If number of turns in primary and secondary coils is increased to two times each, the mutual inductance

Options

A. becomes 4 times
B. becomes 2 times
C. becomes 1/4 times
D. remains unchanged

Explanation

To solve the problem of how mutual inductance changes when the number of turns in both primary and secondary coils is doubled, we can follow these steps:

### Step 1: Understand the formula for mutual inductance
The mutual inductance \( M \) between two coils is given by the formula:
\[ M = \frac{\mu_0 N_p N_s A}{l} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( N_p \) is the number of turns in the primary coil,
- \( N_s \) is the number of turns in the secondary coil,
- \( A \) is the area of the coils,
- \( l \) is the length of the magnetic path.

### Step 2: Determine the effect of doubling the number of turns
If we double the number of turns in both the primary and secondary coils, we have:
\[ N_p' = 2N_p \quad \text{and} \quad N_s' = 2N_s \]

### Step 3: Substitute the new values into the mutual inductance formula
Now, substituting \( N_p' \) and \( N_s' \) into the mutual inductance formula, we get:
\[ M' = \frac{\mu_0 (2N_p)(2N_s) A}{l} \]

### Step 4: Simplify the expression
This simplifies to:
\[ M' = \frac{\mu_0 \cdot 4N_p N_s A}{l} = 4 \cdot \frac{\mu_0 N_p N_s A}{l} = 4M \]

### Conclusion
Thus, when the number of turns in both the primary and secondary coils is doubled, the mutual inductance becomes four times the original mutual inductance:
\[ M' = 4M \]

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