Direct current The most fundamental formula for Joule heating is the generalized power equation: P = I (V_{A} - V_{B}) where • P is the
power (energy per unit time) converted from electrical energy to thermal energy, • I is the current travelling through the resistor or other element, • V_{A}-V_{B} is the
voltage drop across the element. The explanation of this formula (P = IV) is: Assuming the element behaves as a perfect resistor and that the power is completely converted into heat, the formula can be re-written by substituting
Ohm's law, V = I R , into the generalized power equation: P = IV = I^2R = V^2/R where
R is the
resistance.
Alternating current When current varies, as it does in AC circuits, P(t) = U(t) I(t) where
t is time and
P is the instantaneous active power being converted from electrical energy to heat. Far more often, the
average power is of more interest than the instantaneous power. For an ideal resistor, with zero
reactance, the average joule-heating power is P_{\rm avg} = U_\text{rms} I_\text{rms} = (I_\text{rms})^2 R = (U_\text{rms})^2 / R where "avg" denotes
average (mean) over one or more cycles, and "rms" denotes
root mean square. If the reactance is nonzero, the formulas are modified. The average joule-heating power is P_{\rm avg} = U_\text{rms}I_\text{rms}\cos\phi = (I_\text{rms})^2 \operatorname{Re}(Z) = (U_\text{rms})^2 \operatorname{Re}(Y^*) where \phi is phase difference between current and voltage, \operatorname{Re} means
real part,
Z is the
complex impedance, and
Y* is the
complex conjugate of the
admittance (equal to 1/
Z*). Note that \operatorname{Re}(Z) = R, and \operatorname{Re}(Y^*) = \operatorname{Re} \frac{1}{Z^*} = \operatorname{Re} \frac{Z}{|Z|^2} = \frac{R}{|Z|^2}. For more details in the reactive case, see
AC power.
Differential form Joule heating can also be calculated at a particular location in space. The differential form of the Joule heating equation gives the power per unit volume. \frac{\mathrm{d}P}{\mathrm{d}V} = \mathbf{J} \cdot \mathbf{E} Here, \mathbf{J} is the current density, and \mathbf{E} is the electric field. For a material with a conductivity \sigma, \mathbf{J}=\sigma \mathbf{E} and therefore \frac{\mathrm{d}P}{\mathrm{d}V} = \mathbf{J} \cdot \mathbf{E} = \mathbf{J} \cdot \mathbf{J}\frac{1}{\sigma} = J^2\rho where \rho = 1/\sigma is the
resistivity. This directly resembles the "I^2R" term of the macroscopic form. In the harmonic case, where all field quantities vary with the angular frequency \omega as e^{-\mathrm{i} \omega t}, complex valued
phasors \hat\mathbf{J} and \hat\mathbf{E} are usually introduced for the current density and the electric field intensity, respectively. The Joule heating then reads \frac{\mathrm{d}P}{\mathrm{d}V} = \frac{1}{2}\hat\mathbf{J} \cdot \hat\mathbf{E}^* = \frac{1}{2}\hat\mathbf{J} \cdot \hat\mathbf{J}^*/\sigma = \frac{1}{2}J^2\rho, where \bullet^* denotes the
complex conjugate. == Electricity transmission ==