Consider three lenses with focal lengths of 25.0 cm, –15.0

Problem 113IP IP Referring to Example 27–3 Suppose a person’s near-point distance is 67.0 cm. (a) Is the refractive power of the eyeglasses that allow this person to focus on an object just 25.0 cm from the eye greater than or less than 2.53 diopters, which is the refractive power when the near-point distance is 57.0 cm? The glasses are worn 2.00 cm in front of the eyes. (b) Find the required refractive power for this person’s eyeglasses.

Problem 1CQ Why is it restful to your eyes to gaze off into the distance?

Problem 2P Your friend is 1.9 m tall. (a) When she stands 3.2 m from you, what is the height of her image formed on the retina of your eye? (Consider the eye to consist of a thin lens 2.5 cm from the retina.) (b) What is the height of her image when she is 4.2 m from you?

Problem 2CQ If a lens is cut in half through a plane perpendicular to its surface, does it show only half an image?

Problem 1P · CEPredict/Explain BIO Octopus Eyes To focus its eyes, an octopus does not change the shape of its lens, as is the case in humans. Instead, an octopus moves its rigid lens back and forth, as in a camera. This changes the distance from the lens to the retina and brings an object into focus. (a) If an object moves closer to an octopus, must the octopus move its lens closer to or farther from its retina to keep the object in focus? (b) Choose the best explanation from among the following: I. The lens must move closer to the retina—that is, farther away from the object—to compensate for the object moving closer to the eye. II. When the object moves closer to the eye, the image produced by the lens will be farther behind the lens; therefore, the lens must move farther from the retina.

Problem 3CQ If your near-point distance is N, how close can you stand to a mirror and still be able to focus on your image?

Problem 4CQ When you open your eyes underwater, everything looks blurry. Can this be thought of as an extreme case of nearsightedness or farsightedness? Explain.

Problem 3P Which forms the larger image on the retina of your eye: a 43-ft tree seen from a distance of 210 ft, or a 12-in. flower viewed from a distance of 2.0 ft?

Problem 4P Approximating the eye as a single thin lens 2.60 cm from the retina, find the eye’s near-point distance if the smallest focal length the eye can produce is 2.20 cm.

Problem 5CQ Would you benefit more from a magnifying glass if your near point distance is 25 cm or if it is 15 cm? Explain.

Problem 5P Referring to Problem 4, what is the focal length of the eye when it is focused on an object at a distance of (a) 285 cm and (b) 28.5 cm?

Problem 6CQ When you use a simple magnifying glass, does it matter whether you hold the object to be examined closer to the lens than its focal length or farther away? Explain.

Problem 6P Four camera lenses have the following focal lengths and f-numbers: Lens Focal length (mm) f-number A 150 f/1.2 B 150 f/5.6 C 35 f/1.2 D 35 f/5.6 Rank these lenses in order of increasing aperture diameter. Indicate ties where appropriate.

Problem 7CQ Is the final image produced by a telescope real or virtual? Explain.

Problem 7P BIO The focal length of the human eye is approximately 1.7 cm. (a) What is the f-number for the human eye in bright light, when the pupil diameter is 2.0 mm? (b) What is the f-number in dim light, when the pupil diameter has expanded to 7.0 mm?

Problem 8P IP A camera with a 55-mm-focal-length lens has aperture settings of 2.8, 4, 8, 11, and 16. (a) Which setting has the largest aperture diameter? (b) Calculate the five possible aperture diameters for this camera.

Problem 8CQ Does chromatic aberration occur in mirrors? Explain.

Problem 9P The actual frame size of “35-mm” film is 24 mm × 36 mm. You want to take a photograph of your friend, who is 1.9 m tall. Your camera has a 55-mm-focal-length lens. How far from the camera should your friend stand in order to produce a 36-mm- tall image on the film?

Problem 10P To completely fill a frame of “35-mm” film, the image produced by a camera must be 36 mm high. If a camera has a focal length of 150 mm, how far away must a 2.0-m-tall person stand to produce an image that fills the frame?

Problem 11P · · You are taking a photograph of a poster on the wall of your dorm room, so you can’t back away any farther than 3.0 m to take the shot. Tire poster is 0.80 m wide and 1.2 m tall, and you want the image to fit in the 24-mm × 36-mm frame of the film in your camera. What is the longest focal length lens that will work?

Problem 12P A photograph is properly exposed when the aperture is set to f/8 and the shutter speed is 125. Find the approximate shutter speed needed to give the same exposure if the aperture is changed to f/2.4.

Problem 13P You are taking pictures of the beach at sunset. Just before the Sun sets, a shutter speed of f/11 produces a properly exposed picture. Shortly after the Sun sets, however, your light meter indicates that the scene is only one-quarter as bright as before. (a) If you don't change the aperture, what approximate shutter speed is needed for your second shot? (b) If, instead, you keep the shutter speed at 1/100 s, what approximate f-stop will be needed for the second shot?

Problem 14P · · IP You are taking a photograph of a horse race. A shutter speed of 125 at f/5.6 produces a properly exposed image, but the running horses give a blurred image. Your camera has f-stops of 2, 2.8, 4, 5.6, 8, 11, and 16. (a) To use the shortest possible exposure time (i.e., highest shutter speed), which f-stop should you use? (b) What is the shortest exposure time you can use and still get a properly exposed image?

Problem 15P The Hale Telescope The 200-in. (5.08-m) diameter mirror of the Hale telescope on Mount Palomar has a focal length ƒ = 16.9 m. (a) When the detector is placed at the focal point of the mirror (the “prime focus”), what is the f-ratio for this telescope? (b) The coudé focus arrangement uses additional mirrors to bend the light path and increase the effective focal length to 155.4 m. What is the f-ratio of the telescope when the coudé focus is being used? (Coudé is French for “elbow,” since the light path is “bent like an elbow.” This arrangement is useful when the light needs to be focused onto a distant instrument.)

Problem 16P · CEPredict/Explain Two professors arc stranded on a deserted island. Both wear glasses, though one is nearsighted and the other is farsighted. (a) Which person’s glasses should be used to focus the rays of the Sun and start a fire? (b) Choose the best explanation from among the following: I. A nearsighted person can focus close, so that person’s glasses should be used to focus the sunlight on a piece of moss at a distance of a couple inches. II. A farsighted person can’t focus close, so the glasses to correct that person’s vision are converging. A converging lens is what you need to concentrate the rays of the Sun.

Problem 17P · CE A clerk at the local grocery store wears glasses that make her eyes look larger than they actually are. Is the clerk nearsighted or farsighted? Explain.

Problem 18P CE The umpire at a baseball game wears glasses that make his eyes look smaller than they actually are. Is the umpire nearsighted or farsighted? Explain.

Construct a ray diagram for Active Example 27–2.C

Problem 20P The cornea of a normal human eye has an optical power of +43.0 diopters. What is its focal length?

Problem 21P A myopic student is shaving without his glasses. If Iris eyes have a far point of 1.6 m, what is the greatest distance he can stand from the mirror and still see his image clearly?

Problem 22P An eyeglass prescription calls for a lens with an optical power of +2.7 diopters. What is the focal length of this lens?

Problem 23P Two thin lenses, with f1 = +25.0 cm and f2 = —42.5 cm, are placed hr contact. What is the focal length of this combination?

Problem 24P Two thin lenses have refractive powers of +4.00 diopters and —2.35 diopters. What is the refractive power of the two if they are placed in contact? (Note that these are the same two lenses described in the previous problem.)

Problem 25P Two concave lenses, each with ƒ = —12 cm, are separated by 6.0 cm. An object is placed 24 cm in front of one of the lenses. Find (a) the location and (b) the magnification of the final image produced by this lens combination.

Problem 26P IP BIO The focal length of a relaxed human eye is approximately 1.7 cm. When we focus our eyes on a close-up object, we can change the refractive power of the eye by about 16 diopters. (a) Does the refractive power of our eyes increase or decrease by 16 diopters when we focus closely? Explain. (b) Calculate the focal length of the eye when we focus closely.

Problem 27P IP BIO Diopter Change in Diving Cormorants Double- crested cormorants (Phalacrocorax auritus) are extraordinary birds—they can focus on objects in the air, just like we can, but they can also focus underwater as they pursue their prey. To do so, they have one of the largest accommodation ranges in nature— that is, they can change the focal length of their eyes by amounts that are greater than is possible in other animals. When a cormorant plunges into the ocean to catch a fish, it can change the refractive power of its eyes by about 45 diopters, as compared to only 16 diopters of change possible in the human eye. (a) Should this change of 45 diopters be an increase or a decrease? Explain. (b) If the focal length of the cormorant’s eyes is 4.2 mm before it enters the water, what is the focal length after the refractive power changes by 45 diopters?

Problem 29P Repeat Problem 28, this time with the coin placed 18.0 cm to the right of the diverging lens.

Problem 30P Find the focal length of contact lenses that would allow a farsighted person with a near-point distance of 176 cm to read a book at a distance of 10.1 cm.

Problem 31P Find the focal length of contact lenses that would allow a nearsighted person with a 135-cm far point to focus on the stars at night.

Problem 28P A converging lens of focal length 8.000 cm is 20.0 cm to the left of a diverging lens of focal length –6.00 cm. A coin is placed 12.0 cm to the left of the converging lens. Find (a) the location and (b) the magnification of the coin’s final image.

Problem 32P What focal length should a pair of contact lenses have if they are to correct the vision of a person with a near point of 56 cm?

Problem 33P A nearsighted person wears contacts with a focal length of –8.5 cm. Tf this person’s far-point distance with her contacts is 8.5 m, what is her uncorrected far-point distance?

Problem 35P A person whose near-point distance is 49 cm wears a pair of glasses that are 2.0 cm from her eyes. With the aid of these glasses, she can now focus on objects 25 cm away from her eyes. Find the focal length and refractive power of her glasses.

Problem 34P Without Iris glasses, Isaac can see objects clearly only if they are less than 4.5 m from his eyes. What focal length glasses worn 2.1 cm from his eyes will allow Isaac to see distant objects clearly?

Problem 37P IP Your favorite aunt can read a newspaper only if it is within 15.0 cm of her eyes. (a) Is your aunt nearsighted or farsighted? Explain. (b) Should your aunt wear glasses that are converging or diverging to improve her vision? Explain. (c) How many diopters of refractive power must her glasses have if they are worn 2.00 cm from the eyes and allow her to read a newspaper at a distance of 25.0 cm?

Problem 36P A pair of eyeglasses is designed to allow a person with a far- point distance of 2.50 m to read a road sign at a distance of 25.0 m. Find the focal length required of these glasses if they are to be worn (a) 2.00 cm or (b) 1.00 cm from the eyes.

Problem 38P IP The relaxed eyes of a patient have a refractive power of 48.5 diopters. (a) Is this patient nearsighted or farsighted? Explain. (b) If this patient is nearsighted, find the far point. If this person is farsighted, find the near point. (For the purposes of this problem, treat the eye as a single-lens system, with the retina 2.40 cm from the lens.)

Problem 39P IP You are comfortably reading a book at a distance of 24 cm. (a) What is the refractive power of your eyes? (b) Does the refractive power of your eyes increase or decrease when you move the book farther away? Explain. (For the purposes of this problem, treat the eye as a single-lens system, with the retina 2.40 cm from the lens.)

Problem 40P Without glasses, your Uncle Albert can sec things clearly only if they are between 25 cm and 170 cm from his eyes. (a) What power eyeglass lens will correct your uncle’s myopia? Assume the lenses will sit 2.0 cm from his eyes. (b) What is your uncle’s near point when wearing these glasses?

Problem 43P IP With unaided vision, a librarian can focus only on objects that lie at distances between 5.0 m and 0.50 m. (a) Which type of lens (converging or diverging) is needed to correct his nearsightedness? Explain. (b) Which type of lens will correct his farsightedness? Explain. (c) Find the refractive power needed for each part of the bifocal eyeglass lenses that will give the librarian normal visual acuity from 25 cm out to infinity. (Assume the lenses rest 2.0 cm from his eyes.)

Problem 44P IP With unaided vision, a physician can focus only on objects that lie at distances between 5.0 m and 0.50 m. (a) Which type of lens (converging or diverging) is needed to correct her nearsightedness? Explain. (b) Which type of lens will correct her farsightedness? Explain. (c) Find the refractive power needed for each part of the bifocal contact lenses that will give the physician normal visual acuity from 25 cm out to infinity.

Problem 42P A simple camera telephoto lens consists of two lenses. The objective lens has a focal length f1 = +39.0 cm. Precisely 36.0 cm behind this lens is a concave lens with a focal length f2 = –10.0 cm. The object to be photographed is 4.00 m in front of the objective lens. (a) How far behind the concave lens should the film be placed? (b) What is the linear magnification of this lens combination?

Problem 41P A 2.05-cm-tall object is placed 30.0 cm to the left of a converging lens with a focal length f1 = 20.5 cm. A diverging lens, with a focal length f2 = –42.5 cm, is placed 30.0 cm to the right of the first lens. How tall is the final image of the object?

Problem 45P A person’s prescription for her new bifocal glasses calls for a refractive power of –0.445 diopter in the distance-vision part, and a power of +1.85 diopters in the close-vision part. What are the near and far points of this person’s uncorrected vision? Assume the glasses are 2.00 cm from the person’s eyes, and that the person’s near-point distance is 25.0 cm when wearing the glasses.

Problem 46P A person’s prescription for his new bifocal eyeglasses calls for a refractive power of –0.0625 diopter in the distance-vision part and a power of +1.05 diopters in the close-vision part. Assuming the glasses rest 2.00 cm from Iris eyes and that the corrected near-point distance is 25.0 cm, determine the near and far points of this person’s uncorrected vision.

Two lenses, with \(f_{1}=+20.0 \mathrm{~cm} \text { and } f_{2}=+30.0 \mathrm{~cm}\), are placed on the axis, as shown in Figure An object is fixed \(50.0 \mathrm{~cm}\) to the left of lens 1 , and lens 2 is a variable distance to the right of lens 1. Find the lateral magnification and location of the final image relative to lens 2 for the following cases: (a) \(x=115 \mathrm{~cm}\); b) \(x=30.0 \mathrm{~cm}\); (c) \(x=0\) (d) Show that your result for part (c) agrees with the relation for the effective focal length of two lenses in contact, \(1 / f_{\text {eff }}=1 / f_{1}+1 / f_{2}\) Equation Transcription: Text Transcription: f1=+20.0 cm and f2=+30.0 cm 50.0 cm x=115 cm x=30.0 cm x=0 1/feff =1/f1 +1/f2

Problem 48P A converging lens with a focal length of 4.0 cm is to the left of a second identical lens. When a feather is placed 12 cm to the left of the first lens, the final image is the same size and orientation as the feather itself. What is the separation between the lenses?

Problem 50P A magnifying glass is a single convex lens with a focal length of ƒ = +14.0 cm. (a) What is the angular magnification when this lens forms a (virtual) image at —?? How far from the object should the lens be held? (b) What is the angular magnification when this lens forms a (virtual) image at the person’s near point (assumed to be 25 cm)? How far from the object should the lens be held in this case?

Problem 51P IP A student has two lenses, one of focal length f1 = 5.0 cm and the other with focal length f2 = 13 cm. (a) When used as a simple magnifier, which of these lenses can produce the greater magnification? Explain, (b) Find the maximum magnification produced by each of these lenses.

Problem 52P A beetle 4.73 mm long is examined with a simple magnifier of focal length f = 10.1 cm. If the observer’s eye is relaxed while using the magnifier, and has a near-point distance of 25.0 cm, what is the apparent length of the beetle?

Problem 49P The Moon is 3476 km in diameter and orbits the Earth at an average distance of 384,400 km. (a) What is the angular size of the Moon as seen from Earth? (b) A penny is 19 mm in diameter. How far from your eye should the penny be held to produce the same angular diameter as the Moon?

Problem 53P To engrave wishes of good luck on a watch, an engraver uses a magnifier whose focal length is 8.65 cm. If the image formed by the magnifier is at the engraver’s near point of 25.6 cm, find (a) the distance between the watch and the magnifier and (b) the angular magnification of the engraving. Assume the magnifying glass is directly in front of the engraver’s eyes.

Problem 54P A jeweler examines a diamond with a magnifying glass. If the near-point distance of the jeweler is 20.8 cm, and the focal length of the magnifying glass is 7.50 cm, find the angular magnification when the diamond is held at the focal point of the magnifier. Assume the magnifying glass is directly in front of the jeweler’s eyes.

Problem 55P in Problem 54, find the angular magnification when the diamond is held 5.59 cm from the magnifying glass.

Problem 57P CE You have two lenses: lens 1 with a focal length of 0.45 cm and lens 2 with a focal length of 1.9 cm. If you construct a microscope with these lenses, which one should you use as the objective? Explain.

Problem 58P A compound microscope has an objective lens with a focal length of 2.2 cm and an eyepiece with a focal length of 5.4 cm. If the image produced by the objective is 12 cm from the objective, what magnification does this microscope produce?

Problem 59P BIO A typical red blood cell subtends an angle of only 1.9 × 10–5 rad when viewed at a person’s near-point distance of 25 cm. Suppose a red blood cell is examined with a compound microscope in which the objective and eyepiece are separated by a distance of 12.0 cm. Given that the focal length of the eyepiece is 2.7 cm, and the focal length of the objective is 0.49 cm, find the magnitude of the angle subtended by the red blood cell when viewed through this microscope.

Problem 56P A person with a near-point distance of 25 cm finds that a magnifying glass gives an angular magnification that is 1.5 times larger when the image of the magnifier is at the near point than when the image is at infinity. What is the focal length of the magnifying glass?

Problem 60P The medium-power objective lens in a laboratory microscope has a focal length f objective = 4.00 mm. (a) If this lens produces a lateral magnification of –40.0, what is its “working distance”; that is, what is the distance from the object to the objective lens? (b) What is the focal length of an eyepiece lens that will provide an overall magnification of 125?

Problem 61P A compound microscope has the objective and eyepiece mounted in a tube that is 18.0 cm long. The focal length of the eyepiece is 2.62 cm, and the near-point distance of the person using the microscope is 25.0 cm. If the person can view the image produced by the microscope with a completely relaxed eye, and the magnification is –4525, what is the focal length of the objective?

Problem 62P In Problem 61, what is the distance between the objective lens and the object to be examined?

Problem 63P The barrel of a compound microscope is 15 cm in length. The specimen will be mounted 1.0 cm from the objective, and the eyepiece has a 5.0-cm focal length. Determine the focal length of the objective lens.

Problem 64P A compound microscope uses a 75.0-mm lens as the objective and a 2.0-cm lens as the eyepiece. The specimen will be mounted 122 mm from the objective. Determine (a) the barrel length and (b) the total magnification produced by the microscope.

Problem 65P The “tube length” of a microscope is defined to be the difference between the (objective) image distance and objective focal length: L = di – fobjective. Many microscopes are standardized to a tube length of L = 160 mm. Consider such a microscope whose objective lens has a focal length f objective = 7.50 mm. (a) How far from the object should this lens be placed? (b) What focal length eyepiece would give an overall magnification of –55? (c) What focal length eyepiece would give an overall magnification of –110?

Problem 66P CE Two telescopes of different length produce the same angular magnification. Is the focal length of the long telescope’s eyepiece greater than or less than the focal length of the short telescope’s eyepiece? Explain.

Problem 67P CE To construct a telescope, you are given a lens with a focal length of 32 mm and a lens with a focal length of 1600 mm. (a) On the basis of focal length alone, which lens should be the objective and which the eyepiece? Explain. (b) What magnification would this telescope produce?

Problem 69P A 55-power refracting telescope has an eyepiece with a focal length of 5.0 cm. How long is the telescope?

Problem 68P A grade school student plans to build a 35-power telescope as a science fair project. She starts with a magnifying glass with a focal length of 5.0 cm as the eyepiece. What focal length is needed for her objective lens?

Problem 70P An amateur astronomer wants to build a small refracting telescope. The only lenses available to him have focal lengths of 5.00 cm, 10.0 cm, 20.0 cm, and 30.0 cm. (a) What is the greatest magnification that can be obtained using two of these lenses? (b) How long is the telescope with the greatest magnification?

Problem 71P A pirate sights a distant ship with a spyglass that gives an angular magnification of 22. If the focal length of the eyepiece is 11 mm, what is the focal length of the objective?

Problem 72P A telescope has lenses with focal lengths f1 = +30.0 cm and f2 = +5.0 cm. (a) What distance between the two lenses will allow the telescope to focus on an infinitely distant object and produce an infinitely distant image? (b) What distance between the lenses will allow the telescope to focus on an object that is 5.0 m away and to produce an infinitely distant image?

Problem 73P Jason has a 25-power telescope whose objective lens has a focal length of 120 cm. To make his sister appear smaller than normal, he turns the telescope around and looks through the objective lens. What is the angular magnification of his sister when viewed through the “wrong” end of the telescope?

Problem 74P Roughing It with Science A professor shipwrecked on Hooligan’s Island decides to build a telescope from his eyeglasses and some coconut shells. Fortunately, the professor’s eyes require different prescriptions, with the left lens having a power of +5.0 diopters and the right lens having a power of +2.0 diopters. (a) Which lens should he use as the objective? (b) What is the angular magnification of the professor’s telescope?

Problem 76P The Moon has an angular size of 0.50° when viewed with unaided vision from Earth. Suppose the Moon is viewed through a telescope with an objective whose focal length is 53 cm and an eyepiece whose focal length is 25 mm. What is the angular size of the Moon as seen through this telescope?

Problem 77P In Problem 76, an eyepiece is selected to give the Moon an angular size of 15°. What is the focal length of this eyepiece?

Galileo's Telescope Galileo's first telescope used a convex objective lens with a focal length \(f=1.7 \mathrm{~m}\) and a concave eyepiece, as shown in Figure . When this telescope is focused on an infinitely distant object, and produces an infinitely distant image, its angular magnification is \(+3.0\). (a) What is the focal length of the eyepiece? (b) How far apart are the two lenses? Equation Transcription: Text Transcription: f=1.7 m +3.0

Problem 78P A telescope is 275 mm long and has an objective lens with a focal length of 257 mm. (a) What is the focal length of the eyepiece? (b) What is the magnification of this telescope?

Problem 79GP CE Predict/Explain BIO Intracorneal Ring An intracomeal ring is a small plastic device implanted in a person’s cornea to change its curvature. By changing the shape of the cornea, the intracorneal ring can correct a person’s vision. (a) If a person is nearsighted, should the ring increase or decrease the cornea’s curvature? (b) Choose the best explanation from among the following: I. The intracorneal ring should increase the curvature of the cornea so that it bends light more. This will allow it to focus on light coming from far away. II. The intracorneal ring should decrease the curvature of the cornea so it’s flatter and bends light less. This will allow parallel rays from far away to be focused.

Problem 80GP · CE BIO The lens in a normal human eye, with aqueous humor on one side and vitreous humor on the other side, has a refractive power of 15 diopters. Suppose a lens is removed from an eye and surrounded by air. In this case, is its refractive power greater than, less than, or equal to 15 diopters? Explain.

Problem 81GP CE An optical system consists of two lenses, one with a focal length of 0.50 cm and the other with a focal length of 2.3 cm. If the separation between the lenses is 12 cm, is the instrument a microscope or a telescope? Explain.

Problem 82GP CE Air optical system consists of two lenses, one with a focal length of 50 cm and the other with a focal length of 2.5 cm. If the separation between the lenses is 52.5 cm, is the instrument a microscope or a telescope? Explain.

Problem 84GP IP Tire greatest refractive power a patient’s eyes can produce is 44.1 diopters. (a) Is this patient nearsighted or farsighted? Explain. (b) If this patient is nearsighted, find the far point. If this person is farsighted, find the near point. (For the purposes of this problem, treat the eye as a single-lens system, with the retina 2.40 cm from the lens.)

Problem 83GP CE Predict/Explain BIO Treating Cataracts When the lens in a person’s eye becomes clouded by a cataract, the lens can be removed with a process called phacoemulsification and replaced with a man-made intraocular lens. The intraocular lens restores clear vision, but its focal length cannot be changed to allow the user to focus at different distances. In most cases, the intraocular lens is adjusted for viewing of distant objects, and corrective glasses are worn when viewing nearby objects. (a) Should the refractive power of the corrective glasses be positive or negative? (b) Choose the best explanation from among the following: I. The refractive power should be positive—converging— because the intraocular lens will make the person farsighted. II. A negative refractive power is required to bring the focal point of the intraocular lens in from infinity to a finite value.

Problem 85GP IP You are observing a rare species of bird in a distant tree with your unaided eyes. (a) What is the refractive power of your eyes? (b) Does the refractive power of your eyes increase or decrease when you shift your view to the guidebook in your hands? Explain. (For the purposes of this problem, treat the eye as a single-lens system, with the retina 2.40 cm from the lens.)

Problem 86GP Galileo’s original telescope (Figure 27–23) used a convex objective and a concave eyepiece. Use a ray diagram to show that this telescope produces an upright image when a distant object is being viewed. Assume that the eyepiece is to the right of the object and that the right-hand focal point of the eyepiece is just to the left of the objective’s right-hand focal point. hr addition, assume that the focal length of the eyepiece has a magnitude that is about one-quarter the focal length of the objective.

Problem 87GP IP For each of the following cases, use a ray diagram to show that the angular sizes of tire image and the object are identical if both angles are measured from the center of the lens. (a) A convex lens with the object outside the focal length. (b) A convex lens with the object inside the focal length. (c) A concave lens with the object outside the focal length. (d) Given that the angular size does not change, how does a simple magnifier work? Explain.

Problem 89GP BIO Tire eye is actually a multiple-lens system, but we can approximate it with a single-lens system for most of our purposes. When the eye is focused on a distant object, the optical power of the equivalent single lens is +41.4 diopters. (a) What is the effective focal length of the eye? (b) How far in front of the retina is this “equivalent lens” located?

Problem 90GP BIO Fitting Contact Lenses with a Keratometer When a patient is being fitted with contact lenses, the curva true of the patient’s cornea is measured with an instrument known as a keratometer. A lighted object is held near the eye, and the keratometer measures the magnification of the image formed by reflection from the front of the cornea. If an object is held 10.0 cm in front of a patient’s eye, and the reflected image is magnified by a factor of 0.035, what is the radius of curvature of the patient’s cornea?

Problem 88GP IP You have two lenses, with focal lengths f1 = +2.60 cm and f2 = +20.4 cm. (a) How would you arrange these lenses to form a magnified image of the Moon? (b) What is the maximum angular magnification these lenses could produce? (c) How would you arrange the same two lenses to form a magnified image of an insect? (d) If you use the magnifier of part (c) to view an insect, what is the angular magnification when the insect is held 2.90 cm from the objective lens?

Problem 91GP Pricey Stamp A rare 1918 “Jenny” stamp, depicting a misprinted, upside-down Curtiss JN-4 “Jenny” airplane, sold at auction for $525,000. A collector uses a simple magnifying glass to examine the “Jenny,” obtaining a linear magnification of 2.5 when the stamp is held 2.76 cm from the lens. What is the focal length of the magnifying glass?

Problem 92GP IP A person needs glasses with a refractive power of –1.35 diopters to be able to focus on distant objects. (a) Is this person nearsighted or farsighted? Explain. (b) What is this person’s (unaided) far point?

Consider a Galilean telescope, as illustrated in Figure 27–23, constructed from two lenses with focal lengths of \(75.6 \mathrm{~cm} \text { and }-18.0 \mathrm{~mm}\). (a) What is the distance between these lenses if an infinitely distant object is to produce an infinitely distant image? * * (b) What is the angular magnification when the lenses are separated by the distance calculated in part (a)? * Equation Transcription: Text Transcription: 75.6 cm and -18.0 mm

Problem 96GP A farsighted person uses glasses with a refractive power of 3.6 diopters. The glasses are worn 2.5 cm from his eyes. What is this person’s near point when not wearing glasses?

Problem 95GP A converging lens forms a virtual object 12 cm to the right of a second lens that has a refracting power of 3.75 diopter. (a) Where is the image? (b) Is the image real or virtual? Explain.

Problem 93GP IP BIO A Big Eye The largest eye ever to exist on Earth belonged to an extinct species of ichthyosaur, Temnodontosaurus platyodon. This creature had an eye that was 26.4 cm in diameter. It is estimated that this ichthyosaur also had a relatively large pupil, giving it an effective aperture setting of about f/1.1. (a) Assuming its pupil was one-third the diameter of the eye, what was the approximate focal length of the ichthyosaur’s eye? (b) When the ichthyosaur narrowed its pupil in bright light, did its f-number increase or decrease? Explain.

Problem 98GP When using a telescope to photograph a faint astronomical object, you need to maximize the amount of light energy that falls on each square millimeter of the image on the film. For a given telescope and object, the total light that falls on the film is proportional to the length of the exposure, so a long exposure will reveal fainter objects than a short exposure. Show that for a given length of exposure, the brightness of the image is inversely proportional to the square of the f-number of the telescope system.

A Cassegrain astronomical telescope uses two mirrors to form the image. The larger (concave) objective mirror has a focal length \(f_{1}=+50.0 \mathrm{~cm}\). A small convex secondary mirror is mounted \(43.0 \mathrm{~cm}\) in front of the primary. As shown in Figure 27-25, light is reflected from the secondary through a hole in the center of the primary, thereby forming a real image \(8.00 \mathrm{~cm}\) behind the primary mirror. What is the radius of curvature of the secondary mirror? Equation Transcription: Text Transcription: f1=+50.0 cm 43.0 cm 8.00 cm

Landing on an Aircraft Carrier The Long-Range Lineup System (LRLS) used to ensure safe landings on aircraft carriers consists of a series of Fresnel lenses of different colors. Each lens focuses light in a different, specific direction, and hence which light a pilot sees on approach determines whether the plane is above, below, or on the proper landing path. The basic idea behind a Fresnel lens, which has the same optical properties as an ordinary lens, is shown in Figure , along with a photo of the LRLS. Suppose an object (a lightbulb in this case) is \(17.1 \mathrm{~cm}\) behind a Fresnel lens, and that the corresponding image is a distance \(d_{i}=d\) in front of the lens. If the object is moved to a distance of \(12.0 \mathrm{~cm}\) behind the lens, the image distance doubles to \(d_{i}=2 d\). In the LRLS, it is desired to have the image of the lightbulb at infinity. What object distance will give this result for this particular lens? (a) A lens causes light to refract at its surface; therefore, the interior glass can be removed without changing its optical properties. This produces a Fresnel lens, which is much lighter than the original lens. (b) If an airplane is on the correct approach path, the pilot will see an amber light, called the “meatball,” in line with the row of blue lights. Equation Transcription: Text Transcription: 17.1 cm di=d 12.0 cm di=2d

Repeat Problem 100 for the case where the converging lens is replaced with a diverging lens with \(f=-20.0 \mathrm{~cm}\) Everything else in the problem remains the same. Equation Transcription: Text Transcription: f=-20.0 cm

IP convex lens \((f=20.0 \mathrm{~cm})\) is placed \(10.0 \mathrm{~cm}\) in front of a plane mirror. A matchstick is placed \(25.0 \mathrm{~cm}\) in front of the lens, as shown in Figure (a) If you look through the lens toward the mirror, where will you see the image of the matchstick? (b) Is the image real or virtual? Explain. (c) What is the magnification of the image? (d) Is the image upright or inverted? Equation Transcription: Text Transcription: (f=20.0 cm) 10.0 cm 25.0 cm

Problem 102GP · · · Repeat Problem 47 for the case where lens 1 is replaced with a diverging lens with f1 = –20.0 cm. Everything else in the problem remains the same.

The diameter of a collimated laser beam can be expanded or reduced by using two converging lenses, with focal lengths \(f_{1} \text { and } f_{2}\), mounted a distance \(f_{1}+f_{2}\) from each other, as shown in Figure What is the ratio of the two beam diameters, \(\left(d_{1} / d_{2}\right)\), expressed in terms of the focal lengths? Equation Transcription: Text Transcription: f1 and f2 f1+f2 (d1/d2)

Problem 104GP Consider three lenses with focal lengths of 25.0 cm, –15.0 cm, and 11.0 cm positioned on the x axis at x = 0,x = 0.400 m, and x = 0.500 m, respectively. An object is at x = –122 cm. Find (a) the location and (b) the orientation and magnification of the final image produced by this lens system.

Problem 105GP · · · Because a concave lens cannot form a real image of a real object, it is difficult to measure its focal length precisely. One method uses a second, convex, lens to produce a virtual object for the concave lens. Under the proper conditions, the concave lens will form a real image of the virtual object! A student conducting a laboratory project on concave lenses makes the following observations: When a lamp is placed 42.0 cm to the left of a particular convex lens, a real (inverted) image is formed 37.5 cm to the right of the lens. The lamp and convex lens are kept in place while a concave lens is mounted 15.0 cm to the right of the convex lens. A real image of the lamp is now formed 35.0 cm to the right of the concave lens. What is the focal length of each lens?

Problem 106GP · · · A person with a near-point distance N uses a magnifying glass with a focal length f. Show that the greatest magnification that can be achieved with this magnifier is M = 1 + N/f.

Problem 107PP A patient receives a rigid IOL whose focus cannot be changed—it is designed to provide clear vision of objects at infinity. The patient will use corrective contacts to allow for close vision. Should the refractive power of the corrective contacts be positive or negative?

Problem 108PP Referring to the previous problem, find the refractive power of contacts that will allow the patient to focus on a book at a distance of 23.0 cm. A. 0.0435 diopter B. 0.230 diopter C. 4.35 diopters D. 8.70 diopters

Problem 110IP IP Referring to Example 27–2 Suppose a person’s eyeglasses have a focaL length of ?301 cm, are 2.00 cm in front of the eyes, and allow the person to focus on distant objects. (a) Is this person’s far point greater than or less than 323 cm, which is the far point for glasses the same distance from the eyes and with a focal length of ?321 cm? Explain. (b) Find the far point for this person.

Problem 111IP IP Referring to Example 27–2 hi Example 27–2, a person has a far-point distance of 323 cm. If this person wears glasses 2.00 cm in front of the eyes with a focal length of ?321 cm, distant objects can be brought into focus. Suppose a second person’s far point is 353 cm. (a) Is the magnitude of the focal length of the eyeglasses that allow this person to focus on distant objects greater than or less than 321 cm? Assume the glasses are 2.00 cm in front of the eyes. (b) Find the required focal length for the second person’s eyeglasses.

Problem 109PP Suppose a flexible, adaptive IOL has a focal length of 3.00 cm.How far forward must the IOL move to change the focus ofthe eye from an object at infinity to an object at a distance of50.0 cm? A. 1.9 mm B. 2.8 mm C. 3.1 mm D. 3.2 mm

Problem 112IP IP Referring to Example 27–3 Suppose a person’s eyeglasses have a refractive power of 2.75 diopters and that they allow the person to focus on an object that is just 25.0 cm from the eye. The glasses are 2.00 cm in front of the eyes. (a) Is this person’s near point greater than or less than 57.0 cm, which is the near-point distance when the glasses have a refractive power of 2.53 diopters? Explain. (b) Find the near point for this person.