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C-62 Appendix C r A s¡ s B ∫ Figure B (i) (ii) striking the flat surface of the lens at a distance r from the center of the lens and, after refraction at that surface, striking the slanted groove surface where it is again refracted. The groove angle b is measured from the vertical as is shown in an enlarged view in Figure B(ii). Given s 1 , s 2 , and n, we need to show how, for each value of r, we can find a groove angle b such that after refraction the ray passes through point B. Figure C(i) shows a simplified version of Figure B, in which we ignore the thickness of the lens since it is usually negligible compared to s 1 and s 2 . We also introduce the angles u 1 and u 2 , and (using what theorem from geometry?) show them, measured from the horizontal, in Figure C(ii). As in Figure B, Figure C(ii) is an enlarged view of the region in which the ray indicated in Figure C(i) passes through the lens. We are now ready to derive a formula for b. Figure D shows the path of the ray through the lens with all the relevant angles labeled. Angles w 1 and w œ 1 ¨¡ r ¨ ¨¡ ¨ A s¡ s B ∫ Figure C (i) (ii) Index of refraction: 1 ϕª¡ ϕª ¨ ϕ ϕ¡ ¨¡ Index of refraction: n ∫ Index of refraction: 1 Figure D
Appendix C C-63 denote the angles of incidence and refraction at the first surface, and angles and w 2 denote the angles of incidence and refraction at the second surface. Exercise 1 (a) Given r, s 1 , and s 2 , explain how to use a calculator to find u 1 and u 2 . (b) Explain why w 1 u 1 , w 2 b u 2 , and w1 œ w2 œ b. (c) Using Snell’s law and part (b) show that sin u 1 n sin w1 œ n sin w2 œ sin(b u 2 ) n sin w2 œ n sin(b w1) œ and conclude that n sin(b w1) œ sin(b u 2 ) (4) (d) Use an addition formula for sine and a Pythagorean identity to show that n sin(b w1) œ (sin b)1n21 sin 2 w12 œ (cos b)(n sin w1) œ (5) Use equation (1) to obtain n sin(b w1) œ (sin b) 2n 2 sin 2 u 1 cos b sin u 1 (6) Then use equations (4) and (6) and an addition formula for sine to obtain (sin b) 2n 2 sin 2 u 1 cos b sin u 1 sin b cos u 2 cos b sin u 2 (7) (e) It is surprising that equation (7) can be be solved for b in a few steps. Divide both sides by cos b and solve for tan b to derive the Fresnel lens groove angle formula w œ 2 (1) (2) (3) tan b sin u 1 sin u 2 2n 2 sin 2 u 1 cos u 2 (8) Exercise 2 An optical designer must design a Fresnel lens 80 millimeters in diameter to gather light from a lamp 100 mm from the lens and cause it to pass through a small opening 400 mm from the lens on the side opposite the lamp. The index of refraction of the lens material is 1.50. She lets s 1 100 mm and s 2 400 mm, decides to make the width of each groove 0.4 mm, and calculates that there will be 100 grooves. (How?) She writes a computer program to generate a file containing the 100 groove angles, in degrees, which will be the input to another computer program that will generate commands for a computer-controlled machine (a lathe in this case) that will produce the lens. Before trying to make the lens, she decides to calculate the groove angle for grooves numbered 1, 25, 50, and 100, and use the calculated values to check the output of her program. She makes a table with column headings r in mm and u 1 , u 2 , and b in degrees. Her first r-value is 0.200 mm and her second is 9.800 mm, and she calculates u 1 , u 2 , and b to four decimal places. Your job as her assistant is to produce your own version of the table as another check of her work. Use the results of Exercise 1 parts (a) and (e) to produce the table.
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APPENDIX C Title and Page Number Pr
Page 3 and 4:
Appendix C C-3 MINI PROJECT Discuss
Page 5 and 6:
Appendix C C-5 MINI PROJECT Thinkin
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Appendix C C-7 1. equilateral trian
Page 9 and 10:
Appendix C C-9 MINI PROJECT Flying
Page 11 and 12: Appendix C C-11 MINI PROJECT An Ine
Page 13 and 14: Appendix C C-13 (a) The two data po
Page 15 and 16: Appendix C C-15 and http://www.svob
Page 17 and 18: Appendix C C-17 For Figure D we can
Page 19 and 20: 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: Appendix C C-61 PROJECT The Design
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 and 72: Appendix C C-71 PROJECT Snell’s L
Page 73 and 74: Appendix C C-73 vertically downward
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 (
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Appendix C C-115 4. Simplify the fo
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