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Question
Four identical monochromatic sources A, B, C, D as shown in the (Figure) produce waves of the same wavelength λ and are coherent. Two receiver R1 and R2 are at great but equal distances from B.
- Which of the two receivers picks up the larger signal?
- Which of the two receivers picks up the larger signal when B is turned off?
- Which of the two receivers picks up the larger signal when D is turned off?
- Which of the two receivers can distinguish which of the sources B or D has been turned off?
Solution
i. Consider the disturbances at R1 which is a distance d from A. Let the wave at R1 because of A be YA = a cos ωt. 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 cos ωt)`
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 yc = a cos ωt.
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^2/(8d^2))^(1/2) - d + lambda/2`
If d >>λ the path difference `∼ λ/2` and hence the phase difference is π.
∴ `y_D = - a cos ωt`
Thus, the signal picked up at R1 is `y_A + y_B + y_C + y_D` = 0
Let the signal picked up at R2 from B be `y_B = a_1 cos ωt`
The path difference between signal at D and that at B is λ/2.
∴ `y_D = - a_1 cos ωt`
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 ∼ 1/8 lambda^2/d^2`
∴ The phase difference is `(2pi)/(8λ) * λ^2/d^2 = (piλ)/(4d) = phi ∼ 0`.
Hence, `y_A = a_1 cos (ωt - phi)`
Similarly, `y_C = a_1 cos (ωt - phi)`
∴ Signal picked up by R2 is `y_A + y_B + y_C + y_D = y = 2a_1 cos (ωt - phi)`
∴ `|y|^2 = 4a_1^2 cos^2 (ωt - phi)`
∴ `<I> = 2a_1^2`
Thus R1 picks up the larger signal.
ii. If B is switched off
R1 picks up y = a cos ωt
∴ `<I_(R_1)> = 1/2 a^2`
R2 picks up y = a cos ωt
∴ `<I_(R_2)> = 1/2 a_1^2`
Thus R1 and R2 pick up the same signal.
iii. If D is switched off
R1 picks up y = a cos ω t
∴ `<I_(R_1)> = 1/2 a^2`
R2 picks up y = 3a cos ωt
∴ `<I_(R_2)> = 1/2 9a^2`
Thus R2 picks up a 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.
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