Question

Two thin similar convex glass pieces are joined together front to front with its rear portion silvered such that a sharp image is formed $20\,cm$ from the mirror. When the air between the glass pieces is replaced by water $\left( {{\mu _w} = \dfrac{4}{3}} \right)$ , then the image formed from the mirror is at a distance of
A) $8\,cm$
B) $10\,cm$
C) $6\,cm$
D) $12\,cm$

Answer 1

Hint: Use the formula (2) to find the radius of curvature. Substitute the radius of curvature and the refractive index in the formula (1) to find the value of focal length. Then substitute this in the other formula, to find the value of distance of image from mirror.

Formula used:
(1) The relation of the concave lenses is
$\dfrac{1}{F} = \dfrac{{4\left( {\mu - 1} \right)}}{R} + \dfrac{2}{R}$
Where $F$ is the focal length of the mirror, $\mu $ is the refractive index of the mirror, $R$ is the radius of the curvature of the mirror.
(2) For the convex mirror,
$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{2}{R}$
Where $v$ is the distance of the image, $u$n is the distance of the object and $R$ is the radius of curvature.

Complete step by step solution:
It is given that the
Distance of the image formed from the object, $R = 20\,cm$
Before inserting water, it behaves as a convex mirror. Hence using formula (2)
$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{2}{R}$
Substituting the values.
\[\Rightarrow \dfrac{1}{{10}} + \dfrac{1}{{10}} = \dfrac{2}{R}\]
$\Rightarrow R = 20\,cm$
After converting into concave mirror by introducing water,
$\Rightarrow \dfrac{1}{F} = \dfrac{{4\left( {\mu - 1} \right)}}{R} + \dfrac{2}{R}$
Substituting the known values in the above equation, we get
$\Rightarrow \dfrac{1}{F} = \dfrac{{4\left( {\dfrac{4}{3} - 1} \right)}}{{20}} + \dfrac{2}{{20}}$
By simplifying the above equation, we get
$\Rightarrow \dfrac{1}{F} = \dfrac{1}{6}$
We know that $\dfrac{1}{F} = \dfrac{1}{u} + \dfrac{1}{v}$ , hence substituting the values in it, we get
$\Rightarrow \dfrac{2}{u} = \dfrac{1}{F} = \dfrac{1}{6}$
$\Rightarrow u = 12\,cm$

Thus the option (D) is correct.

Note: When the air is introduced between two glass pieces, the glass pieces convert into and act as a convex mirror. But if some amount of water is placed between the two glass pieces instead of air, then it might behave as a concave mirror.