An inductor 200 mH, capacitor `500mu F` and resistor `10 Omega` are connected in series with a 100 V variable frequency ac source. What is the frequency at which the power factor of the circuit is unity?

Options

A. `10.22 Hz`
B. `12.4 Hz`
C. `19.2 Hz`
D. `15.9 Hz`

Explanation

To find the frequency at which the power factor of the circuit is unity, we need to analyze the given LCR circuit consisting of an inductor (L), capacitor (C), and resistor (R) connected in series with an AC source.

### Step-by-Step Solution:

1. **Identify the given values:**
- Inductance, \( L = 200 \, \text{mH} = 200 \times 10^{-3} \, \text{H} \)
- Capacitance, \( C = 500 \, \mu F = 500 \times 10^{-6} \, \text{F} \)
- Resistance, \( R = 10 \, \Omega \)

2. **Understand the condition for unity power factor:**
- The power factor is unity when the circuit is in resonance, which occurs when the inductive reactance (\( X_L \)) equals the capacitive reactance (\( X_C \)).
- Mathematically, this is expressed as:
\[ X_L = X_C \]

3. **Express the reactances:**
- The inductive reactance is given by:
\[ X_L = \omega L = 2\pi f L \]
- The capacitive reactance is given by:
\[ X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} \]

4. **Set the reactances equal to each other:**
\[ 2\pi f L = \frac{1}{2\pi f C} \]

5. **Rearranging the equation:**
- Multiply both sides by \( 2\pi f C \):
\[ (2\pi f)^2 L C = 1 \]
- This simplifies to:
\[ (2\pi f)^2 = \frac{1}{LC} \]

6. **Solve for frequency \( f \):**
\[ f = \frac{1}{2\pi \sqrt{LC}} \]

7. **Substitute the values of L and C:**
\[ f = \frac{1}{2\pi \sqrt{(200 \times 10^{-3}) (500 \times 10^{-6})}} \]

8. **Calculate \( LC \):**
- First, calculate \( LC \):
\[ LC = 200 \times 10^{-3} \times 500 \times 10^{-6} = 100 \times 10^{-9} = 10^{-7} \]

9. **Now substitute \( LC \) back into the frequency formula:**
\[ f = \frac{1}{2\pi \sqrt{10^{-7}}} \]

10. **Calculate \( \sqrt{10^{-7}} \):**
\[ \sqrt{10^{-7}} = 10^{-3.5} = 3.162 \times 10^{-4} \]

11. **Final calculation for frequency:**
\[ f = \frac{1}{2\pi (3.162 \times 10^{-4})} \approx \frac{1}{1.973 \times 10^{-3}} \approx 15.92 \, \text{Hz} \]

### Conclusion:
The frequency at which the power factor of the circuit is unity is approximately **15.92 Hz**.