1. The shape of the interface pictured in Fig. P.5.1 is known as a
Cartesian oval after Rene Descartes who studied it in the early 1800s.
It's the perfect configuration to carry any ray from S to the interface
to P. Prove that the defining equation isℓ0n1 + ℓin2 = constantShow that
this is equivalent ton1(x2+y2)1/2+n2[y2+(s0+si-x2)]1/2 = constantFigure
P.5.1... Get solution
2. Construct a Cartesian oval such that the conjugate points will be separated by 11 cm when the object is 5 cm from the vertex. If n1 | = 1 and n2 = 3/2 draw several points on the required surface. Get solution
3. Use Fig. P.5.3 to show that if a point source is placed at the focus Fv of the ellipsoid, plane waves will emerge from the far side. Remember that the defining requirement for an ellipse is that the net distance from one focus to the curve and back to the other focus is constant.Figure P.5.3... Get solution
4. Diagrammatically construct both a sphero-clliptic positive lens and an elliplo-spheric negative lens, showing rays and wavefronts as they pass through the lens. Do the same for an oval-spheric positive lens. Get solution
5. Making use of Fig. P.5.5, Snell's Law, and the fact that in the paraxial region α = h/s0, φ ≈ h/R, and β ≈ h/si derive Eq. (5.8).Figure P.5.5... Get solution
6. Show that, in the paraxial domain, the magnification produced by a single spherical interface between two continuous media, as shown in Fig'. P.5.6, is given by...Use the small-angle approximation for Snell's Law and approximate the angles by their tangents.Figure P.5.6... Get solution
7. Imagine a hemispherical interface, with a radius of curvature of radius 5.00 cm, separating two media: air on the left, water on the right. A 3.00-cm-tall toad is on the central axis, in air, facing the convex interface and 30.0 cm from its vertex. Where in the water will it Re imaged? How big will it appear to a fish in the water? Use the results of the previous problem, even though our frog is pushing the paraxial approximation. Get solution
8. Locate the image of an object placed 1.2 m from the vertex of a gypsy's crystal ball, which has a 20-cm diameter (n1 = 1.5). Make a sketch (of the rays, not the gypsy). Get solution
9. Return to Problem 5.7 and suppose we cut off the medium on the right foiTning a thick water biconvex lens, with each surface having a radius of curvature of 5.00 cm. If the lens is 10.0 cm thick, determine the total magnification and everything you can about the toad's image. Get solution
10. A biconvex glass (n1 = 1.5) thin lens is to have a +10.0-cm focal length. If the radius of curvature of each surface is measured to 'be the same, what must it be? Show that a spider standing 1.0 cm frcm the lens will be imaged at -1.1 cm. Describe that image and draw a ray diagram. Get solution
11. Going back to Section 5.2.3, prove that for a thin lens mmersed in a medium of index nm....That done, imagine a double-concave air lens surrounded by water; determine if it's converging or diverging. Get solution
12. A meniscus concave glass (nl = 1.5) thin lens (see Fig. 5.12) has radii of curvature of +20.0 cm and +10.0 cm. If an object is placed 20.0 cm in front of the lens, show that the image distance will be -13.3 cm. Describe that image and draw a ray diagram. Get solution
13. A biconcave lens (n1, = 1.5) has radii of 20 cm and 10 cm and an axial thickness of 5 cm. Describe the image of an object 1-inch tall placed 8 cm from the first vertex. Use the thin-lens equation to see how far off it is in determining the final-image location. Get solution
14. A 35-mm camera has a single thin lens having a 50.0-mm focal length. A woman 1.7 m tall stands 10.0 m in front of the camera, (a) Show that the lens-film distance must be 50.3 mm. (b) How tall is her image on the film? Get solution
15. Prove that the minimum separation between conjugate real object and image points for a thin positive lens is 4f. Get solution
16. An object 2 cm high is positioned 5 cm to the right of a positive thin lens with a focal length of 10 cm. Describe the resulting image completely, using both the Gaussian and Newtonian equations. Get solution
17. Make a rough graph of the Gaussian Lens Equation; that is, plot Si versus s0, using unit intervals of f along each axis. (Get both segments of the curve.) Get solution
18. A parallel bundle of rays from a very distant point source is incident on a thin negative lens having a focal length of -50.0 cm. The rays make an angle of 6.0° with the optical axis of the lens. Locate the image of the source. Get solution
19. Get solution
20. What must the focal length of a thin negative lens be for it to form a virtual image 50 cm away (measured from the lens) of an ant located 100 cm away (measured from the lens)? Given that the ant is to the right of the lens, locate and describe its image. Get solution
21. Get solution
22. Compute the focal length in air of a thin biconvex lens (nl = 1.5) having radii of 20 and 40 cm. Locate and describe the image of an object 40 cm from the lens. Get solution
23. Determine the focal length of a planar-concave lens (nl = 1.5) having a radius of curvature of 10 cm. What is its power in diopters? Get solution
24. Get solution
25. Get solution
26. Determine the focal length in air of a thin spherical planar-convex lens having a radius of curvature of 50.0 mm and an index of 1.50. What, if anything, would happen to the focal length if the lens were placed in a tank of water? Get solution
27. Get solution
28. Get solution
29. Get solution
30. We would like to place an object 45 cm in front of a lens and have its image appear on a screen 90 cm behind the lens. What must be the focal length of the appropriate positive lens? Get solution
31. The horse in Fig. 5.26 is 2.25 m tall, and it stands with its face 15.0 m from the plane of the thin lens whose focal length is 3.00 m.(a) Determine the location of the image of the equine nose.(b) Describe the image in detail—type, orientation, and magnification.(c) How tall is the image?(d) If the horse's tail is ! 7.5 m from the lens, how long, nose-to-tail, is the image of the beast? Get solution
32. A candle that is 6.00 cm tall is standing 10 cm from a thin concave lens whose focal length is —30 cm. Determine the location of the image and describe it in detail. Draw an appropriate ray diagram. Get solution
33. The image projected by an cquiconvex lens (n = 1.50) of a frog 5.0 cm tall, who is located 0.60 m from a screen, is to be 25 cm high. Please compute the necessary radii of the lens. Get solution
34. Get solution
35. Get solution
36. Get solution
37. A thin piece of wire 4.00 mm long is located in a plane perpendicular to the optical axis and 60.0 cm in front of a thin lens. The sharp image of the wire formed on a screen is 2.00 mm long. What is the focal length of the lens? When the screen is moved farther from the lens by 10.0 mm, the image blurs to a width of 0.80 mm. What is the diameter of the lens? [Hint: Image a source point on the axis.] Get solution
38. A thin double-convex glass lens (with an index of 1.56) while surrounded by air has a 10-cm focal length. If it is placed under water (having an index of 1.33) 100 cm beyond a small fish, where will the guppy's image be formed? Get solution
39. Consider a homemade television projection system that uses a large positive lens to cast the image of the TV screen onto a wall. The projected picture is enlarged three times, and although dim, it's nice and clear. If the lens has a focal length of 60 cm, what should be the distance between the screen and the wall? Why use a large lens? How should we mount the set with respect to the lens? Get solution
40. Write an expression for the focal length (fw) of a thin lens immersed in water (nw = 4/3) in terms of its focal length when it's in air (fa) Get solution
41. Observe the three vectors ..., ...and ... in Fig. P.5.31, each of which has a length of 0.10 f where f is the focal length of the thin positive lens. The plane formed by ... and ... is at a distance of 1.10 f from (he lens. Describe the image of each vector.Figure P.5.31... Get solution
42. A convenient way to measure the focal length of a positive lens makes use of the following fact. If a pair of conjugate object and (real) image points (S and P) are separated by a distance L > 4f, there will be two locations of the lens, a distance d apart, for which the same pair of conjugates obtain. Show that...Note that this avoids measurements made specifically from the vertex, which are generally not easy to do. Get solution
43. Two positive lenses with focal lengths of 0.30 m and 0.50 m are separated by a distance of 0.20 m. A small butterfly rests on the central axis 0.50 m in front of the first lens. Locate the resulting image with respect to the second lens. Get solution
44. In the process of constructing a doublet, an equieonvex thin lens L2 is positioned in intimate contact with a thin negative lens, such that the combination has a focal length of 50 cm in air. If their indices are 1.50 and 1.55, respectively, and if the focal length of L2 —50 cm, determine all the radii of curvature. Get solution
45. Verify Eq. (5.34), which gives MT for a combination of two thin lenses. Get solution
46. A blade of grass standing 10.0 mm tall is 150 mm in from of a thin positive lens having a 100 mm focal length; 250 mm behind that first lens is a thin negative lens with a focal length of -75.0 mm. (a) Show that the first lens forms an image 300 mm behind it. (b) Describe that image, (c) What's its magnification? (d) Prove that the final image formed by both lenses is located 150 mm belund the negative lens, (e) What is the total magnification of the combination? Get solution
47. Compute the image location and magnification of an object 30 cm from the front doublet of the thin-lens combination in Fig. P.5.37. Do the calculation by finding the effect of each lens separately. Make a sketch of appropriate rays.Figure P.5.37... Get solution
48. Two thin lenses having focal lengths of + 15.0 cm and -15.0 cm are positioned 60.0 cm apart. A page of print is held 25.0 cm in front of the positive lens. Describe, in detail, the image of the print (i.e., insofar as it's paraxial). Get solution
49. Draw a ray diagram for the combination of two positive lenses wherein their separation equals the sum of their respective focal lengths. Do the same thing for the case in which one of the lenses is negative. Get solution
50. Two positive lenses are to be used as a laserbeam expander. An axial 1.0-mm diameter beam enters a short focal length positive lens, which is followed by a somewhat longer focal length positive lens from which it emerges with a diameter of 8.0 mm. Given that the first lens has a 50.0 mm focal length, determine the focal length of the second lens and the separation between the lenses. Draw a diagram. Get solution
51. Redraw the ray diagram for a compound microscope (Fig. 5.99), but this time treat the intermediate image as if it were a real object. This approach should be a bit simpler. Get solution
52. Consider a thin positive lens L1 and using a ray diagram, show that if a second lens L2 is placed at the focal point of L2, the magnification does not change. That's a good reason to wear eyeglasses, whose lenses are different, al the correct distance from the eye. Get solution
53. Figures P.5.43a and P.5.43b are taken from an introductory •physics book. What's wrong with them?Figure P.5.43a...Figure P.5.43b... Get solution
54. Get solution
55. Consider the case of two positive thin lenses, L1, and L2, separated by 5 cm. Their diameters are 6 and 4 cm, respectively, and their focal lengths are f1 = 9 cm and f2 = 3 cm. If a diaphragm with a hole 1 cm in diameter is located between them, 2 cm from L2, find (a) the aperture stop and (b) the locations and sizes of the pupils for an axial point, S, 12 cm in front of (to the left of) L1. Get solution
56. Get solution
57. Make a sketch roughly locating the aperture stop and entrance and exit pupils for the lens in Fig. P.5.45... Get solution
58. Make a sketch roughly locating the aperture stop and entrance and exit pupils for the lens in Fig. P.5.46, assuming the object point to be beyond (to the left of) F01.Figure P.5.46... Get solution
59. Get solution
60. Figure P.5.47 shows a lens system, an object, and the appropri-"ate pupils. Diagrammalically locate the image.... Get solution
61. Draw a ray diagram locating the images of a point source as formed by a pair of mirrors at 90° (Fig. P.5.48a). Now create a ray diagram locating the images of the arrow shown in Fig. P.5.486.Figure P.5.48a. ...b. ... Get solution
62. Examine Velasquez's painting of Venus and Cupid (Fig. P.5.49). Is Venus looking at herself in the mirror? Explain.Figure P.5.49 The Toilet of Venus by Diego Rodriguez de Silva y Veläquez (Courtesy of the Trustees, The National Gallery, London)... Get solution
63. Manet's painting The Bar at the Folies Bergeres (Fig. P.5.50) shows a girl standing in front of a large planar mirror. Reflected in it is her back and a man in evening dress with whom she appears to be talking. It would seem that Manet's intent was to give the uncanny feeling that the viewer is standing where that gentleman must be. From the laws of Geometrical Optics, what's wrong?Figure P.5.50 The Bar at (tie Folies Bergere by Edouard Manet (Courtesy of the Courtauld Institute Galleries, London. Courtauld Collection.)... Get solution
64. Show that Eq. (5.48) for a spherical surface is equally applicable to a plane mirror. Get solution
65. Get solution
66. Get solution
67. Get solution
68. Get solution
69. Get solution
70. Get solution
71. Locate the image of a paperclip 100 cm away from a convex spherical mirror having a radius of curvature of 80 cm. Get solution
72. Imagine that you are standing 5 feet from, and looking directly toward, a brass ball 1 foot in diameter hanging in front of a pawn shop. Describe the image you would see in the ball. Get solution
73. A thin lens having a focal length of +50.0 cm is positioned 250 cm in front of (i.e., to the left of) a plane mirror. An ant sits on the central axis 250 cm in front of (i.e., to the left of) the lens. Locate the three images of the ant. Get solution
74. The image of a red rose is formed by a concave spherical mirror on a screen 100 cm away. If the rose is 25 cm from the mirror, determine its radius of curvature. Get solution
75. From the image configuration determine the shape of the mirror hanging on the back wall in van Eyck's painting of John Amulfini and His Wife (Fig. P.5.56).Figure P.5.56 Detail of John Amolfini and His Wife by Jan van Eyck. (Courtesy of the Trustees, The National Gallery, London... Get solution
76. Get solution
77. There are several varieties of retro-reflector that are commercially available; one type is comprised of transparent spheres, the backs of which arc silvered. Light is refracted at the front surface, focused onto the rear surface, and there reflected back out in the direction it came. Determine the necessary index of refraction of the spheres. Assume the incident light is collimated. Get solution
78. Design an eye for a robot using a concave spherical mirror such that the image of an object 1.0 m tall and 1.0 cm away fills its 1.0-cm-square photosensitive detector (which is movable for focusing purposes). Where should this detector be located with respect to the mirror? What should be the focal length of the mirror? Draw a ray diagram. Get solution
79. Get solution
80. Design a little dentist's mirror to be fixed at the end of a shaft for use in the mouth of some happy soul. The requirements are (1) that the image be erect as seen by the dentist and (2) that when held 1.5 cm from a tooth the mirror produces an image twice life-size. Get solution
81. An object is located at a distance s0 from a spherical mirror of radius R. Show that the resulting image will be magnified by an amount... Get solution
82. A device used to measure the radius of curvature of the cornea of the eye is called a keratomeler. This is useful information when tit-ting contact lenses. In effect, an illuminated object is placed a known distance from the eye, and the image reflected off the cornea is observed. The instrument allows the operator to measure the size of that virtual image. If the magnification is found to be 0.037× when the object distance is set at 100 mm, what is the radius of curvature? Get solution
83. Considering the operation of a spherical mirror, prove that the locations of the object and image are given bys0 =f(MT - 1)/MT and si = -f(MT - 1) Get solution
84. Aman whose face is 25 cm away looks into the bowl of a soup-spoon and sees his image reflected with a magnification of —0.064. Determine the radius of curvature of the spoon. Get solution
85. In an amusement park a large upright convex spherical mirror is facing a plane minor 10.0 m away. A girl 1.0 m tall standing midway between the two sees herself twice as tall in the plane mirror as in the spherical one. In other words, the angle subtended at the observer by the image in the plane mirror is twice the angle subtended by the image in the spherical mirror. What is the focal length of the latter? Get solution
86. A homemade telephoto "lens" (Fig. P.5.65) consists of two spherical mirrors. The radius of curvature is 2.0 m for the primary and 60 em for the secondary. How far from the smaller mirror should the film plane be located if the object is a star? What is the effective focal length of the system?... Get solution
87. A point source S sitting on the central axis of a positive thin lens is located (to the left) between one and two focal lengths from the lens. A concave spherical mirror is to be positioned to the right of the lens so that the final real image also lies at point S. Where should the mirror be placed? Where should a convex spherical mirror be located to accomplish the same feat? Get solution
88. Suppose you have a concave spherical mirror with a focal length of 10 cm. At what distance must an object be placed if its image is to be erect and one and a half times as large? What is the radius of curvature of the mirror? Cheek with Table 5.5. Get solution
89. Describe the image that would result for an object 3 inches tall placed 20 cm from a spherical concave shaving mirror having a radius of curvature of -60 cm. Get solution
90. Get solution
91. Parallel rays along the central axis enter a biconcave lens, both of whose radii of curvature are equal. Some of the light is reflected from the first surface, and the remainder passes through the lens. Show that, if the index of refraction of the lens (which is surrounded by air) is 2.00, the reflected image will fall at the same point as the image formed by the lens. Get solution
92. Referring to the dove prism in Fig. 5.63, rotate it through 90° about an axis along the ray direction. Sketch the new configuration and determine the angle through which the image is rotated. Get solution
93. Determine the numerical aperture of a single clad optical fiber given that the core has an index of 1.62 and the clad 1.52. When immersed in air, what is its maximum acceptance angle? What would happen to a ray incident at, say, 45°? Get solution
94. Get solution
95. Given a fused silica fiber with an attenuation of 0.2 dB/km how far can a signal travel along it before the power level drops by half? Get solution
96. Get solution
97. Get solution
98. Get solution
99. Get solution
100. Determine the intermodal delay (in us/km) for a stepped-index fiber with a cladding of index 1.485 and a core of index 1.500. Get solution
101. Using the information on the eye in Section 5.7.1, compute the approximate size (in milhmeters) of the image of the Moon as cast on the retina. The Moon has a diameter of 2160 miles and is roughly 230 000 miles from here, although tins, of course, varies. Get solution
102. Figure P.5.76 shows an arrangement in which the beam is deviated through a constant angle σ, equal to twice the angle β between the plane mirrors, regardless of the angle-of-incidence. Prove that this is indeed the case.Figure P.5.76... Get solution
103. An object 20 m from the objective (f0 = 4 m) of an astronomical telescope is imaged 30 cm from the eyepiece (/;, = 60 cm). Find the total linear magnification of the scope. Get solution
104. Figure P.5.78, which purports to show an erecting lens system, is taken from an old, out-of-print optics text. What's wrong with it?... Get solution
105. Figure P.5.79 shows a pin hole in an opaque screen being used for something practical. Explain what's happening and why it works. Try it.Figure P.5.79... Get solution
106. If a photograph of a moving merry-go-round is perfectly exposed, but blurred, at ... s and f/11, what must the diaphragm setting be if the shutter speed is raised to ...s in order to "stop" the motion? Get solution
107. The field of view of a simple two-element astronomical telescope is restricted by the size of the eye-lens. Make a ray sketch showing the vignetting that arises. Get solution
108. A field-lens, as a rule, is a positive lens placed at (or near) the intermediate image plane in order to collect the rays that would otherwise miss the next lens in the system. In effect, it increases the field of view without changing the power of the system. Redraw the ray diagram of the previous problem to include a field-iens. Show that as a consequence the eye relief is reduced somewhat. Get solution
109. Describe completely the image that results when a bug sits at the vertex of a thin positive lens. How does this relate directly to the manner in which a field-lens works? (See Problem 5.82.) Get solution
110. It is determined that a patient has a near point at 50 cm. If the eye is approximately 2.0 cm long(a) How much power does the refracting system have when focused on an object at infinity? when focused at 50 cm?(b) How much accommodation is required to see an object at a distance of 50 cm?(c) What power must the eye have to see clearly an object at the standard near-point distance of 25 cm?(d) How much power should be added to the patient's vision system by a correcting lens? Get solution
111. An optometrist finds that a farsighted person has a near point at 125 cm. What power will be required for contact lenses if they are effectively to move that point inward to a more workable distance of 25 cm so that a book can be read comfortably? Use the fact that if the object is imaged at the near point, it can be seen clearly. Get solution
112. Get solution
113. Get solution
114. Get solution
115. Get solution
116. Get solution
117. A farsighted person can see very distant mountains with relaxed eyes while wearing +3.2-D contact lenses. Prescribe spectacle lenses that will serve just as well when worn 17 mm in front of the cornea. Locate and compare the far* point in both cases. Get solution
118. A jeweler is examining a diamond 5.0 mm in diameter with a loupe having a focal length of 25.4 mm.(a) Determine the maximum angular magnification of the loupe.(b) How big does the stone appeal- through the magnifier?(c) What is the angle subtended by the diamond at the unaided eye when held at the near point?(d) What angle does it subtend at the aided eye? Get solution
119. Suppose we wish to make a microscope (that can be used with a relaxed eye) out of two positive lenses, both with a focal length of 25 mm. Assuming the object is positioned 27 mm from the objective, (a) how far apart should the lenses be, and (b) what magnification can we expect? Get solution
120. Figure P.5.89 shows a glancing-incidcnce X-ray focusing system designed in 1952 by Hans Wolter. How does it work? Microscopes with this type of system have been used to photograph, in X-rays, the implosion of fuel pellet targets in laser fusion research. Similar X-ray optical arrangements have been used in astronomical telescopes (see photos on p. 79).Figure P.5.89(a) ...(b) ... Get solution
121. The two glancing-incidence aspherical mirror systems depicted in Fig. P.5.90 are designed to focus X-rays. Explain how each works: identify the shapes of the mirrors, discuss the locations of their various foci, and so on.Figure P.5.90(a)...(b) ... Get solution
122. The orbiting Hubble Space Telescope has a 2.4-m primary, which we will assume to be diffraction limited. Suppose we wanted to use it to read the print on the side of a distant Russian satellite. Assuming that a resolution of 1.0 cm at the satellite will do, how far away could it be from the HST? Get solution
2. Construct a Cartesian oval such that the conjugate points will be separated by 11 cm when the object is 5 cm from the vertex. If n1 | = 1 and n2 = 3/2 draw several points on the required surface. Get solution
3. Use Fig. P.5.3 to show that if a point source is placed at the focus Fv of the ellipsoid, plane waves will emerge from the far side. Remember that the defining requirement for an ellipse is that the net distance from one focus to the curve and back to the other focus is constant.Figure P.5.3... Get solution
4. Diagrammatically construct both a sphero-clliptic positive lens and an elliplo-spheric negative lens, showing rays and wavefronts as they pass through the lens. Do the same for an oval-spheric positive lens. Get solution
5. Making use of Fig. P.5.5, Snell's Law, and the fact that in the paraxial region α = h/s0, φ ≈ h/R, and β ≈ h/si derive Eq. (5.8).Figure P.5.5... Get solution
6. Show that, in the paraxial domain, the magnification produced by a single spherical interface between two continuous media, as shown in Fig'. P.5.6, is given by...Use the small-angle approximation for Snell's Law and approximate the angles by their tangents.Figure P.5.6... Get solution
7. Imagine a hemispherical interface, with a radius of curvature of radius 5.00 cm, separating two media: air on the left, water on the right. A 3.00-cm-tall toad is on the central axis, in air, facing the convex interface and 30.0 cm from its vertex. Where in the water will it Re imaged? How big will it appear to a fish in the water? Use the results of the previous problem, even though our frog is pushing the paraxial approximation. Get solution
8. Locate the image of an object placed 1.2 m from the vertex of a gypsy's crystal ball, which has a 20-cm diameter (n1 = 1.5). Make a sketch (of the rays, not the gypsy). Get solution
9. Return to Problem 5.7 and suppose we cut off the medium on the right foiTning a thick water biconvex lens, with each surface having a radius of curvature of 5.00 cm. If the lens is 10.0 cm thick, determine the total magnification and everything you can about the toad's image. Get solution
10. A biconvex glass (n1 = 1.5) thin lens is to have a +10.0-cm focal length. If the radius of curvature of each surface is measured to 'be the same, what must it be? Show that a spider standing 1.0 cm frcm the lens will be imaged at -1.1 cm. Describe that image and draw a ray diagram. Get solution
11. Going back to Section 5.2.3, prove that for a thin lens mmersed in a medium of index nm....That done, imagine a double-concave air lens surrounded by water; determine if it's converging or diverging. Get solution
12. A meniscus concave glass (nl = 1.5) thin lens (see Fig. 5.12) has radii of curvature of +20.0 cm and +10.0 cm. If an object is placed 20.0 cm in front of the lens, show that the image distance will be -13.3 cm. Describe that image and draw a ray diagram. Get solution
13. A biconcave lens (n1, = 1.5) has radii of 20 cm and 10 cm and an axial thickness of 5 cm. Describe the image of an object 1-inch tall placed 8 cm from the first vertex. Use the thin-lens equation to see how far off it is in determining the final-image location. Get solution
14. A 35-mm camera has a single thin lens having a 50.0-mm focal length. A woman 1.7 m tall stands 10.0 m in front of the camera, (a) Show that the lens-film distance must be 50.3 mm. (b) How tall is her image on the film? Get solution
15. Prove that the minimum separation between conjugate real object and image points for a thin positive lens is 4f. Get solution
16. An object 2 cm high is positioned 5 cm to the right of a positive thin lens with a focal length of 10 cm. Describe the resulting image completely, using both the Gaussian and Newtonian equations. Get solution
17. Make a rough graph of the Gaussian Lens Equation; that is, plot Si versus s0, using unit intervals of f along each axis. (Get both segments of the curve.) Get solution
18. A parallel bundle of rays from a very distant point source is incident on a thin negative lens having a focal length of -50.0 cm. The rays make an angle of 6.0° with the optical axis of the lens. Locate the image of the source. Get solution
19. Get solution
20. What must the focal length of a thin negative lens be for it to form a virtual image 50 cm away (measured from the lens) of an ant located 100 cm away (measured from the lens)? Given that the ant is to the right of the lens, locate and describe its image. Get solution
21. Get solution
22. Compute the focal length in air of a thin biconvex lens (nl = 1.5) having radii of 20 and 40 cm. Locate and describe the image of an object 40 cm from the lens. Get solution
23. Determine the focal length of a planar-concave lens (nl = 1.5) having a radius of curvature of 10 cm. What is its power in diopters? Get solution
24. Get solution
25. Get solution
26. Determine the focal length in air of a thin spherical planar-convex lens having a radius of curvature of 50.0 mm and an index of 1.50. What, if anything, would happen to the focal length if the lens were placed in a tank of water? Get solution
27. Get solution
28. Get solution
29. Get solution
30. We would like to place an object 45 cm in front of a lens and have its image appear on a screen 90 cm behind the lens. What must be the focal length of the appropriate positive lens? Get solution
31. The horse in Fig. 5.26 is 2.25 m tall, and it stands with its face 15.0 m from the plane of the thin lens whose focal length is 3.00 m.(a) Determine the location of the image of the equine nose.(b) Describe the image in detail—type, orientation, and magnification.(c) How tall is the image?(d) If the horse's tail is ! 7.5 m from the lens, how long, nose-to-tail, is the image of the beast? Get solution
32. A candle that is 6.00 cm tall is standing 10 cm from a thin concave lens whose focal length is —30 cm. Determine the location of the image and describe it in detail. Draw an appropriate ray diagram. Get solution
33. The image projected by an cquiconvex lens (n = 1.50) of a frog 5.0 cm tall, who is located 0.60 m from a screen, is to be 25 cm high. Please compute the necessary radii of the lens. Get solution
34. Get solution
35. Get solution
36. Get solution
37. A thin piece of wire 4.00 mm long is located in a plane perpendicular to the optical axis and 60.0 cm in front of a thin lens. The sharp image of the wire formed on a screen is 2.00 mm long. What is the focal length of the lens? When the screen is moved farther from the lens by 10.0 mm, the image blurs to a width of 0.80 mm. What is the diameter of the lens? [Hint: Image a source point on the axis.] Get solution
38. A thin double-convex glass lens (with an index of 1.56) while surrounded by air has a 10-cm focal length. If it is placed under water (having an index of 1.33) 100 cm beyond a small fish, where will the guppy's image be formed? Get solution
39. Consider a homemade television projection system that uses a large positive lens to cast the image of the TV screen onto a wall. The projected picture is enlarged three times, and although dim, it's nice and clear. If the lens has a focal length of 60 cm, what should be the distance between the screen and the wall? Why use a large lens? How should we mount the set with respect to the lens? Get solution
40. Write an expression for the focal length (fw) of a thin lens immersed in water (nw = 4/3) in terms of its focal length when it's in air (fa) Get solution
41. Observe the three vectors ..., ...and ... in Fig. P.5.31, each of which has a length of 0.10 f where f is the focal length of the thin positive lens. The plane formed by ... and ... is at a distance of 1.10 f from (he lens. Describe the image of each vector.Figure P.5.31... Get solution
42. A convenient way to measure the focal length of a positive lens makes use of the following fact. If a pair of conjugate object and (real) image points (S and P) are separated by a distance L > 4f, there will be two locations of the lens, a distance d apart, for which the same pair of conjugates obtain. Show that...Note that this avoids measurements made specifically from the vertex, which are generally not easy to do. Get solution
43. Two positive lenses with focal lengths of 0.30 m and 0.50 m are separated by a distance of 0.20 m. A small butterfly rests on the central axis 0.50 m in front of the first lens. Locate the resulting image with respect to the second lens. Get solution
44. In the process of constructing a doublet, an equieonvex thin lens L2 is positioned in intimate contact with a thin negative lens, such that the combination has a focal length of 50 cm in air. If their indices are 1.50 and 1.55, respectively, and if the focal length of L2 —50 cm, determine all the radii of curvature. Get solution
45. Verify Eq. (5.34), which gives MT for a combination of two thin lenses. Get solution
46. A blade of grass standing 10.0 mm tall is 150 mm in from of a thin positive lens having a 100 mm focal length; 250 mm behind that first lens is a thin negative lens with a focal length of -75.0 mm. (a) Show that the first lens forms an image 300 mm behind it. (b) Describe that image, (c) What's its magnification? (d) Prove that the final image formed by both lenses is located 150 mm belund the negative lens, (e) What is the total magnification of the combination? Get solution
47. Compute the image location and magnification of an object 30 cm from the front doublet of the thin-lens combination in Fig. P.5.37. Do the calculation by finding the effect of each lens separately. Make a sketch of appropriate rays.Figure P.5.37... Get solution
48. Two thin lenses having focal lengths of + 15.0 cm and -15.0 cm are positioned 60.0 cm apart. A page of print is held 25.0 cm in front of the positive lens. Describe, in detail, the image of the print (i.e., insofar as it's paraxial). Get solution
49. Draw a ray diagram for the combination of two positive lenses wherein their separation equals the sum of their respective focal lengths. Do the same thing for the case in which one of the lenses is negative. Get solution
50. Two positive lenses are to be used as a laserbeam expander. An axial 1.0-mm diameter beam enters a short focal length positive lens, which is followed by a somewhat longer focal length positive lens from which it emerges with a diameter of 8.0 mm. Given that the first lens has a 50.0 mm focal length, determine the focal length of the second lens and the separation between the lenses. Draw a diagram. Get solution
51. Redraw the ray diagram for a compound microscope (Fig. 5.99), but this time treat the intermediate image as if it were a real object. This approach should be a bit simpler. Get solution
52. Consider a thin positive lens L1 and using a ray diagram, show that if a second lens L2 is placed at the focal point of L2, the magnification does not change. That's a good reason to wear eyeglasses, whose lenses are different, al the correct distance from the eye. Get solution
53. Figures P.5.43a and P.5.43b are taken from an introductory •physics book. What's wrong with them?Figure P.5.43a...Figure P.5.43b... Get solution
54. Get solution
55. Consider the case of two positive thin lenses, L1, and L2, separated by 5 cm. Their diameters are 6 and 4 cm, respectively, and their focal lengths are f1 = 9 cm and f2 = 3 cm. If a diaphragm with a hole 1 cm in diameter is located between them, 2 cm from L2, find (a) the aperture stop and (b) the locations and sizes of the pupils for an axial point, S, 12 cm in front of (to the left of) L1. Get solution
56. Get solution
57. Make a sketch roughly locating the aperture stop and entrance and exit pupils for the lens in Fig. P.5.45... Get solution
58. Make a sketch roughly locating the aperture stop and entrance and exit pupils for the lens in Fig. P.5.46, assuming the object point to be beyond (to the left of) F01.Figure P.5.46... Get solution
59. Get solution
60. Figure P.5.47 shows a lens system, an object, and the appropri-"ate pupils. Diagrammalically locate the image.... Get solution
61. Draw a ray diagram locating the images of a point source as formed by a pair of mirrors at 90° (Fig. P.5.48a). Now create a ray diagram locating the images of the arrow shown in Fig. P.5.486.Figure P.5.48a. ...b. ... Get solution
62. Examine Velasquez's painting of Venus and Cupid (Fig. P.5.49). Is Venus looking at herself in the mirror? Explain.Figure P.5.49 The Toilet of Venus by Diego Rodriguez de Silva y Veläquez (Courtesy of the Trustees, The National Gallery, London)... Get solution
63. Manet's painting The Bar at the Folies Bergeres (Fig. P.5.50) shows a girl standing in front of a large planar mirror. Reflected in it is her back and a man in evening dress with whom she appears to be talking. It would seem that Manet's intent was to give the uncanny feeling that the viewer is standing where that gentleman must be. From the laws of Geometrical Optics, what's wrong?Figure P.5.50 The Bar at (tie Folies Bergere by Edouard Manet (Courtesy of the Courtauld Institute Galleries, London. Courtauld Collection.)... Get solution
64. Show that Eq. (5.48) for a spherical surface is equally applicable to a plane mirror. Get solution
65. Get solution
66. Get solution
67. Get solution
68. Get solution
69. Get solution
70. Get solution
71. Locate the image of a paperclip 100 cm away from a convex spherical mirror having a radius of curvature of 80 cm. Get solution
72. Imagine that you are standing 5 feet from, and looking directly toward, a brass ball 1 foot in diameter hanging in front of a pawn shop. Describe the image you would see in the ball. Get solution
73. A thin lens having a focal length of +50.0 cm is positioned 250 cm in front of (i.e., to the left of) a plane mirror. An ant sits on the central axis 250 cm in front of (i.e., to the left of) the lens. Locate the three images of the ant. Get solution
74. The image of a red rose is formed by a concave spherical mirror on a screen 100 cm away. If the rose is 25 cm from the mirror, determine its radius of curvature. Get solution
75. From the image configuration determine the shape of the mirror hanging on the back wall in van Eyck's painting of John Amulfini and His Wife (Fig. P.5.56).Figure P.5.56 Detail of John Amolfini and His Wife by Jan van Eyck. (Courtesy of the Trustees, The National Gallery, London... Get solution
76. Get solution
77. There are several varieties of retro-reflector that are commercially available; one type is comprised of transparent spheres, the backs of which arc silvered. Light is refracted at the front surface, focused onto the rear surface, and there reflected back out in the direction it came. Determine the necessary index of refraction of the spheres. Assume the incident light is collimated. Get solution
78. Design an eye for a robot using a concave spherical mirror such that the image of an object 1.0 m tall and 1.0 cm away fills its 1.0-cm-square photosensitive detector (which is movable for focusing purposes). Where should this detector be located with respect to the mirror? What should be the focal length of the mirror? Draw a ray diagram. Get solution
79. Get solution
80. Design a little dentist's mirror to be fixed at the end of a shaft for use in the mouth of some happy soul. The requirements are (1) that the image be erect as seen by the dentist and (2) that when held 1.5 cm from a tooth the mirror produces an image twice life-size. Get solution
81. An object is located at a distance s0 from a spherical mirror of radius R. Show that the resulting image will be magnified by an amount... Get solution
82. A device used to measure the radius of curvature of the cornea of the eye is called a keratomeler. This is useful information when tit-ting contact lenses. In effect, an illuminated object is placed a known distance from the eye, and the image reflected off the cornea is observed. The instrument allows the operator to measure the size of that virtual image. If the magnification is found to be 0.037× when the object distance is set at 100 mm, what is the radius of curvature? Get solution
83. Considering the operation of a spherical mirror, prove that the locations of the object and image are given bys0 =f(MT - 1)/MT and si = -f(MT - 1) Get solution
84. Aman whose face is 25 cm away looks into the bowl of a soup-spoon and sees his image reflected with a magnification of —0.064. Determine the radius of curvature of the spoon. Get solution
85. In an amusement park a large upright convex spherical mirror is facing a plane minor 10.0 m away. A girl 1.0 m tall standing midway between the two sees herself twice as tall in the plane mirror as in the spherical one. In other words, the angle subtended at the observer by the image in the plane mirror is twice the angle subtended by the image in the spherical mirror. What is the focal length of the latter? Get solution
86. A homemade telephoto "lens" (Fig. P.5.65) consists of two spherical mirrors. The radius of curvature is 2.0 m for the primary and 60 em for the secondary. How far from the smaller mirror should the film plane be located if the object is a star? What is the effective focal length of the system?... Get solution
87. A point source S sitting on the central axis of a positive thin lens is located (to the left) between one and two focal lengths from the lens. A concave spherical mirror is to be positioned to the right of the lens so that the final real image also lies at point S. Where should the mirror be placed? Where should a convex spherical mirror be located to accomplish the same feat? Get solution
88. Suppose you have a concave spherical mirror with a focal length of 10 cm. At what distance must an object be placed if its image is to be erect and one and a half times as large? What is the radius of curvature of the mirror? Cheek with Table 5.5. Get solution
89. Describe the image that would result for an object 3 inches tall placed 20 cm from a spherical concave shaving mirror having a radius of curvature of -60 cm. Get solution
90. Get solution
91. Parallel rays along the central axis enter a biconcave lens, both of whose radii of curvature are equal. Some of the light is reflected from the first surface, and the remainder passes through the lens. Show that, if the index of refraction of the lens (which is surrounded by air) is 2.00, the reflected image will fall at the same point as the image formed by the lens. Get solution
92. Referring to the dove prism in Fig. 5.63, rotate it through 90° about an axis along the ray direction. Sketch the new configuration and determine the angle through which the image is rotated. Get solution
93. Determine the numerical aperture of a single clad optical fiber given that the core has an index of 1.62 and the clad 1.52. When immersed in air, what is its maximum acceptance angle? What would happen to a ray incident at, say, 45°? Get solution
94. Get solution
95. Given a fused silica fiber with an attenuation of 0.2 dB/km how far can a signal travel along it before the power level drops by half? Get solution
96. Get solution
97. Get solution
98. Get solution
99. Get solution
100. Determine the intermodal delay (in us/km) for a stepped-index fiber with a cladding of index 1.485 and a core of index 1.500. Get solution
101. Using the information on the eye in Section 5.7.1, compute the approximate size (in milhmeters) of the image of the Moon as cast on the retina. The Moon has a diameter of 2160 miles and is roughly 230 000 miles from here, although tins, of course, varies. Get solution
102. Figure P.5.76 shows an arrangement in which the beam is deviated through a constant angle σ, equal to twice the angle β between the plane mirrors, regardless of the angle-of-incidence. Prove that this is indeed the case.Figure P.5.76... Get solution
103. An object 20 m from the objective (f0 = 4 m) of an astronomical telescope is imaged 30 cm from the eyepiece (/;, = 60 cm). Find the total linear magnification of the scope. Get solution
104. Figure P.5.78, which purports to show an erecting lens system, is taken from an old, out-of-print optics text. What's wrong with it?... Get solution
105. Figure P.5.79 shows a pin hole in an opaque screen being used for something practical. Explain what's happening and why it works. Try it.Figure P.5.79... Get solution
106. If a photograph of a moving merry-go-round is perfectly exposed, but blurred, at ... s and f/11, what must the diaphragm setting be if the shutter speed is raised to ...s in order to "stop" the motion? Get solution
107. The field of view of a simple two-element astronomical telescope is restricted by the size of the eye-lens. Make a ray sketch showing the vignetting that arises. Get solution
108. A field-lens, as a rule, is a positive lens placed at (or near) the intermediate image plane in order to collect the rays that would otherwise miss the next lens in the system. In effect, it increases the field of view without changing the power of the system. Redraw the ray diagram of the previous problem to include a field-iens. Show that as a consequence the eye relief is reduced somewhat. Get solution
109. Describe completely the image that results when a bug sits at the vertex of a thin positive lens. How does this relate directly to the manner in which a field-lens works? (See Problem 5.82.) Get solution
110. It is determined that a patient has a near point at 50 cm. If the eye is approximately 2.0 cm long(a) How much power does the refracting system have when focused on an object at infinity? when focused at 50 cm?(b) How much accommodation is required to see an object at a distance of 50 cm?(c) What power must the eye have to see clearly an object at the standard near-point distance of 25 cm?(d) How much power should be added to the patient's vision system by a correcting lens? Get solution
111. An optometrist finds that a farsighted person has a near point at 125 cm. What power will be required for contact lenses if they are effectively to move that point inward to a more workable distance of 25 cm so that a book can be read comfortably? Use the fact that if the object is imaged at the near point, it can be seen clearly. Get solution
112. Get solution
113. Get solution
114. Get solution
115. Get solution
116. Get solution
117. A farsighted person can see very distant mountains with relaxed eyes while wearing +3.2-D contact lenses. Prescribe spectacle lenses that will serve just as well when worn 17 mm in front of the cornea. Locate and compare the far* point in both cases. Get solution
118. A jeweler is examining a diamond 5.0 mm in diameter with a loupe having a focal length of 25.4 mm.(a) Determine the maximum angular magnification of the loupe.(b) How big does the stone appeal- through the magnifier?(c) What is the angle subtended by the diamond at the unaided eye when held at the near point?(d) What angle does it subtend at the aided eye? Get solution
119. Suppose we wish to make a microscope (that can be used with a relaxed eye) out of two positive lenses, both with a focal length of 25 mm. Assuming the object is positioned 27 mm from the objective, (a) how far apart should the lenses be, and (b) what magnification can we expect? Get solution
120. Figure P.5.89 shows a glancing-incidcnce X-ray focusing system designed in 1952 by Hans Wolter. How does it work? Microscopes with this type of system have been used to photograph, in X-rays, the implosion of fuel pellet targets in laser fusion research. Similar X-ray optical arrangements have been used in astronomical telescopes (see photos on p. 79).Figure P.5.89(a) ...(b) ... Get solution
121. The two glancing-incidence aspherical mirror systems depicted in Fig. P.5.90 are designed to focus X-rays. Explain how each works: identify the shapes of the mirrors, discuss the locations of their various foci, and so on.Figure P.5.90(a)...(b) ... Get solution
122. The orbiting Hubble Space Telescope has a 2.4-m primary, which we will assume to be diffraction limited. Suppose we wanted to use it to read the print on the side of a distant Russian satellite. Assuming that a resolution of 1.0 cm at the satellite will do, how far away could it be from the HST? Get solution
