Understanding Image Formation by a Convex Lens

When dealing with optics, specifically convex lenses, the position of an object relative to the lens determines the characteristics of the image formed. A convex lens is a converging lens, meaning it converges parallel rays of light at a point called the principal focus, which is located at a distance equal to the focal length ($\mathbf{f}$).

The question asks about forming a magnified real image using a convex lens of focal length $\mathbf{f}$. A real image is one that can be projected onto a screen, and it is typically inverted. A magnified image is larger than the object.

Object Positions and Image Characteristics for a Convex Lens

Let's examine how the image characteristics change as we move the object closer to a convex lens from a large distance. The key positions are the optical centre (O), the principal focus ($\mathbf{f}$), and twice the focal length ($\mathbf{2f}$), which is also the centre of curvature for the lens surfaces.

Here's a summary of image formation for different object positions:

• Object at infinity: The image is formed at the principal focus ($\mathbf{f}$) on the opposite side. It is real, inverted, and highly diminished (a point image).
• Object beyond $\mathbf{2f}$: The image is formed between $\mathbf{f}$ and $\mathbf{2f}$ on the opposite side. It is real, inverted, and diminished.
• Object at $\mathbf{2f}$: The image is formed at $\mathbf{2f}$ on the opposite side. It is real, inverted, and the same size as the object.
• Object between $\mathbf{f}$ and $\mathbf{2f}$: The image is formed beyond $\mathbf{2f}$ on the opposite side. It is real, inverted, and magnified.
• Object at $\mathbf{f}$: The image is formed at infinity. It is real, inverted, and highly magnified.
• Object between the optical centre (O) and $\mathbf{f}$: The image is formed on the same side as the object. It is virtual, erect, and magnified. This is how a convex lens is used as a magnifying glass.

We can also represent this information in a table for clarity:

Finding the Position for a Magnified Real Image

Looking at the summary above, we need an image that is both real and magnified. The cases that produce real images are when the object is at infinity, beyond $\mathbf{2f}$, at $\mathbf{2f}$, between $\mathbf{f}$ and $\mathbf{2f}$, or at $\mathbf{f}$.

Out of these real image cases, the ones that produce a magnified image are when the object is placed:

• Between $\mathbf{f}$ and $\mathbf{2f}$ (image is magnified and real).
• At $\mathbf{f}$ (image is highly magnified but formed at infinity).

The case when the object is placed between $\mathbf{f}$ and $\mathbf{2f}$ is the standard scenario that produces a magnified real image at a finite distance (beyond $\mathbf{2f}$). Placing the object exactly at $\mathbf{f}$ produces an image at infinity, which, while highly magnified and real, is often considered a limiting case.

Comparing this with the given options, placing the object between $\mathbf{f}$ and $\mathbf{2f}$ clearly results in a magnified real image.

Placing the object "between f and optical centre" produces a magnified virtual image, not a real one. Placing it "anywhere beyond 2f" produces a diminished or same-sized real image, not a magnified one.

Therefore, to obtain a magnified real image with a convex lens, the object must be placed between the focal length ($\mathbf{f}$) and twice the focal length ($\mathbf{2f}$).