Object O1 is 15.0 cm to the left of a converging lens with

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 that you specify.

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.

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 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 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.]

At an intersection of hospital hallways, a convex mirror is mounted high on a wall to help people avoid collisions. The mirror has a radius of curvature of 0.550m. Locate and describe the image of a patient 10.0m from the mirror. Determine the magnication of the image.

To 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 reects some light from the object, forming an image of it. The magnication 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 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.

A concave spherical mirror has a radius of curvature of 20.0 cm. Locate the images for object distances of (a) 40.0 cm, (b) 20.0 cm, and (c) 10.0 cm. In each case, state whether the image is real or virtual and upright or inverted, and nd the magnication.

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 magnication of the image.

A large church has a niche in one wall. On the oor plan it appears as a semicircular indentation of radius 2.50 m. A worshiper stands on the centerline of the niche, 2.00 m out from its deepest point, and whispers a prayer. Where is the sound concentrated after it reects from the back wall of the niche?"

While looking at her image in a cosmetic mirror, Dina notes that her face is highly magnied when she is close to the mirror, but as she backs away from the mirror, her image 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 magnied by a factor of two. What is the radius of curvature of the mirror?

A certain Christmas tree ornament is a silver sphere having a diameter of 8.50 cm. Determine an object location for which the size of the reected image is three-fourths the size of the object. Use a principal-ray diagram to arrive at a description of the image.

A man standing 1.52m in front of a shaving mirror produces an inverted image 18.0cm 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 spherical mirror with a radius of curvature of 10.0 cm produces a virtual image one-third the size of the real object. Where is the object?

A child holds a candy bar 10.0 cm in front of a convex mirror and notices that the image is only one-half the size of the candy bar. What is the radius of curvature of the mirror?

It is observed that the size of a real image formed by a concave mirror is four times the size of the object when the object is 30.0 cm in front of the mirror. What is the radius of curvature of this mirror?

A spherical mirror is to be used to form an image, 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 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.

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 lled with water? (b) the pool is lled halfway with water?

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 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?

A 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 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 goldsh is swimming at 2.00 cm/s toward the front wall of a rectangular aquarium. What is the apparent speed of the 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 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 nd the magnication.

Where must an object be placed to have unit magnication (M  1.00) (a) for a converging lens of focal length 12.0 cm? (b) for a diverging lens of focal length 12.0 cm?

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 nd the magnication.

The use of a lens in a certain situation is described by the equation Determine (a) the object distance and (b) the image distance. (c) Use a ray diagram to obtain a description of the image. (d) Identify a practical device described by the given equation, and write the statement of a problem having a solution that contains the equation.

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 P23.34 has twice the diameter of the nickel when the lens is 2.84 cm from the nickel. Determine the focal length of the lens.

A certain LCD projector contains a single thin lens. An object 24.0 mm high is to be projected so that its image 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 in order to form the image on the screen?

A person uses a converging lens that has a focal length of 12.5 cm to inspect a gem. The lens forms a virtual image 30.0 cm away. Determine the magnication. Is the image upright or inverted?

A diverging lens is to be used to produce a virtual image one-third as tall as the object. Where should the object be placed?

An object is 5.00 m to the left of a at screen. A converging lens for which the focal length is f  0.800 m is placed between object and screen. (a) Show that there are two lens positions that form an image on the screen, and determine how far these positions are from the object. (b) How do the two images differ from each other?

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.

An object is placed 20.0 cm to the left of a converging lens of focal length 25.0 cm. A diverging lens of focal length 10.0cm is 25.0cm to the right of the converging lens. Find the position and magnication of the nal image.

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 rst lens. Where is the nal image formed, and what is the magnication 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 rst lens and is observed to form a 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 magnication of this system? (c) What is the nature (i.e., real or virtual, upright or inverted) of the 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 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 nal image is to be located between the lenses, at the position indicated. (a) How far to the left of the rst lens should the object be positioned? (b) What is the overall magnication of the system? (c) Is the nal image upright or inverted?

Lens L1 in Figure P23.45 has a focal length of 15.0 cm and is located a xed distance in front of the lm plane of a camera. Lens L2 has a focal length of 13.0 cm, and its distance d from the 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 lm.

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 P23.46. 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 that the thicknesses of the lenses can be ignored in comparison to the other distances involved.

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 nal image real or virtual?

An object is placed 12 cm to the left of a diverging lens of focal length 6.0 cm. A converging lens of focal length 12 cm is placed a distance d to the right of the diverging lens. Find the distance d that places the nal image at innity.

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

The object in Figure P23.50 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 rst towards the mirror, locate the nal image formed by this system. Is the image real or virtual? Is it upright or inverted? What is the overall magnication of the image?

The lens and 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, locate the nal image. State whether the image is upright or inverted, and determine the overall magnication.

A diverging lens (n  1.50) is shaped like that in Active Figure 23.25c. The radius of the 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) innity, (c) 3 f , (d) f , and (e)  f /2.

A parallel beam of light enters a glass hemisphere perpendicular to the 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 raysthat is, assume that 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 nal image of an object placed 40.0 cm in front of the rst lens. (b) If the height of the object is 2.00 cm, what is the height of the 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 46.)

To work this problem, use the fact that the image formed by the 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.

In a darkened room, a burning candle is placed 1.50 m from a white wall. A lens is placed between candle and wall at a location that causes a larger, inverted image to form on the wall. When the lens is moved 90.0 cm toward the wall, another image of the candle is formed. Find (a) the two object distances that produce the images just described and (b) the focal length of the lens. (c) Characterize the second image.

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 nal position and magnication of the 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 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 nal image forms at that location, and describe its characteristics. [Note: A ashlight beam shone on these images has a very startling effect: Even at a glancing angle, the incoming light beam is seemingly reected 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, determine 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, determine the position of the nal image and its magnication as seen by the eye in the gure. (c) Is the 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; (b) the image is virtual and the image distance is three times the focal length. (c) Calculate the magnication 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 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 that the 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.

This experiment will enable you to examine some of the properties of images formed by at mirrors. As shown in Figure A23.1, place a clear plastic sheet between two small candles of the same height. Light one candle placed about 6 inches from the sheet and observe the reected ame from the front side. Move the unlit candle until it also appears to be lit. The candles should be approximately equidistant from the sheet at this point. Why?When you place your nger on the wick of the unlit candle, you will get the illusion that your nger is burning. Explain your observation. A similar experiment can be performed with a at mirror and two pencils. Hold one of the pencils in front of the mirror at a distance of about 10 inches from the mirror and at a position such that about half of the image appears in the mirror. Move the second pencil back and forth behind the mirror until it appears to align perfectly with the image in the mirror. When the two pencils are aligned, measure the distance from the mirror to the location of the second pencil. This pencil is situated at the apparent position of the virtual image. Why? The two pencils should be at equal distances from the mirror. Why?

Fill a clear glass tumbler with water and place a pencil or straw into the tumbler as in Figure A23.2. Now observe the pencil from the side at an angle of about 45 to the surface, and note that the line of the portion of the pencil under water is not parallel with the line of the portion in air. That is, the pencil appears to be bent at the point where it enters the water. Use the techniques of Section 23.4 to explain this observation.

View yourself in a full-length mirror. Stand close to the mirror, and place one piece of tape at the top of the image of your head and another piece at the very bottom of the image of your feet. Now step back a few meters and observe your image. How big is it relative to the original size? How does the distance between the pieces of tape compare with your actual height? Move to a position in front of the mirror such that you can see a full image of yourself with the top of your head just level with the top of the mirror. Have a friend gradually block off the lower portion of the mirror with a sheet or newspaper page until you can see your complete imagebut no more! Measure the length of the mirror and compare this measurement with your height. How do the two compare?

Compare the images formed of your face when you look rst at the front side and then at the back side of a shiny soupspoon. For each side, observe the change in the image of your face as you move closer to and farther away from the spoon.

Draw a sketch of you and your image in a plane mirror when you are walking away from it with velocity . Indicate on your diagram the velocity vectors for you and your image. Keep the relative lengths of the two velocities as you believe they will be. Explain your answer. (b) Repeat the procedure in part (a), but now do it as you are walking towards the mirror. Explain why you have the vectors drawn as you do.