Two radio antennas separated by d = 286 m as shown in the figure below simultane

Question

Two radio antennas separated by d = 286 m as shown in the figure below simultaneously broadcast identical signals at the same wavelength. A car travels due north along a straight line at position x = 1 270 m from the center point between the antennas, and its radio receives the signals. (a) If the car is at the position of the second maximum after that at point O when it has traveled a distance y = 400 m northward, what is the wavelength of the signals? (b) How much farther must the car travel from this position to encounter the next minimum in reception?

Explanation / Answer

a) m = 2, tan theta = 400 / 1270 m

theta = 17.48 deg

d sin theta = m lambda

lambda = 286 * sin 17.48 / 2 = 42.95 m

b) d sin theta = (m + 1/2) lambda [m = 2]

d sin theta = 5/2 * lambda

sin theta = 5 lambda / 2 d = (5 * 42.95) / (2 * 286) = 0.375

theta = 22.05 deg

tan theta = ymin / 1270

ymin = 1270 * tan 22.05 = 514.4 m

The car should move additional = 514.4 - 400 = 114.4 m

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