A resistor of resistance 50 `Omega ` is connected to a power source at 220 V. Calculate the peak voltage and average voltage over half cycle if frequency of source is 50 Hz. Also calculate the rms value of current.

Answer

To solve the problem step by step, we will calculate the peak voltage, average voltage over half cycle, and the RMS value of current for the given resistor connected to an AC power source.

### Step 1: Calculate the Peak Voltage (V₀)

The relationship between the RMS voltage (V_rms) and the peak voltage (V₀) for a sinusoidal wave is given by:

\[ V_{rms} = \frac{V_0}{\sqrt{2}} \]

Given that the RMS voltage (V_rms) is 220 V, we can rearrange the formula to find V₀:

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

Substituting the given value:

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

Calculating this:

\[ V_0 = 220 \times 1.414 \approx 311 \, V \]

### Step 2: Calculate the Average Voltage over Half Cycle

The average voltage (V_avg) over half a cycle of a sinusoidal wave is given by:

\[ V_{avg} = \frac{2V_0}{\pi} \]

Now substituting the value of V₀ we found:

\[ V_{avg} = \frac{2 \times 311}{\pi} \]

Calculating this:

\[ V_{avg} \approx \frac{622}{3.1416} \approx 197.2 \, V \]

### Step 3: Calculate the RMS Value of Current (I_rms)

The RMS value of current (I_rms) can be calculated using Ohm's law:

\[ I_{rms} = \frac{V_{rms}}{R} \]

Where R is the resistance. Given R = 50 Ω:

\[ I_{rms} = \frac{220}{50} \]

Calculating this:

\[ I_{rms} = 4.4 \, A \]

### Summary of Results

1. Peak Voltage (V₀) = 311 V
2. Average Voltage over Half Cycle (V_avg) ≈ 197.2 V
3. RMS Value of Current (I_rms) = 4.4 A