A ray of light is incident grazingly of face AB to a right-angled prism as shown. It emerges out of face AC, as shown , $e = angle of emergence$ . Refractive indices of different media have been shown in the diagram. Choose the correct alternatives

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If the ray just fails to emerge out of the face AC of the prism, for $μ_{2} = \sqrt{3}$ , angle of refraction of face AB is $\sin^{- 1} \sqrt{\frac{2}{3}}$

$μ_{2}^{2} + \sin^{2} e = μ_{1}^{2}$

$μ_{1}^{2} + \sin^{2} e = μ_{2}^{2}$

$μ_{1}^{2} + \cos^{2} e = μ_{2}^{2}$

answer is A, D.

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Detailed Solution

By Snell’s law,
$\frac{μ_{2}}{μ_{1}} = \frac{\sin 90^{\circ}}{\sin r_{1}}$
or $\sin r_{1} = (\frac{μ_{1}}{μ_{2}}) - - - - (i)$
Similarly, $\frac{\sin e}{\sin r_{2}} = μ_{2} - - - - (ii)$

$r_{1} + r_{2} = 90^{o} - - - - (iii)$

from (i), (ii) and (iii),

$μ_{1}^{2} + \sin^{2} e = μ_{2}^{2}$

For TIR at face AC,

using eqn (ii) and (iii),

$\begin{array}{l} \frac{sine}{\sin r_{2}} = μ_{2} \\ \cos r_{1} = \frac{1}{\sqrt{3}} \\ \sin r_{1} = \sqrt{\frac{2}{3}} \\ r_{1} = \sin^{- 1} \sqrt{\frac{2}{3}} \end{array}$