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 $\mu=\sqrt{1+\sin ^{2} \theta}$
$\Rightarrow \mu=\sqrt{1+\sin ^{2} 45^{\circ}}$
$=\mu=\sqrt{1+\frac{1}{2}}$
$=\sqrt{\frac{3}{2}}$