Question

A transparent cube of side $d$, made of a material of refractive index ${\mu }_2$, is immersed in a liquid of refractive index ${\mu }_1\left({\mu }_1<{\mu }_2\right)$. A ray is incident on the face $AB$ at an angle $\theta$ (shown in the figure). Total internal reflection takes place at the point $E$ on the face $BC$.

• A. $\theta <si{n}^{-1}\sqrt{\frac{{\mu }_2^2}{{\mu }_1^2}-1}$
• B. $\theta >si{n}^{-1}\frac{{\mu }_1}{{\mu }_2}$
• C. $\theta <si{n}^{-1}\frac{{\mu }_1}{{\mu }_2}$
• D. $\theta >si{n}^{-1}\sqrt{\frac{{\mu }_2^2}{{\mu }_1^2}-1}$

Answer 1

Answer: (A)

At surface (1)
${\mu }_1\sin \theta ={\mu }_2\sin r$
At surface (2)
$\begin{array}{l}{\mu }_1={\mu }_2\sin (90-r)\\ {\mu }_1={\mu }_2\cos r\dots \text{ (2) }\end{array}$
From equation (1) and (2)
$\begin{array}{l}\Rightarrow \sin \theta =\frac{\sin r}{\cos r}\\ \Rightarrow \sin \theta =\frac{\frac{\left(\sqrt{{\mu }_2^2-{\mu }_1^2}\right)}{{\mu }_2}}{\left(\frac{{\mu }_1}{{\mu }_2}\right)}\\ \Rightarrow \sin \theta =\sqrt{\frac{{\mu }_2^2}{{\mu }_1^2}-1}\end{array}$
$\theta$ must satisfy,
$\Rightarrow \theta <{\sin }^{-1}\left(\sqrt{\frac{{\mu }_2^2}{{\mu }_1^2}-1}\right)$