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Power from Alternating Current
Energy, (W), is needed to move a charge, (Q), through a potential difference, (V).

W = Q×V

Electric current, (I), is the charge, (Q), flowing per unit time, (t).

=> Q = I×t

Combining these equations together gives

=> W = I×t×V

Power, (P) is the energy, (W), per unit time, (t)

=> P = I×V


For a resistor

Ohm's Law gives V = I×R

=> P = I×V   = I2×R   = V2/R


Introduction to the sine wave formula


For a sine wave alternating voltage, V = V0sin(ωt), and the power dissipated in a resistor R over one time period, (T) is

Using the cos(2θ) formula, this can be rearranged to give

Substituting and simplifying using T = 2π/ω gives

Averaging the power over one time period (dividing by 2π/ω) gives that the mean power is

Examining this formula shows that the quantity V0/√2 is the steady (direct) voltage that would dissipate the same power.
This quantity is known as the root mean square (rms) voltage.


For a capacitor

Introduction to capacitors in AC circuits

Using P = I × V and summing over one time period

Using the sin(2θ) formula, this can be rearranged to give

Therefore the mean power dissipated by an ideal capacitor over one time period is zero.