Two radio antennas separated by d = 304 m as shown in the figure below simultane

Question

Two radio antennas separated by d = 304 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 = 1 040 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.)tan(theta)=400/1040

So theta=21.04

d*sin(theta)=m*lambda

m=2

lambda=d*sin(theta)/2=304*sin(21.04)/2= 54.56 m

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

m=2

d*sin(theta)=5*lambda/2

sin(theta)=5*lambda/2d=5*54.56/(2*304)= 0.4487

So theta(min)=26.67

tan(theta)=y(min)/1040

So y(min)=tan(26.67)*1040= 522.2 m

So, the car will have to move additional 522.2 - 400 = 122.2 m

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