An AC circuit drawing power from a source of angular frequency `"50 rad s"^(-1)` has a power factor of 0.6. In this condition, a resistance of `100Omega` is present and the current is lagging behind the voltage. If a capacitor is connected in series, then the required capacitance that will result in a power factor of unity is

Options

A. `30muF`
B. `150muF`
C. `50muF`
D. `200muF`

Answer

To solve the problem, we need to find the required capacitance that will result in a power factor of unity when a capacitor is connected in series with a resistive-inductive circuit. Here are the steps to arrive at the solution:

### Step 1: Understand the Given Information
- Angular frequency, \( \omega = 50 \, \text{rad/s} \)
- Power factor, \( \text{pf} = 0.6 \)
- Resistance, \( R = 100 \, \Omega \)

### Step 2: Calculate the Inductive Reactance
The power factor is given by:
\[ \cos \phi = \frac{R}{Z} \]
where \( Z \) is the impedance of the circuit. Since the power factor is 0.6, we can find the angle \( \phi \):
\[ \cos \phi = 0.6 \implies \phi = \cos^{-1}(0.6) \approx 53.13^\circ \]

### Step 3: Use the Power Factor to Find the Relationship Between Reactance and Resistance
From the power factor, we can express the tangent of the angle:
\[ \tan \phi = \frac{X_L - X_C}{R} \]
where \( X_L \) is the inductive reactance and \( X_C \) is the capacitive reactance. We know:
\[ \tan(53.13^\circ) = \frac{4}{3} \]
Thus,
\[ X_L - X_C = \frac{4}{3} R = \frac{4}{3} \times 100 = \frac{400}{3} \, \Omega \]

### Step 4: Express Inductive and Capacitive Reactance
The inductive reactance is given by:
\[ X_L = \omega L \]
The capacitive reactance is given by:
\[ X_C = \frac{1}{\omega C} \]

### Step 5: Set Up the Equation for Power Factor Unity
To achieve a power factor of unity, we need:
\[ X_L = X_C \]
Substituting the expressions for \( X_L \) and \( X_C \):
\[ \omega L = \frac{1}{\omega C} \]
This implies:
\[ X_L = X_C \implies \frac{400}{3} + X_C = X_C \]
Thus,
\[ X_L = \frac{400}{3} + \frac{1}{\omega C} \]

### Step 6: Solve for Capacitance
Rearranging gives:
\[ \frac{1}{\omega C} = \frac{400}{3} \implies C = \frac{1}{\omega \cdot \frac{400}{3}} = \frac{3}{400 \cdot 50} \]
Calculating this gives:
\[ C = \frac{3}{20000} = 1.5 \times 10^{-4} \, \text{F} \]

### Step 7: Convert to Microfarads
To convert farads to microfarads:
\[ C = 1.5 \times 10^{-4} \, \text{F} = 150 \, \mu F \]

### Final Answer
The required capacitance that will result in a power factor of unity is:
\[ \boxed{150 \, \mu F} \]