The magnification of an image by a convex lens is positive only when the object is placed

Options

A. At its focus F
B. Between F and 2F
C. At 2F
D. Between F and optical centre

Answer

To solve the question regarding the magnification of an image by a convex lens, we will follow these steps:

### Step 1: Understand the Magnification Formula
The magnification (M) of a lens is given by the formula:
\[ M = \frac{V}{U} \]
where:
- \( V \) is the image distance from the optical center of the lens,
- \( U \) is the object distance from the optical center of the lens.

### Step 2: Use the Lens Formula
The lens formula for a convex lens is:
\[ \frac{1}{F} = \frac{1}{V} - \frac{1}{U} \]
Rearranging this gives us:
\[ \frac{1}{V} = \frac{1}{F} + \frac{1}{U} \]
From this, we can express \( V \):
\[ V = \frac{UF}{U + F} \]

### Step 3: Substitute into the Magnification Formula
Substituting \( V \) into the magnification formula:
\[ M = \frac{UF}{U + F} \div U \]
This simplifies to:
\[ M = \frac{F}{U + F} \]

### Step 4: Analyze the Signs
For a convex lens:
- The focal length \( F \) is positive.
- The object distance \( U \) is negative (as per sign conventions).

Thus, we can rewrite the magnification as:
\[ M = \frac{F}{-U + F} \]

### Step 5: Determine Conditions for Positive Magnification
For the magnification \( M \) to be positive, the denominator must also be positive:
\[ -U + F > 0 \]
This implies:
\[ F > U \]
Since \( U \) is negative, this means:
\[ |U| < F \]
This indicates that the object must be placed between the optical center (O) and the focus (F) of the lens.

### Conclusion
Therefore, the magnification of an image by a convex lens is positive only when the object is placed **between the optical center and the focus**.

### Final Answer
The correct option is: **between F and the optical center.**

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