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

Answer

b

Explanation

Intensity at O is proportional to intensity at `S_(3)` and at `S_(4)`
`I = k cos^(2) ((delta)/(2))`, where k is constant and `delta` is the phase
difference.
`delta = (2pi)/(beta) xx (Z)/(2) = (pi Z)/(beta)`, where `beta` is fringe width
`beta = (lambda D)/(d)`
When `Z = lambda D // 2d:`
`delta = (pi d)/(lambda D) xx (lambda D)/(2d) = (pi)/(2)`
`I = I_(0) = k cos^(2) ((pi)/(4))`
`k = 2 I_(0)`
When `Z' = (2 lambda D)/(d):`
Path difference `delta' = (2pi)/(beta) (Z')/(2) = (2 pi d)/(lambda d) xx (2 pi D)/(lambda d) = 2pi`
Required intensity at O is
`I' = k cos^(2).(delta)/(2) = 2 I_(0) cos^(2) (pi) = 2 I_(0)`