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

Options

A. `l` and a increase
B. `l` decreases and A increases
C. `l` increases and A decreases
D. both `l` and A decrease

Explanation

To solve the problem regarding the self-inductance \( L \) of a solenoid, we will derive the relationship step by step.

### Step-by-Step Solution:

1. **Understanding Self-Inductance**:
The self-inductance \( L \) of a solenoid is a measure of how effectively it can induce an electromotive force (emf) in itself due to a change in current. The formula for the self-inductance of a solenoid is given by:
\[ L = \frac{\mu_0 N^2 A}{l} \]
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. **Identifying the Variables**:
- \( N \): Fixed number of turns (constant).
- \( A \): Area of cross-section (can be varied).
- \( l \): Length of the solenoid (can be varied).

3. **Analyzing the Relationship**:
From the formula, we see that:
- Self-inductance \( L \) is **directly proportional** to the area \( A \) (as \( A \) increases, \( L \) increases).
- Self-inductance \( L \) is **inversely proportional** to the length \( l \) (as \( l \) increases, \( L \) decreases).

4. **Conclusion**:
To increase the self-inductance \( L \) of the solenoid:
- The area \( A \) must increase.
- The length \( l \) must decrease.

Thus, we conclude that the self-inductance \( L \) increases as the area \( A \) increases and the length \( l \) decreases.

### Final Answer:
The self-inductance \( L \) of a solenoid increases as the area of cross-section \( A \) increases and the length \( l \) decreases.

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