Two coils of self-inductance `L_(1)` and `L_(2)` are placed closed to each other so that total flux in one coil is completely linked with other. If `M` is mutual inductance between them, then

Options

A. `M=L_(1)L_(2)`
B. `M=L_(1)//L_(2)`
C. `M=sqrt(L_(1)L_(2))`
D. `M=(L_(1)L_(2))^(2)`

Explanation

To solve the problem, we need to understand the relationship between self-inductance, mutual inductance, and the coupling factor of two coils.

### Step-by-Step Solution:

1. **Understanding Self-Inductance and Mutual Inductance**:
- Self-inductance (L1 and L2) refers to the ability of a coil to induce an electromotive force (emf) in itself due to a change in current.
- Mutual inductance (M) is the ability of one coil to induce an emf in another coil due to a change in current in the first coil.

2. **Condition of Complete Linking**:
- The problem states that the total flux in one coil is completely linked with the other coil. This means that all the magnetic flux produced by coil 1 passes through coil 2.

3. **Coupling Factor (K)**:
- The coupling factor (K) is a measure of how effectively the magnetic flux from one coil links with the other coil. Since the problem states that the total flux is completely linked, we have K = 1.

4. **Formula for Mutual Inductance**:
- The mutual inductance (M) can be expressed in terms of the self-inductances and the coupling factor:
\[ M = K \sqrt{L_1 L_2} \]
- Since K = 1, we can simplify this to:
\[ M = \sqrt{L_1 L_2} \]

5. **Conclusion**:
- Therefore, the mutual inductance M between the two coils is given by:
\[ M = \sqrt{L_1 L_2} \]

### Final Answer:
The mutual inductance \( M \) between the two coils is given by:
\[ M = \sqrt{L_1 L_2} \]