Three coherent point sources `S_(1),S_(2) and S_(3)` are placed on a line perpendicular to the screen as shown in the figure. The wavelength of the light emitted by the sources is lambda. The distance between adjacent sources is `d=3lambda` The distance of `S_(2)` from the screen is D`(gtgt lambda)`. Find the minimum (non zero) distance x of a point P on the screen at which complete darkness is obtained.

Options

A. `(2sqrt(2)D)/7`
B. `sqrt((17)D)/8`
C. `4sqrt(5)D`
D. `(4sqrt(2)D)/7`

Answer

B

Explanation

for nearest minima
`dsintheta=8lambda//3 (i.e.2lambda//3+lambda+lambda)Rightarrow3lambdasintheta=8lambda//3 Rightarrow sintheta=8//9`
`Rightarrow tantheta = D//x =8/sqrt(17)Rightarrow x=sqrt((17)D)/8`