In the arrangement shown in Fig., slits `S_(1)` and `S_(4)`are having a variable separation Z. Point O on the screen is at the common perpendicular bisector of `S_(1) S_(2)` and `S_(3) S_(4)`.

When `Z = (lambda D)/(2d)`, the intensity measured at O is `I_(0)`. The intensity at O When `Z = (2 lambda D)/(d)` is

Options

A. `I_(0)`
B. `2I_(0)`
C. `3I_(0)`
D. `4I_(0)`

Answer

B

Explanation

intensity at 'O' is proportional to intensity at `S_(3)` and at `S_(4)`. `I = k cos^(2) (phi//2)` where k is constant, `phi` is the phase difference.
`phi = (2pi)/(beta)xx(Z)/(2)=(pi z)/(beta)` where `beta` is the fringe width
when `Z = (lambda D)/(2d) pgi = (pi Z)/(beta) = (pi)/(lambda D)xxdxx(lambda D)/(2d) = (pi)/(2)`
`I = I_(0) = k cos^(2) (pi//4) = kxx(1)/(2) , k = 2I_(0)`
When `Z' = (2lambda D)/(d)` Phase difference
`phi' = (2 pi)/(beta)xx(Z')(2)= (2pi)/(lambda D)dxx (2lambda D)/(2d) = 2pi`
Required intensity at 'O'
`I' k cos^(2)(phi'//2)= 2I_(0)cos^(2)((2 pi)/(2)) = 2I_(0) (-1)^(2)`
`I' = 2I_(0)`