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solutions_chapter26

2670 set up for destructive interference the net

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order that displays the entire visible spectrum. 26.70. Set Up: For destructive interference the net phase difference must be which is one-half a period, or In phase, A is or ahead of B . At points above the centerline, points are closer to A than to B and the signal from A gains phase relative to B because of the path difference. Destructive interference will occur when 1, At points at an angle below the centerline, the signal from B gains phase relative to A because of the phase difference. Destructive interference will occur when 1, Solve: Points above the centerline: Points below the centerline: 26.71. Set Up: The waves reflected from the surface of the water undergo a half-cycle phase shift and the waves that travel directly to the plane have no reflection and therefore no shift. The condition for constructive interference therefore is where is the path difference for the two sets of waves. Let y be the height of the airplane above the water. Let be the height of the cliffs. The two paths are sketched in Figure 26.71. The law of reflection says that the two angles labeled are equal. The diagram is not to scale, since is actually much larger than h or y . Figure 26.71 Solve: We first solve for the path difference where and and r 2 b 5 y sin u 5 y " D 2 1 1 h 1 y 2 2 h 1 y . r 2 a 5 h sin u 5 h " D 2 1 1 h 1 y 2 2 h 1 y . sin u 5 h 1 y " D 2 1 1 h 1 y 2 2 . sin 2 u 5 tan 2 u 1 1 tan 2 u 5 1 h 1 y 2 2 D 2 1 1 h 1 y 2 2 tan u 5 h 1 y D . D a 1 D b 5 h 1 y tan u D 5 D b 5 y tan u . D a 5 h tan u . r 2 5 r 2 a 1 r 2 b . r 2 2 r 1 , u u D a r 1 r 2 a r 2 b D b h h 2 y y Cliffs D 5 D a 1 D b 5 10.0 3 10 3 m u h 5 200 m r 2 2 r 1 r 2 2 r 1 5 A m 1 1 2 B l , u 5 55.2°. m 5 3: u 5 37.0°; m 5 2: u 5 22.5°; m 5 1: u 5 9.45°; m 5 0: sin u 5 A m 1 3 4 B l d 5 A m 1 3 4 B 1 0.766 m 3.50 m 2 5 0.219 A m 1 3 4 B . u 5 68.6°. m 5 4: u 5 45.4°; m 5 3: u 5 29.5°; m 5 2: u 5 15.9°; m 5 1: u 5 3.14°; m 5 0: sin u 5 A m 1 1 4 B l d 5 A m 1 1 4 B 1 0.766 m 3.50 m 2 5 0.219 A m 1 1 4 B . l 5 340 m / s 444 Hz 5 0.766 m. l 5 c f . 2, c . m 5 0, d sin u 5 A m 1 3 4 B l , u 2, c m 5 0, d sin u 5 A m 1 1 4 B l , l / 4, 90°, l / 2. 180°, 2 l 2 d . 1 1 d 5 1230 lines / mm. d 5 812 nm 5 8.12 3 10 2 4 mm. 700 nm 2 1 0.866 21 400 nm 2 5 0.500 " d 2 2 1 400 nm 2 2 0.866 l 1 d 1 0.500 Å 1 2 l 1 2 d 2 5 l 2 d . cos u 5 " 1 2 sin 2 u 5 Å 1 2 l 1 2 d 2 , 1 cos u 21 0.500 2 5 l 2 d . 1 sin u 21 0.866 2 1 sin 1 u 1 30.0° 2 5 sin u cos 30.0° 1 cos u sin 30.0°, sin 1 u 1 30.0° 2 5 l 2 d . sin u 5 l 1 d . u 1 30.0°. l 2 5 700 nm u , l 1 5 400 nm u . l 1 / d . m 5 1. d sin u 5 m l . 26-16 Chapter 26
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The path difference is For constructive interference and The altitudes are 50 m, 150 m, 250 m, Reflect: Once y is calculated, and can be calculated directly and you can verify that gives gives etc. r 2 2 r 1 5 3 l / 2, y 5 150 m r 2 2 r 1 5 l / 2, y 5 50 m r 2 r 1 c y 5 D l 2 h A m 1 1 2 B 5 1 10.0 3 10 3 m 21 4.00 m 2 2 1 200 m 2 A m 1 1 2 B 5 1 100 m 2 A m 1 1 2 B 5 1 100 m 2 m 1 50 m. 2 hy D 5 A m 1 1 2 B l r 2 2 r 1 5 1 h 1 y 2 2 2 1 h 2 y 2 2 2 D 5 2 hy D . r 1 5 " D 2 1 1 h 2 y 2 2 5 D Å 1 1 1 h 2 y 2 2 D 2 < D 1 1 1 1 2 1 h 2 y 2 2 D 2 2 5 D 1 1 h 2 y 2 2 2 D . r 2 5 r 2 a 1 r 2 b 5 " D 2 1 1 h 1 y 2 2 5 D Å 1 1 1 h 1 y 2 2 D 2 < D 1 1 1 1 2 1 h 1 y 2 2 D 2 2 5 D 1 1 h 1 y 2 2 2 D .
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