Two identical solenoid coils, each of self inductancen L are connected in series. Their turns are in the same sense, and the distance between them is such that the coefficient of coupling is half . Then the equivalent inductance of the combination is ___

Options

A. L
B. 2L
C. 3L
D. L/2

Answer

To solve the problem of finding the equivalent inductance of two identical solenoid coils connected in series with a coefficient of coupling of 0.5, we can follow these steps:

### Step 1: Understand the given information
We have two identical solenoid coils, each with a self-inductance \( L \). They are connected in series, and the coefficient of coupling \( k \) between them is given as \( \frac{1}{2} \).

### Step 2: Recall the formula for equivalent inductance in series
When two inductors are connected in series, the equivalent inductance \( L_{eq} \) can be calculated using the formula:
\[ L_{eq} = L_1 + L_2 + 2M \]
where \( L_1 \) and \( L_2 \) are the self-inductances of the coils, and \( M \) is the mutual inductance.

### Step 3: Identify the values
Since both coils are identical:
- \( L_1 = L \)
- \( L_2 = L \)

The mutual inductance \( M \) can be expressed in terms of the coefficient of coupling \( k \) and the self-inductance \( L \):
\[ M = k \sqrt{L_1 L_2} = k \sqrt{L \cdot L} = kL \]
Given \( k = \frac{1}{2} \):
\[ M = \frac{1}{2} L \]

### Step 4: Substitute values into the formula
Now substituting \( L_1 \), \( L_2 \), and \( M \) into the equivalent inductance formula:
\[ L_{eq} = L + L + 2M = L + L + 2\left(\frac{1}{2}L\right) \]
\[ L_{eq} = 2L + L = 3L \]

### Step 5: Conclusion
Thus, the equivalent inductance of the combination is:
\[ \boxed{3L} \]