Question

The plane face of a planoconvex lens is silvered. If $\mu$ 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. $\frac{R}{\mu}$
• D. $\left[\frac{\left(\mu+1\right)}{\left(\mu-1\right)}\right]R$

Answer 1

Answer: (B)

$\frac{R}{(\mu - 1)}$

Answer 2

For planoconvex lens (without its plane surface silvered),
$\frac{1}{f_{L}} =\left(\mu-1\right)\left(\frac{1}{R}-\frac{1}{\infty}\right)=\frac{\mu-1}{R}$
or $f_{L}=\frac{R}{\left(\mu-1\right)}$
When an object is placed in front of the planoconvex lens with its plane face silvered, light rays are : (i) refracted at the convex surface (ii) reflected at the silvered surface and (iii) refracted again at convex surface. If $F$ is the effective focal length of the combination, then
$\frac{1}{F} =\frac{1}{f_{L}}+\frac{1}{f_{M}}+\frac{1}{f_{L}}=\frac{2}{f_{L}}$ (as $f_M = 8$)
or $F =\frac{f_{L}}{2}=\frac{R}{2\left(\mu-1\right)}$
$\therefore$ Radius of curvature of the concave mirror
$= 2F =\left(\frac{R}{\mu-1}\right)$