Square root ~ 7
With more intervals the rms average turns out to be ( peak value ) / √2 = peak value / = peak value
2 For those who are familiar with the graphs of sine and cosine functions, then the following algebraic method can be attempted.
I = I o sin ωt and I 2 = I o 2 sin 2 ωt
The heating effect depends on I 2 R , and so an average of I 2 is needed and not an average of I .
To find the rms value, you need the average value of sin 2 as time runs on and on.
The graph of sin ωt and the graph of cos ωt look the same, except for a shift of origin. Because they are the same pattern, sin 2 ωt and cos 2 ωt have the same average as time goes on.
But sin 2 ωt + cos 2 ωt = 1. Therefore the average values of either of them must be 1/2.
Therefore the rms value of I o sinω t must be I o / √2
The rms value is times the peak value, and the peak value is times the value the voltmeter shows. The peak value for 230 V mains is 325 V.
3 Alternatively: Plot a graph of sin 2 θ . Cut the graph in half and turn one half upside down, or copy onto a transparency and fit together. The two halves fit together exactly, showing that the mean value is 1/2.