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 ?

Options

A. `R_(1)`
B. `R_(2)`
C. `R_(1)` and `R_(2)`
D. None of these.

Answer

B

Explanation

Consider the disturbances at `R_(1)` which is at distance d from A. Let the wave at `R_(1)` because of A be `y_(A)=a" cos" wt`. 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"(wt-pi)= -a" cos"wt.`
The path difference fo the signal from C with that from A is `lambda` adn hence, the phase difference is `2 pi`.
Thus, the wave at `R_(1)` because of C is `y_(c)=a" cos"wt.`
The path difference between the signals from D with that of A is
`P=R_(1)D-R_(1)A=sqrt((R_(1)B)^(2)+(BD)^(2))-(R_(1)B-AB)`

`=sqrt(d^(2)+((lambda)/(2))^(2))-(d-lambda//2)=d(1+(lambda^(2))/(4d^(2)))^(1//2)-d+(lambda)/(2)`
`=d(1+(lambda^(2))/(8d^(2)))-d+(lambda)/(2)=(lambda^(2))/(8d)+(lambda)/(2)`
If `d gt gt lambda,` the path difference `~(lambda)/(2)` and hence the phase difference is `pi`.
` :. y_(D)= -a" cos"wt.`
Thus, the signal picked up at `R_(1)` is
` "y"="y"_(A)+"y"_(B)+"y"_(C)+"y"_(D)=0`
Let the signal picked up at `R_(2)` from B be ` "y"_(B)^(')=a_(1)" cos"wt.`
The path difference between signal at D and that at B is `lambda//2.`
` :. "y"_(D)^(')= -a_(1)" cos"wt`
The path difference between signal at A and B is
`sqrt((d)^(2)+((lambda)/(2))^(2))-d=d(1+(lambda^(2))/(4d^(2)))^(1//2)-d=(lambda^(2))/(8d)`
` :. ` The phase difference is
`(2pi)/(lambda)*(lambda^(2))/(8d)=(pi lambda)/(4d)=phi~0.`
Hence, ` "y"_(A)^(')=a_(1)" cos"(wt-phi)`
Similarly, ` "y"_(C)^(')=a_(1)" cos"(wt-phi)`
` :.` Signal picked up by `R_(2)` is
` "y"_(A)^(')+"y"_(B)^(')+"y"_(C)^(')+"y"_(D)^(')="y"^(')=2a" cos"(wt-phi)`
` " " [ :' "y"_(B)^(')+"y"_(D)^(')=0]`
` |"y"|^(2)=4a_(1)^(2)" cos"^(2)(wt-phi)`
` :. lt I gt =2a_(1)^(2)`
Thus `R_(2)` picks up the large signal.