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Question

Consider two coherent monochromatic sources S1 and S2, each of wavelength λ and separated by a distance d. The ratio of the intensity of S1 and S2 is 4. A detector moves on the line perpendicular to S1S2 as shown in the figure. If the resultant intensity at point P is equal to 94 times intensity of S1, then the distance of P from S1 is,
(Given : d>0 and n is a positive integer)

A
d2n2λ22nλ
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B
d2+n2λ22nλ
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C
nλdd2n2λ2
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D
2nλdd2n2λ2
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Solution

The correct option is A d2n2λ22nλ

Let the intensity of source S1 and S2 is I1 and I2 respectively.
Let resultant intensity at P is IR
IR=94I1
and I1I2=4 (Given)

The resultant intensity is given by,
IR=I1+I2+2I1I2 cos ϕ,
(ϕ=phase difference)
94I1=I1+I14+2I1×I14 cos ϕ

94I1=5I14+I1 cos ϕ

9454=cos ϕ

cos ϕ=1
ϕ=2nπ
Where n=0,1,2,.....

Therefore, at point P, constructive interference takes place.

Let the path difference at P is Δx.

The relation between path and phase difference is,
ϕ=2nπ=2πλ×Δx

Where, λ=wavelength.
Now, for constructive interference,
Δx=nλ ...(1)


Also, from the diagram, Δx=S2PS1P
From right angled triangle,
Δx=d2+x2x
Δx+x=d2+x2
(Δx+x)2=d2+x2
Δx2+x2+2xΔx=d2+x2
Δx2+2xΔx=d2

Substituting Δx from equation (1),
(nλ)2+2nλx=d2

x=d2n2λ22nλ

Where n=0,1,2,3,...

Hence, (A) is the correct answer.

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