Vacuum electromagnetic energy density and energy flux (the Poynting vector)

Let $U\cong I\times \mathrm{\Omega }$ be the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, so that $I\subseteq \mathbb{R}$ is a nonempty open interval, $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ is nonempty open, $x=(x^1,x^2,x^3)$ is the position in the spatial chart, $(e_1,e_2,e_3)$ is the frame basis of the spatial slice, and the spatial inner product is the Euclidean form $\delta$; let the units and the constants $c>0$, ${\mu }_0>0$ and ${\epsilon }_0:=({\mu }_0{c}^2{)}^{-1}$ be those of Framework-local SI units, the vacuum constants and the exact/measured status of their values; let $k\ge 1$, and let $E,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ be an electric and a magnetic field of class $C^k$ of The electric and magnetic fields as fixed-time spatial vector fields. Write $⟨\cdot ,\cdot ⟩$ and $\mathrm{\mid }\cdot \mathrm{\mid }$ for the Euclidean inner product and norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and $\times$ for the cross product of The cross product in ${\mathbb{R}}^3$.

(a) [Energy density and flux] The electromagnetic energy density $u:\mathrm{\Omega }\times I\to \mathbb{R}$ and the energy flux density (the Poynting vector) $S:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ are defined by

$u(x,t):=\frac{{\epsilon }_0}{2}⟨E(x,t),E(x,t)⟩+\frac{1}{2{\mu }_0}⟨B(x,t),B(x,t)⟩,S(x,t):=\frac{1}{{\mu }_0}E(x,t)\times B(x,t)\mathrm{.}$

At fixed $t\in I$ the map $u(\cdot ,t):\mathrm{\Omega }\to \mathbb{R}$ is a real-valued function on $\mathrm{\Omega }$ and $S(\cdot ,t):\mathrm{\Omega }\to {\mathbb{R}}^3$ is a spatial vector field on $\mathrm{\Omega }$, read in the basis $(e_1,e_2,e_3)$. The energy density carries $\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$ and the energy flux density carries $\mathrm{W}{\mathrm{m}}^{-2}=\mathrm{k}\mathrm{g}{\mathrm{s}}^{-3}$.

(b) [Regularity] If $E,B$ are of class $C^k$, then $u$ and $S$ are of class $C^k$: they are the components of a polynomial expression in the components of $E,B$, and the verification below exhibits the composite map. In particular $u$ and $S$ are continuous, and the definitions use no derivative of the fields.

(c) [Pointwise reading, quadratic character] The definitions use only the values of $E$ and $B$ at the same point $(x,t)$, and they are quadratic in the fields: for every real $\lambda$ one has $u(\lambda E,\lambda B)={\lambda }^{2}u(E,B)$ and $S(\lambda E,\lambda B)={\lambda }^{2}S(E,B)$, and $u\ge 0$. Consequently decompositions of the fields do not make $u$ or $S$ additive; cross terms generally contribute. No convention beyond the fixed laboratory frame enters the definitions.

(d) [Products and singular support] If $E,B$ are only locally integrable (the weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches), the products $E_i{E}_j$, $B_i{B}_j$ and the components of $E\times B$ need not be locally integrable, and no product of distributions is defined. The displayed formulas then define $u$ and $S$ pointwise almost everywhere, and the framework uses them only where the local integrability of the products, respectively of the component being integrated, is separately established. No value of $u$ or $S$ is assigned at a point of singular source support.

(e) [Status] This item defines quantities only. The interpretation of $u$ as an energy density and of $S$ as an energy flux density, and every balance identity involving them, is the content of Local and fixed-volume electromagnetic energy balance and of the other balance items of this page; no identification of $(u,S)$ with components of a covariant tensor is made here, and the relation $S=c^{2}g$ to the momentum density $g$ of Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum is a consequence of those definitions, not an identification made at this point.

• Non-additivity, explicitly. The quadratic character of clause (c) is not a technicality: at a point where $E=e_1$, $E^{\mathrm{\prime }}=e_1$ and $B=0$ one has $u(E+E^{\mathrm{\prime }},0)=2{\epsilon }_0$ but $u(E,0)+u(E^{\mathrm{\prime }},0)={\epsilon }_0$, because $u(e_1,0)={\epsilon }_0\mathrm{/}2$. Every balance identity of this page is therefore stated for the total fields, never for a sum of separately balanced parts without the cross terms.