Determine the thickness of a plate required to produce a change in...

Let the thickness of the plate be $t$.

The optical path of light travelling through the plate of thickness $t$ and refractive index $\mu$ is $\mu t$.

The optical path of light travelling through the same distance $t$ in vacuum is $t$.

The change in the optical path introduced by the plate is $\Delta x = \mu t - t = (\mu - 1)t$.

Given that the change in the optical path is equal to half the wavelength $\lambda$, we have:

$(\mu - 1)t = \dfrac{\lambda}{2}$

$t = \dfrac{\lambda}{2(\mu - 1)}$