## Formulas and proof[edit]

### Direct current[edit]

The most general and fundamental formula for Joule heating is:

where

*P* is the power (energy per unit time) converted from electrical energy to thermal energy,*I* is the current traveling through the resistor or other element,*V* is the voltage drop across the element.

The explanation of this formula (*P=VI*) is:^{[1]}

- (
*Energy dissipated per unit time*) = (*Energy dissipated per charge passing through resistor*) × (*Charge passing through resistor per unit time*)

When Ohm's law is also applicable, the formula can be written in other equivalent forms:

where *R* is the resistance.

### Alternating current[edit]

When current varies, as it does in AC circuits,

where *t* is time and *P* is the instantaneous power being converted from electrical energy to heat. Far more often, the *average* power is of more interest than the instantaneous power:

where "avg" denotes average (mean) over one or more cycles, and "rms" denotes root mean square.

These formulas are valid for an ideal resistor, with zero reactance. If the reactance is nonzero, the formulas are modified:

where is the phase difference between current and voltage, means real part, *Z* is the complex impedance, and *Y** is the complex conjugate of the admittance (equal to 1/*Z**).

For more details in the reactive case, see AC power.

### Differential Form[edit]

In plasma physics, the Joule heating often needs to be calculated at a particular location in space. The differential form of the Joule heating equation gives the power per unit volume.

Here, is the current density, and is the electric field.