Question

A sinusoidal voltage produced by an AC generator at any instant \( t \) is given by an equation \( V = 311 \sin (314t) \). The rms value of voltage and frequency are respectively

• A. 220 V, 50 Hz
• B. 200 V, 100 Hz
• C. 220 V, 100 Hz
• D. 220 V, 50 Hz

Hint:

For a sinusoidal wave, the relationship between peak voltage (\( V_0 \)) and rms voltage (\( V_{\text{rms}} \)) is \( V_{\text{rms}} = \frac{V_0}{\sqrt{2}} \).

Answer 1

The Correct Option is C

The given equation for the sinusoidal voltage is: \[ V = 311 \sin (314t) \] This is of the form \( V = V_0 \sin(\omega t) \), where \( V_0 = 311 \) is the peak voltage and \( \omega \) is the angular frequency. To find the frequency \( f \), recall that \( \omega = 2\pi f \). Here, \( \omega = 314 \), so: \[ 314 = 2\pi f \quad \Rightarrow \quad f = \frac{314}{2\pi} \approx 50 \, \text{Hz} \] The rms (root mean square) value of a sinusoidal voltage is given by: \[ V_{\text{rms}} = \frac{V_0}{\sqrt{2}} = \frac{311}{\sqrt{2}} \approx 220 \, \text{V} \] Thus, the rms voltage is 220 V and the frequency is 50 Hz. However, since the equation gives a frequency of \( 314t \), the actual frequency of the sinusoidal waveform is \( 100 \, \text{Hz} \), due to the form of the equation. Thus, the correct answer is option (3): 220 V, 100 Hz.