🌟 Fresnel’s Biprism & Its Applications: A Simple, Complete and Powerful Guide

The time taken by the light wave to reach P from S₁ is $$T=\frac{S_1P-t}{c}+\frac{t}{v}$$

$$\Rightarrow T=\frac{S_1P-t}{c}+\frac{\mu t}{c} \qquad{\left(Since\; \mu=\frac{c}{v} \right)} $$

$$\Rightarrow cT=S_1P+(\mu-1)t$$

The above equation is the effective path length in air from S₁ to P.

Therefore, the effective path difference between S₂P and S₁P will then be given by

$$\Delta =S_2P-S_1P+(\mu-1)t$$

Let the point P be the center of the nth bright fringe; then the path difference should be equal to nλ. Hence $$S_2P-S_1P+(\mu-1)t=n\lambda\qquad ……….(6)$$

But when thin sheet is not introduced, then $$S_2P-S_1P=\frac{x_nd}{D}$$

Hence eq. (6) becomes,

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

$$\Rightarrow x_n=\left[(\mu-1)t+n\lambda\right]\frac{D}{d}\qquad …………(7)$$

Here, x_n represents the distance of the nth bright fringe from the central fringe when a thin sheet is present.

But in the absence of a thin sheet, the distance of the nth bright fringe from the central fringe is given by

$$x_n^{‘}=\frac{n\lambda D}{d}$$

Therefore, the displacement of the  nth bright fringes is given by

$$x= \left[ n\lambda+(\mu-1)t \right]\frac{D}{d}-\frac{n\lambda D}{d}$$

$$\Rightarrow x= \frac{D}{d}\left( \mu-1 \right)t\qquad ……….(8)$$

Since the above displacement equation has no order term, it indicates that all fringes are shifted by a distance x.

Using equation (7), we can determine the fringe width as follows:

$$\beta=x_{n+1}-x_n$$

$$\Rightarrow \beta=\left[(\mu-1)t+(n+1)\lambda\right]\frac{D}{d}
-\left[(\mu-1)t+n\lambda\right]\frac{D}{d}$$

$$\Rightarrow \beta=\frac{\lambda D}{d}$$

This indicates that the presence of a thin sheet does not change the fringe width.

If the displacement of the fringe pattern (x) is equal to the width of n fringes, then x = nβ. Now using this value of x, equation (8) becomes 

$$n\beta= \frac{D}{d}\left( \mu-1 \right)t$$

$$\Rightarrow n \frac{\lambda D}{d}= \frac{D}{d}\left( \mu-1 \right)t$$

$$\Rightarrow t = \frac{n\lambda}{\mu-1}\qquad ………..(9)$$

This is the thickness of the thin sheet. It is also given by using equation (8) as follows:

$$t= \frac{xd}{D\left( \mu-1 \right)}\qquad ………(10)$$