Electromagnetic radiation

Definition

Let $\rho$ and $J$ be twice continuously differentiable charge and current densities with $\partial_t \rho + \nabla \cdot J = 0$. Assume that they vanish outside a fixed ball around the origin of $\R^3$. Let $E$, $B$ be the retarded solution of Maxwell's equations with these sources, given by Jefimenko's equations. Let $S = E \times B / \mu_0$ be the Poynting vector and let $c$ be the speed of light. For $R > 0$ let $S_R$ be the sphere of radius $R$ about the origin, with outward unit normal $n$. The radiated power of the sources at time $t$ is $$P(t) = \lim_{R \to \infty} \oint_{S_R} S(x, t + R/c) \cdot n \, dA(x),$$ when the limit exists. The sources radiate at time $t$ if $P(t) \neq 0$. The radiation field is the part of $(E, B)$ that decays like $1/R$ at distance $R$; this part alone determines $P$.