Question

A source of light is located at a double focal length from a convergent lens. The focal length of the lens is $f=30cm$. At what distance from the lens should a flat mirror be placed so that the rays reflected from the mirror are parallel after passing through the lens for the second time?

Answer 1

Step 1: Given data

The focal length of the lens is $f=30cm$.

The light source is placed at a distance $2f=60cm$.

Step 2: Observation

1. We know in a convergent lens, the parallel light rays passing through the lens are converged at the focus of the lens and vice versa.
2. In this case, the reflected light rays pass through the focus and again incident on the lens and after passing the lens it becomes parallel to the lens axis.

Step 3: Diagram

Step 4: Finding the location of the mirror

1. From the given figure, the mirror is placed between the double focal length and focal length. This potion is clearly mentioned in the given figure.
2. If the mirror had not been there then the image would have formed at $\mathrm{I}$ which is at a distance of 2f from the lens because the source is at 2f.
3. When the mirror is placed then the ray coming from the lens after reflection will pass through f . so Af will be the location of the image from the plane mirror.
4. We know that distance of an object placed in front of a plane mirror is equal to the distance of the image formed inside the mirror.
5. we can say that: $\mathrm{Af}=\mathrm{AI}$

$$\begin{gathered} \because \mathrm{KI}=60\mathrm{cm} \\ \Rightarrow \mathrm{Kf}+\mathrm{fI}=60 \\ \Rightarrow \mathrm{Kf}+(\mathrm{fA}+\mathrm{AI})=60\left\{\mathrm{from}\mathrm{diagram}\mathrm{fI}=\mathrm{fA}+\mathrm{AI}\right\} \\ \Rightarrow \mathrm{Kf}+\mathrm{fA}+\mathrm{AI}=60-------\left(\mathrm{i}\right) \end{gathered}$$

Also we established that:

$\mathrm{Af}=\mathrm{AI}$

$$\begin{gathered} \because \mathrm{fI}=30 \\ \Rightarrow \mathrm{fA}+\mathrm{AI}=30 \\ \Rightarrow \mathrm{AI}+\mathrm{AI}=30\left\{\mathrm{Af}=\mathrm{AI}\right\} \\ \Rightarrow \mathrm{AI}=15\mathrm{cm}-----\left(\mathrm{ii}\right) \end{gathered}$$

Substituting eq(ii) into eq(i)

$$\begin{gathered} \Rightarrow \mathrm{Kf}+\mathrm{fA}+\mathrm{AI}=60 \\ \Rightarrow \mathrm{KA}+15=60 \\ \Rightarrow \mathrm{KA}=45\mathrm{cm} \end{gathered}$$

Therefore, the position of the mirror is $45cm$.