In a wire of length L, the increase in its length is `l`. If the length is reduced to half, the increase in its length will be

Options

A. Reduced by half
B. Increase by half
C. Remains same
D. None of these

Explanation

To solve the problem, we need to understand the relationship between the length of a wire, the increase in its length, and the factors that affect elasticity.

### Given:
- Original length of the wire = \( L \)
- Increase in length = \( l \)

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- The original length of the wire is \( L \) and it stretches by an amount \( l \).
- The new length of the wire after stretching becomes \( L + l \).

2. **Applying Hooke's Law**:
- According to Hooke's Law, the extension (increase in length) of a wire is directly proportional to the applied force, provided the limit of elasticity is not exceeded.
- Mathematically, this can be expressed as:
\[ l \propto L \]
- This means that if the length of the wire changes, the increase in length will also change proportionally.

3. **Reducing the Length to Half**:
- If we reduce the length of the wire to half, the new length \( L' \) becomes:
\[ L' = \frac{L}{2} \]

4. **Finding the New Increase in Length**:
- Since the increase in length is proportional to the original length, we can express the new increase in length \( l' \) as:
\[ l' \propto L' \]
- Therefore, if the original increase in length was \( l \) for length \( L \), the new increase in length when the length is halved will be:
\[ l' = \frac{l}{2} \]

### Conclusion:
- The increase in length when the wire's length is reduced to half will be \( \frac{l}{2} \).

### Final Answer:
- The increase in length when the length is reduced to half is \( \frac{l}{2} \).

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