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

A, B, C

Explanation

Let the two radio waves be represented by the equations
`y_(1) = A sin 2piv_(1)t`
`y_(2) = A sin 2piv_(2)t`
The equation of resultant wave according to superposition principle
` y = y_(1) + y_(2) = A sin 2piv_(1)t + A sin 2piv_(2)t`
`= A [sin 2piv_(1)t + sin 2piv_(2)t]`
`= A xx 2 sin ((2piv_(1) + 2piv_(2))t/(2) cos ((2piv_(1) + 2piv_(2))t/(2)`
`=2A sin pi(v_(1) + v_(2))tcos pi((v_(1) - v_(2))t`
where the amplitude `A' = 2A cospi((v_(1) - v_(2))t`
Now, intensity `prop (Amplitude)^(2)`
rArr `I prop A'^(2)`
`I prop 4A^(2) cos^(2)(v_(1) - v_(2))t`
The intensity will be maximum when
`cos^(2)pi(v_(1) - v_(2))t = 1`
or, `cospi(v_(1) - v_(2))t = 1`
or, pi(v_(1) - v_(2))t = npi`
rArr `((omega_(1) - omega_(2))/(2)t = npi` or, `t = (2npi)/(omega_(1) - omega_(2))`
:. Time interval between two maxima
or, (2npi)/(omega_(1) - omega_(2)) - (2(n - 1)pi)/(omega_(1) - omega_(2))` or, (2pi)/(omega_(1) - omega_(2)) = (2pi)/(10^(3))sec`
Time interval between two successive maximas is `2pi xx 10^(-3)sec`
(ii) For the detector to sence the radio waves, the resultant intensity`ge 2A^(2)`
:. or, `2A cospi(v_(1) - v_(2))tgesqrt(2A)`
or, `cospi(v_(1) - v_(2))t ge (1)/sqrt(2)` or, `cos[((omega_(1) - omega_(2))t] ge (1)/(sqrt(2)`
The detector lies idle when the values of `cos[((omega_(1) - omega_(2))t]`
is between `0 and (1)/sqrt(2)`
:. `((omega_(1) - omega_(2))t/(2)` is between `(pi)/(2)` and `(pi)/(4)`
:. `t_(1) = (pi)/(omega_(1) - omega_(2))` and `t_(2) = (pi)/(2(omega_(1) - omega_(2))`
:. `= (pi)/(omega_(1) - omega_(2)) - (pi)/(2(omega_(1) - omega_(2)) = (pi)/(2(omega_(1) - omega_(2))`
`= (pi)/(2) xx 10^(-3)sec`