text: 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.
  Here,  is the current density, and  is the electric field. For a material with a conductivity ,  and therefore
  where  is the resistivity.  This directly resembles the "" term of the macroscopic form.
  In the harmonic case, where all field quantities vary with the angular frequency  as , complex valued phasors  and  are usually introduced for the current density and the electric field intensity, respectively. The Joule heating then reads
  where  denotes the complex conjugate.