Physics Question from JEE Main 2023 · 1 February, Shift 2

In Young's double slit experiment the two slits are $S_1$ (upper) and $S_2$ (lower), cut in a barrier, and the pattern is observed at a point $P$ on a screen placed parallel to the barrier. A thin plate of thickness $t = 10 \, \mu\text{m}$ and refractive index $\mu = 1.2$ is inserted immediately in front of slit $S_1$. The experiment is conducted in air ($\mu = 1$) and uses a monochromatic light of wavelength $\lambda = 500 \, \text{nm}$. 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 _____.