The equivalent inductance of two inductances is `2.4` henry when connected in parallel and `10` henry when connected in series. The difference between the two inductance is

Options

A. 8 H, 2H
B. 6H, 4H
C. 5H, 5H
D. 7H, 3H

Explanation

To solve the problem of finding the difference between two inductances \( L_1 \) and \( L_2 \) given their equivalent inductances in parallel and series, we can follow these steps:

### Step 1: Define the equations for inductance
1. When connected in parallel, the equivalent inductance \( L_p \) is given by:
\[ \frac{1}{L_p} = \frac{1}{L_1} + \frac{1}{L_2} \]
For this problem, \( L_p = 2.4 \, \text{H} \).

2. When connected in series, the equivalent inductance \( L_s \) is given by:
\[ L_s = L_1 + L_2 \]
For this problem, \( L_s = 10 \, \text{H} \).

### Step 2: Set up the equations
From the parallel combination:
\[ \frac{1}{2.4} = \frac{1}{L_1} + \frac{1}{L_2} \]
This can be rearranged to:
\[ L_1 L_2 = 2.4 (L_1 + L_2) \]

From the series combination:
\[ L_1 + L_2 = 10 \]

### Step 3: Substitute and simplify
Let \( L_1 + L_2 = 10 \) (Equation 1) and \( L_1 L_2 = 2.4 \times 10 = 24 \) (Equation 2).

### Step 4: Solve the equations
From Equation 1, we can express \( L_2 \) in terms of \( L_1 \):
\[ L_2 = 10 - L_1 \]

Substituting \( L_2 \) in Equation 2:
\[ L_1(10 - L_1) = 24 \]
Expanding this gives:
\[ 10L_1 - L_1^2 = 24 \]
Rearranging leads to:
\[ L_1^2 - 10L_1 + 24 = 0 \]

### Step 5: Solve the quadratic equation
Using the quadratic formula \( L_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -10, c = 24 \):
\[ L_1 = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} \]
\[ L_1 = \frac{10 \pm \sqrt{100 - 96}}{2} \]
\[ L_1 = \frac{10 \pm 2}{2} \]
Thus, we have:
\[ L_1 = 6 \quad \text{or} \quad L_1 = 4 \]

### Step 6: Find \( L_2 \)
Using \( L_1 + L_2 = 10 \):
- If \( L_1 = 6 \), then \( L_2 = 4 \).
- If \( L_1 = 4 \), then \( L_2 = 6 \).

### Step 7: Find the difference
The difference between the two inductances is:
\[ |L_1 - L_2| = |6 - 4| = 2 \, \text{H} \]

### Final Answer
The difference between the two inductances is \( 2 \, \text{H} \).

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