Electrical Energy and Power

Consider a resistor $AB$ connected to a cell in a circuit (Fig. 11.6), with current flowing from $A$ to $B$. The cell maintains a potential difference $V$ between the two terminals of the resistor, with $A$ at the higher potential and $B$ at the lower. If a charge $Q$ flows through the resistor from $A$ to $B$ in time $\Delta t$, then since $V$ (by definition) is the work done per unit positive charge moved from $A$ to $B$,

As charge $Q$ moves from the higher-potential point $A$ to the lower-potential point $B$ -- i.e. through a DECREASE in potential of value $V$ -- its potential energy decreases by an amount

By conservation of energy, this lost potential energy does not simply vanish; it is converted into some other form of energy by the resistor (or whatever device sits in place of it). Taking the limit $\Delta t\to 0$,

Here $dU/dt$ is the POWER -- the time rate at which energy is transferred -- so

This is the general expression for the electrical power delivered by a cell to a resistor, or indeed to any other device (a motor, a rechargeable battery being charged, etc.) that takes its place. …