Animated Solution for Physics - Optics: A point like object is placed at a distance of $1\text{ m}$ in front of a convex lens of focal length $0.5\text{ m}$. A plane mirror is placed at a distance of $2\text{ m}$ behind the lens. The position and nature of the final image formed by the system is

Lens, Mirror, and Lens Again!

Imagine a fascinating optical setup: a convex lens and a plane mirror placed $2\text{ m}$ apart. We place a point object $1\text{ m}$ in front of the lens. To find the final image, we must trace the journey of light through three distinct optical events: a refraction, a reflection, and a final refraction.

The First Refraction

First, the light rays from the object travel towards the convex lens. To find where the lens forms the image, we use the thin lens formula:

The object is $1\text{ m}$ in front of the lens, so $u_1=-1\text{ m}$. The focal length of the convex lens is $f=+0.5\text{ m}$. Substituting these values:

The lens converges the rays to form a real image, $I_1$, exactly $1\text{ m}$ behind the lens.

The Reflection

But the light doesn't stop there! These converging rays continue and hit the plane mirror located $2\text{ m}$ from the lens. Since $I_1$ is $1\text{ m}$ from the lens, it is exactly $2-1=1\text{ m}$ in front of the mirror.

A plane mirror always forms a virtual image at the same distance behind it as the object is in front. So, it forms a virtual image, $I_2$, exactly $1\text{ m}$ behind the mirror.

The Second Refraction

Now, the reflected rays travel back towards the lens. For this second encounter with the lens, the virtual image $I_2$ acts as the new object. How far is $I_2$ from the lens? It's $2\text{ m}$ to the mirror plus $1\text{ m}$ behind it, making it $3\text{ m}$.

Applying the lens formula again, we must be careful with our sign convention. The light is now travelling from right to left. We measure object distance against the direction of incident light, so our new object distance $u_2$ is $-3\text{ m}$. The focal length remains $+0.5\text{ m}$.

Locating the Final Image

The positive sign means the image is formed in the direction of the light's travel, which is to the left of the lens.

Finally, let's find the distance of this final real image from the mirror. It's $2\text{ m}$ from the mirror to the lens, plus $0.6\text{ m}$ from the lens to the image. That's a total of $2.6\text{ m}$ from the mirror. And since the rays actually converge after passing through the lens, the final image is real!