Question

Light is incident at an angle $\alpha$ on one planer end of a transparent cylindrical rod of refractive index $n$. Determine the least value of $n$ so that the light entering the rod does not emerge from the curved surface of the rod irrespective of the value of $\alpha$

Answer 1

The light entering the rod does nor emerge from the curved surface of rod when angle $90-r$ is greater than critical angle.

$\mu \le \frac{1}{\sin c}$ ,c is critical angle.

Here, $c=90-r$

$\mu \le \frac{1}{\sin (90-1)}$

$\mu \le \frac{1}{\cos r}$

As a limiting case , $\mu =\frac{1}{\cos r}$

By Snell's law at A,

$\mu =\frac{\sin alpha}{\sin r}$

$\sin r=\frac{\sin \alpha }{\mu }$

Smallest angle of incident on curved surface is when $\alpha =\frac{\pi }{2}$

$\sin r=\frac{\sin \left(\pi /2\right)}{\mu }$

$\mu =\frac{1}{\sin r}$

$\sin r=\cos r$

$\mu =\frac{1}{\cos 45}=\frac{1/1}{1/\sqrt{2}}$

$\mu =\sqrt{2}$