Question

The rms and the average value of an ac voltage \[ V=V_0\sin\omega t \] over a cycle respectively will be:

• A. \(\dfrac{V_0}{2},\dfrac{V_0}{2}\)
• B. \(\dfrac{V_0}{\pi},\dfrac{V_0}{2}\)
• C. \(\dfrac{V_0}{\sqrt{2}},0\)
• D. \(V_0,\dfrac{V_0}{2}\)

Hint:

For a sinusoidal AC quantity: \[ V_{\text{rms}}=\frac{V_0}{\sqrt2} \] \[ I_{\text{rms}}=\frac{I_0}{\sqrt2} \] and the average value over one complete cycle is always \[ 0. \] Do not confuse this with the average value over only one half-cycle, which is \(\frac{2V_0}{\pi}\).

Answer 1

The Correct Option is C

Concept: An alternating voltage continuously changes its magnitude and direction with time. For a sinusoidal alternating voltage \[ V=V_0\sin\omega t, \] two important quantities are frequently used:
• Root Mean Square (RMS) Value
• Average Value over a complete cycle The RMS value represents the equivalent DC voltage that would produce the same heating effect in a resistor. The average value over one complete cycle is obtained by averaging all instantaneous values over the full period.

Step 1: Determine the RMS value of the alternating voltage. For a sinusoidal voltage \[ V=V_0\sin\omega t, \] the RMS value is given by the standard relation \[ V_{\text{rms}} = \frac{V_0}{\sqrt2}. \] This result follows from the definition \[ V_{\text{rms}} = \sqrt{\frac{1}{T}\int_0^T V^2\,dt}. \] Substituting \[ V=V_0\sin\omega t, \] one obtains \[ V_{\text{rms}} = \frac{V_0}{\sqrt2}. \] Thus, \[ \boxed{V_{\text{rms}}=\frac{V_0}{\sqrt2}} \]

Step 2: Determine the average value over one complete cycle. The average value is \[ V_{\text{avg}} = \frac{1}{T} \int_0^T V_0\sin\omega t\,dt. \] Over one complete cycle, the positive half-cycle and negative half-cycle are equal in magnitude but opposite in sign. Therefore, the positive area exactly cancels the negative area. Hence, \[ V_{\text{avg}}=0. \] Thus, \[ \boxed{V_{\text{avg}}=0} \]

Step 3: Compare with the given options. We have obtained \[ V_{\text{rms}} = \frac{V_0}{\sqrt2} \] and \[ V_{\text{avg}} = 0. \] This corresponds exactly to Option (C).

Step 4: State the final answer. Therefore, \[ \boxed{ V_{\text{rms}}=\frac{V_0}{\sqrt2}, \qquad V_{\text{avg}}=0 } \] Hence, the correct answer is \[ \boxed{\text{(C)}}. \]