When the number of turns in a coil is doubled without any change in the length of the coil, its self-inductance becomes

Options

A. four times
B. `doubled
C. halved
D. unchanged

Explanation

To determine how the self-inductance of a coil changes when the number of turns is doubled while keeping the length of the coil constant, we can follow these steps:

### Step 1: Understand the formula for self-inductance
The self-inductance \( L \) of a coil is given by the formula:
\[ L = \frac{\mu N^2 A}{l} \]
where:
- \( L \) is the self-inductance,
- \( \mu \) is the permeability of the core material,
- \( N \) is the number of turns,
- \( A \) is the cross-sectional area of the coil,
- \( l \) is the length of the coil.

### Step 2: Analyze the effect of doubling the number of turns
If the number of turns \( N \) is doubled, we can express the new number of turns as:
\[ N' = 2N \]

### Step 3: Substitute the new number of turns into the formula
Now, substituting \( N' \) into the self-inductance formula:
\[ L' = \frac{\mu (N')^2 A}{l} = \frac{\mu (2N)^2 A}{l} \]

### Step 4: Simplify the expression
Now simplifying the expression:
\[ L' = \frac{\mu (4N^2) A}{l} = 4 \left(\frac{\mu N^2 A}{l}\right) = 4L \]

### Step 5: Conclusion
Thus, when the number of turns in a coil is doubled without any change in the length of the coil, the self-inductance becomes:
\[ L' = 4L \]

### Final Answer
The self-inductance becomes four times the original self-inductance.

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