Two ideal slits S1 and S2 are at a distance d apart, and illuminated by light of wavelength l 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

Path difference between the waves reaching at  P, Δ=Δ1+Δ2where Δ1= Initial path difference Δ2=Path difference between the waves after emerging from slits.Δ1=S S1−S S2=D2+d2−Dand  Δ2=S1O−S2O=D2+d2−D∴   Δ=2(D2+d2)12−D=2D(1+d22D2)−D   =d2D             (From Binomial expansion)For obtaining dark at O,  Δ must be equals to (2n−1)λ2  i.e. d2D=(2n−1)λ2 ⇒d=(2n−1)λ D2 For minimum distance  n=1 so  d=λ D2