Two radio antennas separated by d = 296 m, as shown in the figure below, simulta

Question

Two radio antennas separated by d = 296 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 = 1,200 m

from the center point between the antennas, and its radio receives the signals. Hint: Do not use the small-angle approximation in this problem.

(a) If the car is at the position of the third maximum after that at point O when it has traveled a distance of

y = 400 m

northward, what is the wavelength of the signals?
___ m

(b) How much farther must the car travel to encounter the next minimum in reception?
____ m

Explanation / Answer

tan theta = 400 / 1200

theta = 18.42 deg

d sin theta = m lambda = 3 lambda

lambda = d sin theta / 3 = (296 * sin 18.42) / (3)

lambda = 31.18 m

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

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

d sin theta = [3 + 1/2 ] lambda = 7/2 lambda

sin theta = 7 lambda / 2 d = (7 * 31.18) / (2 * 296) = 0.369

thetamin = 21.65 deg

tan theta = ymin / 1200

ymin = 1200 * tan 21.65 = 476 m

The car will have to move additonal = 476 - 400 = 76 m

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