question_answer

The radius of curvature of a thin plano-convex lens is 10 cm (of curved surface) and the refractive index is\[1.5\]. If the plane surface is silvered, then it behaves like a concave mirror of focal length

A)  \[10\,cm\]                    

B)  \[15\,cm\]

C)  \[20\,cm\]                    

D)  \[5\,cm\]

Correct Answer: C

Solution :

 The silvered piano convex lens behaves as a concave mirror; whose focal length is given by \[\frac{1}{F}=\frac{2}{{{f}_{1}}}+\frac{1}{{{f}_{m}}}\] If plane surface is silvered \[{{f}_{m}}=\frac{{{R}_{2}}}{2}=\frac{\infty }{2}=\infty \] \[\therefore \] \[\frac{1}{{{f}_{1}}}=(\mu -1)\left( \frac{1}{{{R}_{1}}}-\frac{1}{{{R}_{2}}} \right)=(\mu -1)\left( \frac{1}{R}-\frac{1}{\infty } \right)=\frac{\mu -1}{R}\] \[\therefore \] \[\frac{1}{F}=\frac{2(\mu -1)}{R}+\frac{1}{\infty }=\frac{2(\mu -1)}{R}\Rightarrow F=\frac{R}{2(\mu -1)}\] Here \[R=20\,cm,\,\mu =1.5\] \[\therefore \] \[F=\frac{20}{2(1.5-1)}=20\,cm\]