What Is the Difference Between I²R, V²/R, and VI for Power?

The formulas $P=VI$, $P=I^2R$, and $P=V^2/R$ can all describe electrical power, but they apply using different quantities. The most general starting point for a circuit element is voltage across that element times current through it. The other two forms follow by substituting Ohm’s law for a resistor.

The key is to use the voltage, current, and resistance that belong to the same component and operating condition. Mixing a source voltage with the resistance of one component, for example, can give the wrong power.

The general formula: $P=VI$

For a two-terminal element, electrical power is

$$P=VI,$$

where $V$ is the potential difference across the element and $I$ is the current through it. With the passive sign convention—current entering the terminal marked positive—positive power means the element absorbs energy. A source delivering energy may have negative power under that convention.

This formula does not require the element to be an ordinary resistor. It applies to lamps, motors, capacitors, and other components as an instantaneous relationship, with instantaneous voltage and current:

$$p(t)=v(t)i(t).$$

For steady DC, the quantities are constant, so the power is constant too. For varying signals, instantaneous power can vary over time; average power may be the quantity of interest.

How the resistor formulas follow

For an ohmic resistor, voltage and current are related by Ohm’s law:

$$V=IR.$$

Substitute this relation into $P=VI$ in either of two ways. Replacing $V$ gives

$$P=VI=(IR)I=I^2R.$$

Replacing $I$ with $V/R$ gives

$$P=VI=V\left(\frac{V}{R}\right)=\frac{V^2}{R}.$$

Thus, the formulas are equivalent when Ohm’s law applies and the quantities refer to the same resistor. Choose the version that matches the values you know: use $I^2R$ when current and resistance are known, or $V^2/R$ when voltage across the resistor and resistance are known.

Worked example

A $6\,\Omega$ resistor has $12\,\text{V}$ across it. Its current is

$$I=\frac{V}{R}=\frac{12}{6}=2\,\text{A}.$$

Calculate its power using each expression:

$$\begin{aligned} P&=VI=(12)(2)=24\,\text{W},\\ P&=I^2R=(2)^2(6)=24\,\text{W},\\ P&=\frac{V^2}{R}=\frac{12^2}{6}=24\,\text{W}. \end{aligned}$$

All three give the same result because the voltage, current, and resistance satisfy Ohm’s law for that resistor. The resistor absorbs $24\,\text{J}$ of electrical energy per second, which it typically dissipates as heat.

What changes for AC circuits?

For a pure resistor in sinusoidal AC, the voltage and current are in phase. Using RMS voltage and current, average power is

$$P_{\text{avg}}=V_{\text{rms}}I_{\text{rms}}=I_{\text{rms}}^2R=\frac{V_{\text{rms}}^2}{R}.$$

For a circuit with reactance, voltage and current can be out of phase. For sinusoidal voltage and current, average real power is

$$P_{\text{avg}}=V_{\text{rms}}I_{\text{rms}}\cos\phi,$$

where $\phi$ is the phase angle between voltage and current. If the circuit is represented by a series impedance with resistance $R$ and reactance, its real power is $I_{\text{rms}}^2R$; using the total RMS voltage in $V^2/R$ would generally be incorrect. The simple $V^2/R$ form applies to the voltage across a resistor, not automatically to the total voltage across a reactive circuit.

Common mistakes

• Using voltage from the wrong place: In a series circuit, the source voltage is not necessarily the voltage across an individual resistor. Use that resistor’s own voltage drop.
• Combining mismatched quantities: The current must be the current through the component whose voltage and resistance you use.
• Treating every component as an ideal resistor: A non-ohmic component does not have one fixed resistance across all operating conditions. $P=VI$ still applies, but a fixed-resistance version may not.
• Ignoring AC phase: In AC, multiplying RMS voltage by RMS current gives apparent power. For average real power in a reactive load, include the power factor $\cos\phi$.

If you know voltage and current across a component, $P=VI$ is the safest starting point. Use the resistance forms when the resistor model and the relevant values are clear.