Question

A plano-convex lens when silvered on the plane side behaves like a concave mirror of focal length $60cm$. However, when silvered on the convex side, it behaves like a concave mirror of focal length $20cm$.Then, the refractive index of the lens is

• A. 3
• B. 1.5
• C. 1
• D. 2

Answer 1

The correct option is B 1.5

Given: A plano-convex lens when silvered on the plane side behaves like a concave mirror of focal length 60cm. However, when silvered on the convex side, it behaves like a concave mirror of focal length 20cm.

To find the refractive index of the lens

Solution:

We know,

$\frac{1}{F}=\frac{2}{f}+\frac{1}{f_m}............\left(i\right)$, where

When plane surface is silvered it becomes concave mirror of focal length 60cm,

and $f_m=\infty$

So the eqn(i) becomes,

$\frac{1}{60}=\frac{2}{f}+\frac{1}{\infty }⟹f=120cm$

When convex side is silvered it becomes concave mirror of focal length 10cm,

and $f_m=\frac{R}{2}$

So the eqn(i) becomes,

$\frac{1}{20}=\frac{2}{f}+\frac{2}{R}⟹\frac{2}{R}=\frac{1}{20}-\frac{2}{f}⟹\frac{2}{R}=\frac{1}{20}-\frac{2}{120}⟹\frac{2}{R}=\frac{6-2}{120}⟹R=\frac{120\times 2}{4}⟹R=60cm$

Now using the focal length formula of the plano-convex lens, we get

$\frac{1}{f}=(\mu -1)\times \frac{1}{R}⟹\frac{1}{120}=(\mu -1)\times \frac{1}{60}⟹\mu -1=\frac{60}{120}⟹\mu =\frac{1}{2}+1⟹\mu =1.5$

is the refractive index of the lens.