Light of wavelength $500 \, nm$ is used to form an interference pattern in Young's double-slit experiment. $A$ uniform glass plate of refractive index $1.5$ and thickness $0.1 \, mm$ is introduced in the path of one of the interfering beams. The number of fringes by which the central maximum shifts 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. . . . .

In a double slit arrangement,fringes are produced using light of wavelength $4800 \ \mathring A$. One slit is covered by a thin plate of glass of refractive index $1.4$ and the other with another glass plate of the same thickness but of refractive index $1.7$. By doing so,the central bright fringe shifts to the original fifth bright fringe from the center. The thickness of the glass plate is ...... $\mu m$.

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 Young's double-slit experiment,if one slit is made twice as wide as the other instead of having equal widths,then in the interference pattern:

The figure shows a Young's double slit experimental setup. It is observed that when a thin transparent sheet of thickness $t$ and refractive index $\mu$ is placed in front of one of the slits,the central maximum shifts by a distance equal to $n$ fringe widths. If the wavelength of light used is $\lambda$,then $t$ will be: