# Average power associated with a resistor derivation

In this short piece article, we will discuss the Average power associated with a resistor, so let’s get started…

## Average power associated with a resistor derivation

In the case of a pure resistor, the voltage and current are always in the same phase. So we can write the instantaneous values of voltage and current as follows :
$$V=V_0 \sin \omega t\: \text{and} \:I=I_0 \sin \omega t$$
Work done in small time $d t$ will be

\begin{aligned}d W &=P d t=V I d t=V_0 I_0 \sin ^2 \omega t d t\\
&=\frac{V_0 I_0}{2}(1-\cos 2 \omega t) d t\end{aligned}

The average power dissipated per cycle in the resistor will be