Question

Obtain the conditions for observing maxima and minima in a Young two-slit
interference pattern. Show that these conditions change when a thin transparent
sheet of thickness t and refractive index μ is introduced in the path of one of the
superposing beams.

Answer 1

1 Obtain the conditions for observing maxima and minima in a Young two-slit interference pattern.

Let S₁ and S2 be two slits separated by a distance d. Consider a point P on XY plane such that CP = x. The nature of interference between two waves reaching point P depends on the path difference S₂P-S₁P.

We using figure:

$$S1P^2=D^2+(x-\frac{d}{2})^2 (1)$$ $$S2P^2=D^2+(x+\frac{d}{2})^2 (2)$$

We get using (1) and (2):

$$S2P^2 - S1P^2=2xd (3)$$

$$(S2P - S1P)= \frac{2xd}{ S2P + S1P } (4)$$

for x, d<<< D , S₁P+S₂P =2D with negligible error included , path difference would be

$$(S2P - S1P)= \frac{xd}{D} (5)$$

Phase difference between wave for constructive interference is equal to

$$n \lambda$$

In this case, we can write

$$n \lambda=\frac{xd}{D} (6)$$

where

$$n=0, \pm1, \pm2, \pm3, ...$$

Phase difference between wave for destructive interference is equal to

$$\frac{(2n+1)\lambda }{2}$$

Similarly, for destructive interference,

$$\frac{(2n+1)\lambda }{2} =\frac{xd}{D} (7)$$

2 Show that these conditions change when a thin transparent sheet of thickness t and refractive index μ is introduced in the path of one of the superposing beams.

Let a thin transparent sheet of thickness t and refractive index μ be introduced in the path of wave from one slit S1. It is seen from the figure that light reaching the point P from source S1 has to traverse a distance t in the sheet and a distance (S1P−t) in the air. If c and v are velocities of light in air and in transparent sheet respectively, then the time taken by light to reach from S1 to P is given by

$$\frac{(S1P-t) } {c } + \frac{t} {v}=\frac{(S1P-t) } {c } + \frac{\mu t} {c} (8)$$

The effective path difference at any point P on the screen

$$Δ= S2P - (S1P+(\mu-1)t) (9)$$

Using (5) and (9) we can write for constructive interference

$$n \lambda=\frac{xd}{D} - (\mu-1)t (10)$$

for destructive interference

$$\frac{(2n+1)\lambda }{2}=\frac{xd}{D} - (\mu-1)t (11)$$