Consider the situation shown in figure. The two slits `S_1 and S_2` placed symmetrically around the central line are illuminated by a monochromatic light of wavelength `lamda`. The separation between the slits is d. The light transmitted by the slits falls on a screen `Sigma_1` placed at a distance D from the slits. The slit `S_3` is at the placed central line and the slit `S_4`, is at a distance z from `S_3`. Another screen`Sigma_2` is placed a further distance D away from 1,1. Find the ratio of the maximum to minimum intensity observed on `Sigma_2` if z is equal to a. `z=(lamdaD)/(2d)` b. `(lamdaD)/d` c. `(lamdaD)/(4d)`

Options

A. `[3 - 2 sqrt 2]^(2)`
B. `[3 + sqrt 2]^(2)`
C. `[3 - sqrt 2]^(2)`
D. `[3 + 2 sqrt 2]^(2)`

Answer

d

Explanation

`z = (lambda D)/(4d)`
`Delta x = y (d)/(D) = (lambda D)/(4 d) (d)/(D) = (lambda)/(4)`
`phi = (2 pi)/(lambda) ((lambda)/(4)) = (pi)/(4)`
Intensity at `S_(4): I_(4) = 2 I_(0) (1 + cos ((pi)/(2))) = 2 I_(0)`
Intensity at `S_(3): I_(3) = 4 I_(0)`
`:. (I_(max))/(I_(min))=([(4I_(0))^(1//2)+(2I_(0))^(1//2)]^(2))/([(4I_(0))^(1//2)-(2I_(0))^(1//2)]^(2))=[((2+sqrt(2)))/((2-sqrt(2)))]^(2)`
`= [((2 + sqrt 2)^(2))/(4 - 2)]^(2) = (1)/(4) [4 + 2 + 4 sqrt 2]^(2)`
`= (1)/(4) [ 6 + 4 sqrt 2]^(2) = [3 + 2 sqrt 2]^(2)`