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If the sum of the heights of transmitting and receiving antennas in line of sight of communication is fixed at h, show that the range is maximum when the two antennas have a height `h//2` each. |
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Answer» The range (r) of line of sight communication between two antennas (during space wave propagation) is given by `r = sqrt(2Rh_(T)) + sqrt(2Rh_(R))` where `h_(T) and h_(R)` are the height of transmitting and receiving antennas. Here, `h_(T) + h_(R) = h. If h_(T) = H, then h_(R) = h-H` Then `r = sqrt(2R)[sqrt(H) + sqrt(h-H)]` For r to be maximum, `(dr)/(dH) = 0` `:. (dr)/(dH) = 0 = sqrt(2R) [ 1/(2sqrtH) + (1)/(2sqrt (h-H)xx(-1))]` or `1/(2sqrtH) - 1/(2sqrt(h-H)) = 0` or `2sqrt(H) = 2 sqrt(h-H)` or `H = h-H or h = 2H` or `H = h//2` Hence, for maximum range, `h_(T) = h_(R) = h//2`. |
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