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

Salamat A.

Numerade Educator

Suppose you stand in front of a flat mirror and focus a camera on your image. If the camera is in focus when set for a distance of 3.00 m, how far are you standing from the mirror?

Zachary W.

Numerade Educator

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 first 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.

Salamat A.

Numerade Educator

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 flat 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.

Zachary W.

Numerade Educator

A periscope (Fig. P23.5) is useful for viewing objects that cannot be seen directly. It can be used in submarines and when watching golf matches or parades from behind a crowd of people. Suppose the object is a distance $p_{1}$ from the upper mirror and the centers of the two flat mirrors are

separated by a distance h. (a) What is the distance of the final image from the lower mirror? (b) Is the final image real or virtual? (c) Is it upright or inverted? (d) What is its magnification? (e) Does it appear to be left–right reversed?

Salamat A.

Numerade Educator

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 mirror’s radius of curva ture and (b) the magnification of the image.

Zachary W.

Numerade Educator

A convex spherical mirror, whose focal length has a magnitude of 15.0 cm, is to form an image 10.0 cm behind the mirror. (a) Where should the object be placed? (b) What is the magnification of the mirror?

Salamat A.

Numerade Educator

To fit a contact lens to a patient’s eye, a keratometer can be used to measure the curvature of the cornea—the front surface of the eye. This instrument places an illuminated object of known size at a known distance p from the cornea, which then reflects some light from the object, forming an image of it. The magnification 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 field of view by a prism arrangement. Determine the radius of curvature of the cornea when p = 30.0 cm and M = 0.013 0.

Zachary W.

Numerade Educator

A virtual image is formed 20.0 cm from a concave mirror having a radius of curvature of 40.0 cm. (a) Find the position of the object. (b) What is the magnification of the mirror?

Salamat A.

Numerade Educator

While looking at her image in a cosmetic mirror, Dina notes that her face is highly magnified when she is close to the mirror, but as she backs away from the mirror, her image first 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?

Zachary W.

Numerade Educator

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?

Salamat A.

Numerade Educator

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 magnitude of the radius of curvature of the hubcap?

Zachary W.

Numerade Educator

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

Salamat A.

Numerade Educator

A 1.80 - m - tall person stands 9.00 m in front of a large, concave spherical mirror having a radius of curvature of 5.00 m. Determine (a) the mirror’s focal length, (b) the image distance, and (c) the magnification. (d) Is the image real or virtual? (e) Is the image upright or inverted?

Zachary W.

Numerade Educator

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 chin’s actual size?

Salamat A.

Numerade Educator

When an object is placed 40.0 cm in front of a convex spherical mirror, a virtual image forms 15.0 cm behind the mirror. Determine (a) the mirror’s focal length and (b) the magnification.

Zachary W.

Numerade Educator

At an intersection of hospital hallways, a convex spherical mirror is mounted high on a wall to help people avoid collisions. The magnitude of the mirror’s radius of curvature is 0.550 m. (a) Locate the image of a patient located 10.0 m from the mirror. (b) Indicate whether the image is upright or inverted. (c) Determine the magnification of the image.

Salamat A.

Numerade Educator

The mirror of a solar cooker focuses the Sun’s rays on a point 25.0 cm in front of the mirror. What is the mirror’s radius?

Zachary W.

Numerade Educator

A spherical mirror is to be used to form an image, five 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 object be positioned relative to the mirror?

Salamat A.

Numerade Educator

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 ball’s image in the mirror. (b) At what time do the ball and its image coincide?

Zachary W.

Numerade Educator

A cubical block of ice 50.0 cm on an edge is placed on a level floor 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.

Salamat A.

Numerade Educator

A goldfish is swimming inside a spherical bowl of water having an index of refraction $n=1.333$ . Suppose the goldfish is $p=10.0 \mathrm{cm}$ from the wall of a bowl of radius $|R|=15.0 \mathrm{cm},$ as in Figure P23.22. Neglecting the refraction of light caused by the wall of the bowl, determine the apparent distance of the goldfish from the wall according to an observer outside the bowl.

Zachary W.

Numerade Educator

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 flat 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?

Salamat A.

Numerade Educator

The top of a swimming pool is at ground level. If the pool is 2.00 m deep, how far below ground level does the bottom of the pool appear to be located when (a) the pool is completely filled with water? (b) When it is filled halfway with water?

Zachary W.

Numerade Educator

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?

Salamat A.

Numerade Educator

A man inside a spherical diving bell watches a fish through a window in the bell, as in Figure P23.26. If the diving bell has radius R 5 1.75 m and the fish is a distance p 5 1.00 m from the window, calculate (a) the image distance and (b) the magnification. Neglect the thickness of the window.

Zachary W.

Numerade Educator

A jellyfish is floating in a water - filled aquarium 1.00 m behind a flat pane of glass 6.00 cm thick and having an index of refraction of 1.50. (a) Where is the image of the jellyfish 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)?

Salamat A.

Numerade Educator

Figure P23.28 shows a curved surface separating a material with index of refraction $n_{1}$ from a material with index $n_{2}$ . The surface forms an image $I$ of object $O$ . The ray shown in red passes through the surface along a radial line. Its angles of incidence and refraction are both zero, so its direction does not change at the surface. For the ray shown in blue, the direction changes according to $n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2} .$ For paraxial rays, we assume $\theta_{1}$ and $\theta_{2}$ are small, so we may write $n_{1}$ tan $\theta_{1}=n_{2} \tan \theta_{2}$ The magnification is defined as $M=h^{\prime} / h .$ Prove that the magnification is given by $M=-n_{1} q / n_{2} p .$

Zachary W.

Numerade Educator

A contact lens is made of plastic with an index of refraction of 1.50. The lens has an outer radius of curvature of 12.00 cm and an inner radius of curvature of 12.50 cm. What is the focal length of the lens?

Salamat A.

Numerade Educator

A thin plastic lens with index of refraction $n=1.67$ has radii of curvature given by $R_{1}=-12.0 \mathrm{cm}$ and $R_{2}=40.0 \mathrm{cm} .$ Determine (a) the focal length of the lens, (b) whether the lens is converging or diverging, and the image distances for object distances of (c) infinity, (d) 5.00 cm, and (e) 50.0 cm.

Zachary W.

Numerade Educator

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 find the magnification.

Salamat A.

Numerade Educator

An object is placed 20.0 cm from a concave spherical mirror having a focal length of magnitude 40.0 cm. (a) Use graph paper to construct an accurate ray diagram for this situation. (b) From your ray diagram, determine the location of the image. (c) What is the magnification of the image? (d) Check your answers to parts (b) and (c) using the mirror equation.

Zachary W.

Numerade Educator

A diverging lens has a focal length of magnitude 20.0 cm. (a) Locate the images for object distances of (i) 40.0 cm, (ii) 20.0 cm, and (iii) 10.0 cm. For each case, state whether the image is (b) real or virtual and (c) upright or inverted. (d) For each case, find the magnification.

Salamat A.

Numerade Educator

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 magnification, 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?

Zachary W.

Numerade Educator

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

Salamat A.

Numerade Educator

The nickel’s 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.

Zachary W.

Numerade Educator

An object of height 8.00 cm is placed 25.0 cm to the left of a converging lens with a focal length of 10.0 cm. Determine (a) the image location, (b) the magnification, and (c) the image height. (d) Is the image real or virtual? (e) Is the image upright or inverted?

Salamat A.

Numerade Educator

An object is located 20.0 cm to the left of a diverging lens having a focal length f = 232.0 cm. Determine (a) the location and (b) the magnification of the image. (c) Construct a ray diagram for this arrangement.

Zachary W.

Numerade Educator

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.

Salamat A.

Numerade Educator

(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?

Zachary W.

Numerade Educator

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 first lens. Where is the final image formed, and what is the magnification of the system?

Salamat A.

Numerade Educator

A converging lens is placed at $x=0,$ a distance $d=10.0 \mathrm{cm}$ to the left of a diverging lens as in Figure $\mathrm{P} 23.42$ (where $F_{C}$ and $F_{D}$ locate the focal points for the converging and the diverging lens, respectively). An object is located at $x=-2.00 \mathrm{cm}$ to the left of the converging lens and the focal lengths of the converging and diverging lenses are 4.00 cm and 28.00 cm,

respectively. (a) Determine the x - location of the final image and (b) determine its overall magnification.

Zachary W.

Numerade Educator

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 216.00 cm is 6.00 cm to the right of the converging lens. Find the position and height of the final image. Is the image inverted or upright? Real or virtual?

Salamat A.

Numerade Educator

Two converging lenses having focal lengths of $f_{1}=10.0 \mathrm{cm}$ and $f_{2}=20.0 \mathrm{cm}$ are placed $d=50.0 \mathrm{cm}$ apart, as shown in Figure P23.44. The final image is to be located between the lenses, at the position x 5 31.0 cm indicated. (a) How far to the left of the first lens should the object be positioned? (b) What is the overall magnification of the system? (c) Is the final

image upright or inverted?

Zachary W.

Numerade Educator

Lens $L_{1}$ in Figure $P 23.45$ has a focal length of 15.0 $\mathrm{cm}$ and is located a fixed distance in front of the film plane of a camera. Lens $L_{2}$ has a focal length of $13.0 \mathrm{cm},$ and its distance d from the film 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 film.

Salamat A.

Numerade Educator

An object is placed 15.0 cm from a first 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 first converging lens. (a) Find the position $q_{1}$ of the image formed by the first converging lens. (b) How far from the second lens is the image of the first lens? (c) What is the value of $p_{2},$ the object position for the second lens? (d) Find the position $q_{2}$ of the image formed by the second lens. (e) Calculate the magnification of the first lens. (f) Calculate the magnification of the second lens. (g) What is the total magnification for the system? (h) Is the final image real or virtual? Is it upright or inverted (compared to the original object for the lens system)?

Zachary W.

Numerade Educator

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 final image real or virtual?

Salamat A.

Numerade Educator

A real object’s distance from a converging lens is five times the focal length. (a) Determine the location of the image q in terms of the focal length f. (b) Find the magnification 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?

Zachary W.

Numerade Educator

The magnitudes of the radii of curvature are 32.5 cm and 42.5 cm for the two faces of a biconcave lens. The glass has index of refraction 1.53 for violet light and 1.51 for red light. For a very distant object, locate (a) the image formed by violet light and (b) the image formed by red light.

Salamat A.

Numerade Educator

A diverging lens (n 5 1.50) is shaped like that in Figure 23.25c. The radius of the first 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) infinity, (c) $3|f|,(\mathrm{d})|f|,$ and $(\mathrm{e})|f| / 2$

Zachary W.

Numerade Educator

The lens and the mirror in Figure P23.51 are separated by 1.00 m and have focal lengths of 180.0 cm and 250.0 cm, respectively. If an object is placed 1.00 m to the left of the lens, where will the final image be located? State whether the image is upright or inverted, and determine the overall magnification.

Salamat A.

Numerade Educator

The object in Figure $P 23.52$ is mid- way between the lens and the mirror, which are separated by a distance $d=$ 25.0 $\mathrm{cm} .$ The magnitude of the mirror’s radius of curvature is 20.0 cm,

and the lens has a focal length of 216.7 cm. (a) Considering only the light that leaves the object and travels first toward the mirror, locate the final image formed by this system. (b) Is the image real or virtual? (c) Is it upright or inverted? (d) What is the overall magnification of the image?

Zachary W.

Numerade Educator

A parallel beam of light enters a glass hemisphere perpendicular to the flat face, as shown in Figure $P 23.53$ . The radius of the hemisphere is $R=6.00 \mathrm{cm},$ and the index of refraction is $n=1.56 .$ Determine the point at which the beam is focused. (Assume paraxial rays; i.e., assume all rays are located close to the principal axis.)

Salamat A.

Numerade Educator

Two rays traveling parallel to the principal axis strike a large plano - convex lens having a refractive index of 1.60 (Fig. P23.54). If the convex face is spherical, a ray near the edge does not pass through the focal point (spherical aberration occurs). Assume this face has a radius of curvature of $R=20.0 \mathrm{cm}$ and the two rays are at distances $h_{1}=0.500 \mathrm{cm}$ and $h_{2}=12.0 \mathrm{cm}$ from the principal axis. Find the difference $\Delta x$ in the positions where each crosses the principal axis.

Zachary W.

Numerade Educator

To work this problem, use the fact that the image formed by the first surface becomes the object for the second surface. Figure $\mathrm{P} 23.55$ shows a piece of glass with index of refraction $n=$ 1.50 surrounded by air. The ends are hemispheres with radii $R_{1}=2.00 \mathrm{cm}$ and $R_{2}=4.00 \mathrm{cm},$ and the centers of the hemispherical ends are separated by a distance of $d=8.00 \mathrm{cm} .$ A point object is in air, a distance $p=1.00 \mathrm{cm}$ from the left end of the glass. (a) Locate the image of the object due to refraction at the two spherical surfaces. (b) Is the image real or virtual?

Salamat A.

Numerade Educator

Consider two thin lenses, one of focal length $f_{1}$ and the other of focal length $f_{2},$ 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 $1 / f=1 / f_{1}+1 / f_{2}$ Assume the thicknesses of the lenses can be ignored in comparison to the other distances involved.

Zachary W.

Numerade Educator

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 220.0 cm is placed 110 cm to the right of the converging lens. (a) Determine the final position and magnification of the final image. (b) Is the image upright

or inverted? (c) Repeat parts (a) and (b) for the case in which the second lens is a converging lens having a focal length of 120.0 cm.

Salamat A.

Numerade Educator

A “floating 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 final image forms at that location and describe its characteristics. Note: A flashlight beam shone on these images has a very startling effect: Even at a glancing angle, the incoming light beam is seemingly reflected off the images of the strawberry! Do you understand why?

Zachary W.

Numerade Educator

Figure $\mathrm{P} 23.59$ shows a converging lens with radii $R_{1}=9.00 \mathrm{cm}$ and $R_{2}=-11.00 \mathrm{cm},$ in front of a concave spherical mirror of radius $R=8.00 \mathrm{cm} .$ The focal points $\left(F_{1} \text { and } F_{2}\right)$ for the thin lens and the center of curvature (C ) of the mirror are also shown. (a) If the focal points $F_{1}$ and $F_{2}$ are 5.00 $\mathrm{cm}$ 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 final image and its magnification as seen by the eye in the figure? (c) Is the final image inverted or upright? Explain.

Salamat A.

Numerade Educator

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 absolute value of the image distance is three times the focal length. (c) Calculate the magnification of the lens for cases (a) and (b).

Zachary W.

Numerade Educator

The lens-maker's equation for a lens with index $n_{1}$ immersed in a medium with index $n_{2}$ takes the form

$$\frac{1}{f}=\left(\frac{n_{1}}{n_{2}}-1\right)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)$$

A thin diverging glass (index $=1.50 )$ lens with $R_{1}=-3.00 \mathrm{m}$ and $R_{2}=-6.00 \mathrm{m}$ 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 $=1.33 )$ and (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?

Salamat A.

Numerade Educator

An observer to the right of the mirror–lens 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. (Don’t assume the figure is drawn to scale.)

Zachary W.

Numerade Educator

The lens-maker's equation applies to a lens immersed in a liquid if $n$ in the equation is replaced by $n_{1} / n_{2} .$ Here $n_{1}$ refers to the refractive index of the lens material and $n_{2}$ 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.

Salamat A.

Numerade Educator

A certain Christmas tree ornament is a silver sphere having a diameter of 8.50 cm. (a) If the size of an image created by reflection in the ornament is three - fourth’s the reflected object’s actual size, determine the object’s location. (b) Use a principal - ray diagram to determine whether the image is

upright or inverted.

Zachary W.

Numerade Educator

A glass sphere $(n=1.50)$ with a radius of 15.0 $\mathrm{cm}$ has a tiny air bubble 5.00 cm above its center. The sphere is viewed looking down along the extended radius containing the bubble. What is the apparent depth of the bubble below the surface of the sphere?

Salamat A.

Numerade Educator

An object 10.0 cm tall is placed at the zero mark of a meterstick. A spherical mirror located at some point on the meterstick creates an image of the object that is upright, 4.00 cm tall, and located at the 42.0 - cm mark of the meterstick. (a) Is the mirror convex or concave? (b) Where is the mirror? (c) What is the mirror’s focal length?

Zachary W.

Numerade Educator