Question

A slab of refractive index $\mu$ is placed in air and light is incident at maximum angle ${\theta }_0$ from vertical. Find minimum value of $\mu$ for which total internal reflection takes place at the vertical surface.

Answer 1

Using Snell’s law at the surface S₁:

$\mu \sin r=\sin {\theta }_0............\left(1\right)$

For TIR at vertical surface:

$\mu \sin {\theta }_c=\sin 90$

$\mu \sin {\theta }_c=1$

$\because {\theta }_c=90-r$

$\mu \sin (90-r)=1$

$\mu \cos r=\mathrm{1......}\left(2\right)$

Solving equation (1) and (2)

${\mu }^2({\sin }^{2}r+{\cos }^{2}r)={\sin }^2{\theta }_0+1$

${\mu }^2={\sin }^2{\theta }_0+1$

$\mu =\sqrt{{\sin }^2{\theta }_0+1}$