Obtain the formula of path difference at a point on the screen in Young's double slit experiment in term of x, d and D.

Explanation

One wavefront out of wavefronts emerging from slit S incident on slits S, and S, at the same time hence `S_(1)` and `S_(2)` act as coherent sources.
Suppose distance between `S_(1)` and `S_(2)` is d which in the order to mm and screen GG. is placed at distance D from perpendicular bisector of `S_(1)S_(2)`. D is in the order of meter.
Interference obtain at a point P on a screen. The maximum or minimum intensity of light at P depend on the path difference of `S_(1)` and `S_(2)`. Let OP=x.
`S_(1)MbotGG.` and draw `S_(2)NbotGG.`. So `S_(1)M=D` and `PM=x-(d)/(2)andS_(2)N=DandPN=x+(d)/(2)`
From `DeltaPMS_(1)`
`S_(1)P^(2)=S_(1)M^(2)+PM^(2)`
`=D^(2)+(x-(d)/(2))^(2)" "......(1)`
and from `DeltaPNS_(2)`
`S_(2)P^(2)=S_(2)M^(2)+PN^(2)`
`=D^(2)+(x+(d)/(2))^(2)" "......(2)`
`S_(2)P^(2)-S_(1)P^(2)=D^(2)+(x+(d)/(2))^(2)-D^(2)+(x-(d)/(2))^(2)`
`:.S_(2)P^(2)-S_(1)P^(2)=(x+(d)/(2))^(2)-(x-(d)/(2))^(2)`
`=x^(2)+xd+(d^(2))/(4)-x^(2)+xd-(d^(2))/(4)`
`:.(S_(2)P-S_(1)P)(S_(2)P+S_(1)P)=2xd`
But x and d are very small compare to D so
`S_(1)P=S_(2)P=D` can be taken
`:.(S_(2)P-S_(1)P)(D+D)=2xd`
`:.S_(2)P-S_(1)P=(2xd)/(2D)`
`:.S_(2)P-S_(1)P=(xd)/(D)`
Hence, path difference of `S_(1)` and `S_(2)` from `P=(xd)/(D)`
If `S_(2)P-S_(1)P` (path difference) `=nlamda` where n=0, 1, 2, ......, the intensity of light at point P is maximum (constructive interence).
Hence, `nlamda=(x_(n)d)/(D)`
`:.xorx_(n)=(nlamdaD)/(d)` where `n=0,pm1,pm2,......,`
constructive interference obtained of `n^(th)` order.
If `S_(2)P-S_(1)P` (path difference ) `=(n+(1)/(2))lamda`
`:.(x_(n)d)/(D)=(n+(1)/(2))lamda`
`:.x_(n)=(n+(1)/(2))(lamdaD)/(d)`
where `n=0,pm,1pm2,......,` at point P then intensity of light will be minimum (zero) and destructive interference obtained.
The distance of bright fringe from the mid point of central bright fringe is x but minimum of zeroth order cannot be obtained hence the formula of destructive interference is `(n-(1)/(2))lamda` where `n=pm1,pm2,......,`