At what distance from a convex lens of focal length `30cm` an object should be placed so that the size of image be `(1)/(4) `that of object?

Options

A. 30 cm
B. 60 cm
C. 15 cm
D. 90 cm

Answer

To solve the problem step by step, we need to find the object distance (u) from a convex lens given the focal length (f) and the magnification (m).

### Step 1: Understand the relationship between magnification and image distance
The magnification (m) is given by the formula:
\[ m = -\frac{v}{u} \]
where:
- \( v \) is the image distance,
- \( u \) is the object distance.

Given that \( m = \frac{1}{4} \), we can express this relationship as:
\[ \frac{1}{4} = -\frac{v}{u} \]

### Step 2: Rearrange the magnification formula
From the magnification formula, we can express \( v \) in terms of \( u \):
\[ v = -\frac{u}{4} \]

### Step 3: Use the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( f = 30 \, \text{cm} \) and \( v = -\frac{u}{4} \) into the lens formula:
\[ \frac{1}{30} = \frac{1}{-\frac{u}{4}} + \frac{1}{u} \]

### Step 4: Simplify the equation
Substituting \( v \):
\[ \frac{1}{30} = -\frac{4}{u} + \frac{1}{u} \]
Combining the terms on the right side:
\[ \frac{1}{30} = -\frac{4}{u} + \frac{1}{u} = -\frac{4 - 1}{u} = -\frac{3}{u} \]

### Step 5: Solve for \( u \)
Now, we can solve for \( u \):
\[ \frac{1}{30} = -\frac{3}{u} \]
Cross-multiplying gives:
\[ u = -90 \, \text{cm} \]

### Conclusion
The object should be placed at a distance of \( 90 \, \text{cm} \) from the lens (the negative sign indicates that the object is placed on the same side as the incoming light).

### Final Answer
The object distance \( u = -90 \, \text{cm} \).

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