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. 0.5 H

Answer

To solve the problem, we will use the formula for the self-inductance \( L \) of a solenoid, which is given by:

\[ L = \mu_0 n^2 A l \]

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.

### Step-by-Step Solution:

1. **Identify the given values**:
- Initial self-inductance \( L = 2 \, H \)
- New length \( l' = 2l \) (the length is doubled)
- Turn density \( n \) and area \( A \) remain constant.

2. **Write the formula for the initial self-inductance**:
\[ L = \mu_0 n^2 A l \]

3. **Substitute the new length into the inductance formula**:
The new self-inductance \( L' \) when the length is doubled becomes:
\[ L' = \mu_0 n^2 A (2l) \]

4. **Factor out the constants**:
\[ L' = 2 \cdot (\mu_0 n^2 A l) \]

5. **Relate it back to the initial self-inductance**:
Since \( \mu_0 n^2 A l = L \):
\[ L' = 2L \]

6. **Substitute the initial self-inductance value**:
\[ L' = 2 \cdot 2 \, H = 4 \, H \]

### Final Answer:
The new self-inductance \( L' \) is \( 4 \, H \).

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