Two coherent point sources `S_(1)` and `S_(2)` vibrating in phase emit light of wavelength `lambda`. The separation between the sources is `2lambda`. Consider a line passing through `S_(2)` and perpendicular to line `S_(1) S_(2)`. Find the position of farthest and nearest minima.
.

Explanation

Visualise a point just close to `S_2`. At this point, path difference between two waves would be `2 lambda` and hence there will be constructive interference at this point. When we move further on line, then path difference between waves decreases. At a point where path difference reduces to `3lambda//2`, destructive interference takes place and this is nearest minima. And if we go on moving on the same line, then at a point path difference will reduce to `lambda//2` and so destructive interference will take place at this point. This is farthest minima on this line.
Let there be a point P on the line at a distance x from `S_2` , then path diffrence at this point can be written as follows:
`Delta x = S_1P - S_2P`
`Dleta x = sqrt(x^2 + 4 lambda^2) - x`
1. Nearest minima
`Deltax = sqrt(x^2 + 4 lambda^2) - x = (3 lambda)/2`
`implies sqrt(x^2 + 4 lambda^2) = (3lambda)/(2) + x`
`implies x^2 + 4lambda^2 = (9 lambda^2)/(4) + x^2 + 3 lambda x`
`implies 3 lambdax = 4 lambda^2 - (9 lambda^2)/(4)`
`implies 3x = 4lambda - (9 lambda)/4`
`implies 3x = (7 lambda)/4`
`implies x = ( 7lambda)/12`
2. Farthest minima
`Dletax = sqrt(x^2 + 4 lambda^2) - x = lambda/2`
`implies sqrt(x^2 + 4lambda^2) = lambda/2 + x`
`implies x^2 + 4lambda^2 = (lambda^2)/4 + x^2 + lambdax`
`implies 4 lambda^2 = (lambda^2)/4 + lambda x`
`implies x = 4 lambda - lambda/4`
`implies x = (15 lambda)/4`.