Question

One of the refracting surfaces of a prism, of refractive index 2 is silvered. The angle of the prism is equal to the critical angle of a medium of refractive index 2. A ray of light incident on the unsilvered surface passes through the prism and retraces its path after reflection at the silvered face. Then the angle of incidence of the unsilvered surface is

• A. 0${}^{\circ }$
• B. 30${}^{\circ }$
• C. 45${}^{\circ }$
• D. 90${}^{\circ }$

Answer 1

The correct option is D 90${}^{\circ }$

Since, ray of light retraces its path, it must incident on mirror normally.

Now, from geometry of $\Delta APQ$,

using exterior angle property:-

${90}^o-r+A={90}^o$

$⟹r=A$

But, $A=$critical angle=$si{n}^{-1}\left(\frac{1}{\mu }\right)$

$⟹r={\sin }^{-1}\left(\frac{1}{\mu }\right)$

$⟹\sin \left(r\right)=\frac{1}{\mu }$

Now, at refracting surface:-

$\frac{\sin i}{\sin r}=\mu$

$⟹\sin i=\mu \sin r=\mu \times \frac{1}{\mu }$

$⟹\sin i=1$

$⟹i={90}^o$

Hence, answer is option-(D).