Fig. shows an experimental set up simillar to Young's double slit experiment to observe interference of light. Here `SS_(2) - SS_(1) = lambda//4`. Write down the conditions of (i) Contstructive interference
(ii) Destructive interference at any point `P` in terms of path diff. `(S_(2)P - S_(1)P)`. Does the central fringe observed in the above set up lie above or below `O` ? Give reason.

Explanation

Initial path difference between sources `S_1 and S_2`
`SS_2 - SS_1 = lambda/4`
Let `OP = x`, the path difference between disturbances
from `S_1 and S_2 = (xd)/(D)`
Total path difference = `lambda/4 + (xd)/D`
For constructive interference,
`lambda/4 + (xd)/D = n lambda`
`= (xd)/(D) = lambda (n - 1/4) " ".......(i)`
For destructive interference,
`lambda/4 + (xd)/D = (2n - 1) lambda/2`
`(xd)/D = ((2 n lambda)/2 - lambda/2 - lambda/4)`
`= lambda (n - 1/2 - 1/4) = lambda (n - 3/4)`
(b) For central fringe putting n = 0 in (1) we get
`(xd)/D = lambda ((-1)/(4))`
`x = (lambda D)/(4d)`
The negative sign indicates that the central fringe lies below the central point.