Voltage measures energy transferred per unit charge, while current measures charge transferred per unit time. Their product therefore gives a rate of energy transfer.

For a circuit element with voltage $v$ across it and current $i$ through it,

$$\boxed{P=vi}.$$

The SI unit is the watt:

$$1,\mathrm W=1,\mathrm{J/s}.$$

Passive sign convention

A useful convention defines the current as positive when it enters the terminal labeled at higher potential. Under this passive sign convention,

$$P=vi>0$$

means the element is absorbing electrical energy, while

$$P=vi<0$$

means it is delivering electrical energy to the rest of the circuit.

The sign is bookkeeping, not a new physical law. It depends on the chosen reference directions for voltage and current.

Energy transferred over time

If power varies with time, the net electrical energy transferred to the element from $t_1$ to $t_2$ is

$$\boxed{\Delta E=\int_{t_1}^{t_2}P(t),dt =\int_{t_1}^{t_2}v(t)i(t),dt}.$$

For constant voltage and current,

$$\Delta E=VI\Delta t.$$

Power in an ohmic resistor

For a resistor obeying

$$V=IR,$$

substitution gives

$$\boxed{P=I^2R=\frac{V^2}{R}}.$$

With the passive sign convention, this power is nonnegative: the resistor absorbs electrical energy and converts it mainly into internal thermal energy.

Worked example

A resistor of

$$R=24,\Omega$$

has

$$V=12,\mathrm V$$

across it. The current is

$$I=\frac{V}{R}=0.50,\mathrm A.$$

The absorbed power is

$$P=VI=(12)(0.50)=\boxed{6.0,\mathrm W}.$$

Equivalently,

$$P=I^2R=(0.50)^2(24)=6.0,\mathrm W.$$

If this condition lasts for $5.0,\mathrm{min}=300,\mathrm s$, the energy transferred is

$$\Delta E=P\Delta t=(6.0)(300)=1.8\times10^3,\mathrm J.$$

Sources can deliver power

A battery or other source can have current leave its higher-potential terminal while supplying a load. With the passive sign convention applied to the source itself, its power is then negative: electrical energy is leaving the source.

This distinguishes power from voltage or current alone. A source can maintain a large voltage while supplying almost no power if the current is nearly zero; likewise, substantial current at very low voltage can correspond to modest power.

The relation $P=vi$ is a general circuit energy balance and is not specific to resistors. Resistor formulas such as $I^2R$ are special cases obtained by combining it with a component's voltage-current relation.