$\text{Intensity at a point where phase difference is }\Delta \phi \text{is given as } I = 4I_0 cos^2\frac{\Delta \phi}{2} = 2I_0, \Rightarrow cos\frac{\Delta \phi}{2} = \frac{1}{\sqrt 2}$

$\Rightarrow \frac{\Delta \phi}{2}= \frac{\pi}{4}, \Rightarrow \Delta\phi = \frac{\pi}{2}$

$\text{Path difference is }\Delta x = \frac{\lambda}{4} = (\mu -1)t$

$\Rightarrow t = \frac{\lambda /4}{(3/2 -1)} = \frac{\lambda}{2}$