We know that time average value of AC over one cycle is zero and it can be proved easily
Instantaneous current I and time average of AC over half cycle could be positive for one half cycle and negative for another half cycle but quantity i^{2} would always remain positive
So time average of quantity i^{2} is
This is known as the mean square current
The square root of mean square current is called root mean square current or rms current.
Thus,
thus ,the rms value of AC is .707i_{0} of the peak value of alternating current
Similarly rms value of alternating voltage or emf is
If we allow the AC current represented by i=i_{0}sin(ωt+φ) to pass through a resistor of resistance R,the power dissipated due to flow of current would be
P=i^{2}R
Since magnitude of current changes with time ,the power dissipation in circuit also changes
The average Power dissipated over one complete current cycle would be
If we pass direct current of magnitude i_{rms} through the resistor ,the power dissipate or rate of production of heat in this case would be
P=(i_{rms})^{2}R
Thus rms value of AC is that value of steady current which would dissipate the same amount of power in a given resistance in a given tine as would gave been dissipated by alternating current
This is why rms value of AC is also known as virtual value of current