A capacitor of capacitance 10` muF ` is connected to a 220 V , 30 Hz alternating current source . Calculate the peak value of voltage across the capacitor . Also calculate the rms value of current in the circuit .

Answer

To solve the problem step by step, we will first calculate the peak value of voltage across the capacitor and then calculate the RMS value of current in the circuit.

### Step 1: Calculate the Peak Value of Voltage Across the Capacitor

The relationship between the peak voltage (\( V_0 \)) and the RMS voltage (\( V_{RMS} \)) for a sinusoidal AC supply is given by the formula:

\[ V_0 = V_{RMS} \times \sqrt{2} \]

Given:
- \( V_{RMS} = 220 \, V \)

Substituting the value into the formula:

\[ V_0 = 220 \, V \times \sqrt{2} \]

Calculating \( \sqrt{2} \):

\[ \sqrt{2} \approx 1.414 \]

Now, substituting this value:

\[ V_0 = 220 \times 1.414 \approx 311.08 \, V \]

Thus, the peak value of voltage across the capacitor is approximately:

\[ \boxed{311.08 \, V} \]

### Step 2: Calculate the RMS Value of Current in the Circuit

To find the RMS current (\( I_{RMS} \)), we first need to calculate the capacitive reactance (\( X_C \)). The formula for capacitive reactance is:

\[ X_C = \frac{1}{2 \pi f C} \]

Where:
- \( f = 30 \, Hz \)
- \( C = 10 \, \mu F = 10 \times 10^{-6} \, F \)

Substituting the values into the formula:

\[ X_C = \frac{1}{2 \pi \times 30 \times 10 \times 10^{-6}} \]

Calculating \( 2 \pi \):

\[ 2 \pi \approx 6.2832 \]

Now substituting this value:

\[ X_C = \frac{1}{6.2832 \times 30 \times 10 \times 10^{-6}} \approx \frac{1}{1.884 \times 10^{-4}} \approx 5307.96 \, \Omega \]

Now that we have \( X_C \), we can calculate the RMS current using the formula:

\[ I_{RMS} = \frac{V_{RMS}}{X_C} \]

Substituting the values:

\[ I_{RMS} = \frac{220 \, V}{5307.96 \, \Omega} \approx 0.0415 \, A \]

Thus, the RMS value of current in the circuit is approximately:

\[ \boxed{0.0415 \, A} \]

### Summary of Results
- Peak Value of Voltage: \( 311.08 \, V \)
- RMS Value of Current: \( 0.0415 \, A \)