Using an AC voltmeter, the potential difference in the electrical line in a house is read to be 234 volt. If the line frequency is known to be 50 cycle per sec, the equation for the line voltage is

Options

A. `165 sin (100pit)`
B. `234 sin (10pit)`
C. `331 sin (100pit)`
D. `440 sin (100pit)`

Answer

To derive the equation for the line voltage given the RMS voltage and frequency, we can follow these steps:

### Step 1: Understand the given values
We are given:
- The RMS voltage (Vrms) = 234 volts
- The frequency (f) = 50 cycles per second (Hz)

### Step 2: Relate RMS voltage to peak voltage
The relationship between the RMS voltage and the peak voltage (V0) in an AC circuit is given by the formula:
\[ V_0 = V_{rms} \times \sqrt{2} \]

### Step 3: Calculate the peak voltage
Substituting the given RMS voltage into the formula:
\[ V_0 = 234 \times \sqrt{2} \]
Calculating this:
\[ V_0 \approx 234 \times 1.414 \approx 331 \text{ volts} \]

### Step 4: Write the equation for the line voltage
The general equation for an AC voltage is given by:
\[ V(t) = V_0 \sin(2 \pi f t) \]
Substituting the peak voltage and frequency into this equation:
\[ V(t) = 331 \sin(2 \pi \times 50 t) \]

### Step 5: Simplify the equation
Calculating \(2 \pi \times 50\):
\[ 2 \pi \times 50 = 100 \pi \]
Thus, the equation for the line voltage becomes:
\[ V(t) = 331 \sin(100 \pi t) \]

### Final Answer
The equation for the line voltage is:
\[ V(t) = 331 \sin(100 \pi t) \]

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