Two ideal slits `S_(1)` and `S_(2)` are at a distance `d` apart, and illuninated by light of wavelength `lambda` 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 `(dlt lt D)`

Options

A. `sqrt((3lambdaD)/(2))`
B. `sqrt(lambdaD)`
C. `sqrt((lambdaD)/(2))`
D. `sqrt(3lambdaD)`

Answer

C

Explanation

Path difference = `(SS_(1)+ S_(1)O)- SO`
= `2sqrt(D^(2)+ d^(2))- 2D`
`= d[(D^(2)+ (d^(2))/(2D))-D]= (d^(2))/(D)` [Using Binomial expansion ]

For dark at O
`(d^(2))/(D)= (2n-1)(lambda)/(2) implies d= sqrt((lambdaD)/(2)), n=1 `