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10hw10 - Physics 315 Oscillations and Waves Homework 10 Due...

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Physics 315: Oscillations and Waves Homework 10: Due in class on Wednesday, Dec. 2nd 1. (a) Consider the geometric series S = summationdisplay n =0 ,N - 1 z n , where z is a complex number. Demonstrate that S = 1 - z N 1 - z . (b) Suppose that z = e i θ , where θ is real. Employing the well-known identity sin θ 1 2 i ( e i θ - e - i θ ) , show that S = e i( N - 1) θ/ 2 sin( N θ/ 2) sin( θ/ 2) . (c) Finally, making use of de Moivre’s theorem, e i n θ cos( n θ ) + i sin( n θ ) , demonstrate that C = summationdisplay n =1 ,N cos( α y n ) , where y n = [ n - ( N + 1) / 2] d, evaluates to C = sin( N α d/ 2) sin( α d/ 2) .
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2. Suppose that a monochromatic laser of wavelength 632 . 8 nm emits a diffrac- tion-limited beam of initial diameter 2 mm. Estimate how large a light spot the beam would produce on the surface of the Moon (which is a mean distance 3 . 76 × 10 5 km from the surface of the Earth). Neglect any effects of the Earth’s atmosphere. 3. Estimate how far away an automobile is when you can only just barely resolve the two headlights with your eyes.
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