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 find the relationship between the mutual inductance \( M \) and the self-inductances \( L_1 \) and \( L_2 \) of two coils placed close to each other, we can follow these steps:

### Step 1: Understanding Mutual Inductance
Mutual inductance \( M \) is defined as the ratio of the induced electromotive force (EMF) in one coil to the rate of change of current in the other coil. For two coils, we can express this relationship mathematically.

### Step 2: Write the Induced EMF Equations
For coil 1, the induced EMF \( E_1 \) due to the change in current \( I_2 \) in coil 2 is given by:
\[ E_1 = -M \frac{dI_2}{dt} \]
For coil 2, the induced EMF \( E_2 \) due to the change in current \( I_1 \) in coil 1 is given by:
\[ E_2 = -M \frac{dI_1}{dt} \]

### Step 3: Relate Induced EMF to Self-Inductance
The induced EMF can also be expressed in terms of self-inductance. For coil 1:
\[ E_1 = -L_1 \frac{dI_1}{dt} \]
For coil 2:
\[ E_2 = -L_2 \frac{dI_2}{dt} \]

### Step 4: Equate the Expressions
From the equations for \( E_1 \) and \( E_2 \), we can equate the expressions for the induced EMF:
1. From coil 1:
\[ -M \frac{dI_2}{dt} = -L_1 \frac{dI_1}{dt} \]
2. From coil 2:
\[ -M \frac{dI_1}{dt} = -L_2 \frac{dI_2}{dt} \]

### Step 5: Rearranging the Equations
From the first equation, we can express \( M \):
\[ M = \frac{L_1}{\frac{dI_1}{dt}} \cdot \frac{dI_2}{dt} \]
From the second equation:
\[ M = \frac{L_2}{\frac{dI_2}{dt}} \cdot \frac{dI_1}{dt} \]

### Step 6: Finding the Relationship
By equating the two expressions for \( M \):
\[ \frac{L_1}{\frac{dI_1}{dt}} \cdot \frac{dI_2}{dt} = \frac{L_2}{\frac{dI_2}{dt}} \cdot \frac{dI_1}{dt} \]

### Step 7: Simplifying the Relationship
Cross-multiplying gives:
\[ M^2 = L_1 \cdot L_2 \]

### Step 8: Final Expression for Mutual Inductance
Taking the square root of both sides, we find:
\[ M = \sqrt{L_1 L_2} \]

### Summary
Thus, the relationship between the mutual inductance \( M \) and the self-inductances \( L_1 \) and \( L_2 \) is given by:
\[ M = \sqrt{L_1 L_2} \]