The total number of turns and cross-section area in a solenoid is fixed. However, its length L is varied by adjusting the separation between windings. The inductance of solenoid will be proportional to :

Options

A. L
B. `L^(2)`
C. `1//L^(2)`
D. `1//L`

Answer

To solve the problem, we need to understand the relationship between the inductance of a solenoid and its physical parameters, specifically when the length of the solenoid is varied while keeping the number of turns and the cross-sectional area constant.

### Step-by-Step Solution:

1. **Understanding Inductance of a Solenoid**:
The 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 total number of turns,
- \( A \) is the cross-sectional area,
- \( l \) is the length of the solenoid.

2. **Identifying Fixed Parameters**:
In this problem, we are told that:
- The total number of turns \( N \) is fixed.
- The cross-sectional area \( A \) is fixed.
- The length \( l \) is varied.

3. **Analyzing the Effect of Length on Inductance**:
Since \( N \) and \( A \) are constant, we can see that the inductance \( L \) is inversely proportional to the length \( l \). This means that as the length increases, the inductance decreases, and vice versa.

4. **Expressing the Relationship**:
Therefore, we can express the relationship as:
\[ L \propto \frac{1}{l} \]

5. **Conclusion**:
The inductance of the solenoid will be proportional to \( \frac{1}{L} \). Thus, the correct answer is:
\[ \text{Inductance } L \propto \frac{1}{l} \]

### Final Answer:
The inductance of the solenoid will be proportional to \( \frac{1}{L} \).

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