Question

The angle of a prism is ${}^{\prime }{A}^{\prime }$. One of its refracting surface is silvered. Light rays falling at an angle of incident $2A$ 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

• 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.