Consider a road that runs parallel to the line connecting a pair of radio towers that transmit the same station (assume that their transmissions are synchronized), which has an AM frequency of 1000 kilohertz. If the road is 5 kilometers from the towers and the towers are separated by 400 meters, find the angle θ to the first point of minimum signal (m=0). Hint: A frequency of 1000 kilohertz corresponds to a wavelength of 300 meters for radio waves.
Question
Answer:
θ = 0.384 rad = 22 degrees
Explanation:
Given:
– The wavelength of the source λ = 300 meters
– The distance between towers d = 400 m
– The order of minimum signal m = 0
Find:
find the angle θ to the first point of minimum signal (m=0)
Solution:
– The given problem is analogous to Young’s double slit experiment where the split through which the waves are initiated are towers with slit separation d = 400 m ( i.e distance between towers ).
– For the minimum signal, we are asked for destructive interference for which the formula is given as:
d*sin(θ) = ( m + 0.5 )*λ
For m = 0,
sin(θ) = 0.5*λ / d
sin(θ) = 0.5*300 / 400
θ = arc sin ( 3/8)
θ = 0.384 rad = 22 degrees