question_answer

The plane face of a planoconvex lens is silvered. If u be the refractive index and R, the radius of curvature of curved surface, then the system will behave like a concave mirror of radius of curvature

A) \[\mu R\]                                           

B) \[\frac{R}{(\mu -1)}\]

C) \[\frac{{{R}^{2}}}{\mu }\]                            

D)        \[\left[ \frac{(\mu +1)}{(\mu -1)} \right]R\]

Correct Answer: B

Solution :

When an object is placed in front of such a lens, the rays are first of all refracted from the convex surface, then reflected from the polished plane surface and again refracted from convex surface. Let\[{{f}_{l}},\,\,{{f}_{m}}\]be focal lengths of convex surface and mirror (plane polished surface) respectively, then effective focal length is                 \[\frac{1}{F}=\frac{1}{{{f}_{l}}}+\frac{1}{{{f}_{m}}}+\frac{1}{{{f}_{l}}}=\frac{2}{{{f}_{l}}}+\frac{1}{{{f}_{m}}}\] Since,    \[{{f}_{m}}=\frac{R}{2}=\infty \] \[\therefore \]  \[\frac{1}{F}=\frac{2}{{{f}_{l}}}\] From lens formula                 \[\frac{1}{{{f}_{l}}}=(\mu -1)\left( \frac{1}{R} \right)\] \[\therefore \]  \[\frac{1}{F}=\frac{2(\mu -1)}{R}\] \[\Rightarrow \]               \[F=\frac{R}{2(\mu -1)}\] or            \[{{R}_{eq}}=2\,\,F=\frac{R}{(\mu -1)}\]