Question

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

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

Hint:

AC voltmeters and ammeters always measure and display the \textbf{RMS} value unless otherwise specified.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In Alternating Current (AC), the voltage varies sinusoidally. The RMS value represents the effective DC equivalent voltage, while the average value is the arithmetic mean of the voltage over time.

Step 2: Key Formula or Approach:
1. \(V_{rms} = \frac{V_0}{\sqrt{2}}\)
2. \(V_{avg} \text{ (Full Cycle)} = \frac{1}{T} \int_{0}^{T} V(t) dt\)

Step 3: Detailed Explanation:
For a complete sine wave cycle, the area under the curve for the positive half-cycle is exactly equal to the area under the curve for the negative half-cycle. Thus, they cancel out, making the average value over a full cycle zero. The RMS value is mathematically derived as the peak voltage divided by the square root of 2.

Step 4: Final Answer:
The rms value is \(\frac{V_0}{\sqrt{2}}\) and the average value is \(0\).