To make the central fringe appear at the centre $O$ of the screen,a mica sheet of refractive index $\mu = 1.5$ is introduced in front of one of the slits. Choose the correct statement.

In the figure shown,if a parallel beam of white light is incident on the plane of the slits,then the distance of the white spot on the screen from $O$ is [Assume $d << D, \lambda << d$].

In the figure shown in $YDSE$, a parallel beam of light is incident on the slits from a medium of refractive index $n_1$. The wavelength of light in this medium is $\lambda_1$. $A$ transparent slab of thickness $t$ and refractive index $n_3$ is placed in front of one slit. The medium between the screen and the plane of the slits is $n_2$. The phase difference between the light waves reaching point $O$ (symmetrical, relative to the slits) is:

In a Young's double slit experiment,each of the two slits $A$ and $B$,as shown in the figure,are oscillating about their fixed center with a mean separation of $0.8 \ mm$. The distance between the slits at time $t$ is given by $d = (0.8 + 0.04 \sin \omega t) \ mm$,where $\omega = 0.08 \ rad \ s^{-1}$. The distance of the screen from the slits is $1 \ m$ and the wavelength of the light used to illuminate the slits is $6000 \ \mathring A$. The interference pattern on the screen changes with time,while the central bright fringe (zeroth fringe) remains fixed at point $O$.
$(1)$ The $8^{\text{th}}$ bright fringe above the point $O$ oscillates with time between two extreme positions. The separation between these two extreme positions,in micrometer $(\mu m)$,is. . . . .
$(2)$ The maximum speed in $\mu m/s$ at which the $8^{\text{th}}$ bright fringe will move is. . . . .

Two coherent point sources $S_1$ and $S_2$ vibrating in phase emit light of wavelength $\lambda$. The separation between them is $2 \lambda$ as shown in the figure. The first bright fringe is formed at $P$ due to interference on a screen placed at a distance $D$ from $S_1$ $(D >> \lambda)$. Find the distance $OP$.

On placing a thin film of mica of thickness $12 \times 10^{-5} \text{ cm}$ in the path of one of the interfering waves in Young's double slit experiment using monochromatic light,the fringe pattern shifts through a distance equal to the width of a bright fringe. If the wavelength used is $\lambda = 6 \times 10^{-5} \text{ cm}$,the refractive index of mica is: