Physics Principles with Applications1912 solutions
The Galilean Telescope. Figure P34.100 is a diagram of a Galilean telescope, or opera glass, with both the object and its final image at infinity. The image serves as a virtual object for the eyepiece. The final image is virtual and erect. (a) Prove that the angular magnification is M=-f1/f2. (b) A Galilean telescope is to be constructed with the same objective lens as in Exercise 34.61. What focal length should the eyepiece have if this telescope is to have the same magnitude of angular magnification as the one in Exercise 34.61? (c) Compare the lengths of the telescopes. Figure P34.100
In setting up an experiment for a high school biology lab, you use a concave spherical mirror to produce real images of a 4.00 - mm -tall firefly. The firefly is to the right of the mirror, on the mirror’s optic axis, and serves as a real object for the mirror. You want to determine how far the object must be from the mirror’s vertex (that is, object distance ) to produce an image of a specified height. First, you place a square of white cardboard to the right of the object and find what its distance from the vertex needs to be so that the image is sharply focused on it. Next you measure the height of the sharply focused images for five values of . For each value, you then calculate the lateral magnification . You find that if you graph your data with on the vertical axis and on the horizontal axis, then your measured points fall close to a straight line. (a) Explain why the data plotted this way should fall close to a straight line. (b) Use the graph in Fig. P34.102 to calculate the focal length of the mirror. (c) How far from the mirror’s vertex should you place the object in order for the image to be real 8.00 mm, tall, and inverted? (d) According to Fig. P34.102, starting from the position that you calculated in part (c), should you move the object closer to the mirror or farther from it to increase the height of the inverted, real image? What distance should you move the object in order to increase the image height from to 12.00 mm? (e) Explain why approaches zero as approaches . Can you produce a sharp image on the cardboard when s = 25 cm ? (f) Explain why you can’t see sharp images on the cardboard when s < 25 cm (and is positive).
DATA It is your first day at work as a summer intern at an optics company. Your supervisor hands you a diverging lens and asks you to measure its focal length. You know that with a converging lens, you can measure the focal length by placing an object a distance to the left of the lens, far enough from the lens for the image to be real, and viewing the image on a screen that is to the right of the lens. By adjusting the position of the screen until the image is in sharp focus, you can determine the image distance and then use Eq. (34.16) to calculate the focal length f of the lens. But this procedure won’t work with a diverging lens—by itself, a diverging lens produces only virtual images, which can’t be projected onto a screen. Therefore, to determine the focal length of a diverging lens, you do the following: First you take a converging lens and measure that, for an object 20.0 cm to the left of the lens, the image is 29.7 cm to the right of the lens. You then place a diverging lens to the right of the converging lens and measure the final image to be 42.8 cm to the right of the converging lens. Suspecting some inaccuracy in measurement, you repeat the lens-combination measurement with the same object distance for the converging lens but with the diverging lens 20.0 cm to the right of the converging lens. You measure the final image to be 31.6 cm to the right of the converging lens. (a) Use both lens-combination measurements to calculate the focal length of the diverging lens. Take as your best experimental value for the focal length the average of the two values. (b) Which position of the diverging lens, 20.0 cm to the right or 25.0 cm to the right of the converging lens, gives the tallest image? Answer
The science museum where you work is constructing a new display. You are given a glass rod that is surrounded by air and was ground on its left-hand end to form a hemispherical surface there. You must determine the radius of curvature of that surface and the index of refraction of the glass. Remembering the optics portion of your physics course, you place a small object to the left of the rod, on the rod’s optic axis, at a distance s from the vertex of the hemispherical surface. You measure the distance of the image from the vertex of the surface, with the image being to the right of the vertex. Your measurements are as follows:
CALC (a) For a lens with focal length f, find the smallest distance possible between the object and its real image. (b) Graph the distance between the object and the real image as a function of the distance of the object from the lens. Does your graph agree with the result you found in part (a)?
An Object at an Angle. A 16.0cm long pencil is placed at a 45.0° angle, with its center 15.0cm above the optic axis and 45.0 cm from a lens with a 20.0cm focal length as shown in Fig. P34.106. (Note that the figure is not drawn to scale.) Assume that the diameter of the lens is large enough for the paraxial approximation to be valid. (a) Where is the image of the pencil? (Give the location of the images of the points and on the object, which are located at the eraser, point, and center of the pencil, respectively.) (b) What is the length of the image (that is, the distance between the images of points and )? (c) Show the orientation of the image in a sketch.
People with normal vision cannot focus their eyes underwater if they aren’t wearing a face mask or goggles and there is water in contact with their eyes (see Discussion Question Q34.23). (a) Why not? (b) With the simplified model of the eye described in Exercise 34.50, what corrective lens (specified by focal length as measured in air) would be needed to enable a person underwater to focus an infinitely distant object? (Be careful—the focal length of a lens underwater is not the same as in air! See Problem 34.92. Assume that the corrective lens has a refractive index of and that the lens is used in eyeglasses, not goggles, so there is water on both sides of the lens. Assume that the eyeglasses are 2.00cm in front of the eye.)
The eyes of amphibians such as frogs have a much flatter cornea but a more strongly curved (almost spherical) lens than do the eyes of air-dwelling mammals. In mammalian eyes, the shape (and therefore the focal length) of the lens changes to enable the eye to focus at different distances. In amphibian eyes, the shape of the lens doesn’t change. Amphibians focus on objects at different distances by using specialized muscles to move the lens closer to or farther from the retina, like the focusing mechanism of a camera. In air, most frogs are near-sighted; correcting the distance vision of a typical frog in air would require contact lenses with a power of about -6.0 D .A frog can see an insect clearly at a distance of 10cm. At that point the effective distance from the lens to the retina is 8 mm. If the insect moves farther from the frog, by how much and in which direction does the lens of the frog’s eye have to move to keep the insect in focus? (a) 0.02 cm toward the retina; (b)0.02 cm, away from the retina; (c)0.06 cm, toward the retina; (d)0.06 cm, away from the retina.
The eyes of amphibians such as frogs have a much flatter cornea but a more strongly curved (almost spherical) lens than do the eyes of air-dwelling mammals. In mammalian eyes, the shape (and therefore the focal length) of the lens changes to enable the eye to focus at different distances. In amphibian eyes, the shape of the lens doesn’t change. Amphibians focus on objects at different distances by using specialized muscles to move the lens closer to or farther from the retina, like the focusing mechanism of a camera. In air, most frogs are near-sighted; correcting the distance vision of a typical frog in air would require contact lenses with a power of about -6.0 D .What is the farthest distance at which a typical “near-sighted” frog can see clearly in air? (a) 12 m; (b) 6.0 m; (c) 80 cm (d) 17 cm.
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