In the arrangment shown in fin.

For what minimum value of d is there a dark bant at point O on the screen?

Options

A. `sqrt((D lambda)/(4))`
B. `sqrt((3D lambda)/(4))`
C. `sqrt((D lambda)/(8))`
D. `sqrt((2D lambda)/(43)`

Answer

a

Explanation

For dark spot at O,
`Delta x = (AB + BO) - AO`
`= 2(D^(2) + d^(2))^(1//2) - 2 D = 2D [(1 + (d^(2))/(D^(2)))^(1//2)-1]`
`implies Delta x =(d^(2))/(D)`
`(d^(2))/(D) = (lambda)/(2) implies d = sqrt((D lambda)/(4)`
`Detla x` at P:
`Delta x = (AB + BP) - (AC + PC)`
`= (AB - AC) + (BP - PC)`
`Delta x` at O: `Delta x = (lambda)/(2)`
The path difference at P will be zero if `x = d`. Hence next maxima will front at P.
Fringe width = distance between two consecutive dark to brigth fringes
`= 2 xx` distance between two consective bright dark fringes
`= 2 xx d`