Using an ac voltmeter, the potential difference in the electrical line in a house is read to be 234 V. If the line freqency is known to be 50 cycles per second, the equation for the line voltage is

Options

A. 165 sin `(200pit)`
B. 234 sin `(100pit)`
C. 331 sin `(100pit)`
D. 440 sin `(200pit)`

Explanation

To derive the equation for the line voltage based on the given information, we will follow these steps:

### Step 1: Identify the given values
- The RMS voltage (V_RMS) is given as 234 V.
- The frequency (f) is given as 50 Hz.

### Step 2: Calculate the peak voltage (V_0)
The relationship between the RMS voltage and the peak voltage is given by the formula:
\[ V_{RMS} = \frac{V_0}{\sqrt{2}} \]
Rearranging this formula to find the peak voltage (V_0):
\[ V_0 = V_{RMS} \times \sqrt{2} \]
Substituting the given RMS voltage:
\[ V_0 = 234 \times \sqrt{2} \]
Calculating this gives:
\[ V_0 \approx 234 \times 1.414 \approx 331 \text{ V} \]

### Step 3: Calculate the angular frequency (ω)
The angular frequency (ω) is related to the frequency (f) by the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \times 50 = 100\pi \text{ rad/s} \]

### Step 4: Write the equation for the line voltage
The general equation for an AC voltage can be expressed as:
\[ V(t) = V_0 \sin(\omega t) \]
Substituting the values of V_0 and ω:
\[ V(t) = 331 \sin(100\pi t) \]

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

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