Question

A plano-convex lens (f=20cm) is silvered at the plane surface. The new focal length will be:
A) 20cm
B) 40cm
C) 30cm
D) 10cm

Answer 1

Hint: A Plano-convex lens is a lens which is made by joining a flat plane and a convex lens.
Silvering of flat plane converts it into a mirror as now the light rays will be reflected by the lens instead of passing through after refraction.
The power of a lens is the ability of a lens to converge or diverge a ray of light falling on it.

Complete step by step solution:
The following images show how a Plano-convex lens looks like

Now when its plane part is silvered so it acts as a mirror now, refer the image

Here we can say that it is now a combination of a convex lens and a plane mirror both of which have their powers
The power of a Plano-convex lens is equal to the sum of twice the power of lens and mirror
i.e.,$P_{total}=2P_{lens}+P_{mirror}$
We know that power of a lens is given by $P_{lens}=\left({\displaystyle \frac{1}{f_{lens}}}\right)$
Where $f_{lens}$is the focal length of the mirror
And the power of a plane mirror is given by $P_{mirror}=\left({\displaystyle \frac{1}{f_{mirror}}}\right)$
Where $f_{mirror}$ is the focal length of the mirror
So total power of the setup will be
$$\begin{gathered} \because P_{total}=2P_{lens}+P_{mirror} \\ \therefore P_{total}=2\times \left({\displaystyle \frac{1}{f_{lens}}}\right)+{\displaystyle \frac{1}{f_{mirror}}} \end{gathered}$$
Here the focal length of the lens is given as $20cm$
And we know that the focal length of a plane mirror is $\mathrm{\infty }$
Substituting these values in the above equation
$$\begin{gathered} P_{total}=2\times \left({\displaystyle \frac{1}{20}}\right)+{\displaystyle \frac{1}{\mathrm{\infty }}} \\ \Rightarrow P_{total}={\displaystyle \frac{1}{10}} \end{gathered}$$
Here we get to know that the power of a plane mirror is zero because a plane mirror neither converges or diverges any ray of light falling on it just reflects them back.
Now as we know that the power is the reciprocal of the focal length so the new focal length corresponding to this power will be,
Let new focal length be $f‘$
We know that
$$\begin{gathered} P_{total}={\displaystyle \frac{1}{f‘}} \\ \therefore f‘={\displaystyle \frac{1}{P_{total}}} \\ \because P_{total}={\displaystyle \frac{1}{10}} \\ \therefore f‘=10cm \end{gathered}$$

The new focal length of the Plano-convex mirror will be $10cm$.

Note: 1) The power of a lens depends upon its focal length
2) The focal length of a plane mirror is infinity and thus its power is zero
3) Silvering of any lens makes it a mirror because now it will reflect the light rays falling on it