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) `10^(-3) sec, (ii) 0.5 xx10^(-3)sec`

Explanation

(i) Let the signal be given by `y_1=A sin 2omega_1t,y_2=A sin 2pi omega_2t` at position x=0
(For simplicity of calculation x=0 position has been taken as observer position)
`therefore` The resultant disturbance at x=0 position is given by
=`y=y_1+y_2=A sin 2 omega_1t+A sin 2pi omega_2t`
`2A sin (2pi (omega_1+omega_2)t)/(2) cos.(2pi(omega_2-omega_1))/(2)2A cos {2pi((omega_2-omega_1))/(2)t}sin {2pi((omega_2+omega_1)t)/(2)}`
Let `omega_1=omega,omega_2=omega+triangleomega`
Therefore, `omega_1+omega_2 approx 2omega`
`y=2A cos pi (omega_2-omega_1) tsin2 pi omegat`
Thus the resultant disturbance has amplitude `2A cos{pi(omega_2-omega_1)t}`
`therefore` For maximum resultant amplitude `cos{pi(omega_2-omega_1)t}=pm1`
or `pi (omega_2-omega_1)t=npi,n=0,1,2,3`
`t=(n)/(omega_2-omega_1)=0,(1)/(omega_2-omega_1),2/(omega_2-omega_1),`.....
Clearly time interval between successive maxima `=1/(omega_2-omega_1)=1/(10^(3))=10^(-3) s`
(ii) For the detector to sense the radio waves,the resultant intensity `ge2A^(2)`
`therefore`The detector will be idle when the resultant intensity `lt2 A^(2)`
`therefore`[Resultant amplitude] `ltsqrt(2A)`
or `|2 A cos {pi(omega_2-omega_1)t}lt sqrt(2A) implies 0 lt cos{pi(omega_2-omega_1)t}lt1/sqrt2`
`therefore pi/4ltpi (omega_2-omega_1)tlt(3pi)/4`(For one cycle of resultant intensity)
`implies 1/(4(omega_2-omega_1))lttlt3/(4(omega_2-omega_1))`
The time gap `=3/(4(omega_2-omega_1))-1/(4(omega_2-omega_1))=1/(2(omega_1-omega_2))=0.5xx10^(-3)`