Two radio stations broadcast their programmes at the same amplitude `A` and at slightly different frequencies `omega_(1)` and `omega_(2)` respectively, where `omega_(1) - omega_(2) = 10^(3) Hz`. A detector receives the signals from the two stations simultaneously, it can only detect signals of intensity `ge 2A^(2)`.
(i) Find the time interval between successive maxima of the intensity of the signal received by the detector.
(ii) Find the time for which the detector remains idle in each cycle of the intensity of the signal.

Explanation

(a) The intensity will be maximum at time
`t = 0, (1)/(omega_(2) - omega_(1)) , (2)/(omega_(2) - omega_(1)) ,…`
Time interval between successive maxima
`T = (1)/(omega_(2) - omega_(1)) = (1)/(10^(3)) = 10^(-3)s`
(b) Resultant amplitude
`A_(r)^(2) = a_(1)^(2) + a_(2)^(2) + 2a _(1) a_(2) cos phi`
`= A^(2) + A^(2) + 2A^(2) cos phi`
`= 2A^(2)(1 + cos phi) = 2A^(2)(1 + 2 cos^(2) phi//2 - 1)`
`= 4A^(2) cos^(2) (phi//2)`
Intensity `prop A_(r)^(2)`
`|{:(phi=0,,I=4A^(2)),(phi=(pi)/(2),,I=2A^(2)),(phi=pi,,I=0),(phi=(3pi)/(2),,I=2A^(2)),(phi=2pi,,I=4A^(2)):}|`
The detector remaining idle from `phi = (pi)/(2)` to `(3pi)/(2)`
i.e. for time,`(T)/(2) = (10^(-3))/(2) = 5 xx 10^(-4)s`