In YDSE, $β$ is the fringe width M and N are the two slits. The central maximum is formed at a point O. After some time, the central maximum is found to be shifted by $t (μ - 1)\frac{β}{λ}$ towards A. Which of the followings is correct ?

A thin film of thickness t and refractive index $μ$ is placed in front of A

Path difference due to film = $t (μ - 1)$

Path difference at a point x from the centre of screen = $x\frac{d}{D}$

$∴t (μ - 1)=x\frac{d}{D}$   $⇒x=t (μ - 1)\frac{D}{d}$

$∵β=\frac{λD}{d}$   $∴\frac{D}{d}=\frac{B}{λ}$

$∴x=t(μ - 1)\frac{β}{λ}$