In the harmonic case, where all field quantities vary with the angular frequency ω ω as e − − i ω ω t , complex valued phasors J ^ ^ and E ^ ^ are usually introduced for the current density and the electric field intensity, respectively. The Joule heating then reads d P d V = 1 2 J ^ ^ ⋅ ⋅ E ^ ^ ∗ ∗ = 1 2 J ^ ^ ⋅ ⋅ J ^ ^ ∗ ∗ / σ σ = 1 2 J 2 ρ ρ , where ∙ ∙ ∗ ∗ denotes the complex conjugate.