In Young’s double slit experiment, two slits $S_1$ and $S_2$ are $d$ distance apart and the separation from slits to screen is $D$ (as shown in figure). Now if two transparent slabs of equal thickness $0.1 \mathrm{mm}$ but refractive index $1.51$ and $1.55$ are introduced in the path of beam $\left(\mathrm{\lambda }=4000 \mathrm{\AA }\right)$ from $S_1$ and $S_2$ respectively. The central bright fringe spot will shift by ________ number of fringes.

 

The displacement equations of two interfering waves are given by $y_1=10 \sin \left(\omega t+\frac{\pi }{3}\right) \mathrm{cm}$, $y_2=5\left[\sin \left(\omega t\right)+\sqrt{3} \cos \omega t\right] \mathrm{cm}$ respectively. The amplitude of the resultant wave is _____ $\mathrm{cm}$.

As shown in the figure, in Young's double slit experiment, a thin plate of thickness $t=10 \mathrm{\mu m}$ and refractive index $\mu =1.2$ is inserted infront of slit $S_1$. The experiment is conducted in air ($\mu =1$) and uses a monochromatic light of wavelength $\lambda =500 \mathrm{nm}$. Due to the insertion of the plate, central maxima is shifted by a distance of $x{\beta }_0$. ${\beta }_0$ is the fringe-width before the insertion of the plate. The value of the $x$ is ______.