Mutual inductance of two coils depends on their self inductance `L_(1)` and `L_(2)` as :

Options

A. `M_(12) = L_(1)//L_(2)`
B. `M_(12)=L_(2)//L_(1)`
C. `M_(12) = sqrt(L_(1)L_(2))`
D. `M_(12)=sqrt(L_(1)//L_(2))`

Answer

To find the relationship between the mutual inductance \( M \) of two coils and their self-inductances \( L_1 \) and \( L_2 \), we can follow these steps:

### Step 1: Define the Self-Inductance
The self-inductance \( L_1 \) of the first coil is defined as the ratio of the magnetic flux \( \Phi_{11} \) linked with the first coil due to its own current \( I_1 \):
\[ \Phi_{11} = L_1 I_1 \]

### Step 2: Define the Mutual Inductance
The mutual inductance \( M \) is defined as the ratio of the magnetic flux \( \Phi_{12} \) linked with the first coil due to the current \( I_2 \) in the second coil:
\[ \Phi_{12} = M I_2 \]

### Step 3: Write the Flux Relationships
For the second coil, we can similarly define the self-inductance \( L_2 \) and the mutual inductance \( M \):
\[ \Phi_{21} = M I_1 \quad \text{(flux in the second coil due to the first coil)} \]
\[ \Phi_{22} = L_2 I_2 \quad \text{(self-flux in the second coil)} \]

### Step 4: Set Up the Equations for Maximum Flux Linkage
At maximum flux linkage, the self-flux in each coil is equal to the mutual flux due to the other coil:
\[ \Phi_{11} = \Phi_{12} \quad \Rightarrow \quad L_1 I_1 = M I_2 \quad \text{(1)} \]
\[ \Phi_{22} = \Phi_{21} \quad \Rightarrow \quad L_2 I_2 = M I_1 \quad \text{(2)} \]

### Step 5: Multiply the Two Equations
Now, we multiply equations (1) and (2):
\[ (L_1 I_1)(L_2 I_2) = (M I_2)(M I_1) \]
This simplifies to:
\[ L_1 L_2 I_1 I_2 = M^2 I_1 I_2 \]

### Step 6: Cancel Common Terms
Assuming \( I_1 \) and \( I_2 \) are not zero, we can cancel \( I_1 I_2 \) from both sides:
\[ L_1 L_2 = M^2 \]

### Step 7: Solve for Mutual Inductance
Taking the square root of both sides gives us the expression for mutual inductance:
\[ M = \sqrt{L_1 L_2} \]

### Final Result
Thus, the mutual inductance \( M \) of the two coils depends on their self-inductances \( L_1 \) and \( L_2 \) as:
\[ M = \sqrt{L_1 L_2} \]

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