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.

Answer

(i) `6.28xx10^(-3)S`
(ii) `1.57xx10^(-3)S`

Explanation

`I=4I_(0)cos^(2)theta` where `theta =(omega_(1)-omega_(2))t=10^(3)t`
(i) For successive maxima
`Deltat=(2pi)/(10^(3))=6.28xx10^(-3)sec`
(ii) For detection of sound
`2A^(2)=4A^(2)cos^(2)theta`
`rArr cos theta = pm(1)/(sqrt(2))rArrtheta = (pi)/(4),(pi)/(4)`
`rArr 10^(3)t=2((pi)/(4))=(pi)/(2)`
`rArr t=(pi)/(2)xx10^(-3)=1.57xx10^(-3)s`.