Question

The angle of a prism is $A$. One of its refracting surfaces is silvered. Light rays falling at an angle of incidence $2A$ on the first surface returns back through the same path after suffering reflection at the silvered surface. The refractive index $\mu$, of the prism is
[Assume, surrounding medium to be air]

• A. $2\sin A$
• B. $2\cos A$
• C. $\frac{1}{2}\cos A$
• D. $\tan A$

Answer 1

The correct option is B $2\cos A$

On reflection from the silvered surface, the incident ray will retrace its path, if it falls normally on the surface.


In the $\Delta \text{PMN}$,

$\angle M=({180}^{\circ }-{90}^{\circ }-A)=({90}^{\circ }-A$)

Since, $({90}^{\circ }-A)+r_1={90}^{\circ }$

$\therefore r_1=A$

Applying Snell's law at the surface $\text{PQ}$,

${\mu }_{air}\sin i={\mu }_{prism}\sin r_1$

Substituting the given data we have ,

$1\sin i=\mu \sin r_1$

$\Rightarrow \mu =\frac{\sin 2A}{\sin A}\because [i=2A]$

$\Rightarrow \mu =\frac{2\sin A\cos A}{\sin A}\because [\sin 2A=2\sin A\cos A]$

$\therefore \mu =2\cos A$

Hence, option (b) is the correct answer.

Why this Question? This question tests the understanding of concept of refraction through prism when one of the face of the prism reflecting instead of refracting.