Displacement of Fringes Due to Glass Slab

The speed of light in any transparent material medium depends upon the refractive index of that material.

The speed of light in air, $c$ is equal to $3\times 10^8$ m/s.

Suppose $v$ is the speed of light in another material having refractive index $\mu$.

The refractive index $(\mu )$ is defined as the ratio of speed of the light in vacuum to the speed of light in any specific medium.

Now, with the help of this refractive index concept, we will discuss the displacement of fringe in Young's double slit experiment.

Let's discuss the displacement of fringes due to glass slab.

Suppose that the two waves from the slits $S_1$ and $S_2$ meet at a point $P$ on the screen.

Now, the path difference will be $(S_{2}P-S_{1}P)$. Here the medium is air.

Now, consider a situation when we change the medium.

We have put a glass slab of refractive index $\mu$ and thickness of $t$ in the path of one of the two sources.

Then the center of the fringes is shifted to a new position.

The shifting of the center of fringes happens due to the difference in the optical path.

Now, we assume the same thickness as a glass tube in an air medium.

The optical distance is the distance traveled in air multiplied by the refractive index.

Now, the optical distance in the first case will be $t$ because the $\mu$ of air is one. while in the second case the optical distance will be $\mu t$.

If we replace the air medium by glass medium then the difference in the path is $\mu -1$.

Now, if point $P$ is the $3^{rd}$ bright fringe in the first case then the point $P$ in the second case will be $5^{th}$ bright fringe.

Because any point on the screen will have its nature depending on the path difference.

And, path difference will change by the $(\mu -1)t$. If $(\mu -1)t$ is equal to the bandwidth then only fringe width will be shifted.

If $(\mu -1)t=2$ then twice fringe will be shifted.

Let, the distance of $P$ from the center of the bright fringe be $x$.

Where, $d$ is the distance between two slits and $D$ is the distance of the screen from the slit.

Now we are interested in finding the shift or displacement of fringes $S$ from $3B$ to $5B$.

Because the path difference is $(\mu -1)t$ so the shift or displacement can be calculated as,

This is the general formula to calculate the shift when both mediums having refractive index ${\mu }_1$ and ${\mu }_2$.

Revision

Two waves traveling through the air and meet at a point $P$.

When we change the medium as a glass tube in the path of one ray then its fringes will shift to the new position.

And, displacement of the fringes $S$ can be calculated as,

The End