Two radio antennas are 120 m apart on a north-south line,and they radiate in phase at a frequency of 3.4 MHz.All radio measurements are made far from the antennas.If the east-west reference line passes midway between the two antennas,what is the smallest angle from the antennas,measured north of east,at which constructive interference of two radio waves occurs? (c = 3.00 × 10⁸ m/s) A) 43° B) 22° C) 68° D) 47° E) 30°

Question

Quizplus · Accepted Answer

The condition for constructive interference is that the path difference between the two waves from the antennas should be an integer multiple of the wavelength. Let's assume that the angle we are looking for is θ.

The path difference between the two waves arriving at a point P north of the antennas is given by:
Δx = d sinθ
where d is the distance between the antennas, and θ is the angle between the line connecting the antennas and the line connecting one of the antennas to point P.

The wavelength of the radio waves is given by:
λ = c/f = 88 m
where c is the speed of light and f is the frequency of the waves.

Constructive interference occurs when:
Δx = nλ
where n is an integer.

Substituting the values we have:
120 sinθ = n × 88
or
sinθ = n×88/120

Since we want the smallest angle, we take n = 1, which gives:
sinθ = 88/120
or
θ = sin−1(88/120) = 47° (rounded to the nearest degree) north of east.

Therefore, the correct answer is D.

Two Radio Antennas Are 120 M Apart on a North-South