Question

Four identical monochromatic sources A,B,C,D as shown in the (Fig.10.7) produce waves of the same wavelength λ and are coherent. Two receiver R₁ and R₂ are at great but equal distaces from B. 

(i) Which of the two receivers picks up the larger signal? 

(ii) 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 signal 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

(i) Consider the disturbances at R₁ which is a distance d from A. Let the wave at R₁ because of A be Y_A = a cosωt. The path difference of the signal from A with that from B is λ/2 and hence the phase difference is π. 

Thus the wave at R₁ because of B is 

y_B = a cos(ωt – π ) = –a cosωt. 

The path difference of the signal from C with that from A is λ and hence the phase difference is 2π. 

Thus the wave at R₁ because of C is y_c = a cosωt.

The path difference between the signal from D with that of A is

Let the signal picked up at R₂ from B be y_B = a₁ cosωt.

The path difference between signal at D and that at B is λ/2.

The path difference between signal at A and that at B is

Thus R₂ picks up larger signal compared to R₁.

(iv) Thus a signal at R₁ indicates B has been switched off and an enhanced signal at R₂ indicates D has been switched off.