Two coils of self-inductance `2mH` and `8 mH` are placed so close together that the effective flux in one coil is completely linked with the other. The mutual inductance between these coil is

Options

A. 4 mH
B. 16 mH
C. 10 mH
D. 6 mH

Explanation

To find the mutual inductance between two coils with given self-inductances, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Self-inductance of coil 1, \( L_1 = 2 \, \text{mH} = 2 \times 10^{-3} \, \text{H} \)
- Self-inductance of coil 2, \( L_2 = 8 \, \text{mH} = 8 \times 10^{-3} \, \text{H} \)

2. **Understand the Concept of Mutual Inductance:**
- When two coils are placed close together, the mutual inductance \( M \) can be calculated using the formula:
\[ M = K \sqrt{L_1 L_2} \]
- Here, \( K \) is the coupling coefficient, which indicates how effectively the magnetic flux of one coil links with the other. Since the problem states that the effective flux in one coil is completely linked with the other, we have \( K = 1 \).

3. **Substitute the Values into the Formula:**
- Using the values of \( L_1 \) and \( L_2 \):
\[ M = 1 \cdot \sqrt{(2 \times 10^{-3}) \cdot (8 \times 10^{-3})} \]

4. **Calculate the Product Inside the Square Root:**
- Calculate \( L_1 \times L_2 \):
\[ L_1 \times L_2 = 2 \times 10^{-3} \times 8 \times 10^{-3} = 16 \times 10^{-6} \, \text{H}^2 \]

5. **Take the Square Root:**
- Now, take the square root:
\[ \sqrt{16 \times 10^{-6}} = 4 \times 10^{-3} \, \text{H} = 4 \, \text{mH} \]

6. **Final Calculation of Mutual Inductance:**
- Since \( K = 1 \):
\[ M = 4 \, \text{mH} \]

### Conclusion:
The mutual inductance between the two coils is \( M = 4 \, \text{mH} \).

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