ph2a_quiz3_soln

# ph2a_quiz3_soln - Quiz 3 Solutions by Jonathan Arnold...

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Quiz 3: Solutions by Jonathan Arnold November 17, 2009 Problem 1 A ”phased-array” of radio antennas can be electronically “pointed” by introduc- ing regular phase delays between successive antennas. Assume that the system operates at frequency ν and wavelength λ . The antenna are “line” shaped, much longer than a single wavelength in the dimension out of the plane. (a) [2 pts] Determine the smallest phase delay Δ φ (in radians) that causes the antenna to direct its transmitted beam into the angle θ , in terms of θ , λ , and d . Solution: We want to find the phase difference between neighboring antennas such that along parallel rays, each with an angle θ as drawn, the waves end up in-phase along the dotted lines. We first note that for positive θ as drawn, the source to the right in each pair should lag the source to the left. The phase lag needed to produce rays in-phase at angle θ is given by Δ φ = k Δ x. (1) 1

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Using the definition k = 2 π/λ and Δ x = d sin θ from the geometry of the figure, we find Δ φ = 2 πd sin θ λ . (2) This should not be surprising; it is the phase difference we associated with a given angle for the double-slit goemetry. In this case, we need our initial phase Δ φ to cancel the phase coming from the difference in path length. This tells us that
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ph2a_quiz3_soln - Quiz 3 Solutions by Jonathan Arnold...

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