RMS Value of AC

Ordinary moving-coil meters (ammeters and voltmeters) work by responding to the average value of whatever current or voltage is applied to them -- but since the true (full-cycle) average of AC is always zero, such an instrument would show a reading of exactly zero when connected to an AC supply, which is obviously not a useful measurement. What is needed instead is some property of the current that does NOT depend on its instantaneous direction, so that it does not average away to nothing.

The heating effect of a current is exactly such a property: the heat produced by a current i flowing through a resistance R over time t depends on $i^2R$, and since $i^2$ is always positive regardless of the actual sign/direction of i, the heating effect never cancels out over a cycle. This motivates the definition of the RMS (root-mean-square) value: $i_{rms}$ (also called the effective or virtual value) is defined as the value of a STEADY (DC) current that would produce exactly the same amount of heat, in the same resistance, over the same time, as the given alternating current actually does. Carrying out this comparison -- equating the heat produced by the AC current over one cycle to the heat a steady current $i_{rms}$ would produce over the same time -- gives $i_{rms}=\dfrac{i_0}{\sqrt2}\approx0.707\,i_0$, and by an identical argument, $e_{rms}=\dfrac{e_0}{\sqrt2}\approx0.707\,e_0$. …