For a sinusoidal voltage with peak Vp, what is the RMS value?

For a sinusoidal voltage with peak Vp, what is the RMS value?

• A. Vp
• B. Vp/√2
• C. Vp/2
• D. Vp√2

Answer: (B)

The key idea is that RMS is the effective value of a sine wave for delivering power to a resistor. For a voltage v(t) = Vp sin(ωt), the instantaneous power is proportional to v^2, so the RMS value is the square root of the average of v^2 over one period. Compute ⟨v^2⟩ = (1/T) ∫_0^T Vp^2 sin^2(ωt) dt. Since the average of sin^2 over a full cycle is 1/2, ⟨v^2⟩ = Vp^2/2, and Vrms = sqrt(Vp^2/2) = Vp/√2. So the RMS value is Vp/√2, about 0.707 times the peak value. This distinguishes it from the peak value (Vp), a value like Vp/2, or a larger value like Vp√2.

The key idea is that RMS is the effective value of a sine wave for delivering power to a resistor. For a voltage v(t) = Vp sin(ωt), the instantaneous power is proportional to v^2, so the RMS value is the square root of the average of v^2 over one period. Compute ⟨v^2⟩ = (1/T) ∫_0^T Vp^2 sin^2(ωt) dt. Since the average of sin^2 over a full cycle is 1/2, ⟨v^2⟩ = Vp^2/2, and Vrms = sqrt(Vp^2/2) = Vp/√2. So the RMS value is Vp/√2, about 0.707 times the peak value. This distinguishes it from the peak value (Vp), a value like Vp/2, or a larger value like Vp√2.