Question

The plane face of a plano-convex lens is silvered. If $\mu$ be the refractive index and $R$ be the radius of curvature of curved surface, then system will behave like a concave mirror of radius of curvature

• A. $\mu R$
• B. $R^2/\mu$
• C. $R/\left(\mu -1\right)$
• D. $\left[\left(\mu +1\right)/\left(\mu -1\right)\right]R$

Answer 1

Answer: (C)

When an object is placed in front of such a lens, the rays first of all refracted from the convex surface, then reflect from the polished plane surface and again refracts from convex surface. If $f_l$ and $f_m$ be the focal lengths of lens (convex surface) and mirror (plane polished surface) respectively, then effective focal length $F$ is given by
$\frac{1}{F}=\frac{1}{f_l}+\frac{1}{f_m}+\frac{1}{f_l}$
$=\frac{2}{f_l}+\frac{1}{f_m}=\frac{2}{f_l}$ $\left(\because f_m=\frac{R}{2}=\infty \right)$
Now, $\frac{1}{f_l}=\left(\mu -1\right)\left(\frac{1}{R}\right)$
$\therefore \frac{1}{F}=\frac{2\left(\mu -1\right)}{R}$
or, $F=\frac{R}{2\left(\mu -1\right)}$
As, $R=2F=\frac{R}{\left(\mu -1\right)}$