The inductance of a coil in which a current of 0.1 A increasing at the rate of 0.5A/s represents a power flow of `(1)/(2)` watt ,is

Options

A. 2 H
B. 8 H
C. 20 H
D. 10 H

Answer

To find the inductance of the coil, we can follow these steps:

### Step 1: Identify the given values
- Current (I) = 0.1 A
- Rate of change of current (dI/dt) = 0.5 A/s
- Power (P) = 1/2 W

### Step 2: Use the formula for power in an inductor
The power (P) in an inductor can be expressed as:
\[ P = E \cdot I \]
Where \( E \) is the induced electromotive force (emf).

### Step 3: Relate induced emf to inductance
The induced emf (E) in an inductor is given by:
\[ E = L \cdot \frac{dI}{dt} \]
Where \( L \) is the inductance.

### Step 4: Substitute the expression for E into the power formula
Substituting \( E \) in the power formula, we get:
\[ P = L \cdot \frac{dI}{dt} \cdot I \]

### Step 5: Rearrange the equation to solve for L
Rearranging the equation gives:
\[ L = \frac{P}{I \cdot \frac{dI}{dt}} \]

### Step 6: Substitute the known values into the equation
Now, substituting the known values:
- \( P = \frac{1}{2} \) W
- \( I = 0.1 \) A
- \( \frac{dI}{dt} = 0.5 \) A/s

We get:
\[ L = \frac{\frac{1}{2}}{0.1 \cdot 0.5} \]

### Step 7: Calculate the inductance
Calculating the denominator:
\[ 0.1 \cdot 0.5 = 0.05 \]
Now substituting this back:
\[ L = \frac{\frac{1}{2}}{0.05} = \frac{1}{2} \cdot \frac{1}{0.05} = \frac{1}{2} \cdot 20 = 10 \, \text{H} \]

### Final Answer
The inductance \( L \) of the coil is **10 Henry**.

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