Two ideal slits S 1 and S 2 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 S 2 , 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

Step-by-step solution

From the above diagram, path difference between the waves reaching at P is, Δ = ( S S 1 - S S 2 ) + S 1 O - S 2 O From the diagram, we can write, S S 1 - S S 2 = D 2 + d 2 - D and S 1 O - S 2 O = D 2 + d 2 - D . So, Δ = 2 D 2 + d 2 - 2 D = 2 D 1 + d 2 D 2 1 2 - 2 D From binomial expansion, as d D < 1 , ∆ = 2 D 1 + d 2 2 D 2 - 2 D = d 2 D To form darkness at O ,   ∆ must be equal to λ 2 (minimum value). Then, d 2 D = λ 2 ⇒ d = λ D 2