The RMS value of an AC voltage with peak value V₀ is:

Solution

Physics sir · 3 steps

RMS means square it, average it over a cycle, then take the root: the square of a sine averages to one half.

1. Write the voltage. A sinusoidal AC voltage is V = V₀ sin ω t. RMS is the square root of the mean of V² over one full cycle.
2. Average the square. V² = V₀² sin² ω t. Since sin² ω t = (1 - cos 2ω t)/2 and the cosine averages to zero over a cycle, the mean of sin² ω t is 1/2.

   Vᵣₘₛ = √(⟨ V² ⟩) = √(V₀²/2) = V₀/√2

3. So. Vᵣₘₛ = V₀/√2 ≈ 0.707 V₀, which is option (b).

The idea to keep. For any sinusoid, RMS is peak divided by √2, because sin² averages to one half. The average of the plain sine over a full cycle is zero, so never confuse mean with RMS.

Answer from the NTA final answer key. Solution by PrepWiser.