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, $R=2F=\frac{R}{\left(\mu - 1\right)}$