Two radio antennas separated by d = 270 m as shown in the figure below simultane

Question

Two radio antennas separated by d = 270 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 = 1 160 m from the center point between the antennas, and its radio receives the signals. Note: Do not use the small-angle approximation in this problem.

(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?

________ m

(b) How much farther must the car travel from this position to encounter the next minimum in reception?

_____ m

Explanation / Answer

a) m = 2

tan theta = 400 / 1160 = 0.344

theta = 19.0 deg

d sin theta = m lambda

lambda = d sin theta / 2 = 270 * sin 19 / 2 = 43.95 m

-------------------------------------------------

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

d sin theta = 5/2 * lambda

sin theta = 5 lambda / 2 d = 5 * 43.95 / 2 * 270 = 0.407

theta = 24 deg

tan theta = ymin / 1160 = tan 24

ymin = 516 m

car travel from this position to encounter the next minimum in reception 516 - 400 = 116 m

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