Find the angular separation between the consecutive bright fringes in a Young's double slit experiment with blue-green light of wavelength $500 \mathrm{nm}.$ The separation between the slits is $2.0\times {10}^{-3} \mathrm{m}$.

A plate of thickness $t$ made of a material of refractive index $\mu$ is placed in front of one of the slits in a double slit experiment.

$\left(a\right)$ Find the change in the optical path due to introduction of the plate.

$\left(b\right)$ What should be the minimum thickness $t$ which will make the intensity at the center of the fringe pattern zero? Wavelength of the light used is $d$. Neglect any absorption of light in the plate.