Chapter 25

\(\cdot\) The focal length of an \(f / 4\) camera lens is 300 \(\mathrm{mm}\) . (a) What is the aperture diameter of the lens? (b) If the correct exposure of a certain scene is \(\frac{1}{250}\) s at \(f / 4,\) what is the correct exposure at \(f / 8 ?\)

\(\cdot\) Camera \(A\) has a lens with an aperture diameter of 8.00 \(\mathrm{mm}\) . It photographs an object, using the correct exposure time of \(\frac{1}{30}\) s. What exposure time should be used with camera \(\mathrm{B}\) in photographing the same object with the same film if camera \(B\) has a lens with an aperture diameter of 23.1 \(\mathrm{mm}\) ?

\(\bullet\) (a) A small refracting telescope designed for individual use has an objective lens with a diameter of 6.00 \(\mathrm{cm}\) and a focal length of 1.325 \(\mathrm{m}\) . What is the \(f\) -number of this instrument? (b) The 200 -inch-diameter objective mirror of the Mt. Palomar telescope has an \(f\) -number of \(3.3 .\) Calculate its focal length. (c) The distance between lens and retina for a normal human eye is about 2.50 \(\mathrm{cm}\) , and the pupil can vary in size from 2.0 \(\mathrm{mm}\) to 8.0 \(\mathrm{mm}\) . What is the range of \(f\) -numbers for the human eye?

. A 135 mm telephoto lens for a 35 mm camera has \(f\) -stops that range from \(f / 2.8\) to \(f / 22\) . (a) What are the smallest and largest aperture diameters for this lens? What is the diameter at \(f / 11 ?\) (b) If a 50 \(\mathrm{mm}\) lens had the same \(f-\) stops as the telephoto lens, what would be the smallest and largest aperture diameters for that lens? (c) At a given shutter speed, what is the ratio of the greatest to the smallest light intensity of the film image? (d) If the shutter speed for correct exposure at \(f / 22\) is 1\(/ 30\) s, what shutter speed is needed at \(f / 2.8 ?\)

A camera lens has a focal length of 200 \(\mathrm{mm}\) . How far from the lens should the subject for the photo be if the lens is 20.4 \(\mathrm{cm}\) from the film?

. A camera with a 90 -mm-focal-length lens is focused on an object 1.30 m from the lens. To refocus on an object 6.50 \(\mathrm{m}\) from the lens, by how much must the distance between the lens and the film be changed? To refocus on the more distant object, is the lens moved toward or away from the film?

\(\cdot\) A certain digital camera having a lens with focal length 7.50 \(\mathrm{cm}\) focuses on an object 1.85 m tall that is 4.25 \(\mathrm{m}\) from the lens. (a) How far must the lens be from the sensor array? (b) How tall is the image on the sensor array? Is it erect or inverted? Real or virtual? (c) A SLR digital camera often has pixels measuring 8.0\(\mu \mathrm{m} \times 8.0 \mu \mathrm{m} .\) How many such pixels does the height of this image cover?

Your digital camera has a lens with a 50 \(\mathrm{mm}\) focal length and a sensor array that measures 4.82 \(\mathrm{mm} \times 3.64 \mathrm{mm}\) . Suppose you're at the zoo, and want to take a picture of a \(4.50-\mathrm{m}-\) tall giraffe. If you want the giraffe to exactly fit the longer dimension of your sensor array, how far away from the animal will you have to stand?

You want to take a full-length photo of your friend who is 2.00 m tall, using a 35 mm camera having a 50.0 -mm- focal-length lens. The image dimensions of 35 \(\mathrm{mm}\) film are \(24 \mathrm{mm} \times 36 \mathrm{mm},\) and you want to make this a vertical photo in which your friend's image completely fills the image area. (a) How far should your friend stand from the lens? (b) How far is the lens from the film?

. Zoom lens, I. A zoom lens is a lens that varies in focal length. The zoom lens on a certain digital camera varies in focal length from 6.50 \(\mathrm{mm}\) to 19.5 \(\mathrm{mm}\) . This camera is focused on an object 2.00 \(\mathrm{m}\) tall that is 1.50 \(\mathrm{m}\) from the camera. Find the distance between the lens and the photo sensors and the height of the image (a) when the zoom is set to 6.50 \(\mathrm{mm}\) focal length and (b) when it is at 19.5 \(\mathrm{mm}\) . (c) Which is the telephoto focal length, 6.50 \(\mathrm{mm}\) or 19.5 \(\mathrm{mm} ?\)

A slide projector uses a lens of focal length 115 \(\mathrm{mm}\) to focus a 35 \(\mathrm{mm}\) slide (having dimensions 24 \(\mathrm{mm} \times 36 \mathrm{mm} )\) on a screen. The slide is placed 12.0 \(\mathrm{cm}\) in front of the lens. (a) Where should you place the screen to view the image of this slide? (b) What are the dimensions of the slide's image on the screen?

An An LCD projector (see Sec. 25.2\()\) has a projection lens with \(f\) -number of 1.8 and a diameter of 46 \(\mathrm{mm}\) . The LCD array measures 3.30 \(\mathrm{cm} \times 3.30 \mathrm{cm}\) and will be projected on a screen 8.00 m from the lens. If the array is \(800 \times 600\) pixels, what will be the dimensions of a single pixel on the screen?

You are designing a projection system for a hall having a screen measuring 4.00 m square. The lens of a 35 \(\mathrm{mm}\) slide projector in the projection booth is 15.0 \(\mathrm{m}\) from this screen. You want to focus the image of 35 \(\mathrm{mm}\) slides (which are 24 \(\mathrm{mm} \times 36 \mathrm{mm}\) ) onto this screen so that you can fill as much of the screen as possible without any part of the image extending beyond the screen. (a) What focal-length lens should you use in the projector? (b) How far from the lens should the slide be placed? (c) What are the dimensions of the slide's image on the screen?

.. In a museum devoted to the history of photography, you are setting up a projection system to view some historical 4.0 inch \(\times 5.0\) inch color slides. Your screen is 6.0 \(\mathrm{m}\) from the projector lens, and you want the image to be 4.0 ft \(\times 5.0\) ft on the screen. (a) What focal-length lens do you need? (b) How far from the lens should you put the slide?

Crystalline lens of the eye. The crystalline lens of the eye is double convex and has a typical index of refraction of \(1.43 .\) At minimum power, the front surface has a radius of 10.0 \(\mathrm{mm}\) and the back surface has a radius of 6.0 \(\mathrm{mm}\) ; at maximum power, these radii are 6.0 \(\mathrm{mm}\) and 5.5 \(\mathrm{mm}\) , respectively (although the values do vary from person to person). (a) Find the maximum and minimum power (in diopters) of the crystalline lens if it were in air. (b) What is the range of focal lengths the eye can achieve? (c) At minimum power, where does it focus the image of a very distant object? (d) At maximum power, where does it focus the image of an object at the near point of 25 \(\mathrm{cm} ?\)

Contact lenses. Contact lenses are placed right on the eyeball, so the distance from the eye to an object (or image) is the same as the distance from the lens to that object (or image).A certain person can see distant objects well, but his near point is 45.0 \(\mathrm{cm}\) from his eyes instead of the usual 25.0 \(\mathrm{cm} .\) (a) Is this person nearson nearsighted or farsighted? (b) What type of lens (con- verging or diverging is needed to correct his vision? (c) If the correcting lenses will be contact lenses, what focal length lens is needed and what is its power in diopters?

\(\cdot\) A person can see clearly up close, but cannot focus on objects beyond 75.0 \(\mathrm{cm}\) . She opts for contact lenses to correct her vision. (a) Is she nearsighted or farsighted? (b) What type of lens (converging or diverging) is needed? (b) What type vision? (c) What focal-length contact lens is needed, and what is its power in diopters?

A student's far point is at \(17.0 \mathrm{cm},\) and she needs glasses to view her computer screen comfortably at a distance of 45.0 \(\mathrm{cm} .\) What should be the power of the lenses for her glasses?

\(\cdot\) You want to view an insect 2.00 \(\mathrm{mm}\) in length through a magnifier. If the insect is to be at the focal point of the magnifier, what focal length will give the image of the insect an angular size of 0.025 radian?

\(\cdot\) A thin lens with a focal length of 6.00 \(\mathrm{cm}\) is used as a sim-ple magnifier. (a) What angular magnification is obtainable with the lens if the object is at the focal point? (b) When an object is examined through the lens, how close can it be brought to the lens? Assume that the image viewed by the eye is at infinity and that the lens is very close to the eye.

. The focal length of a simple magnifier is 8.00 \(\mathrm{cm} .\) Assume the magnifier to be a thin lens placed very close to the eye. (a) How far in front of the magnifier should an object be placed if the image is formed at the observer's near point, 25.0 \(\mathrm{cm}\) in front of her eye? (b) If the object is 1.00 \(\mathrm{mm}\) high, what is the height of its image formed by the magnifier?

A microscope has an objective lens with a focal length of 12.0 \(\mathrm{mm}\) . A small object is placed 0.8 \(\mathrm{mm}\) beyond the focal point of the objective lens. (a) At what distance from the objective lens does a real image of the object form? (b) What is the magnification of the real image? (c) If an eyepiece with a focal length of 2.5 \(\mathrm{cm}\) is used, with a final image at infinity, what will be the overall angular magnification of the object?

\(\cdot\) A compound microscope has an objective lens of focal length 10.0 \(\mathrm{mm}\) with an eyepiece of focal length \(15.0 \mathrm{mm},\) and it produces its final image at infinity. The object to be viewed is placed 2.0 \(\mathrm{mm}\) beyond the focal point of the objective lens. (a) How far from the objective lens is the first image formed? (b) What is the overall magnification of this microscope?

An insect 1.2 \(\mathrm{mm}\) tall is placed 1.0 \(\mathrm{mm}\) beyond the focal point of the objective lens of a compound microscope. The objective lens has a focal length of \(12 \mathrm{mm},\) the eyepiece a focal length of 25 \(\mathrm{mm}\) . (a) Where is the image formed by the objective lens and how tall is it? (b) If you want to place the eye-piece so that the image it produces is at infinity, how far should this lens be from the image produced by the objective lens? (c) Under the conditions of part (b), find the overall magnification of the microscope.

\(\bullet\) The focal length of the eyepiece of a certain microscope is 18.0 \(\mathrm{mm}\) . The focal length of the objective is 8.00 \(\mathrm{mm}\) . The distance between objective and eyepiece is 19.7 \(\mathrm{cm}\) . The final image formed by the eyepiece is at infinity. Treat all lenses as thin. (a) What is the distance from the objective to the object being viewed? (b) What is the magnitude of the linear magnification produced by the objective? (c) What is the overall angular magnification of the microscope?

. A certain microscope is provided with objectives that have focal lengths of \(16 \mathrm{mm}, 4 \mathrm{mm}\) , and 1.9 \(\mathrm{mm}\) and with eye pieces that have angular magnifications of \(5 \times\) and \(10 \times\) Each objective forms an image 120 \(\mathrm{mm}\) beyond its second focal point. Determine (a) the largest overall angular magnification obtainable and (b) the smallest overall angular magnification obtainable.

\(\bullet\) Resolution of a microscope. The image formed by a microscope objective with a focal length of 5.00 \(\mathrm{mm}\) is 160 \(\mathrm{mm}\) from its second focal point. The eyepiece has a focal length of 26.0 \(\mathrm{mm}\) . (a) What is the angular magnification of the microscope? (b) The unaided eye can distinguish two points at its near point as separate if they are about 0.10 \(\mathrm{mm}\) apart. What is the minimum separation that can be resolved with this microscope?

\(\cdot\) A refracting telescope has an objective lens of focal length 16.0 in and eyepieces of focal lengths \(15 \mathrm{mm}, 22 \mathrm{mm}, 35 \mathrm{mm},\) and 85 \(\mathrm{mm}\) . What are the largest and smallest angular magnifications you can achieve with this instrument?

\(\bullet\) Galileo's telescopes, I. While Galileo did not invent the telescope, he was the first known person to use it astronomically, beginning around \(1609 .\) Five of his original lenses have survived (although he did work with others). Two of these have focal lengths of 1710 \(\mathrm{mm}\) and 980 \(\mathrm{mm}\) . (a) For greatest magnification, which of these two lenses should be the eye- piece and which the objective? How long would this telescope be between the two lenses? (b) What is the greatest angular magnification that Galileo could have obtained with these lenses? (Note: Galileo actually obtained magnifications up to about \(30 \times\) but by using a diverging lens as the eye- piece.) (c) The Moon subtends an angle of \(\frac{10}{2}\) when viewed with the naked eye. What angle would it subtend when viewed through this telescope (assuming that all of it could be seen)?

The largest refracting telescope in the world is at Yerkes Observatory in Wisconsin. The objective lens is 1.02 \(\mathrm{m}\) in diameter and has a focal length of 19.4 \(\mathrm{m} .\) Suppose you want to magnify Jupiter, which is \(138,000 \mathrm{km}\) in diameter, so that its image subtends an angle of \(\frac{10}{2}\) (about the same as the moon) when it is \(6.28 \times 10^{8} \mathrm{km}\) from earth. What focal-length eye-piece do you need?

A thin planoconvex lens has a radius of curvature of magnitude 22.5 \(\mathrm{cm}\) on the curved side. When a color chart is placed 48.0 \(\mathrm{cm}\) from the lens, green light of wavelength 550 \(\mathrm{nm}\) is focused 277 \(\mathrm{cm}\) from the lens and blue light of wavelength 450 \(\mathrm{nm}\) is focused 17 \(\mathrm{I} \mathrm{cm}\) from the lens. What are the indices of refraction for these two wavelengths of light?

\(\bullet\) You are examining a flea with a converging lens that has a focal length of 4.00 \(\mathrm{cm}\) . If the image of the flea is 6.50 times the size of the flea, how far is the flea from the lens? Where, relative to the lens, is the image?

\(\bullet\) Physician, heal thyself! (a) Experimentally determine the near and far points for both of your own eyes. Are these points the same for both eyes? (All you need is a tape measure or ruler and a cooperative friend.) (b) Design correcting lenses, as needed, for your closeup and distant vision in one of your eyes. If you prefer contact lenses, design that type of lens. Otherwise design lenses for ordinary glasses, assuming that they will be 2.0 \(\mathrm{cm}\) from your eye. Specify the power (in diopters) of each correcting lens.

\(\bullet\) It's all done with mirrors. A photographer standing 0.750 \(\mathrm{m}\) in front of a plane mirror is taking a photograph of her image in the mirror, using a digital camera having a lens with a focal length of 19.5 \(\mathrm{mm}\) (a) How far is the lens from the light sensors of the camera? (b) If the camera is 8.0 \(\mathrm{cm}\) high, how high is its image on the sensors?

\(\bullet\) During a lunar eclipse, a picture of the moon (which has a diameter of \(3.48 \times 10^{6} \mathrm{m}\) and is \(3.86 \times 10^{8} \mathrm{m}\) from the earth) is taken with a camera whose lens has a focal length of 300 \(\mathrm{mm}\) . (a) What is the diameter of the image on the film? (b) What per- cent is this of the width of a 24 \(\mathrm{mm} \times 36 \mathrm{mm}\) color slide?

\bullet A microscope with an objective of focal length 8.00 \(\mathrm{mm}\) and an eyepiece of focal length 7.50 \(\mathrm{cm}\) is used to project an image on a screen 2.00 \(\mathrm{m}\) from the eyepiece. Let the image distance of the objective be 18.0 \(\mathrm{cm} .\) (a) What is the lateral magnification of the image? (b) What is the distance between the objective and the eyepiece?

. A person with a near point of \(85 \mathrm{cm},\) but excellent distant vision, normally wears corrective glasses. But he loses them while traveling. Fortunately, he has his old pair as a spare. (a) If the lenses of the old pair have a power of \(+2.25\) diopters, what is his near point (measured from his eye) when he is wearing the old glasses if they rest 2.0 \(\mathrm{cm}\) in front of his eye? (b) What would his hear point be if his old glasses were contact lenses instead?

A telescope is constructed from two lenses with focal lengths of 95.0 \(\mathrm{cm}\) and \(15.0 \mathrm{cm},\) the 95.0 -cm lens being used as the objective. Both the object being viewed and the final image are at infinity. (a) Find the angular magnification of the telescope. (b) Find the height of the image formed by the objective of a building 60.0 \(\mathrm{m}\) tall and 3.00 \(\mathrm{km}\) away. (c) What is the angular size of the final image as viewed by an eye very close to the eyepiece?

Water drop magnifier. You can make a pretty good magnifying lens by putting a small drop of water on a piece of transparent kitchen wrap. Suppose your drop has an upper surface with a radius of curvature of 1.6 \(\mathrm{cm}\) and the side on the kitchen wrap is essentially flat. (a) Calculate the focal length of your water lens. (b) What's the angular magnification of the lens? (c) Suppose you place this planoconvex water lens directly onto the surface of a table, so that the tabletop is in effect about half the thickness of the drop. or 1.0 \(\mathrm{mm}_{\text { a away }}\) from the lens. Where does the image of the tabletop form, what type is it, and what is its magnification? (Use the thin lens equation here, even though the small object distance relative to the thickness of the lens makes it a poor approximation in this case.) What does this result tell you about how a simple magnifier works?