An inductor of resistance 80 `Omega` is connected to an AC source of frequency 50 Hz . Calculate the inductance if the impedance of circuit is 150 `Omega` .

Answer

To solve the problem step by step, we need to calculate the inductance of an inductor connected to an AC source, given its resistance and the impedance of the circuit.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Resistance (R) = 80 Ω
- Impedance (Z) = 150 Ω
- Frequency (f) = 50 Hz

2. **Calculate Angular Frequency (ω):**
The angular frequency (ω) is calculated using the formula:
\[ ω = 2πf \]
Substituting the value of frequency:
\[ ω = 2 \times 3.14 \times 50 = 314 \, \text{rad/s} \]

3. **Use the Impedance Formula for RL Circuit:**
The impedance (Z) in an RL circuit is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
Rearranging this formula to find the inductive reactance (X_L):
\[ X_L = \sqrt{Z^2 - R^2} \]

4. **Substitute the Known Values:**
Substitute the values of Z and R into the equation:
\[ X_L = \sqrt{150^2 - 80^2} \]
Calculate \(150^2\) and \(80^2\):
\[ 150^2 = 22500 \]
\[ 80^2 = 6400 \]
Now, substitute these values:
\[ X_L = \sqrt{22500 - 6400} = \sqrt{16100} \]

5. **Calculate X_L:**
Now, calculate the square root:
\[ X_L = \sqrt{16100} \approx 126.89 \, \Omega \]

6. **Relate Inductive Reactance to Inductance:**
The inductive reactance (X_L) is related to the inductance (L) by the formula:
\[ X_L = Lω \]
Rearranging this gives:
\[ L = \frac{X_L}{ω} \]

7. **Substitute the Values of X_L and ω:**
Substitute the calculated value of X_L and the value of ω:
\[ L = \frac{126.89}{314} \]

8. **Calculate Inductance (L):**
Now, perform the division:
\[ L \approx 0.404 \, \text{H} \]

### Final Answer:
The inductance of the circuit is approximately **0.40 H**.