As an object moves from just outside the focal point of a converging lens to just inside it, the image goes from _____ and _____ to _____ and _____. (a) large; inverted; large; upright. (b) large; upright; large; inverted. (c) small; inverted; small; upright. (d) small; upright; small; inverted.

Problem 14P (II) The image of a distant tree is virtual and very small when viewed in a curved mirror. The image appears to be 19.0 cm behind the mirror. What kind of mirror is it, and what is its radius of curvature?

A ray of light is refracted through three different materials (Fig. 23–49). Which material has (a) the largest index of refraction, (b) the smallest? FIGURE 23–49 Question 14.

\(\text { (II) }\) A mirror at an amusement park shows an upright image of any person who stands \(1.9 \mathrm{~m}\) in front of it. If the image is three times the person’s height, what is the radius of curvature of the mirror? (See Fig. 23–50.) Equation Transcription: Text Transcription: (II) 1.9 m

Problem 15Q A child looks into a pool to see how deep it is. She then drops a small toy into the pool to help decide how deep the pool is. After this careful investigation, she decides it is safe to jump in—only to discover the water is over her head. What went wrong with her interpretation of her experiment?

Problem 16P (II) In Example 23–4, show that if the object is moved 10.0 cm farther from the concave mirror, the object’s image size will equal the object’s actual size. Stated as a multiple of the focal length, what is the object distance for this “actual-sized image” situation?

Problem 16Q Can a light ray traveling in air be totally reflected when it strikes a smooth water surface if the incident angle is chosen correctly? Explain.

Problem 17P (II) You look at yourself in a shiny 8.8-cm-diameter Christmas tree ball. If your face is 25.0 cm away from the ball’s front surface, where is your image? Is it real or virtual? Is it upright or inverted?

What type of mirror is shown in Fig. 23–50? Explain. FIGURE 23–50 Question 17 and Problem 15.

Problem 18P (II) 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 mirror’s radius of curvature if cars 16.0 m away appear 0.33 their normal size?

Problem 18Q Light rays from stars (including our Sun) always bend toward the vertical direction as they pass through the Earth’s atmosphere. (a) Why does this make sense? (b)What can you conclude about the apparent positions of stars as viewed from Earth? Draw a circle for Earth, a dot for you, and 3 or 4 stars at different angles.

Problem 19P (II) 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?

Problem 19Q Where must the film be placed if a camera lens is to make a sharp image of an object far away? Explain.

Problem 20P (II) (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?

Problem 21P (II) A shaving or makeup mirror is designed to magnify your face by a factor of 1.40 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.

Problem 21Q Can a diverging lens form a real image under any circumstances? Explain.

Problem 22P (II) 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 (do > r), and is greater than 1 if the object is within C (do < r).

Problem 22Q Light rays are said to be “reversible.” Is this consistent with the thin lens equation? Explain.

EXERCISE A Does the result of Example 23–2 depend on your distance from the mirror? (Try it and see, it’s fun.

EXERCISE B Return to Chapter-Opening Question 1, page 644, and answer it again now. Try to explain why you may have answered differently the first time.

Problem 23EC Light passes from a medium with n = 1.3 (water) into a medium with n = 1.5 (glass). Is the light bent toward or away from the perpendicular to the interface?

Problem 23ED Light traveling in air strikes a glass surface with n = 1.48. For what range of angles will total internal reflection occur?

Return to Chapter-Opening Question 2, page 644, and answer it again now. Try to explain why you may have answered differently the first time.

(III) Show, using a ray diagram, that the magnification m of a convex mirror is \(m=-d_i/d_o\), just as for a concave mirror. [Hint: Consider a ray from the top of the object that reflects at the center of the mirror.]

EXERCISE G If the leaf (object) of Example 23–12 is moved farther from the lens, does the image move closer to or farther from the lens? (Don’t calculate!)

Problem 23Q Can real images be projected on a screen? Can virtual images? Can either be photographed? Discuss carefully.

(III) 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?

Problem 24Q A thin converging lens is moved closer to a nearby object. Does the real image formed change (a) in position, (b) in size? If yes, describe how.

(I) The speed of light in ice is \(2.29 \times 10^8 \ \mathrm {m/s}\). What is the index of refraction of ice?

Problem 25Q If a glass converging lens is placed in water, its focal length in water will be (a) longer, (b) shorter, or (c) the same as in air. Explain.

Problem 26P (I) What is the speed of light in (a) ethyl alcohol, (b) lucite, (c) crown glass?

Problem 26Q Compare the mirror equation with the thin lens equation. Discuss similarities and differences, especially the sign conventions for the quantities involved.

Problem 27P (II) The speed of light in a certain substance is 82% of its value in water. What is the index of refraction of that substance?

Problem 27Q A lens is made of a material with an index of refraction n = 1.25. In air, it is a converging lens.Will it still be a converging lens if placed in water? Explain, using a ray diagram.

Problem 28P (I) A flashlight beam strikes the surface of a pane of glass (n = 1.56) at at a 67° angle to the normal. What is the angle of refraction?

(I) A flashlight beam strikes the surface of a pane of glass (n = 1.56) at a \(67^{\circ}\) angle to the normal. What is the angle of refraction?

Problem 29Q An underwater lens consists of a carefully shaped thinwalled plastic container filled with air. What shape should it have in order to be (a) converging, (b) diverging? Use ray diagrams to support your answer.

Problem 30P (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?

The thicker a double convex lens is in the center as compared to its edges, the shorter its focal length for a given lens diameter. Explain.

Problem 31P (I) Rays of the Sun are seen to make a 36.0° angle to the vertical beneath the water. At what angle above the horizon is the Sun?

A non-symmetrical lens (say, planoconvex) forms an image of a nearby object. Use the lensmaker’s equation to explain if the image point changes when the lens is turned around.

\(\text { (II) }\) An aquarium filled with water has flat glass sides whose index of refraction is 1.54. A beam of light from outside the aquarium strikes the glass at a \(43.5^{\circ}\) angle to the perpendicular (Fig. 23–56). 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 23–56 Problem 32. Equation Transcription: Text Transcription: (II) 43.5°C 43.5°C

Example 23–16 shows how to use a converging lens to measure the focal length of a diverging lens. (a) Why can’t you measure the focal length of a diverging lens directly? (b) It is said that for this to work, the converging lens must be stronger than the diverging lens. What is meant by “stronger,” and why is this statement true?

Problem 33P (II) A beam of light in air strikes a slab of glass and is partially reflected and partially refracted. Determine the angle of incidence if the angle of reflection is twice the angle of refraction.

\(\text { (II) }\) In searching the bottom of a pool at night, a watchman shines a narrow beam of light from his flashlight, \(1.3 \mathrm{~m}\) above the water level, onto the surface of the water at a point \(2.5 \mathrm{~m}\) from his foot at the edge of the pool (Fig. 23–57). Where does the spot of light hit the bottom of the \(2.1-m-\text { deep }\) pool? Measure from the bottom of the wall beneath his foot. FIGURE 23–57 Problem 34. Equation Transcription: Text Transcription: (II) 1.3 m 2.5 m 2.1-m-deep

Problem 35P (I) What is the critical angle for the interface between water and crown glass? To be internally reflected, the light must start in which material?

Problem 36P (I) The critical angle for a certain liquid–air surface is 47.2°. What is the index of refraction of the liquid?

Problem 37P (II) A beam of light is emitted in a pool of water from a depth of 82.0 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?

Problem 38P (II) A beam of light is emitted 8.0 cm beneath the surface of a liquid and strikes the air 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?

Problem 39P (III) (a) What is the minimum index of refraction for a glass or plastic prism to be used in binoculars (Fig. 23–28) 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?

\(\text { (III) }\) A beam of light enters the end of an optic fiber as shown in Fig. 23–58. (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 the light beam reflects back into the material at point A, assuming air outside. (b) What if the fiber were immersed in water? FIGURE 23–58 Problem 40. Equation Transcription: Text Transcription: (III) \alpha \beta \gamma

Problem 41P (I) A sharp image is located 391 mm behind a 215-mmfocal- length converging lens. Find the object distance (a) using a ray diagram, (b) by calculation.

Problem 42P (I) Sunlight is observed to focus at a point 16.5 cm behind a lens. (a) What kind of lens is it? (b) What is its power in diopters?

Problem 44P (II) A certain lens focuses light from an object 1.55 m away as an image 48.3 cm on the other side of the lens. What type of lens is it and what is its focal length? Is the image real or virtual?

Problem 45P (II) A 105-mm-focal-length lens is used to focus an image on the sensor of a camera. The maximum distance allowed between the lens and the sensor plane is 132 mm. (a) How far in front of the sensor should the lens (assumed thin) be positioned if the object to be photographed is 10.0m away? (b) 3.0 m away? (c) 1.0m away? (d) What is the closest object this lens could photograph sharply?

Problem 47P (II) A stamp collector uses a converging lens with focal length 28 cm to view a stamp 16 cm in front of the lens. (a)Where is the image located? (b) What is the magnification?

Problem 47P (II) A stamp collector uses a converging lens with focal length 28 cm to view a stamp 16 cm in front of the lens. (a)Where is the image located? (b) What is the magnification?

Problem 48P (II) It is desired to magnify reading material by a factor of 3.0 X when a book is placed 9.0 cm behind a lens. (a) Draw a ray diagram and describe the type of image this would be. (b) What type of lens is needed? (c) What is the power of the lens in diopters?

Problem 50P (II) An object is located 1.50 m from a 6.5-D lens. By how much does the image move if the object is moved (a) 0.90m closer to the lens, and (b) 0.90 m farther from the lens?

(II) (a) How far from a 50.0-mm-focal-length lens must an object be placed if its image is to be magnified \(2.50 \times\) and be real? (b) What if the image is to be virtual and magnified \(2.50 \times\)?

Problem 52P (II) Repeat Problem 51 for a -50.0 –mm-focal-length lens. [Hint: Consider objects real or virtual (formed by some other piece of optics).]

Problem 53P (II) How far from a converging lens with a focal length of 32 cm should an object be placed to produce a real image which is the same size as the object?

Problem 54P (II) (a) A 2.40-cm-high insect is 1.30 m from a 135-mmfocal- length lens. Where is the image, how high is it, and what type is it? (b) What if f = -135 nm?

(III) A bright object and a viewing screen are separated by a distance of 86.0 cm. At what location(s) between the object and the screen should a lens of focal length 16.0 cm be placed in order to produce a sharp image on the screen? [Hint: First draw a diagram.]

(III) How far apart are an object and an image formed by an 85-cm-focal-length converging lens if the image is \(3.25\times\) larger than the object and is real?

\(\text { (III) }\) In a film projector, the film acts as the object whose image is projected on a screen (Fig. 23–59). If a \(\text { 105-mmfocal-length }\) lens is to project an image on a screen \(25.5 \mathrm{~m}\) away, how far from the lens should the film be? If the film is \(24 \mathrm{~mm}\) wide, how wide will the picture be on the screen? FIGURE 23–59 Film projector, Problem 57. Equation Transcription: Text Transcription: (III) 105-mmfocal-length 25.5 m 24 mm

Problem 58P (II) A diverging lens with F =-36.5 cm is placed 14.0 cm behind a converging lens with f =20.0 cm. Where will an object at infinity be focused?

Problem 60P (II) A 38.0-cm-focal-length converging lens is 28.0 cm behind a diverging lens. Parallel light strikes the diverging lens. After passing through the converging lens, the light is again parallel. What is the focal length of the diverging lens? [Hint: First draw a ray diagram.]

Problem 60P (II) A 38.0-cm-focal-length converging lens is 28.0 cm behind a diverging lens. Parallel light strikes the diverging lens. After passing through the converging lens, the light is again parallel. What is the focal length of the diverging lens? [Hint: First draw a ray diagram.]

Problem 61P (II) Two lenses, one converging with focal length 20.0 cm and one diverging with focal length -10.0 cm. are placed 25.0 cm apart. An object is placed 60.0 cm in front of the converging lens. Determine (a) the position and (b) the magnification of the final image formed. (c) Sketch a ray diagram for this system.

\((I I) \) A lighted candle is placed \(36 \mathrm{~cm} \) in front of a converging lens of focal length \(f_{1}=13 \mathrm{~cm}\), which in turn is \(56 \mathrm{~cm}\) in front of another converging lens of focal length \(f_{2}=16 \mathrm{~cm}\) (see Fig. 23-60). (a) Draw a ray diagram and estimate the location and the relative size of the final image. (b) Calculate the position and relative size of the final image. \(f_{1}=13 \mathrm{~cm} f_{2}=16 \mathrm{~cm}\) FIGURE 23–60 Problem 62 Equation Transcription: Text Transcription: (II) 36 cm f1=13 cm 56 cm f2=16 cm f1=13 cm f2=16 cm

Problem 63P (I) A double concave lens has surface radii of 33.4 cm and 28.8 cm. What is the focal length if n =1.52?

Problem 64P (I) Both surfaces of a double convex lens have radii of 34.1 cm. If the focal length is 28.9 cm, what is the index of refraction of the lens material?

Problem 65P (I) A planoconvex lens (Fig. 23–31a) with n=1.55 is to have a focal length of 16.3 cm. What is the radius of curvature of the convex surface?

Problem 66P (II) A symmetric double convex lens with a focal length of 22.0 cm is to be made from glass with an index of refraction of 1.52. What should be the radius of curvature for each surface?

Problem 67P (II) A prescription for an eyeglass lens calls for +3.50 diopters. The lensmaker grinds the lens from a “blank” with and convex front surface of radius of curvature of 30.0 cm. What should be the radius of curvature of the other surface?

Problem 68P (III) An object is placed 96.5 cm from a glass lens (n =1.52) with one concave surface of radius 22.0 cm and one convex surface of radius 18.5 cm. Where is the final image? What is the magnification?

Sunlight is reflected off the Moon. How long does it take that light to reach us from the Moon?

Problem 71GP We wish to determine the depth of a swimming pool filled with water by measuring the width ( x = 6.50m ) and then noting that the far bottom edge of the pool is just visible at an angle of 13.0° above the horizontal as shown in Fig. 23–62. Calculate the depth of the pool.

Problem 72GP The critical angle of a certain piece of plastic in air is ?c = 37.8°.What is the critical angle of the same plastic if it is immersed in water?

Problem 74GP When an object is placed 60.0 cm from a certain converging lens, it forms a real image. When the object is moved to 40.0 cm from the lens, the image moves 10.0 cm farther from the lens. Find the focal length of this lens.

A 4.5-cm-tall object is placed 32 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?

Problem 78GP (a) A plane mirror can be considered a limiting case of a spherical mirror. Specify what this limit is. (b) Determine an equation that relates the image and object distances in this limit of a plane mirror. (c) Determine the magnification of a plane mirror in this same limit. (d) Are your results in parts (b) and (c) consistent with the discussion of Section 23–2 on plane mirrors?

Problem 79GP An object is placed 18 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?

Light is incident on an equilateral glass prism at a \(45.0^{\circ} \mathrm{C}\) angle to one face, Fig. 23–64. Calculate the angle at which light emerges from the opposite face. Assume that \(n=1.54\) FIGURE 23–64 Problems 80 and 81. Equation Transcription: Text Transcription: 45.0°C n=1.54 45.0°C

Suppose a ray strikes the left face of the prism in Fig. 23–64 at \(45.0^{\circ} \mathrm{C}\) as shown, but is totally internally reflected at the opposite side. If the apex angle (at the top) is \(\theta=65.0^{\circ}\) what can you say about the index of refraction of the prism? Equation Transcription: Text Transcription: 45.0°C \theta =65.0°

Problem 83GP How large is the image of the Sun on a camera sensor with (a) a 35-mm-focal-length lens, (b) a 50-mm-focal-length lens, and (c) a 105-mm-focal-length lens? The Sun has diameter 1.4*106km, and it is 1.5*108 km away.

Problem 82GP (a) An object 37.5 cm in front of a certain lens is imaged 8.20 cm in front of that lens (on the same side as the object). What type of lens is this, and what is its focal length? Is the image real or virtual? (b) If the image were located, instead, 44.5 cm in front of the lens, what type of lens would it be and what focal length would it have?

Figure 23–65 is a photograph of an eyeball with the image of a boy in a doorway. (a) Is the eye here acting as a lens or as a mirror? (b) Is the eye being viewed right side up or is the camera taking this photo upside down? (c) Explain, based on all possible images made by a convex mirror or lens. FIGURE 23–65 Problem 84.

Which of the two lenses shown in Fig. 23–66 is converging, and which is diverging? Explain using ray diagrams and show how each image is formed. FIGURE 23–66 Problem 85.

Figure 23–67 shows a liquid-detecting prism device that might be used inside a washing machine. If no liquid covers the prism’s 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 sensor’s signal decreases. Thus a large signal from the light sensor indicates the absence of liquid in the reservoir. Determine the allowable range for the prism’s index of refraction \(n\). FIGURE 23–67 Problem 86. Equation Transcription: Text Transcription: n 45° n 45°

Problem 87GP (a) Show that if two thin lenses of focal lengths f1 and f2 are placed in contact with each other, the focal length of the combination is given by fT = f1f2/(f1 + f2). (b) Show that the power P of the combination of two lenses is the sum of their separate powers, P = P1 + P2.

Problem 88GP Two converging lenses are placed 30.0 cm apart. The focal length of the lens on the right is 20.0 cm, and the focal length of the lens on the left is 15.0 cm. An object is placed to the left of the 15.0-cm-focal-length lens. A final image from both lenses is inverted and located halfway between the two lenses. How far to the left of the 15.0-cm-focal-length lens is the original object?

Problem 89GP An object is placed 30.0 cm from a +5.0-D lens. A spherical mirror with focal length 25 cm is placed 75 cm behind the lens. Where is the final image? (Note that the mirror reflects light back through the lens.) Be sure to draw a diagram.

A small object is \(25.0 \mathrm{~cm}\) from a diverging lens as shown in Fig. 23–68. A converging lens with a focal length of \(12.0 \mathrm{~cm} \text { is } 30.0 \mathrm{~cm}\) to the right of the diverging lens. The two-lens system forms a real inverted image \(17.0 \mathrm{~cm}\) to the right of the converging lens. What is the focal length of the diverging lens? FIGURE 23–68 Problem 90. Equation Transcription: Text Transcription: 25.0 cm 12.0 cm is 30.0 cm 17.0 cm 25.0 cm 30.0 cm 17.0 cm

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.

Suppose that you want to take a photograph of yourself as you look at your image in a mirror 3.1 m away. For what distance should the camera lens be focused?

Two plane mirrors meet at a 135 angle, Fig. 2352. If light rays strike one mirror at 34 as shown, at what angle do they leave the second mirror?

A person whose eyes are 1.72 m above the floor stands 2.20 m in front of a vertical plane mirror whose bottom edge is 38 cm above the floor, Fig. 2353. 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?

Stand up two plane mirrors so they form a 90.0 angle as in Fig. 2354. 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.

Two plane mirrors, nearly parallel, are facing each other 2.3 m apart as in Fig. 2355. You stand 1.6 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?

Suppose you are 94 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?

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 21.0 cm) must an object be placed if its image is to be at infinity?

A small candle is 38 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?

An object 3.0 mm high is placed 16 cm from a convex mirror of radius of curvature 16 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. 232 using a focal length of (c) Compute the image size, using Eq. 233.

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

You are standing 3.4 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.

The image of a distant tree is virtual and very small when viewed in a curved mirror. The image appears to be 19.0 cm behind the mirror. What kind of mirror is it, and what is its radius of curvature?

(II) A mirror at an amusement park shows an upright image of any person who stands 1.9 m in front of it. If the image is three times the person’s height, what is the radius of curvature of the mirror? (See Fig. 23–50.)

In Example 234, show that if the object is moved 10.0 cm farther from the concave mirror, the objects image size will equal the objects actual size. Stated as a multiple of the focal length, what is the object distance for this actual-sized image situation?

You look at yourself in a shiny 8.8-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?

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 16.0 m away appear 0.33 their normal size

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) 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 shaving or makeup mirror is designed to magnify your face by a factor of 1.40 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

(II) 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 \(\mathrm{C}\left(d_{\mathrm{o}}>r\right)\), and is greater than 1 if the object is within \(\mathrm{C}\left(d_{\mathrm{o}}<r\right)\).

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

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?

The speed of light in ice is What is the index of refraction of ice?

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

The speed of light in a certain substance is 82% of its value in water. What is the index of refraction of that substance?

A flashlight beam strikes the surface of a pane of glass at a 67 angle to the normal. What is the angle of refraction?

A diver shines a flashlight upward from beneath the water at a 35.2 angle to the vertical. At what angle does the light leave the water?

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 airwater interface from below the surface?

Rays of the Sun are seen to make a 36.0 angle to the vertical beneath the water. At what angle above the horizon is the Sun?

An aquarium filled with water has flat glass sides whose index of refraction is 1.54. A beam of light from outside the aquarium strikes the glass at a 43.5 angle to the perpendicular (Fig. 2356). 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?

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

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. 2357). Where does the spot of light hit the bottom of the 2.1-m-deep pool? Measure from the bottom of the wall beneath his foot.

What is the critical angle for the interface between water and crown glass? To be internally reflected, the light must start in which material?

The critical angle for a certain liquidair surface is 47.2. What is the index of refraction of the liquid?

A beam of light is emitted in a pool of water from a depth of 82.0 cm. Where must it strike the airwater interface, relative to the spot directly above it, in order that the light does not exit the water?

(II) A beam of light is emitted 8.0 cm beneath the surface of a liquid and strikes the air 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?

(a) What is the minimum index of refraction for a glass or plastic prism to be used in binoculars (Fig. 2328) so that total internal reflection occurs at 45? (b) Will binoculars work if their prisms (assume ) are immersed in water? (c) What minimum n is needed if the prisms are immersed in water?

A beam of light enters the end of an optic fiber as shown in Fig. 2358. (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 the light beam reflects back into the material at point A, assuming air outside. (b) What if the fiber were immersed in water?

A sharp image is located 391 mm behind a 215-mmfocal-length converging lens. Find the object distance (a) using a ray diagram, (b) by calculation.

(I) Sunlight is observed to focus at a point 16.5 cm behind a lens. (a) What kind of lens is it? (b) What is its power in diopters?

(I) (a) What is the power of a 32.5-cm-focal-length lens? (b) What is the focal length of a -6.75-D lens? Are these lenses converging or diverging?

A certain lens focuses light from an object 1.55 m away as an image 48.3 cm on the other side of the lens. What type of lens is it and what is its focal length? Is the image real or virtual?

A 105-mm-focal-length lens is used to focus an image on the sensor of a camera. The maximum distance allowed between the lens and the sensor plane is 132 mm. (a) How far in front of the sensor should the lens (assumed thin) be positioned if the object to be photographed is 10.0 m away? (b) 3.0 m away? (c) 1.0 m away? (d) What is the closest object this lens could photograph sharply?

(II) Use ray diagrams to show that a real image formed by a thin lens is always inverted, whereas a virtual image is always upright if the object is real.

A stamp collector uses a converging lens with focal length 28 cm to view a stamp 16 cm in front of the lens. (a)Where is the image located? (b) What is the magnification?

It is desired to magnify reading material by a factor of when a book is placed 9.0 cm behind a lens. (a) Draw a ray diagram and describe the type of image this would be. (b) What type of lens is needed? (c) What is the power of the lens in diopters?

(II) A -7.00-D lens is held 12.5 cm from an ant 1.00 mm high. Describe the position, type, and height of the image.

An object is located 1.50 m from a 6.5-D lens. By how much does the image move if the object is moved (a) 0.90 m closer to the lens, and (b) 0.90 m farther from the lens?

(a) How far from a 50.0-mm-focal-length lens must an object be placed if its image is to be magnified and be real? (b) What if the image is to be virtual and magnified 2.50x?

Repeat Problem 51 for a lens. [Hint: Consider objects real or virtual (formed by some other piece of optics).]

How far from a converging lens with a focal length of 32 cm should an object be placed to produce a real image which is the same size as the object?

(a) A 2.40-cm-high insect is 1.30 m from a 135-mmfocal-length lens. Where is the image, how high is it, and what type is it? (b) What if

A bright object and a viewing screen are separated by a distance of 86.0 cm. At what location(s) between the object and the screen should a lens of focal length 16.0 cm be placed in order to produce a sharp image on the screen? [Hint: First draw a diagram.]

How far apart are an object and an image formed by an 85-cm-focal-length converging lens if the image is larger than the object and is real?

In a film projector, the film acts as the object whose image is projected on a screen (Fig. 2359). If a 105-mmfocal-length lens is to project an image on a screen 25.5 m away, how far from the lens should the film be? If the film is 24 mm wide, how wide will the picture be on the screen?

A diverging lens with is placed 14.0 cm behind a converging lens with Where will an object at infinity be focused?

Two 25.0-cm-focal-length converging lenses are placed 16.5 cm apart. An object is placed 35.0 cm in front of one lens. Where will the final image formed by the second lens be located? What is the total magnification?

A 38.0-cm-focal-length converging lens is 28.0 cm behind a diverging lens. Parallel light strikes the diverging lens. After passing through the converging lens, the light is again parallel. What is the focal length of the diverging lens? [Hint: First draw a ray diagram.]

Two lenses, one converging with focal length 20.0 cm and one diverging with focal length are placed 25.0 cm apart. An object is placed 60.0 cm in front of the converging lens. Determine (a) the position and (b) the magnification of the final image formed. (c) Sketch a ray diagram for this system.

A lighted candle is placed 36 cm in front of a converging lens of focal length which in turn is 56 cm in front of another converging lens of focal length (see Fig. 2360). (a) Draw a ray diagram and estimate the location and the relative size of the final image. (b) Calculate the position and relative size of the final image

A double concave lens has surface radii of 33.4 cm and 28.8 cm. What is the focal length if n = 1.52?

Both surfaces of a double convex lens have radii of 34.1 cm. If the focal length is 28.9 cm, what is the index of refraction of the lens material?

A planoconvex lens (Fig. 2331a) with is to have a focal length of 16.3 cm. What is the radius of curvature of the convex surface?

A symmetric double convex lens with a focal length of 22.0 cm is to be made from glass with an index of refraction of 1.52. What should be the radius of curvature for each surface?

A prescription for an eyeglass lens calls for diopters. The lensmaker grinds the lens from a blank with and convex front surface of radius of curvature of 30.0 cm. What should be the radius of curvature of the other surface?

An object is placed 96.5 cm from a glass lens with one concave surface of radius 22.0 cm and one convex surface of radius 18.5 cm. Where is the final image? What is the magnification?

Sunlight is reflected off the Moon. How long does it take that light to reach us from the Moon?

You hold a small flat mirror 0.50 m in front of you and can see your reflection twice in that mirror because there is a full-length mirror 1.0 m behind you (Fig. 2361). Determine the distance of each image from you.

We wish to determine the depth of a swimming pool filled with water by measuring the width and then noting that the far bottom edge of the pool is just visible at an angle of 13.0 above the horizontal as shown in Fig. 2362. Calculate the depth of the pool

The critical angle of a certain piece of plastic in air is What is the critical angle of the same plastic if it is immersed in water?

A pulse of light takes 2.63 ns (see Table 14) to travel 0.500 m in a certain material. Determine the materials index of refraction, and identify this material.

When an object is placed 60.0 cm from a certain converging lens, it forms a real image. When the object is moved to 40.0 cm from the lens, the image moves 10.0 cm farther from the lens. Find the focal length of this lens

A 4.5-cm-tall object is placed 32 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?

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

If the apex angle of a prism is (see Fig. 2363), 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?

(a) A plane mirror can be considered a limiting case of a spherical mirror. Specify what this limit is. (b) Determine an equation that relates the image and object distances in this limit of a plane mirror. (c) Determine the magnification of a plane mirror in this same limit. (d) Are your results in parts (b) and (c) consistent with the discussion of Section 232 on plane mirrors?

An object is placed 18 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?

Light is incident on an equilateral glass prism at a 45.0 angle to one face, Fig. 2364. Calculate the angle at which light emerges from the opposite face. Assume that n = 1.54.

Suppose a ray strikes the left face of the prism in Fig. 2364 at 45.0 as shown, but is totally internally reflected at the opposite side. If the apex angle (at the top) is what can you say about the index of refraction of the prism?

(a) An object 37.5 cm in front of a certain lens is imaged 8.20 cm in front of that lens (on the same side as the object). What type of lens is this, and what is its focal length? Is the image real or virtual? (b) If the image were located, instead, 44.5 cm in front of the lens, what type of lens would it be and what focal length would it have?

How large is the image of the Sun on a camera sensor with (a) a 35-mm-focal-length lens, (b) a 50-mm-focal-length lens, and (c) a 105-mm-focal-length lens? The Sun has diameter and it is away.

Figure 23–65 is a photograph of an eyeball with the image of a boy in a doorway. (a) Is the eye here acting as a lens or as a mirror? (b) Is the eye being viewed right side up or is the camera taking this photo upside down? (c) Explain, based on all possible images made by a convex mirror or lens.

Which of the two lenses shown in Fig. 2366 is converging, and which is diverging? Explain using ray diagrams and show how each image is formed.

Figure 2367 shows a liquid-detecting prism device that might be used inside a washing machine. 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. Determine the allowable range for the prisms index of refraction n.

(a) Show that if two thin lenses of focal lengths \(f_1\) and \(f_2\) are placed in contact with each other, the focal length of the combination is given by \(f_\mathrm T=f_1f_2/{(f_1+f_2)}\). (b) Show that the power P of the combination of two lenses is the sum of their separate powers, \(P=P_1 + P_2\).

Two converging lenses are placed 30.0 cm apart. The focal length of the lens on the right is 20.0 cm, and the focal length of the lens on the left is 15.0 cm. An object is placed to the left of the 15.0-cm-focal-length lens. A final image from both lenses is inverted and located halfway between the two lenses. How far to the left of the 15.0-cm-focal-length lens is the original object?

An object is placed 30.0 cm from a +5.0-D lens. A spherical mirror with focal length 25 cm is placed 75 cm behind the lens. Where is the final image? (Note that the mirror reflects light back through the lens.) Be sure to draw a diagram.

A small object is 25.0 cm from a diverging lens as shown in Fig. 2368. A converging lens with a focal length of 12.0 cm is 30.0 cm to the right of the diverging lens. The two-lens system forms a real inverted image 17.0 cm to the right of the converging lens. What is the focal length of the diverging lens?