In a part of ac circuit shown in the figure, `R=0.2 Omega.` At the instant shown, `V_(A)-V_(B)=0.5V,i=0.5A, (di)/(dt)=8A//s`. The inductance L is

Options

A. 0.01 H
B. 0.02 H
C. 0.05 H
D. 0.5 H

Answer

To find the inductance \( L \) in the given AC circuit, we can follow these steps:

### Step 1: Identify the given values
- Resistance \( R = 0.2 \, \Omega \)
- Voltage difference \( V_A - V_B = 0.5 \, V \)
- Current \( i = 0.5 \, A \)
- Rate of change of current \( \frac{di}{dt} = 8 \, A/s \)

### Step 2: Calculate the voltage across the resistor \( V_R \)
Using Ohm's Law, we can calculate the voltage across the resistor:
\[ V_R = I \times R \]
Substituting the known values:
\[ V_R = 0.5 \, A \times 0.2 \, \Omega = 0.1 \, V \]

### Step 3: Apply Kirchhoff's Voltage Law
According to Kirchhoff's Voltage Law, the sum of the potential differences in a closed loop is equal to zero. Therefore, we can express the voltage across the inductor \( V_L \) as:
\[ V_L = V_A - V_B - V_R \]
Substituting the known values:
\[ V_L = 0.5 \, V - 0.1 \, V = 0.4 \, V \]

### Step 4: Relate the voltage across the inductor to inductance
The voltage across the inductor is given by the formula:
\[ V_L = L \frac{di}{dt} \]
Substituting the values we have:
\[ 0.4 \, V = L \times 8 \, A/s \]

### Step 5: Solve for inductance \( L \)
Rearranging the equation to solve for \( L \):
\[ L = \frac{0.4 \, V}{8 \, A/s} = 0.05 \, H \]

### Conclusion
The inductance \( L \) is \( 0.05 \, H \) (Henry).

---