Question

In the $YDSE$ shown, the two slits are covered with thin sheets having thickness $t$ and $2t$ and refractive index $2\mu \text{and}\mu$ respectively. Find the position $\left(y\right)$ of central maxima:

• A. $\text{zero}$
• B. $\frac{tD}{d}$
• C. $\frac{tD}{2d}$
• D. $\frac{2tD}{d}$

Answer 1

The correct option is B $\frac{tD}{d}$

Optical length will be altered due to placement of transparent sheets.

Thus, the path difference between interfering waves at $P$,

$\Rightarrow \Delta x=[S_{2}P+(\mu -1\left)2t\right]-[S_{1}P+(2\mu -1\left)t\right]$

$\Rightarrow \Delta x=(S_{2}P-S_{1}P)+2\mu t-2\mu t-2t+t...\left(1\right)$

We know that,

$S_{2}P-S_{1}P=\frac{yd}{D}...\left(2\right)$

Therefore, from equation $\left(1\right)\text{\&}\left(2\right)$ :

$\Delta x=\frac{yd}{D}-t$

For obtaining central maxima, path difference must be zero.

$\Rightarrow \Delta x=0$

$\Rightarrow \frac{yd}{D}-t=0$

$\Rightarrow y=\frac{tD}{d}$

Hence, option $\left(\text{B}\right)$ is correct.