Question

For the given incident ray as shown in figure, 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 $\sqrt{\frac{3}{2}}$

For critical reflection

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

$\cos {\theta }_c=\sqrt{1-{\sin }^2{\theta }_c}=\sqrt{1-\frac{1}{{\mu }^2}}$

$\mu \cos {\theta }_c=\sqrt{{\mu }^2-1}........\left(1\right)$

Applying Snell’s law at first incident

${\mu }_o\sin {\theta }_i=\mu \sin {\theta }_r$

$\Rightarrow 1\times \sin {45}^o=\mu \sin (90-{\theta }_c)$

$\Rightarrow \frac{1}{\sqrt{2}}=\mu \cos {\theta }_c=\sqrt{{\mu }^2-1}$

$\Rightarrow \mu =\sqrt{\frac{3}{2}}$