Question

For the given incident ray as shown in figure, the condition of total internal refraction of this ray the required 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

Answer: (C)

For given condition snell's law give
$1\sin 45^{\circ }=\mu \cdot \sin \left(90-{\theta }_c\right)$

$\frac{1}{\sqrt{2}}=\mu \cos {\theta }_c=\sqrt{{\mu }^2-1}$
$\Rightarrow {\mu }^2=1+\frac{1}{2}$
$\Rightarrow {\mu }^2=\frac{3}{2}$
$\Rightarrow \mu =\sqrt{\frac{3}{2}}$

$\sin {\theta }_c=\frac{1}{\mu }$
$\cos {\theta }_c=\sqrt{\frac{{\mu }^2-1}{\mu }}$
Alternate solution (objective method)
for given condition $\mu =\sqrt{1+{\sin }^2\theta }$
$\Rightarrow \mu =\sqrt{1+{\sin }^{2}45^{\circ }}$
$=\mu =\sqrt{1+\frac{1}{2}}$
$=\sqrt{\frac{3}{2}}$