`text(Calculation of the average values)`

For an alternating voltage given by `E = E_0sin(omega t)` and the
corresponding alternating current in
a purely resistive circuit `i = i_0sin( omega t)` . average values of
voltage and current can be calculated as follows
Average voltage,
`vecE = (int_0^T E_0 sin(omega t)dt)/T` , where `T = (2pi)/omega`
`=> vecE = E_0/T[(cos(omega t))/omega]_0^T => vecE = 0`

Similarly it can be derived that the average current over a
cycle `T` is zero. Average current and voltage are zero
essentially due to the 'alternating' nature of the voltage
source and are therefore not of great significance in
analysis and calculations.

However, for calculations of average power dissipation
etc, a more significant physical quantity called 'root mean
square' or rms of voltage and current is calculated

`text(Calculation of the rms values)`

Root mean square of a physical quantity is simply defined
as the square root of the average of it's squared values or
rms Voltage for example `E_(rms) = sqrt(E^2)`

`=> E_(rms) = sqrt((int_0^T E^2dt)/T) = sqrt(((int_0^T E_0^2 sin^2(omega t))dt)/T)`

`=> E_(rms) = E_0 sqrt((int_0^T sin^2(omega t)dt)/T)`

`= E_0 sqrt((int_0^T (1-cos(omega t)^2)dt)/2T)`

`= E_0 sqrt([t- [sin(omega t)]/omega]_0^T/(2T))`

`=> E_(rms) = E_0/sqrt2 approx 0.7 E_0`

For the instantaneous current `i = i_0,sin(omega t)` , the rms value
can be identically calculated as
`=> i_(rms) = i_0/sqrt2 approx 0.7i_0`