The object in Figure P23.52 is midway between the lens and

When an image of a real object is formed by a fl at mirror, which of the following statements are always true? (a) The image is larger than the object. (b) The image is the same size as the object. (c) The image is virtual. (d) The image is smaller than the object. (e) The image is upright.

An object is placed 16.0 cm away from a concave mirror with focal length 6.00 cm. Find the location of the image. (a) 9.60 cm in front of the mirror (b) 4.36 cm in front of the mirror (c) 9.60 cm behind the mirror (d) 4.36 cm behind the mirror (e) 10.0 cm in front of the mirror

An object is placed 16.0 cm away from a convex mirror with a focal length of magnitude 6.00 cm. What is the location of the image? (a) 9.60 cm in front of the mirror (b) 4.36 cm in front of the mirror (c) 9.60 cm behind the mirror (d) 4.36 cm behind the mirror (e) 10.0 cm in front of the mirror

A converging lens has a virtual object located a fi nite distance away. Which of the following must be true? (a) The image is virtual. (b) The image is real. (c) p _ 0 (d) p 0 (e) f q

A thin, convergent lens has a focal length of 8.00 cm. If a real, inverted image is located 12.0 cm to the right of the lens, where is the object located? (a) 12.0 cm to the left of the lens (b) 24.0 cm to the right of the lens (c) 24.0 cm to the left of the lens (d) 18.0 cm to the right of the lens (e) 18.0 cm to the left of the lens

A real object is 10.0 cm to the left of a thin, diverging lens having a focal length of magnitude 16.0 cm. What is the location of the image? (a) 6.15 cm to the right of the lens (b) 6.15 cm to the left of the lens (c) 26.7 cm to the right of the lens (d) 26.7 cm to the left of the lens (e) 6.00 cm to the right of the lens

When the image of a real object is formed by a concave mirror, which of the following statements are true? (a) The image is always real. (b) The image is always virtual. (c) If the object distance is less than the focal length, the image is real. (d) If the object distance is greater than the focal length, the image is real. (e) If q < 0, the image is virtual.

A convex mirror forms an image of a real object. Which of the following statements is always true? (a) The image is real and upright. (b) The image is virtual and larger than the object. (c) The image is virtual and upright. (d) The image is larger than the object. (e) The image is virtual and inverted.

An object is located 50 cm from a converging lens having a focal length of 15 cm. Which of the following is true regarding the image formed by the lens? (a) It is virtual, upright, and larger than the object. (b) It is virtual, inverted, and smaller than the object. (c) It is real, inverted, and smaller than the object. (d) It is real, inverted, and larger than the object. (e) It is real, upright, and larger than the object.

For a real object, is the image formed by a diverging lens always (a) virtual and inverted, (b) virtual and upright, (c) real and upright, (d) real and inverted, or (e) real and magnifi ed?

Tape a picture of yourself on a bathroom mirror. Stand several centimeters away from the mirror. Can you focus your eyes on both the picture taped to the mirror and your image in the mirror at the same time? So where is the image of yourself?

Explain why a mirror cannot give rise to chromatic aberration.

A fl at mirror creates a virtual image of your face. Suppose the fl at mirror is combined with another optical element. Can the mirror form a real image in such a combination?

Why does a clear stream always appear to be shallower than it actually is?

A virtual image is often described as an image through which light rays dont actually travel, as they do for a real image. Can a virtual image be photographed?

A common mirage is formed when the air gets gradually cooler as the height above the ground increases. What might happen if the air grows gradually warmer as the height increases? This often happens over bodies of water or snow-covered ground; the effect is called looming.

Suppose you want to use a converging lens to project the image of two trees onto a screen. One tree is a distance x from the lens; the other is at 2x, as in Figure CQ23.7. You adjust the screen so that the near tree is in focus. If you now want the far tree to be in focus, do you move the screen toward or away from the lens?

Lenses used in eyeglasses, whether converging or diverging, are always designed such that the middle of the lens curves away from the eye. Why?

Can a converging lens be made to diverge light if placed in a liquid? How about a converging mirror?

If a cylinder of solid glass or clear plastic is placed above the words LEAD OXIDE and viewed from the side, as shown in Figure CQ23.10, the word LEAD appears inverted, but the word OXIDE does not. Explain.

In a Jules Verne novel, a piece of ice is shaped into a magnifying lens to focus sunlight to start a fi re. Is that possible?

Light from an object passes through a lens and forms a visible image on a screen. If the screen is removed, would you be able to see the image (a) if you remained in your present position and (b) if you could look at the lens along the its axis, beyond the original position of the screen?

Why does the focal length of a mirror not depend on the mirror material when the focal length of a lens does depend on the lens material?

An inverted image of an object is viewed on a screen from the side facing a converging lens. An opaque card is then introduced covering only the upper half of the lens. What happens to the image on the screen? (a) Half the image would disappear. (b) The entire image would appear and remain unchanged. (c) Half the image would disappear and be dimmer. (d) The entire image would appear, but would be dimmer.

Does your bathroom mirror show you older or younger than your actual age? Compute an order-of- magnitude estimate for the age difference, based on data you specify.

Two plane mirrors stand facing each other, 3.0 m apart, and a woman stands between them. The woman faces one of the mirrors from a distance of 1.0 m, with the palm of her left hand facing the closer mirror. (a) What is the apparent position of the closest image of her left hand, measured from the surface of the mirror in front of her? Does it show the palm of her hand or the back of her hand? (b) What is the position of the next image? Does it show the palm of her hand or the back of her hand? (c) Repeat for the third image. (d) Which of the images are real and which are virtual?

A person walks into a room that has, on opposite walls, two plane mirrors producing multiple images. Find the distances from the person to the fi rst three images seen in the left-hand mirror when the person is 5.00 ft from the mirror on the left wall and 10.0 ft from the mirror on the right wall.

In a church choir loft, two parallel walls are 5.30 m apart. The singers stand against the north wall. The organist faces the south wall, sitting 0.800 m away from it. So that she can see the choir, a fl at mirror 0.600 m wide is mounted on the south wall, straight in front of the organist. What width of the north wall can she see? Hint: Draw a top-view diagram to justify your answer.

Use Active Figure 23.2 to give a geometric proof that the virtual image formed by a plane mirror is the same distance behind the mirror as the object is in front of it. In the following problems, algebraic signs are not given. We leave it to you to determine the correct sign to use with each quantity, based on an analysis of the problem and the sign conventions in Table 23.1.

A dentist uses a mirror to examine a tooth that is 1.00 cm in front of the mirror. The image of the tooth is formed 10.0 cm behind the mirror. Determine (a) the mirrors radius of curvature and (b) the magnifi cation of the image.

An object is placed 40.0 cm from a concave mirror of radius 20.0 cm. (a) Find the location of the image. (b) What is the magnifi cation of the mirror? Is the image real or virtual? Is the image upright or inverted?

To fi t a contact lens to a patients eye, a keratometer can be used to measure the curvature of the corneathe front surface of the eye. This instrument places an illuminated object of known size at a known distance p from the cornea, which then refl ects some light from the object, forming an image of it. The magnifi cation M of the image is measured by using a small viewing telescope that allows a comparison of the image formed by the cornea with a second calibrated image projected into the fi eld of view by a prism arrangement. Determine the radius of curvature of the cornea when p _ 30.0 cm and M _ 0.013 0.

An object of height 2.0 cm is placed 30.0 cm from a convex mirror of focal length having magnitude 10.0 cm. (a) Find the location of the image. (b) Describe the properties of the image.

While looking at her image in a cosmetic mirror, Dina notes that her face is highly magnifi ed when she is close to the mirror, but as she backs away from the mirror, her image fi rst becomes blurry, then disappears when she is about 30 cm from the mirror, and then inverts when she is beyond 30 cm. Based on these observations, what can she conclude about the properties of the mirror?

A 2.00-cm-high object is placed 3.00 cm in front of a concave mirror. If the image is 5.00 cm high and virtual, what is the focal length of the mirror?

A dedicated sports car enthusiast polishes the inside and outside surfaces of a hubcap that is a section of a sphere. When he looks into one side of the hubcap, he sees an image of his face 30.0 cm in back of it. He then turns the hubcap over, keeping it the same distance from his face. He now sees an image of his face 10.0 cm in back of the hubcap. (a) How far is his face from the hubcap? (b) What is the radius of curvature of the hubcap?

A concave makeup mirror is designed so that a person 25 cm in front of it sees an upright image magnifi ed by a factor of two. What is the radius of curvature of the mirror?

An object is placed 10.0 cm from a convex mirror having a focal length of magnitude 15.0 cm. (a) Use graph paper to construct an accurate ray diagram for this situation. From your ray diagram, determine the location of the image, its magnifi cation, and whether it is upright or inverted. (b) Check your answers to part (a) using the mirror equation.

A man standing 1.52 m in front of a shaving mirror produces an inverted image 18.0 cm in front of it. How close to the mirror should he stand if he wants to form an upright image of his chin that is twice the chins actual size?

A convex mirror has a focal length of magnitude 8.0 cm. (a) If the image is virtual, what is the object location for which the magnitude of the image distance is one third the magnitude of the object distance? (b) Find the magnifi cation of the image and state whether it is upright or inverted.

A woman holds a tube of lipstick 10.0 cm from a spherical mirror and notices that the image of the tube is upright and half its normal size. (a) Determine the position of the image. Is it in front of or behind the mirror? (b) Calculate the focal length of the mirror.

A concave mirror has a radius of curvature of 24 cm. (a) Determine the object position for which the resulting image is upright and three times the size of the object. (b) Draw a ray diagram to determine the position of the image. Is the image real or virtual?

A spherical mirror is to be used to form an image, fi ve times as tall as an object, on a screen positioned 5.0 m from the mirror. (a) Describe the type of mirror required. (b) Where should the mirror be positioned relative to the object?

A ball is dropped from rest 3.00 m directly above the vertex of a concave mirror having a radius of 1.00 m and lying in a horizontal plane. (a) Describe the motion of the balls image in the mirror. (b) At what time do the ball and its image coincide?

A cubical block of ice 50.0 cm on an edge is placed on a level fl oor over a speck of dust. Locate the image of the speck, when viewed from directly above, if the index of refraction of ice is 1.309.

One end of a long glass rod (n _ 1.50) has the shape of a convex surface of radius 8.00 cm. An object is positioned in air along the axis of the rod in front of the convex surface. Find the image position that corresponds to each of the following object positions: (a) 20.0 cm, (b) 8.00 cm, (c) 4.00 cm, (d) 2.00 cm.

A paperweight is made of a solid glass hemisphere with index of refraction 1.50. The radius of the circular cross section is 4.0 cm. The hemisphere is placed on its fl at surface, with the center directly over a 2.5-mm-long line drawn on a sheet of paper. What length of line is seen by someone looking vertically down on the hemisphere?

The top of a swimming pool is at ground level. If the pool is 2 m deep, how far below ground level does the bottom of the pool appear to be located when (a) the pool is completely fi lled with water and (b) the pool is fi lled halfway with water?

A transparent sphere of unknown composition is observed to form an image of the Sun on its surface opposite the Sun. What is the refractive index of the sphere material?

A fl int glass plate (n _ 1.66) rests on the bottom of an aquarium tank. The plate is 8.00 cm thick (vertical dimension) and covered with water (n _ 1.33) to a depth of 12.0 cm. Calculate the apparent thickness of the plate as viewed from above the water. (Assume nearly normal incidence of light rays.)

A jellyfi sh is fl oating in a water-fi lled aquarium 1.00 m behind a fl at pane of glass 6.00 cm thick and having an index of refraction of 1.50. (a) Where is the image of the jellyfi sh located? (b) Repeat the problem when the glass is so thin that its thickness can be neglected. (c) How does the thickness of the glass affect the answer to part (a)?

A goldfi sh is swimming at 2.00 cm/s toward the front wall of a rectangular aquarium. What is the apparent speed of the fi sh as measured by an observer looking in from outside the front wall of the tank? The index of refraction of water is 1.333.

A contact lens is made of plastic with an index of refraction of 1.50. The lens has an outer radius of curvature of _2.00 cm and an inner radius of curvature of _2.50 cm. What is the focal length of the lens?

The left face of a biconvex lens has a radius of curvature of 12.0 cm, and the right face has a radius of curvature of 18.0 cm. The index of refraction of the glass is 1.44. (a) Calculate the focal length of the lens. (b) Calculate the focal length if the radii of curvature of the two faces are interchanged.

A converging lens has a focal length of 10.0 cm. Locate the images for object distances of (a) 20.0 cm, (b) 10.0 cm, and (c) 5.00 cm, if they exist. For each case, state whether the image is real or virtual, upright or inverted, and fi nd the magnifi cation.

A converging lens has a focal length of 10.0 cm. Use graph paper to construct accurate ray diagrams for object distances of (a) 20.0 cm and (b) 5.0 cm. From your ray diagrams, determine the location of the image and its properties and compare your results with the values found algebraically in Problem 31. (c) Name at least three errors in constructing the graph that could lead to differences in the fi nal answer.

A diverging lens has a focal length of 20.0 cm. Locate the images for object distances of (a) 40.0 cm, (b) 20.0 cm, and (c) 10.0 cm. For each case, state whether the image is real or virtual and upright or inverted, and fi nd the magnifi cation.

A diverging lens has a focal length of 20.0 cm. Use graph paper to construct accurate ray diagrams for object distances of (a) 40.0 cm and (b) 10.0 cm. In each case determine the location of the image from the diagram and the image magnifi cation, and state whether the image is upright or inverted. (c) Estimate the magnitude of uncertainty in locating the points in the graph. Are your answers and the uncertainty consistent with the algebraic answers found in Problem 33?

A transparent photographic slide is placed in front of a converging lens with a focal length of 2.44 cm. The lens forms an image of the slide 12.9 cm from it. How far is the lens from the slide if the image is (a) real? (b) Virtual?

The nickels image in Figure P23.36 has twice the diameter of the nickel when the lens is 2.84 cm from the nickel. Determine the focal length of the lens. FIGURE P23.36

A certain LCD projector contains a single thin lens. An object 24.0 mm high is to be projected so that its image fi lls a screen 1.80 m high. The object-to-screen distance is 3.00 m. (a) Determine the focal length of the projection lens. (b) How far from the object should the lens of the projector be placed to form the image on the screen?

A convex lens of focal length 18.0 cm is used as a magnifying glass. At what distance from a tiny insect should you hold this lens to get a magnifi cation of _3.00?

A converging lens is placed 30.0 cm to the right of a diverging lens of focal length 10.0 cm. A beam of parallel light enters the diverging lens from the left, and the beam is again parallel when it emerges from the converging lens. Calculate the focal length of the converging lens.

(a) Use the thin-lens equation to derive an expression for q in terms of f and p. (b) Prove that for a real object and a diverging lens, the image must always be virtual. Hint: Set f _ __ f _ and show that q must be less than zero under the given conditions. (c) For a real object and converging lens, what inequality involving p and f must hold if the image is to be real?

Two converging lenses, each of focal length 15.0 cm, are placed 40.0 cm apart, and an object is placed 30.0 cm in front of the fi rst lens. Where is the fi nal image formed, and what is the magnifi cation of the system?

Object O1 is 15.0 cm to the left of a converging lens with a 10.0-cm focal length. A second lens is positioned 10.0 cm to the right of the fi rst lens and is observed to form a fi nal image at the position of the original object O1. (a) What is the focal length of the second lens? (b) What is the overall magnifi cation of this system? (c) What is the nature (i.e., real or virtual, upright or inverted) of the fi nal image?

A 1.00-cm-high object is placed 4.00 cm to the left of a converging lens of focal length 8.00 cm. A diverging lens of focal length _16.00 cm is 6.00 cm to the right of the converging lens. Find the position and height of the fi nal image. Is the image inverted or upright? Real or virtual?

Two converging lenses having focal lengths of 10.0 cm and 20.0 cm are placed 50.0 cm apart, as shown in Figure P23.44. The fi nal image is to be located between the lenses, at the position indicated. (a) How far to the left of the fi rst lens should the object be positioned? (b) What is the overall magnifi cation of the system? (c) Is the fi nal image upright or inverted?

Lens L1 in Figure P23.45 has a focal length of 15.0 cm and is located a fi xed distance in front of the fi lm plane of a camera. Lens L2 has a focal length of 13.0 cm, and its distance d from the fi lm plane can be varied from 5.00 cm to 10.0 cm. Determine the range of distances for which objects can be focused on the fi lm. FIGURE P23.45 L1 L2 Film d 12.0 cm

An object is placed 15.0 cm from a fi rst converging lens of focal length 10.0 cm. A second converging lens with focal length 5.00 cm is placed 10.0 cm to the right of the fi rst converging lens. (a) Find the position q1 of the image formed by the fi rst converging lens. (b) How far from the second lens is the image of the fi rst lens? (c) What is the value of p2, the object position for the second lens? (d) Find the position q2 of the image formed by the sec- ond lens. (e) Calculate the magnifi cation of the fi rst lens. (f) Calculate the magnifi cation of the second lens. (g) What is the total magnifi cation for the system? (h) Is the fi nal image real or virtual? Is it upright or inverted?

An object placed 10.0 cm from a concave spherical mirror produces a real image 8.00 cm from the mirror. If the object is moved to a new position 20.0 cm from the mirror, what is the position of the image? Is the fi nal image real or virtual?

A real objects distance from a converging lens is fi ve times the focal length. (a) Determine the location of the image q in terms of the focal length f. (b) Find the magnifi cation of the image. (c) Is the image real or virtual? Is it upright or inverted? Is the image on the same side of the lens as the object or on the opposite side?

A convergent lens with a 50.0-mm focal length is used to focus an image of a very distant scene onto a fl at screen 35.0 mm wide. What is the angular width a of the scene included in the image on the screen?

A diverging lens (n _ 1.50) is shaped like that in Active Figure 23.25c. The radius of the fi rst surface is 15.0 cm, and that of the second surface is 10.0 cm. (a) Find the focal length of the lens. Determine the positions of the images for object distances of (b) infi nity, (c) 3_f _, (d) _f _, and (e) _f _/2.

The lens and the mirror in Figure P23.51 are separated by 1.00 m and have focal lengths of _80.0 cm and _50.0 cm, respectively. If an object is placed 1.00 m to the left of the lens, where will the fi nal image be located? State whether the image is upright or inverted, and determine the overall magnifi cation.

The object in Figure P23.52 is midway between the lens and the mirror. The mirrors radius of curvature is 20.0 cm, and the lens has a focal length of _16.7 cm. Considering only the light that leaves the object and travels fi rst toward the mirror, locate the fi nal image formed by this system. Is the image real or virtual? Is it upright or inverted? What is the overall magnifi cation of the image?

A parallel beam of light enters a glass hemisphere perpendicular to the fl at face, as shown in Figure P23.53. The radius of the hemisphere is R _ 6.00 cm, and the index of refraction is n _ 1.560. Determine the point at which the beam is focused. (Assume paraxial rays; i.e., assume all rays are located close to the principal axis.)

A converging lens of focal length 20.0 cm is separated by 50.0 cm from a converging lens of focal length 5.00 cm. (a) Find the position of the fi nal image of an object placed 40.0 cm in front of the fi rst lens. (b) If the height of the object is 2.00 cm, what is the height of the fi nal image? Is the image real or virtual? (c) If the two lenses are now placed in contact with each other and the object is 5.00 cm in front of this combination, where will the image be located? (See Problem 56.)

To work this problem, use the fact that the image formed by the fi rst surface becomes the object for the second surface. Figure P23.55 shows a piece of glass with index of refraction 1.50. The ends are hemispheres with radii 2.00 cm and 4.00 cm, and the centers of the hemispherical ends are separated by a distance of 8.00 cm. A point object is in air, 1.00 cm from the left end of the glass. Locate the image of the object due to refraction at the two spherical surfaces.

Consider two thin lenses, one of focal length f1 and the other of focal length f2, placed in contact with each other, as shown in Figure P23.56. Apply the thin-lens equation to each of these lenses and combine the results to show that this combination of lenses behaves like a thin lens having a focal length f given by 1/f _ 1/f1 _ 1/f2. Assume the thicknesses of the lenses can be ignored in comparison to the other distances involved.

An object 2.00 cm high is placed 40.0 cm to the left of a converging lens having a focal length of 30.0 cm. A diverging lens having a focal length of _20.0 cm is placed 110 cm to the right of the converging lens. (a) Determine the fi nal position and magnifi cation of the fi nal image. (b) Is the image upright or inverted? (c) Repeat parts (a) and (b) for the case where the second lens is a converging lens having a focal length of _20.0 cm.

A fl oating strawberry illusion can be produced by two parabolic mirrors, each with a focal length of 7.5 cm, facing each other so that their centers are 7.5 cm apart (Fig. P23.58). If a strawberry is placed on the bottom mirror, an image of the strawberry forms at the small opening at the center of the top mirror. Show that the fi nal image forms at that location and describe its characteristics. Note: A fl ashlight beam shone on these images has a very startling effect: Even at a glancing angle, the incoming light beam is seemingly refl ected off the images of the strawberry! Do you understand why?

Figure P23.59 shows a converging lens with radii R1 _ 9.00 cm and R2 _ _11.00 cm, in front of a concave spherical mirror of radius R _ 8.00 cm. The focal points (F1 and F2) for the thin lens and the center of curvature (C) of the mirror are also shown. (a) If the focal points F1 and F2 are 5.00 cm from the vertex of the thin lens, what is the index of refraction of the lens? (b) If the lens and mirror are 20.0 cm apart and an object is placed 8.00 cm to the left of the lens, what is the position of the fi nal image and its magnifi cation as seen by the eye in the fi gure? (c) Is the fi nal image inverted or upright? Explain.

Find the object distances (in terms of f ) for a thin converging lens of focal length f if (a) the image is real and the image distance is four times the focal length and (b) the image is virtual and the image distance is three times the focal length. (c) Calculate the magnifi cation of the lens for cases (a) and (b).

The lens-makers equation for a lens with index n1 immersed in a medium with index n2 takes the form 1 f 5 a n1 n2 2 1ba 1 R1 2 1 R2 b A thin diverging glass (index _ 1.50) lens with R1 _ _3.00 m and R2 _ _6.00 m is surrounded by air. An arrow is placed 10.0 m to the left of the lens. (a) Determine the position of the image. Repeat part (a) with the arrow and lens immersed in (b) water (index _ 1.33) (c) a medium with an index of refraction of 2.00. (d) How can a lens that is diverging in air be changed into a converging lens?

An observer to the right of the mirrorlens combination shown in Figure P23.62 sees two real images that are the same size and in the same location. One image is upright, and the other is inverted. Both images are 1.50 times larger than the object. The lens has a focal length of 10.0 cm. The lens and mirror are separated by 40.0 cm. Determine the focal length of the mirror. (Dont assume the fi gure is drawn to scale.)

The lens-makers equation applies to a lens immersed in a liquid if n in the equation is replaced by n1/n2. Here n1 refers to the refractive index of the lens material and n2 is that of the medium surrounding the lens. (a) A certain lens has focal length of 79.0 cm in air and a refractive index of 1.55. Find its focal length in water. (b) A certain mirror has focal length of 79.0 cm in air. Find its focal length in water.