In figure, parallel beam of light is incident on the plane of the slits of a Young's double-slit experiment. Light incident on the slit `S_(1)` passes through a medium of variable refractive index `mu = 1 + ax` (where 'x' is the distance from the plane of slits as shown), up to distance 'I' before falling on `S_(1)`. Rest of the space is filled with air. If at 'O' a minima is formed. then the minimum value of the positive constant a (in terms of l and wavelength `lambda` in air) is

Options

A. `(lambda)/(l)`
B. `(lambda)/(l^(2))`
C. `(l^(2))/(lambda)`
D. none of these

Answer

b

Explanation

When light passes through a medium of refractive index `mu`, the optical path it travels is `mu t`
Path difference `= (mu l + b) - (l + b) = (mu -1) l`
For a small element 'dx', path difference,
`Delta x = [(1 + ax) - 1] dx = axdx`
For a minima to be 'O'
`Delta x = (2n + 1) (lambda)/(2)`
`implies (a l^(2))/(2) = (2 n + 1) (lambda)/(2)`
For minimum 'a', `n = 0`
`:. (a l^(2))/(2) = (lambda)/(2) implies a = (lambda)/(l_(2))`