Draw a neat labelled diagram of Young.s Double Slit experiment. Show that `beta= (lamda D)/(d)`, where the terms have their usual meaning (either for bright or dark fringe).

Explanation

For Young.s double slit experiment :

Consider that S is a source of light, `S_1` and `S_2` are two slits which are at equal distances from light source S.
Let , `S_1 S_2 = d`
Path difference between waves at point P
`S_2P - S_1P = S_2A`
`DeltaS_1S_2A` and `DeltaPCO` are similar to each other
`therefore (S_2A)/(S_1S_2) = (OP)/(CP)`
Since `CO` is very much greater than `S_1S_2`,
So, `CP = CO`
`therefore (S_2A)/(S_1S_2) = (OP)/(CO)`
`(S_2A)/d = x/D`
`therefore `Path difference `= (xd)/(D)`
For bright fringes `(xd)/(D) = m lambda`, where `m = 0,1,2,3.....`
`x = m (D lambda)/(d)`
For `m^(th)` and `(m + 1)^(th)` fringe ,
`x_m = m (Dlambda)/d and x_(m + 1) = (m +1) (D lambda)/(d)`
Fringe width, `beta = x_(m + 1) - x_m`
`= (m + 1)(D lambda)/(d)- m(D lambda)/(d)`
`beta = (D lambda)/(d)`