Obtain the expression for fringe width in the case of interference of light waves.

Explanation

Consider Young.s double slit arrangement for obtaining interference fringes as shown in figure. `S_(1) and S_(2)` -two coherent sources (Yong.s double slits)
d= distance between slits
D- distance of screen from coherent sources/slits
O- center point on the screen and is equidistant from `S_(1) and S_(2)`

The path difference between the two light waves from `S_(1) and S_(2)` reaching the point O is zero. Thus the point O has maximum intensity. Consider a point P at a distance x from O.
The path difference between the light waves from `S_(1) and S_(2)` reaching the point P is, `delta = S_(2) P- S_(1)P`
From the figure, `(S_(2)P)^(2)= (S_(2)F)^(2)= (FP)^(2)= D^(2) + (x + (d)/(2))^(2)`
Similarly `(S_(1)P)^(2)= (S_(1)E)^(2)= (EP)^(2)= D^(2) + (x- (d)/(2))^(2)`
`therefore (S_(2)P)^(2)- (S_(1)P)^(2)= [D^(2) + (x+ (d)/(2))^(2)]- [D^(2) + (x + (d)/(2))^(2)] = [D^(2) + x^(2) + (d^(2))/(4) + 2(x) ((d)/(2))]- [D^(2) + x^(2) + (d^(2))/(4) + 2(x) ((d)/(2))]= 2xd`
`(S_(2)P-S_(1)P) (S_(2)P) + S_(1)P)= 2xd`
`(S_(2)P-S_(1)P)= (2xd)/((S_(2)P+ S_(1)P))`
Since P is very close to O and `d lt lt D`, therefore `(S_(2)P + S_(1)P) ~~ 2D`
Path difference, `(S_(2)P-S_(1)P) = (2xd)/(2D)= (xd)/(D)` ....(1)
Equation (1) represents the path difference between light waves from `S_(1) and S_(2)` superposing at the point P.
For constructive interference, `S_(2)P- S_(1)P= n lamda`, where, n= 0, 1, 2........
`(xd)/(D)= n lamda or x= n ((lamda D)/(d))`
i.e., The distance of then `n^(th)` bright fring from the centre O of the screen is `x_(0)=n ((lamda D)/(d))`
The distance of `(n +1)^(th)` bright fringe from the centre of the scrren is `x_(n+1)= (n+1) ((lamda D)/(d))`
The distance between the centers of any two consecutive bright fringes is called the fringe width of bright fringes. The fringe width is given by,
`beta= x_(n+1) -x_(n)= (n+1) ((lamda D)/(d))-n ((lamda D)/(d))= (lamda D)/(d)`
`therefore beta= (lamda D)/(d)`
Similarly for dark fringes also we obtain the same expression for fringe width.