A solenoid have self inductance 2H. If length of the solenoid is double having turn density and area constant then new self inductance is :-

Options

A. 4H
B. 1H
C. 8H
D. 0.5H

Answer

To solve the problem, we need to understand how the self-inductance of a solenoid changes when its length is doubled while keeping the turn density and area constant.

### Step-by-Step Solution:

1. **Understand the formula for self-inductance (L)**:
The self-inductance of a solenoid is given by the formula:
\[ L = \mu_0 n^2 A l \]
where:
- \(L\) is the self-inductance,
- \(\mu_0\) is the permeability of free space,
- \(n\) is the turn density (number of turns per unit length),
- \(A\) is the cross-sectional area of the solenoid,
- \(l\) is the length of the solenoid.

2. **Identify the given values**:
We know:
- Initial self-inductance, \(L = 2 \, \text{H}\)
- The length of the solenoid is doubled, so the new length \(l' = 2l\).
- Turn density \(n\) and area \(A\) remain constant.

3. **Write the expression for the new self-inductance (L')**:
Since \(n\) and \(A\) are constant, we can express the new self-inductance as:
\[ L' = \mu_0 n^2 A (2l) \]

4. **Relate the new self-inductance to the old self-inductance**:
We can factor out the constant terms:
\[ L' = 2 \cdot (\mu_0 n^2 A l) = 2L \]
Since \(L = \mu_0 n^2 A l\), we substitute:
\[ L' = 2L \]

5. **Substitute the known value of L**:
Now substituting \(L = 2 \, \text{H}\):
\[ L' = 2 \cdot 2 \, \text{H} = 4 \, \text{H} \]

6. **Conclusion**:
The new self-inductance \(L'\) is \(4 \, \text{H}\).

### Final Answer:
The new self-inductance of the solenoid is \(4 \, \text{H}\).

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