Question

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

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

Answer 1

Answer: (C)

$R/(\mu -1)$

When an object is placed in front of such a lens, the rays first of all refracted from the convex surface, then refract from the polished plane surface and again refracts from convex surface. If $f_1$ 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
$\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_m}+\frac{1}{f_1}=\frac{2}{f_1}+\frac{1}{f_m}=\frac{2}{f_1}$ $(\because f_m=\frac{R}{2}=\infty )$
$\frac{1}{f_1}=(\mu -1)\left(\frac{1}{R}\right)$
$\therefore \frac{1}{F}=\frac{2(\mu -1)}{R}$ or $F=\frac{R}{2(\mu -1)}$ or $R=2F=\frac{R}{(\mu -1)}$.