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 determine when the magnification of an image formed by a convex lens is positive, we can follow these steps:

### Step 1: Understanding Magnification
The magnification (M) produced by a convex lens is given by the formula:
\[ M = \frac{V}{U} \]
where \( V \) is the image distance and \( U \) is the object distance.

### Step 2: Lens Formula
The lens formula for a convex lens is:
\[ \frac{1}{V} - \frac{1}{U} = \frac{1}{F} \]
where \( F \) is the focal length of the lens. For a convex lens, \( F \) is always positive.

### Step 3: Rearranging the Lens Formula
From the lens formula, we can express \( V \) in terms of \( U \) and \( F \):
\[ \frac{1}{V} = \frac{1}{F} + \frac{1}{U} \]
This can be rearranged to:
\[ V = \frac{UF}{U + F} \]

### Step 4: Substituting into Magnification Formula
Now, substituting \( V \) back into the magnification formula:
\[ M = \frac{UF}{U + F} \cdot \frac{1}{U} = \frac{F}{U + F} \]

### Step 5: Condition for Positive Magnification
For the magnification \( M \) to be positive:
\[ \frac{F}{U + F} > 0 \]
Since \( F \) is positive, \( U + F \) must also be positive:
\[ U + F > 0 \]
This implies:
\[ U > -F \]

### Step 6: Analyzing the Sign of U
Since the object distance \( U \) is measured from the optical center of the lens and is conventionally taken as negative for real objects (as per the sign convention), we have:
\[ U < 0 \]
Thus, for \( U + F > 0 \) to hold true, we can rewrite it as:
\[ -|U| + F > 0 \]
or
\[ F > |U| \]
This means that the magnitude of the object distance must be less than the focal length.

### Step 7: Conclusion
Therefore, the object must be placed between the focal point (F) and the optical center of the lens for the magnification to be positive. This corresponds to the object being placed between the focus and the lens.

### Final Answer
The magnification of an image by a convex lens is positive only when the object is placed between the focal point and the optical center of the lens.

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