Question

A converging and a converging mirror are placed with their principal axis coinciding. Their separation equals $40cm$. A point source $S$ is placed on the principal axis at a distance of $2cm$ from the lens as shown in the figure. It is found that the final beam comes out parallel to the principal axis. The focal length of the lens equals $5cm$. Find the focal length of the mirror.

Answer 1

Given: A converging lens and a converging mirror are placed with their principal axis coinciding. Their separation equals 40cm. A point source S is placed on the principal axis at a distance of 2cm from the lens as shown in the figure. It is found that the final beam comes out parallel to the principal axis. The focal length of the lens equals 5cm.

To find the focal length of the mirror.

Solution:

First consider converging lens,

As per the given criteria,

object distance, $u=-2cm$

focal length of the lens, $f=5cm$

Applying the lens formula, we get

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}⟹\frac{1}{5}=\frac{1}{v}-\frac{1}{(-2)}⟹\frac{1}{v}=\frac{1}{5}-\frac{1}{2}⟹\frac{1}{v}=\frac{2-5}{10}⟹v=-3.33cm$

So the image of the point source object formed by the converging lens is 3.33 cm from the lens on the same side as the object.

This forms object for the converging mirror.

Now will consider converging mirror,

As the distance between the lens and mirror is 40cm.

So the object distance for the mirror is $u^{\prime }=(40+3.33)=43.33cm$

as the final beam comes out is parallel to the principal axis, the image is formed at infinity.

So the image distance, $v^{\prime }=\infty$

Applying the mirror formula, we get

$\frac{1}{f^{\prime }}=\frac{1}{v^{\prime }}+\frac{1}{u^{\prime }}⟹\frac{1}{f^{\prime }}=\frac{1}{\infty }+\frac{1}{43.33}⟹f^{\prime }=43.33cm$

is the focal length of the mirror.