Question

A ray of light is incident at an angle of ${45}_o$ on a square slab of a transparent material. What should be the refractive index of the material of the slab so that total internal reflection occurs at the vertical face AB.

(IIT-JEE 2005)

• A. $\sqrt{2}$
• B. $\sqrt{\frac{3}{2}}$
• C. 1.5
• D. $\sqrt{\frac{5}{3}}$

Answer 1

The correct option is B $\sqrt{\frac{3}{2}}$


At first interface
$\frac{sin\left({46}^{\circ }\right)}{sin\left(r\right)}=x$ ..(1)

At second interface
$cos\left(r\right)=\frac{1}{x}$ ..(2)

$\Rightarrow sin\left(r\right)=\frac{\sqrt{x^2-1}}{x}$
from (1) and (2)
$\frac{1}{\sqrt{2}\times \frac{x^2-1}{x}}=x$

$\Rightarrow \frac{1}{\sqrt{2}}=\sqrt{x^2-1}$
$\Rightarrow x^2-1=\frac{1}{2}$
$\Rightarrow x^2=\frac{3}{2}$
$\Rightarrow x=\sqrt{\frac{3}{2}}$