Question

Four identical monochromatic sources A,B,C,D as shown in the (figure) produce waves of the same wavelength `lambda` and are coherent. Two receiver `R_(1)` and `R_(2)` are at great but equal distances from B.
(i) Which of the two receivers picks up the larger signal when B is turned off?
(iii) Which of the two receivers picks up the larger singnal when D is turned off ?
(iv) Which of the two receivers can distinguish which of the sources B or D has been turned off ?

Answer 1

Consider the disturbances at the reciever `R_(1)` which is at a distance d from B.
Let the wave at `R_(1)` because of A be `y_(A) = a cos omega t` . The path difference of the signal from A with that from B is `lambda//2` and hence, the phase difference is `pi`.
Thus, the wave at `R_(1)` because of B is
`y_(B)=a cos (omega t- pi)= - a cos omegat`.
The path difference of the signal from C with that from A is `lambda ` and hence the phase difference is `2 pi`.
Thus, the wave at `R_(1)` because of C is `y_(C)= a cos (omega t -2 pi)=a cos omega t`

The path difference between the signal from D with that of A is
`sqrt(d^(2)+((lambda^(2))/(2)))-(d-lambda//2)=d(1+(lambda)/(4d^(2)))^(1//2)-d+(lambda)/(2)`
`=d(1+(lambda)/(4d^(2)))^(1//2)-d+(lambda)/(2)~~(lambda)/(2)" "( :. d gt gtlambda)`
Therefore,phase difference is `pi`.
`:. Y_(D)= a cos (omegat-pi)= - a cos omega t`
Thus, the signal picked up at `R_(1)` from all the four sources is `Y_(R_(1))=y_(A)+y_(B)+y_(C)+y_(D)`
`=a cos omegat-a cos omega t +a cos omega t- a cos omega t =0`
(i) Let the signal picked up at `R_(2)` from B be `y_(B)= a_(1) cos omega t`.
The path difference between signal at D and that at B is `lambda//2`.
`: y_(D)= -a_(1) cos omega t`
The path difference between signal at A and that at B is
`sqrt(d^(2)+((lambda^(2))/(2)))-d=d(1+(lambda^(2))/(4d^(2)))^(1//2)-d " " underline~ (1 lambda^(2))/(8d^(2))`
As `d gt gt lambda`, therefore this path difference `rarr0`
and phase difference `=(2 pi)/(lambda)((1)/(8) lambda^(2)/(d^(2))) rarr 0`
Hence, `y_(A)=a_(1) cos (omegat- phi)`
Similarly, `y_(C)=a_(1) cos (omegat- phi)`
`:.` Signal picked up by `R_(2)` is
`y_(A)+y_(B)+y_(C)+y_(D)=y=2a_(1) cos (omegat- phi)`
`:. |y|^(2)=4a_(1)^(2) cos^(2)(omega t- phi)`
`:. lt I gt =2a_(1)^(2)`
Thus, `R_(1)` picks up the larger signal.
(ii) If B is switched off,
`R_(1)` picks up `y= a cos omega t`
`:. (:I_(R_(1)):)=(1)/(2)a^(2)`
`R_(2)` pick up `y=a cos omega t`
`:. (:I_(R_(2)):)=a^(2) lt cos ^(2) omegat gty=(a^(2))/(2)`
(iii) Thus, `R_(1) " and " R_(2)` pick up the same signal
If D is switched off.
`R_(1)` pick up `y= a cos omega t`
`:. (:I_(R_(1)):)=(1)/(2)a^(2)`
`R_(2)` picks up `y= a cos omega t`
`:. (:I_(R_(2)):)=9a^(2) lt cos ^(2) omegat gty=(9a^(2))/(2)`
Thus, `R_(2)` picks up larger signal compared to `R_(1)`.
(iv) Thus, a signal at `R_(1)` indicates B has been switched off and an enhanced signal at `R_(2)` indicates D has been switched off.