Question

For the given incident ray as shown in figure, for the condition of total internal reflection of this ray, the minimum refractive index of prism will be:-

• A. $\frac{\sqrt{3}+1}{2}$
• B. $\frac{\sqrt{2}+1}{2}$
• C. $\sqrt{\frac{3}{2}}$
• D. $\sqrt{\frac{7}{6}}$

Answer 1

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

By Snell's law
$1\times \sin {45}^{\circ }=\mu \sin ({90}^{\circ }-{\theta }_C)$
$\frac{1}{\sqrt{2}}=\mu \cos {\theta }_C...\left(i\right)$
For second surface
$\frac{1}{\mu }=\sin {\theta }_C...\left(2\right)$
Also,
${\sin }^2{\theta }_C+{\cos }^2{\theta }_C=1$

$\cos {\theta }_C=\sqrt{1-{\sin }^2{\theta }_C}$ put in (1)
$\frac{1}{\sqrt{2}}=\mu \times \sqrt{1-\frac{1}{{\mu }^2}}\Rightarrow \frac{1}{\sqrt{2}}=\sqrt{{\mu }^2-1}$
$\sqrt{\frac{1}{2}+1}=\mu \Rightarrow \mu =\sqrt{\frac{3}{2}}$