Question

AB and CD are surfaces of two slabs as shown in Fig. The medium between the slabs has refractive index 2. Refractive index of the slab above AB is $\sqrt{2}$ and below CD is $\sqrt{3}$. The minimum angle of incidence at Q is ($x\times 12$) in degree, so that the ray is totally reflected by both the slabs. The value of $x$ is:

Answer 1

Let $\theta$ be the angle of incidence at face AB, then for total internal reflection at face AB.
$sin\theta >C_1=\frac{{\mu }_1}{{\mu }_2}=\frac{\sqrt{2}}{2}$
$\Rightarrow sin\theta >\frac{1}{\sqrt{2}}$ or $sin\theta >sin{45}^o\Rightarrow \theta >{45}^o$
For total internal reflection at face CD,
$sin\theta >sin{C}_2=\frac{{\mu }_3}{{\mu }_2}=\frac{\sqrt{3}}{2}\Rightarrow sin\theta >\frac{\sqrt{3}}{2}$
or $sin\theta >sin60$ or $\theta >{60}^o$
Therefore, for total internal reflection at both the surfaces,
${\theta }_{min}={60}^o$.