Lesson 2 of 10 · 6 min
RMS values and average power
NCERT §7.2
Riya's multimeter says the mains is 220 V, but her oscilloscope shows peaks of 311 V. Which number is the voltage of the supply?
The lesson in notes
In short
The power in the resistor at any instant is p = i²R = im²R sin²ωt. Since sin²ωt = ½(1 − cos 2ωt) and cos 2ωt averages to zero over a cycle, the average power is ½im²R.
To write this like the dc formula P = I²R, define the root mean square (rms) or effective current I = im/√2 = 0.707 im. Then P = I²R.
Likewise the rms voltage is V = vm/√2 = 0.707 vm, and V = IR, so P = V²/R = IV, exactly as for dc.
The rms current is the steady current that would heat the same resistor at the same average rate.
AC values are quoted as rms values unless stated otherwise. The household 220 V is an rms value; its peak is √2 × 220 = 311 V.
Example 7.1: a bulb rated 100 W for 220 V has R = V²/P = 484 Ω; the source's peak voltage is 311 V; the rms current is P/V = 0.454 A.