Planes bounding the dielectric are as shown below.





Due to dielectric, a light ray incident at $y=0,$ at some angle $\theta$ bends towards normal and it strikes plane $x=a$.

If angle of incidence at $x=a$, is more than angle of critical incidence, $\mathrm{TIR}$ occurs and light ray, then emerges out of plane $y=b$.

For $\mathrm{TIR}$ at

$x=a$ $\sin \left(90°-r\right)=\frac{1}{n}$ or $\cos r=\frac{1}{n}....\left(\mathrm{i}\right)$

Also, $n=\frac{\sin \theta }{\sin r}$

Now $\theta$ at $y=0,$ varies from $0°$ to $a$ maximum of $90°$

So, for limiting incidence,

$n=\frac{\sin 90°}{\sin r}\Rightarrow n=\frac{1}{\sin r}....\left(\mathrm{ii}\right)$

From Eqs. (i) and (ii), we get

$\sin r=\cos r\Rightarrow r=45°$

Substituting $r=45°$ in Eq. (i) or Eq. (ii),

we get $n_{\min }=\frac{1}{\cos 45°}=\sqrt{2}$