Question
A light ray enters through a right-angled prism at point $P$ with the angle of incidence $30^{\circ}$ as shown in figure. It travels through the prism parallel to its base $BC$ and emerges along the face $AC$. The refractive index of the prism is:
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Solution
Sol. [Key Concept: Refraction and total internal reflection in a prism]
$$A = 90^{\circ}$$
In prism, $r_{1} + c = A$
$$\begin{array}{r} r_{1} = 90^{\circ} - c\#(1) \\ sinc = \frac{1}{\mu} \Rightarrow cosc = \frac{\sqrt{\mu^{2} - 1}}{\mu}\#(1) \end{array}$$
⇒ Apply Snell's law, on incidence surface
$$1 \cdot sin30^{\circ} = \mu sin\left( r_{1} \right) \Rightarrow 1 \times \frac{1}{2} = \mu \times sin\left( 90^{\circ} - c \right)$$
$$\frac{1}{2} = \mu \times \frac{\sqrt{\mu^{2} - 1}}{\mu}$$
On squaring $\frac{1}{4} = \mu^{2} - 1$
$$\Rightarrow \mu^{2} = \frac{5}{4} \Rightarrow \mu = \frac{\sqrt{5}}{2} $$
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