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

B

Explanation

Path difference `Deltax=SS_(1)O-SS_(2)O`
`=2SS_(1)-2SS_(2)`
`=2[sqrt(D^(2)+d^(2))-D]`
`=2[D(1+d^(2)/D^(2))^(1/2)-D]`
`=2[D(1+d^(2)/(2D^(2)))-D]`
`=2[d^(2)/(2D^(2))]=d^(2)/(2D)`
For daekness at `O`
`Deltax=(2n+1)lambda/2, n=0, 1,2`.
`d^(2)/(2D)=(0+1)lambda/2`
`d=sqrt(lambdaD)`