The self inductance L of a solenoid of length l and area of cross-section A, with a fixed number of turns N increases as

Explanation

To determine how the self-inductance \( L \) of a solenoid changes with respect to its length \( l \) and area of cross-section \( A \), we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Formula for Self-Inductance**:
The self-inductance \( L \) of a solenoid is given by the formula:
\[ L = \frac{\mu_0 N^2 A}{l} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( N \) is the number of turns,
- \( A \) is the cross-sectional area,
- \( l \) is the length of the solenoid.

2. **Analyzing the Relationship**:
From the formula, we can see that:
- \( L \) is directly proportional to the area \( A \) (i.e., if \( A \) increases, \( L \) increases).
- \( L \) is inversely proportional to the length \( l \) (i.e., if \( l \) increases, \( L \) decreases).

3. **Considering Changes in Length and Area**:
- If the area \( A \) increases while keeping \( N \) constant, \( L \) will increase.
- If the length \( l \) increases while keeping \( N \) constant, \( L \) will decrease.

4. **Evaluating the Options**:
- **Option A**: If both \( l \) and \( A \) increase, \( L \) will depend on which effect is stronger. However, since \( L \) is inversely proportional to \( l \) and directly proportional to \( A \), this option is ambiguous.
- **Option B**: If \( l \) decreases and \( A \) increases, \( L \) will definitely increase. This option is correct.
- **Option C**: If \( l \) increases and \( A \) decreases, \( L \) will definitely decrease. This option is incorrect.

5. **Conclusion**:
The self-inductance \( L \) of the solenoid increases when the area \( A \) increases and/or the length \( l \) decreases. Therefore, the correct answer is **Option B**.