If the number of turns per units length of a coils of solenoid is doubled , the self- inductance of the soleniod will

Options

A. remain unchanged
B. be halved
C. be doubled
D. become four times

Explanation

To solve the problem regarding the change in self-inductance of a solenoid when the number of turns per unit length is doubled, 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:
\[ L = \mu_0 n^2 A \]
where:
- \( L \) is the self-inductance,
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns per unit length,
- \( A \) is the cross-sectional area of the solenoid.

2. **Identifying the Change in Turns**:
If the number of turns per unit length \( n \) is doubled, we can express the new number of turns per unit length as:
\[ n' = 2n \]

3. **Substituting the New Value into the Inductance Formula**:
We substitute \( n' \) into the self-inductance formula:
\[ L' = \mu_0 (n')^2 A = \mu_0 (2n)^2 A \]

4. **Calculating the New Inductance**:
Simplifying the expression:
\[ L' = \mu_0 (4n^2) A = 4 \mu_0 n^2 A \]
Thus, we can express the new inductance in terms of the old inductance:
\[ L' = 4L \]
where \( L \) is the original self-inductance.

5. **Conclusion**:
Therefore, when the number of turns per unit length of the solenoid is doubled, the self-inductance becomes four times the original self-inductance.

### Final Answer:
The self-inductance of the solenoid will become **4 times** the original self-inductance.

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