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C-72 Appendix C an index of refraction very close to 1, for water it’s about 43, and for many glasses and plastics it’s about 32. Note: Although it is often left out of statements of Snell’s Law, part I is essential because light rays travel in three-dimensional space where three lines can intersect at a point, but not lie in the same plane. (You should think of an example.) The third principle concerns vision. The eye can see a light source if and only if light rays pass from the source into the pupil of the eye. In Figure B, the eye labeled E can see the light source S because rays pass from the source to the eye through the gap in the blackened surrounding sphere. However, the eye labeled F can’t see S because light traveling along the line of sight SF is blocked before reaching the eye. E S Figure B F Our fourth principle is that an object appears to be located in the direction determined by the rays entering the eye from the object. This is illustrated in Figure C(i), where one ray from the fish’s mouth at A appears to come from the direction of A. While the single ray in Figure C(i) suffices to give the apparent direction to the fish’s mouth, it is not sufficient to determine the distance to A. To determine this distance we must consider several rays coming from point A. Figure C(ii) shows several such rays which, after bending at the water’s surface, all appear to come from point A. The fact that all of the rays entering the eye from point A appear to come from a single point after refraction is quite surprising. We will examine this in more detail below. Aª Aª A A Figure C (i) (ii) Although the principles we’ve stated do not completely define geometric optics, they are sufficient to allow us to analyze the experiment discussed in the opening quotation. Figure D shows a greatly exaggerated eye looking
Appendix C C-73 vertically downward to the point at the center of a coin lying at the bottom of a cup of water. The solid red rays emerge from this point and enter the eye after bending at the water’s surface. The ray emerging from the center of the coin to the center of the eye is called the chief ray. On the right side of the figure, one ray emerging from the center of the coin is shown refracted at the surface of the water and entering the pupil of the eye. The dashed line extends the part of this ray in air back through the water intersecting the chief ray. We could perform this same construction for any other non-chief ray in Figure D. Our goal is to show that all the dashed extensions would intersect the chief ray at approximately the same point. In Figure E we lay out the geometry for a typical ray in great detail. Air: nª=1 P ˙ª 1 y O yª A ˙ I B 4 Water: n= 3 Figure D Figure E Exercise 1 (a) Show that a ray emerging from the coin making an angle f with the vertical intersects the air-water boundary with angle of incidence f. Then show that angle BPI is equal to the angle of refraction f. (b) Show that the length of line segment IB is sin f, which by Snell’s law is n sin f, then show that the length of PB is 21 n 2 sin 2 f. Caution: Pay attention to the direction of the rays in Figure E. (c) Show that the length of line segment OI is y tan f. (d) Show that triangles IOA and IBP are similar. y tan f21 n 2 sin 2 f (e) Use parts (a) through (d) to show that y¿ and n sin f simplify to show y¿ 21 n2 sin 2 f cos f 1 n y The last formula allows us to compute the point at which the center of the coin would appear to be located if we just consider the chief ray and a particular ray emerging from the center point at an angle f to the vertical. To complete
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APPENDIX C Title and Page Number Pr
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Appendix C C-3 MINI PROJECT Discuss
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Appendix C C-5 MINI PROJECT Thinkin
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Appendix C C-7 1. equilateral trian
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Appendix C C-9 MINI PROJECT Flying
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Appendix C C-11 MINI PROJECT An Ine
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Appendix C C-13 (a) The two data po
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Appendix C C-15 and http://www.svob
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Appendix C C-17 For Figure D we can
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Appendix C C-19 MINI PROJECT A Grap
Page 21 and 22: Appendix C C-21 MINI PROJECT Who Ar
Page 23 and 24: Appendix C C-23 MINI PROJECT 1 How
Page 25 and 26: Appendix C C-25 either the quadrati
Page 27 and 28: Appendix C C-27 PROJECT The Least-S
Page 29 and 30: Appendix C C-29 Then, for any line
Page 31 and 32: Appendix C C-31 PROJECT Finding Som
Page 33 and 34: Appendix C C-33 3. When a person co
Page 35 and 36: Appendix C C-35 2. Use part (b) of
Page 37 and 38: Appendix C C-37 PROJECT Coffee Temp
Page 39 and 40: Appendix C C-39 MINI PROJECT More C
Page 41 and 42: Appendix C C-41 The calculations co
Page 43 and 44: Appendix C C-43 PROJECT Transits of
Page 45 and 46: Appendix C C-45 E A ¨ V ¨ Bª d S
Page 47 and 48: Appendix C C-47 PROJECT A Linear Ap
Page 49 and 50: Appendix C C-49 Continuing with the
Page 51 and 52: Appendix C C-51 PROJECT Constructin
Page 53 and 54: Appendix C C-53 1 y Figure A A grap
Page 55 and 56: Appendix C C-55 PROJECT Fourier Ser
Page 57 and 58: Appendix C C-57 sine and cosine fun
Page 59 and 60: Appendix C C-59 PROJECT The Motion
Page 61 and 62: Appendix C C-61 PROJECT The Design
Page 63 and 64: Appendix C C-63 denote the angles o
Page 65 and 66: Appendix C C-65 inverse trigonometr
Page 67 and 68: Appendix C C-67 PROJECT Inverse Sec
Page 69 and 70: Appendix C C-69 Alternatively, usin
Page 71: Appendix C C-71 PROJECT Snell’s L
Page 75 and 76: 0 0 0 Appendix C C-75 PROJECT Vecto
Page 77 and 78: Appendix C C-77 Exercise 8 We outli
Page 79 and 80: Appendix C C-79 y y=2x+5 Îy=2 A (1
Page 81 and 82: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Appendi
Page 83 and 84: Appendix C C-83 PROJECT Parameteriz
Page 85 and 86: Appendix C C-85 It is worth noting
Page 87 and 88: Appendix C C-87 6. Show that the li
Page 89 and 90: Appendix C C-89 2. An altitude of a
Page 91 and 92: Appendix C C-91 PROJECT The Leontie
Page 93 and 94: Appendix C C-93 n x units of that
Page 95 and 96: Appendix C C-95 him/herself through
Page 97 and 98: Appendix C C-97 PROJECT The Leontie
Page 99 and 100: Appendix C C-99 As in the solution
Page 101 and 102: Appendix C C-101 MINI PROJECT 2 Con
Page 103 and 104: Appendix C C-103 PROJECT Using Hype
Page 105 and 106: Appendix C C-105 PROJECT Two Method
Page 107 and 108: Appendix C C-107 (b) Use Method 1 t
Page 109 and 110: Appendix C C-109 MINI PROJECT An Un
Page 111 and 112: Appendix C C-111 PROJECT More on Su
Page 113 and 114: Appendix C C-113 SOLUTION 25 25 a (
Page 115: Appendix C C-115 4. Simplify the fo
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