Rms And Peak Values

Question

A sinusoidal alternating voltage has a peak value of 311 V, which is typical of a 50 Hz domestic mains supply. What is the corresponding root-mean-square voltage available to appliances?

RankGuru · Accepted Answer

The root-mean-square value of an alternating quantity is the steady direct value that would deliver the same average power to a resistor, and for a pure sinusoid it relates to the peak value through $V_{rms} = V_0/\sqrt{2}$. This factor emerges from averaging $\sin^2$ over a full cycle, which equals one-half. Substituting the peak $V_0 = 311$ V gives $V_{rms} = 311/1.414 \approx 220$ V, exactly the nominal household rating. The option 311 V mistakes the peak for the RMS value. The option 440 V multiplies by $\sqrt{2}$ instead of dividing, inflating the answer. The option 156 V wrongly divides by 2 rather than $\sqrt{2}$. This is the defining relationship NCERT uses to justify why meters and ratings quote RMS rather than peak values. A plausibility check confirms the RMS must lie below the peak, and 220 V matching the stated domestic standard validates the result; the same $1/\sqrt{2}$ factor applies equally to sinusoidal current.

Rms And Peak Values

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