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:

When one of the slits of Young's experiment is covered with a transparent sheet of thickness $4.8 \, mm$,the central fringe shifts to a position originally occupied by the $30^{th}$ bright fringe. What should be the thickness of the sheet if the central fringe has to shift to the position occupied by the $20^{th}$ bright fringe?

While conducting the Young's double slit experiment,a student replaced the two slits with a large opaque plate in the $x-y$ plane containing two small holes that act as two coherent point sources $(S_1, S_2)$ emitting light of wavelength $600 \ nm$. The student mistakenly placed the screen parallel to the $x-z$ plane (for $z>0$) at a distance $D=3 \ m$ from the mid-point of $S_1 S_2$,as shown schematically in the figure. The distance between the sources $d=0.6003 \ mm$. The origin $O$ is at the intersection of the screen and the line joining $S_1 S_2$. Which of the following is(are) true of the intensity pattern on the screen?
$(A)$ Straight bright and dark bands parallel to the $x$-axis
$(B)$ The region very close to the point $O$ will be dark
$(C)$ Hyperbolic bright and dark bands with foci symmetrically placed about $O$ in the $x$-direction
$(D)$ Semi circular bright and dark bands centered at point $O$

Interference fringes were produced in Young's double slit experiment using light of wavelength $5000 \, Å$. When a film of material $2.5 \times 10^{-3} \, cm$ thick was placed over one of the slits, the fringe pattern shifted by a distance equal to $20$ fringe widths. The refractive index of the material of the film is

In $YDSE,$ the source placed symmetrically with respect to the slits is now moved parallel to the plane of the slits so that it is closer to the upper slit,as shown. Then,

In Young's double-slit arrangement,the screen starts moving towards the right with a constant speed $v$. The initial distance between the screen and the plane of the slits is $x$. At $t=0$,the $1^{\text{st}}$ order maxima is located at point $A$. After how much time will the first order minima lie at point $A$?