Rms And Peak Values
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?
Select the correct option:
Solution
220 V
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.
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About This Question
- Subject
- physics
- Chapter
- electromagnetic induction and alternating currents
- Topic
- rms and peak values
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
220 V
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.
This easy difficulty physics question is from the chapter electromagnetic induction and alternating currents, covering the topic of rms and peak values. It appeared in the 2025 exam.
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