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. `(lambdas)/(l^(2))`
C. `(l^(2))/(lambda)`
D. `(l^(2))/(2lambda)`

Answer

B

Explanation

When light travels through medium of R.I `'mu'` , the optical path it travels is `mu t`
path difference `(mu t+b) - (l+b) = (mu-1)l`
for small element
`Delta x = (mu -1)dx`
`=(1+ax-1)dx`
For the whole length
`Delta x = int_(o)^(l)ax dx`
`Delta x = (ax^(2))/(2)|_(O)^(t) = (at^(2))/(2)`
For a minimum to be at 'O'
`Delta x = (2n - 1) (lambda)/(2)`
`(al^(2))/(2) = (lambda)/(2)` for 1st minimum
`a = (lambda)/(l^(2))`