The inductors of self inductance `L_(1)` and `L_(2)` are put in series .

Options

A. `L_(1)+L_(2)`only
B. `L_(1)-L_(2)`only
C. May be in between `L_(1)-L_(2)` and `L_(1) +L_(2)`
D. `L_(1)+2L_(2)`

Answer

To solve the problem of finding the net inductance of two inductors \( L_1 \) and \( L_2 \) connected in series, we can follow these steps:

### Step 1: Understand the Series Inductance Formula
When inductors are connected in series, their total inductance \( L_{net} \) can be expressed as:
\[ L_{net} = L_1 + L_2 + 2M \]
where \( M \) is the mutual inductance between the two inductors.

### Step 2: Apply Kirchhoff's Voltage Law (KVL)
According to KVL, the sum of the voltage drops across the inductors must equal the applied voltage. Thus, we can write:
\[ E_0 = -L_1 \frac{dI}{dt} - L_2 \frac{dI}{dt} \]
This simplifies to:
\[ E_0 = -\left(L_1 + L_2\right) \frac{dI}{dt} \]

### Step 3: Include Mutual Inductance
If there is mutual inductance \( M \) between the inductors, we need to consider its effect. The equation becomes:
\[ E_0 = -\left(L_1 + L_2 - M\right) \frac{dI}{dt} \]

### Step 4: Determine the Value of Mutual Inductance
The mutual inductance \( M \) can be expressed in terms of the inductances \( L_1 \) and \( L_2 \):
\[ M = k \sqrt{L_1 L_2} \]
where \( k \) is the coupling coefficient, which can vary between -1 and 1.

### Step 5: Calculate the Total Inductance
Substituting \( M \) into the equation for \( L_{net} \), we get:
\[ L_{net} = L_1 + L_2 + 2M \]
This means:
\[ L_{net} = L_1 + L_2 + 2k \sqrt{L_1 L_2} \]

### Step 6: Determine the Range of \( L_{net} \)
Depending on the value of \( k \):
- If \( k = 1 \): \( L_{net} = L_1 + L_2 + 2\sqrt{L_1 L_2} \) (maximum coupling)
- If \( k = -1 \): \( L_{net} = L_1 + L_2 - 2\sqrt{L_1 L_2} \) (minimum coupling)

Thus, the net inductance will lie between:
\[ L_{1} + L_{2} - 2\sqrt{L_{1}L_{2}} \quad \text{and} \quad L_{1} + L_{2} + 2\sqrt{L_{1}L_{2}} \]

### Conclusion
The final expression for the net inductance \( L_{net} \) of two inductors \( L_1 \) and \( L_2 \) in series, considering mutual inductance, is:
\[ L_{net} = L_1 + L_2 \pm 2\sqrt{L_1 L_2} \]