In a series RC circuit connected across an AC source of 220 V 50 Hz calculate the rms voltage across resistor and capacitor if values of R and C are 100 `Omega` and `20 μF` respectively .

Answer

To solve the problem of finding the RMS voltage across the resistor and capacitor in a series RC circuit connected to an AC source, we can follow these steps:

### Step 1: Identify Given Values
- Source RMS Voltage, \( V_{\text{rms}} = 220 \, \text{V} \)
- Frequency, \( f = 50 \, \text{Hz} \)
- Resistance, \( R = 100 \, \Omega \)
- Capacitance, \( C = 20 \, \mu F = 20 \times 10^{-6} \, F \)

### Step 2: Calculate the Capacitive Reactance (\( X_c \))
The formula for capacitive reactance is given by:
\[ X_c = \frac{1}{\omega C} \]
where \( \omega = 2\pi f \).

First, calculate \( \omega \):
\[ \omega = 2 \pi \times 50 \approx 314.16 \, \text{rad/s} \]

Now, substitute \( \omega \) and \( C \) into the formula for \( X_c \):
\[ X_c = \frac{1}{314.16 \times 20 \times 10^{-6}} \approx 159.15 \, \Omega \]

### Step 3: Calculate the Phase Angle (\( \phi \))
The phase angle \( \phi \) can be calculated using the formula:
\[ \tan \phi = \frac{R}{X_c} \]
Substituting the values:
\[ \tan \phi = \frac{100}{159.15} \]
Calculating \( \phi \):
\[ \phi = \tan^{-1}\left(\frac{100}{159.15}\right) \approx 32.13^\circ \]

### Step 4: Calculate the RMS Voltage Across the Resistor (\( V_R \))
Using the relationship:
\[ V_R = V_{\text{rms}} \sin \phi \]
Substituting the values:
\[ V_R = 220 \sin(32.13^\circ) \approx 220 \times 0.532 \approx 117.04 \, \text{V} \]

### Step 5: Calculate the RMS Voltage Across the Capacitor (\( V_C \))
Using the relationship:
\[ V_C = V_{\text{rms}} \cos \phi \]
Substituting the values:
\[ V_C = 220 \cos(32.13^\circ) \approx 220 \times 0.847 \approx 186.31 \, \text{V} \]

### Final Results
- RMS Voltage across Resistor, \( V_R \approx 117.04 \, \text{V} \)
- RMS Voltage across Capacitor, \( V_C \approx 186.31 \, \text{V} \)