PHY7C FNT4_MoreInterference

PHY7C FNT4_MoreInterference - FNT#4 More interference...

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FNT #4 More interference August 15, 2005 Question 1: KDVS radio station Suppose that KDVS, the campus radio station, transmitted from two an- tennas placed λ/ 4 apart in the West-East direction. Suppose also that the constant phase of antenna A is π/ 2 (90 ) ahead of antenna B. B N S E W A λ/4 a) For each of the four major compass points, is the interference constructive, destructive or partial? The phase difference in each case is ∆Φ = - 2 π λ ( x A - x B ) + φ A - φ B = - 2 π λ ( x A - x B ) + π 2 as φ A is ahead of (i.e. greater than) φ B . (The minus sign is because the wave is travelling out, so that as t increases the positive distance from both station A and station B increases.) North and south The path difference for a point on the north-south axis is zero, as x A = x B . 1
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B B x A x N S E W A The total phase difference is then ∆Φ = - 2 π λ (0) + π 2 = π 2 which is partial interference. East A B x B x N S E W A For a point in the East we have x A = x B + λ/ 4. This leads to a total phase difference of ∆Φ = - 2 π λ ( x A - x B ) + π 2 = - 2 π λ parenleftbigg a8 a8 x B + λ 4 - a8 a8 x B parenrightbigg + π 2 = - 2 π a19 λ a19 λ 4 + π 2 = - π 2 + π 2 = 0 which means that we have constructive interference in the East. West B B x A x N S E W A 2
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For a point in the West we have x A = x B - λ/ 4, leading to an interference of ∆Φ = - 2 π λ ( x A - x B ) + π 2 = - 2 π λ parenleftbigg a8 a8 x B - λ 4 - a8 a8 x B parenrightbigg + π 2 = + 2 π a19 λ a19 λ 4 + π 2 = + π 2 + π 2 = π which leads to destructive interference. b) Why would they choose to do this? Where should KDVS set up their radio station? We have a given amount of power that we are turning into radio waves. By using this set up, we are generating no signal travelling to the east, but a larger signal going to the west.
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