Wave Optics

As shown in the figure, in Young's double slit experiment, a thin plate of thickness \(t=10 \mu \mathrm{m}\) 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 befor the insertion of the plate. The value of the \(x\) is _____________.