A thin plano-convex lens acts like a concave mirror of focal length $0.2m$ when silvered at its plane surface. Refractive index of the material of the lens is $1.5$. Radius of curvature of convex surface of the lens is:
A) $0.1m$
B) $0.2m$
C) $0.4m$
D) $0.8m$

Hint: In order to solve this question, the knowledge of lens formula and lens maker’s formula is important. The concept and structure of a plano-convex lens is also important. Remember that the focal length and radius of curvature of a plane mirror is always zero. Take care of SI units of the given physical quantities.

Complete step by step solution:
Here we are given the question that the focal length of the mirror is $0.2m$ .
Let it be represented by the variable $f$ so we have $f = 0.2m$ .
In centimetres we have $f = 20cm$ .
 The refractive index of the lens is given as $1.5$ .
Let it be represented by the variable $\mu $ .
So, we have $\mu = 1.5$ .
As we know that,
$\dfrac{1}{f} = \dfrac{2}{{{f_1}}} + \dfrac{1}{{{f_{mirror}}}}$
Here , as we know that in a plano-convex lens one side is a plane mirror and has the focal length as infinity and the other side is the convex surface.
So here, ${f_{mirror}} = \infty $
So we have, $\dfrac{1}{f} = \dfrac{2}{{{f_1}}}$
Here ${f_1}$ is the focal length of the convex side of the plano-convex lens.
On solving we have, ${f_1} = 2f$
Putting the value of $f = 20cm$ . We have,
\[{f_1} = 2 \times 20cm\]
On solving we have,
${f_1} = 40cm$
Now we have lens maker's formula as,
$\dfrac{1}{{{f_1}}} = (\mu - 1)[\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}]$
Here in the plano-convex lens one side is plane and its radius of curvature is infinity.
So, ${R_2} = \infty $
On putting all the values in the lens maker’s formula we have,
$\dfrac{1}{{40}} = (1.5 - 1)[\dfrac{1}{{{R_1}}} - \dfrac{1}{\infty }]$
As we know that, $\dfrac{1}{\infty } = 0$
So we have, ${R_1} = 40 \times (1.5 - 1)$
On solving we have,
${R_1} = 20cm$

So we have the radius of the curve of the convex surface of the lens as $20cm$ or $0.2m$.

Note: The lens maker’s formula is the relation between the focal length of a lens to the refractive index of the material of the lens and the radii of curvature of its two surfaces. This formula is used by lens manufacturers to make the lenses of particular power from the glass of a given refractive index. It is important to note that the lens should be thin so that the separation between the two refracting surfaces should be small. Also, the medium on either side of the lens should be the same.