#### Question

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

#### Solution

a) Let S_{1} and S_{2} be the two coherent monochromatic sources which are separated by short distance d. They emit light waves of wavelength λ

b) Let D = horizontal distance between screen and source

c) Draw S_{1}M and S_{2}N ⊥ AB

OP = perpendicular bisector of slit.

Since S_{1}P = S_{2}P, the path difference between waves reaching P from S_{1} and S_{2} is zero, therefore there is a bright point at P.

d) Consider a point Q on the screen which is at a distance x from the central point P on the screen. Light waves from S_{1} and S_{2} reach at Q simultaneously by covering path S_{1}Q and S_{2}Q, where they superimpose

h). If x << D and d << D then,

S_{1}Q ≈ S_{2}Q ≈ D

S_{2}Q + S_{1}Q = 2D

∴ Equation (3) becomes,

`S_2Q - S_1Q = (2xd)/(2D)`

`:. S_2Q - S_1Q = (xd)/D`

`:.trianglex = (xd)/D` .........4

Equation (4) gives the path difference of two interfering light waves