A light beam is traveling from region I to region IV (refer fig.). The refractive index in region, I, II, $\text{ III and IV are }{\mathrm{n}}_0,\frac{{\mathrm{n}}_0}{2},\frac{{\mathrm{n}}_0}{6},\frac{{\mathrm{n}}_0}{8}$ ….. respectively. The angle of incidence $\theta$ for which the beam just misses entering region IV is ${\sin }^{-1}\left(\frac{1}{\mathrm{p}}\right)$. The value of p is