Using analytical method for interference bands, obtain an expression for path difference between two light waves.

Explanation

Consider S be narrow slit which is illuminated by monochromatic light of wavelength ( `lamda ` ). ` S _ 1 and S _ 2 ` are two parallel narrow slits which are separated by short distance d and are equidistant from S. the light waves from S will arrive at ` S _ 1 and S _ 2 ` in the same phase. Beyond ` S_ 1 and S _ 2 ` the waves will proceed as they had started from ` S _ 1 and S _ 2 ` and will produce an interference pattern on the screen XY kept at distance D from the sources ` S _ 1 and S_ 2`.

Theory of interference bands :
Let the perpendicular bisector OM of the line joning ` S _1 and S _ 2 ` meet the screen at point O. since ` S _ 1 O = S _ 2 O `, the path difference between waves reaching O from ` S _ 1 and S _ 2 ` is zero, therefore there is a bright point at O . The point O is called the centre of the interference pattern. suppose P be any point on the screen at distance x from O. The difference between the waves reaching P from ` S _ 1 and S _ 2 ` is ( ` S _ 2 P - S _ 1 P ` ) . suppose ` S _ 1 M _ 1 and S_ 2 M_ 2 ` are perependiculars on the screen.
We have, ` PM _ 1 = x - ( d ) /( 2) and PM _ 2 = x + ( d ) / ( 2 ) `
In ` Delta S _ 2 M _ 2 P `,
` ( S _ 2 P) ^2 = ( S _ 2 M _ 2 ) ^ 2 + ( PM _ 2 ) ^ 2 `
` therefore ( S _ 2P ) ^2 = D^2 + ( x + ( d ) /( 2 ) ) ^ 2 " " `... (i)
In `Delta S _ 1 M _ 1 P `,
` ( S _ 1 P ) ^ 2 = ( S _ 1 M _ 1 ) ^ 2 + ( P M _ 1 ) ^ 2 `
` therefore ( S _ 1 P ) ^ 2 = D ^ 2 + ( x - ( d ) /( 2 ) ) ^ 2 `... (ii)
` ( S _ 2 P ) ^ 2 - ( S _ 1 P ) ^ 2 = D ^ 2 + ( x + ( d ) /( 2 ) ) ^ 2 - D^2 - ( x - ( d ) / ( 2 ) ) ^ 2 `
` ( S _ 2 P - S _ 1 P ) ( S _ 2P + S _ 1 P ) = x ^ 2 + xd + ( d^ 2 ) /( 4 ) - x ^ 2 + xd - ( d ^ 2 ) /( 4 ) `
` therefore S _ 2 P - S _ 1 P = ( 2 xd ) / ( S _ 2 P +S_ 1 P) `
Now, the distance D is very large as compared to x and d.
` therefore S _ 1 P = S _ 2 P = D or S _2 P + S _ 1 P = 2D `
Therefore , the path difference between two waves is given by
` S _ 2 P - S _ 1 P = ( 2xd ) /( 2D) = ( xd ) /( D) `