Two ideal slits S1 and S2 are at a distance d apart, and illuminated by light of wavelength λ passing through an ideal source slit S placed on the line through S2 as shown. The distance between the planes of slits and the source slit is D. A screen is held at a distance D from the plane of the slits. The minimum value of d for which there is darkness at O is

(1) 3λD2

(2) λD

(3) λD2

(4) 3λD

(3) Path difference between the waves reaching at P, Δ=Δ1+Δ2

where Δ1 = Initial path difference

Δ2 = Path difference between the waves after emerging from slits.

Δ1=SS1SS2=D2+d2D

and Δ2=S1OS2O=D2+d2D

   Δ=2(D2+d2)12D=2D2+d22DD

=d2D(From Binomial expansion)

For obtaining dark at O, Δ must be equals to (2n1)λ2 i.e. d2D=(2n1)λ2d(2n1)λD2

For minimum distance n = 1 so d=λD2

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