Question

A thin mica sheet of thickness $t$ and refractive index ($\mu = 1.5$) is introduced in the path of one of the waves in YDSE. The wavelength of the wave is $5000\mathop {\rm A}\limits^ \circ$. If central maxima is shifted by $2$ fringes then $t$ is:
A. $2\mu m$
B. $6\mu m$
C. $1\mu m$
D. $4\mu m$

Answer 1

When the monochromatic light used in the Young’s Double Slit experiment is hindered by a thin sheet with a small thickness the light suffers partial reflections and refractions when passing through the sheet and hence there is a difference in the interference pattern and the fringes are shifted a little. The formula for the path difference is applied. The equation for the path difference is equated to the fringe shift in-order to find the thickness of the sheet used.

Formula used:
When an extra path difference is observed the thickness $t$ of the thin sheet introduced is given by:
$t = \dfrac{{n\lambda }}{{\left( {\mu - 1} \right)}}$
Where, $t$ is the thickness of the mica sheet, $n$ is the number of fringes shifted, $\lambda$ is the wavelength and $\mu$ is the refractive index.

Complete step by step answer:
In the Young’s double slit experiment a source of monochromatic light is used and this light is made to pass through two slits that are arranged symmetrically to another slit from which the source light appears. This set-up is successful in determining an interference pattern on the screen which consists of alternating dark and bright fringes which appear to be damped or tend to die down as it progresses. The center most fringe is the fringe with the highest intensity among other fringes and is known as the central maxima.

The above question revolves around the concept of introducing a thin mica sheet to the light used in the YDSE experiment. When a thin sheet of negligible thickness is inserted in the path of one of the interfering beams then the interference phenomenon seems to be retained but there occurs a shift of the fringe pattern about its central maxima.

The light is said to suffer certain changes in its path due to the change in medium and hence there is a path difference that is observed. The path length, $p$, is the extra path difference introduced due to the insertion of the mica sheet. We are aware that the sheet is placed in only one path of light. Hence, the path difference is the difference between the lengths of the path of light passing through the sheet and that travelling through air.
Hence the relation will be:
$p = \mu t - t$

Hence this path difference is given by the equation:
$\Rightarrow p = \left( {\mu - 1} \right)t$ ------($1$)
Since there is a shift in the fringes there is a displacement that takes place. This displacement is as such that for the first shift the central maxima shifts a little and at the place of the central maxima the central minima occurs, that is, a dark fringe with the same intensity of that of the central maxima replaces the central maxima.

Here, in the question it is given that the fringe pattern shifts by two fringes which means the bright fringe again replaces the dark fringe, that is, after the fringe shifts twice.
The path difference for a bright fringe is given by the formula:
$p = n\lambda$ -----($2$)
Since equations ($1$) and ($2$) are identical we equate them to get:
$\left( {\mu - 1} \right)t = n\lambda$
By rearranging the terms we get the equation for thickness of the sheet to be:
$\Rightarrow t = \dfrac{{n\lambda }}{{\left( {\mu - 1} \right)}}$ -----($3$)
Now, we extract the data given in the question.
Given, $\lambda = 5000\mathop {\rm A}\limits^ \circ$
$\Rightarrow n = 2$
$\Rightarrow \mu = 1.5$
We are asked to find the thickness of the mica sheet. Hence by substituting these values into the equation ($3$) we get:
$t = \dfrac{{2 \times 5000\mathop {\rm A}\limits^ \circ }}{{\left( {1.5 - 1} \right)}}$

We now convert Armstrong into meters which is the SI unit of wavelength.
$1\mathop {\rm A}\limits^ \circ = {10^{ - 10}}m$
$\Rightarrow 5000\mathop {\rm A}\limits^ \circ = 5000 \times {10^{ - 10}}m$
Hence we substitute this value in the above equation:
$t = \dfrac{{2 \times 5000 \times {{10}^{ - 10}}}}{{\left( {1.5 - 1} \right)}}$
We solve the equation to get the value for the thickness $t$.
$t = \dfrac{{2 \times 5 \times {{10}^3} \times {{10}^{ - 10}}}}{{0.5}}$
$\Rightarrow t = \dfrac{{2 \times 5 \times {{10}^3} \times {{10}^{ - 10}}}}{{\dfrac{1}{2}}}$
$\Rightarrow t = 4 \times 5 \times {10^{ - 7}}$
$\Rightarrow t = 20 \times {10^{ - 7}}$
$\Rightarrow t = 2 \times {10^{ - 6}}m$
This can be converted into micrometers as $1\mu m = {10^{ - 6}}m$
$\therefore t = 2\mu m$
Thus the thickness of the mica sheet introduced in one of the paths of the waves is $2\mu m$.

Hence, the correct option is option A.

Additional information: Interference is the phenomenon which occurs when two light waves of same frequency and having zero or constant phase difference travelling in the same direction superpose each other and the intensity in the region of superposition gets redistributed becoming maximum at certain points and minimum at other points.

According to the principle of superposition when the crest on one wave of light falls on the crest of another wave of light or the troughs of the two waves fall on each other, then the amplitudes are added up giving maximum intensity and this type of interference is called constructive interference leading to the formation of bright fringes. When the crest of one wave falls on the trough of another wave then destructive interference occurs leading to the formation of dark fringes.

Note: Even-though there is a shift in the fringes of the interference pattern the fringe width does not change and remains the same. The entire fringe pattern shifts about the central maxima and this shift always occurs towards the side in which the thin sheet is placed. The displacement and the path difference are not the same which is often a common misconception. The displacement is the amount of shift in the fringes from its original position while path difference is the difference in the wavelengths of the two lights with respect to each other.