Question

A converging lens of focal length $15cm$ and a converging mirror of focal length $20cm$ are placed with their principal axis coinciding. A point source $S$ is placed on the principal axis at a distance of $12cm$ from the lens as shown in the figure. It is found that the final beam comes out parallel to the principal axis. Find the separation between the mirror and the lens.

Answer 1

1). For the redfraction through the lens we get

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$

By using the distance of the object as

$u=-12cm$

$f=15cm$

$\frac{1}{v}=\frac{1}{f}+\frac{1}{u}$

We get the distance of the image formed as $v=-60cm$

Negative as the image is on the left side.

Now this image formed at $I_1$ by the lens will act as a source for the mirror.

The image formed by the mirror acts again as the source for the lens so the image formed by the mirror must be at the focus of the lens to satisfy the given condition of final parallel rays.

The distance between the two images formed by the mirror and the lens will be $75cm$

2). For the redfraction through mirror

Considering the mirror formula we get

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$

Consider the distance between the mirror and the second image$I_2$is $d$

$u=-(75+d)cm$

$v=-dcm$

$f=-20cm$

$\frac{1}{-20}=\frac{1}{-d}+\frac{1}{-\left(75+d\right)}$

By solving the equation we get

$d^2+35d-1500=0$

We get the distance between the mirror and the lens as

$d=25+15=40cm$