Exam 2 - University of Alabama Department of Physics and...

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University of Alabama Department of Physics and Astronomy Department of Electrical and Computer Engineering PH 495/ECE 493 LeClair & Kung Spring 2011 Problem Set 5 Instructions: 1. Answer all questions below. All questions have equal weight. 2. Show your work for full credit. 3. All problems are due Friday 25 March 2011 by 11:59pm. 4. You may collaborate, but everyone must turn in their own work. Problems 9 and 10 involve a take-home experiment, for which you will need 2 glass slides and a plano-convex lens. These items will be provided in lecture on 10 March, or you may pick them up from Dr. LeClair by appointment after that time. 1. Bekefi & Barrett 8.2; Hecht 9.24 A radar antenna operating on a wavelength of 0.10 m is located 8 m above the water line of a torpedo boat. Treat the reflected beam from the water as originating in a source 8 m below the water directly under the radar antenna. The dipole antenna is oriented perpendicular to the plane of the page. 8m R θ x (a) What is the altitude x of an airplane 12 km from the boat if it is to be in the first interference minimum of the radar signal? (b) What is the total number of minima one observes as one scans the sky in the vertical plane as a function of the angle θ , from θ = 0 to θ = π , keeping the distance R fixed? 2. Bekefi & Barrett 8.3 Two dipole radiators (e.g., the oscillating current segments we discussed in class) are separated by a distance λ / 2 along the x axis (half-wave dipole antenna). The dipoles are oriented along z , as in the problem we worked in class. Assume the distance to the observation point r satisfies r λ .
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(a) Plot the intensity of radiation in the x y plane. Note the values of intensity at θ = 0, π / 3, π / 2, π if the oscillators are in phase. i (b) Repeat (a) if the oscillators are 180 out of phase. (c) The oscillators are now spaced by a distance λ / 4 and are 90 out of phase. Repeat (a). Note that this configuration would be very useful for a broadcast station in a coastal city, for example . . . 3. Bekefi & Barrett 8.5 We desire to superpose the oscillations of several simple harmonic oscillators having the same frequency ω and amplitude A , but di ff ering from one another by constant phase increments α ; that is, E ( t ) = A cos ω t + A cos ( ω t + α ) + A cos ( ω t + 2 α ) + A cos ( ω t + 3 α ) + · · · (1) (a) Using graphical phasor addition, find E ( t ) ; that is, writing E ( t )= A o cos ( ω t + ϕ ) , find A o and ϕ for the case when there are five oscillators with A = 3 units and α = π / 9 radians. (b) Study the polygon you obtained in part (a) and, using purely geometrical considerations, show that for N oscillators E ( t ) = ( NA ) sin N α / 2 N sin α / 2 cos ω t + N 1 2 α (2) (c) Sketch the amplitude of E ( t ) as a function of α . The above calculation is the basis of finding radiation from antenna arrays and di ff raction gratings.
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