The minimum value of d os that there is a dark fringe at O is `d_(min)`. For the value of `d_(min)`, the distance at which the next bright fringe is formed is x. Then

Options

A. `d_(min) = sqrt(lambda D)`
B. `d_(min) = sqrt((lambda D)/(2))`
C. `x = (d_(min))/(2)`
D. `x = d_(min)`

Answer

b.,d

Explanation

There is a dark fringe at O if the path difference `delta = ABO - AO'O = (lambda)/(2)`
`implies 2sqrt(D^(2) - d^(2)) - 2D = (2d^(2))/(2D) = (D^(2))/(D) = (lambda)/(2)`
`d_(min) = sqrt((lambda D)/(2))`
The bright fringe is formed at P if the path difference
`delta = A O' P - ABP = lambda`
`= D + sqrt(D^(2) + x^(2)) - sqrt(D^(2) + d^(2)) - sqrt(D^(2) + (x - d)^(2) = lambda`
`= (x^(2))/(2 D) - (d^(2))/(2 D) - ((s^(2) + d^(2) - 2 x d))/(2 D) = lambda`
Given `d = d_(min)`
On solving, `x = d_(min) = sqrt((lambda D)/(2))`