In a series LCR circuit with an AC source `(E_("rms")50V and v=50//pi Hz), R= 300Omega, C=0.02 mF,L=1.0 H` , which of the following is correct

Options

A. the rms current in the circuit is 0.1 A
B. the rms potential difference across the capacitor is 50 V
C. the rms potential difference across the capacitor is 14.1 V
D. the rms current in the circuit is 0.14 A

Answer

To solve the given problem step by step, we will calculate the RMS current in the circuit and the RMS potential difference across the capacitor in the series LCR circuit.

### Given Data:
- \( E_{rms} = 50 \, V \)
- Frequency \( f = \frac{50}{\pi} \, Hz \)
- Resistance \( R = 300 \, \Omega \)
- Capacitance \( C = 0.02 \, mF = 0.02 \times 10^{-3} \, F \)
- Inductance \( L = 1.0 \, H \)

### Step 1: Calculate the Reactance of the Capacitor \( X_C \)
The capacitive reactance \( X_C \) is given by the formula:
\[ X_C = \frac{1}{2 \pi f C} \]
Substituting the values:
\[ f = \frac{50}{\pi} \implies 2 \pi f = 50 \implies X_C = \frac{1}{50 \times 0.02 \times 10^{-3}} = \frac{1}{0.001} = 1000 \, \Omega \]

### Step 2: Calculate the Reactance of the Inductor \( X_L \)
The inductive reactance \( X_L \) is given by the formula:
\[ X_L = 2 \pi f L \]
Substituting the values:
\[ X_L = 2 \pi \left(\frac{50}{\pi}\right) \times 1 = 100 \, \Omega \]

### Step 3: Calculate the Impedance \( Z \)
The impedance \( Z \) in a series LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
Substituting the values:
\[ Z = \sqrt{300^2 + (100 - 1000)^2} = \sqrt{300^2 + (-900)^2} \]
Calculating:
\[ Z = \sqrt{90000 + 810000} = \sqrt{900000} = 300\sqrt{10} \approx 500 \, \Omega \]

### Step 4: Calculate the RMS Current \( I_{rms} \)
The RMS current \( I_{rms} \) is given by:
\[ I_{rms} = \frac{E_{rms}}{Z} \]
Substituting the values:
\[ I_{rms} = \frac{50}{500} = 0.1 \, A \]

### Step 5: Calculate the RMS Potential Difference Across the Capacitor \( V_C \)
The potential difference across the capacitor \( V_C \) is given by:
\[ V_C = I_{rms} \cdot X_C \]
Substituting the values:
\[ V_C = 0.1 \times 1000 = 100 \, V \]

### Final Results:
- **RMS Current in the Circuit**: \( I_{rms} = 0.1 \, A \)
- **RMS Potential Difference Across the Capacitor**: \( V_C = 100 \, V \)