For a pure sine wave, the rms (root-mean-square) value is related to the peak amplitude how?

Introduction / Context:The rms value of a periodic signal represents the DC-equivalent heating value in a resistive load. For sinusoidal waveforms, there is a fixed relationship between the peak value and the rms value that is used extensively in AC circuit analysis and instrumentation.

Given Data / Assumptions:

• Signal is a pure sine: v(t) = Vm * sin(ωt).
• We compare Vrms to Vm (peak amplitude).
• No distortion or DC offset.

Concept / Approach:

• Definition: Vrms = sqrt( (1/T) * ∫ v(t)^2 dt over one period ).
• For a sine: Vrms = Vm / √2 ≈ 0.707 * Vm.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• Number of cycles in one second: That is frequency (Hz), not rms.
• Time for one cycle: That is period T, not rms.
• 1.41 times peak: Inverse relation; peak ≈ 1.414 * Vrms, not rms ≈ 1.41 * peak.
• None of the above: Incorrect because 0.707 * peak is correct.

Common Pitfalls:

• Confusing peak, peak-to-peak, and rms values.
• Applying the sine relation to non-sinusoidal waveforms where the factor differs.

Final Answer: