Question

Comprehension
Two thin similar spherical glass pieces of radius $40\text{cm}$ are joined together front to front with rear portion of one piece silvered and the combination of glass pieces is placed at a distance $60\text{cm}$ from a screen. A point object is placed on the optical axis of the combination at the mid point of the line joining this combination and screen. Initially there is air between glass pieces as well as between combination and screen.


If air between the glass pieces is replaced by water $(\mu =\frac{4}{3})$, the new system (combination of glass pieces and water) behaves like :

• A. Convex mirror of focal length $12\text{cm}$
• B. Convex mirror of focal length $24\text{cm}$
• C. Concave mirror of focal length $12\text{cm}$
• D. Concave mirror of focal length $24\text{cm}$

Answer 1

The correct option is C Concave mirror of focal length $12\text{cm}$


To calculate the focal length of the combination, let us assume that parallel rays are incident on the combination from an doject placed at infinite distance from it. Now, the place where these rays will meet after the referaction and reflaction from the combination will be the focus of the combination.
$\Rightarrow u=-\infty$,
$f_{mirror}=-20\text{cm}$
$R_{lens}=40\text{cm}$
${\mu }_{lens}=\frac{4}{3}$
Image will be formed due to refraction from the lens at a distance of :-
$\frac{1}{v}-\frac{1}{u}=(\mu -1)(\frac{1}{R_1}-\frac{1}{R_2})$
$\Rightarrow \frac{1}{v}-\frac{1}{-\infty }=(\frac{4}{3}-1)(\frac{1}{40}-\frac{1}{-40})$
$\Rightarrow \frac{1}{v}=\frac{1}{3}\times \frac{2}{40}$
$\Rightarrow v=60\text{cm}$
$\therefore$ Image will be formed on the left hand side of the combination at distance of $60\text{cm}$ from the combination due to refraction from the lens.
This image will now act as an object for the mirror.
So, image will be formed by the mirror at a distance of :-
$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$
$\frac{1}{-20}=\frac{1}{v}+\frac{1}{60}$
$\frac{1}{v}=\frac{1}{-20}-\frac{1}{60}=\frac{-4}{60}$
$v=-15\text{cm}$.
Now, this image will act as an object for the lens.
$\Rightarrow$ Final image distance from combination will be:-
$\frac{1}{v}-\frac{1}{u}=(\mu -1)(\frac{1}{R_1}-\frac{1}{R_2})$
$\frac{1}{v}-\frac{1}{-15}=(\frac{4}{3}-1)\left(\frac{-2}{40}\right)$
$[\because R_1=-40\text{cm},R_2=40\text{cm}]$
$\frac{1}{v}=\frac{-1}{60}-\frac{1}{15}=\frac{-5}{60}$
$\Rightarrow v=f=-12\text{cm}$

Thus, the combination will behave as a concave mirror of local length $12\text{cm}$.