Polarization, magnetization, bound and free sources, and the fields D and H

Let the laboratory region $U\cong I\times \mathrm{\Omega }$, its frame basis $(e_1,e_2,e_3)$, its units, its fields $E,B$ and its source densities $\rho ,J$ be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values, The electric and magnetic fields as fixed-time spatial vector fields and Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, and let the averaging convention be that of Declared averaging kernel, macroscopic scale assumptions and averaged fields, in whose sense the objects below are macroscopic. Any convolution used to instantiate that reading is taken on its declared exact interior-footprint domain, or under a declared locally integrable compatible extension covering every used footprint. Its every-time use requires every-time locally integrable spatial slices; joint spacetime local integrability alone supplies only the almost-every-time branch. The joint $C^2$ regularity specified below is a separate model hypothesis, not a consequence of spatial smoothing.

(a) [Dipole moments, polarization and magnetization] Fix the following moment convention in the frame basis. For a supplied closed rectangular box $K\subset {\mathbb{R}}^3$, a reference position $r_0\in {\mathbb{R}}^3$, a continuous charge density $f:{\mathbb{R}}^3\to \mathbb{R}$ supported in the interior of $K$, and a stationary current density $g:{\mathbb{R}}^3\to {\mathbb{R}}^3$ of class $C^1$ supported in the interior of $K$ with $\mathrm{div}g=0$, define the electric and magnetic dipole moments about $r_0$ by

$p[f;r_0]:={\int }_K(y-r_0)f(y)dy,m[g;r_0]:=\frac{1}{2}{\int }_K(y-r_0)\times g(y)dy\mathrm{.}$

Here $f$ carries $\mathrm{C}{\mathrm{m}}^{-3}$, $g$ carries $\mathrm{A}{\mathrm{m}}^{-2}$, $p$ carries $\mathrm{C}\mathrm{m}$ and $m$ carries $\mathrm{A}{\mathrm{m}}^2$; the cross product is that of The cross product in ${\mathbb{R}}^3$. Integrals are componentwise Riemann integrals of The Riemann integral of a bounded function over a bounded Jordan measurable set, finite by A continuous real function on a compact Jordan measurable set is Riemann integrable over that set. These specify the normalization of source moments, with $r_0$ retained as part of the data; they impose no microscopic representation on the material below.

The material model specifies two maps $P,M:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ of class $C^2$ in the sense of $C^k$ Euclidean maps and diffeomorphisms. To say that they are electric and magnetic dipole-moment densities means that, for every compact Jordan-measurable volume $V\subset \mathrm{\Omega }$ and $t\in I$, the model assigns the material moment contents

$\mathcal{P}(V,t):={\int }_{V}P(y,t)dy,\mathcal{M}(V,t):={\int }_{V}M(y,t)dy,$

in $\mathrm{C}\mathrm{m}$ and $\mathrm{A}{\mathrm{m}}^2$, respectively, using the same finite componentwise integrals. Thus $P$ carries $\mathrm{C}{\mathrm{m}}^{-2}$ and $M$ carries $\mathrm{A}{\mathrm{m}}^{-1}$. These contents record the model's assigned material moments; they are not defined by applying the source-moment formulas to the total or bound sources restricted to $V$. Matching them to microscopic charge/current moments requires an additional material identification, not supplied here. The maps are data of the chosen material model, and no relation between $(P,M)$ and $E,B$ is asserted here.

(b) [Bound sources] Write $\mathrm{div}$ and $\mathrm{curl}$ for the divergence and curl of Divergence and curl of a $C^1$ vector field taken at fixed $t$, and ${\mathrm{\partial }}_t$ for the partial derivative of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ at fixed position, and define the bound charge density and bound current density by

${\rho }_b:=-\mathrm{div}P,J_b:={\mathrm{\partial }}_{t}P+\mathrm{curl}M\mathrm{.}$

Since $P,M$ are jointly $C^2$, ${\rho }_b$ and $J_b$ are continuous, indeed $C^1$, by the componentwise sum and product rules of Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$; ${\rho }_b$ carries $\mathrm{C}{\mathrm{m}}^{-3}$ and $J_b$ carries $\mathrm{A}{\mathrm{m}}^{-2}$.

(c) [Free sources] Define ${\rho }_f:=\rho -{\rho }_b$ and $J_f:=J-J_b$, in the same units as $\rho$ and $J$; these are the free charge and free current densities of the model. The decomposition $\rho ={\rho }_f+{\rho }_b$, $J=J_f+J_b$ is part of the chosen material model: the same total pair $(\rho ,J)$ with a different pair $(P,M)$ is a different model.

(d) [Displacement and magnetic intensity] Define the electric displacement and the magnetic intensity by

$D:={\epsilon }_{0}E+P,H:={\mu }_0^{-1}B-M,$

where ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ and ${\mu }_0,c>0$ are the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values; $D$ carries $\mathrm{C}{\mathrm{m}}^{-2}$ and $H$ carries $\mathrm{A}{\mathrm{m}}^{-1}$.

(e) [Status] This item fixes objects and the bound/free convention only. No field equation (the item pthm-em-macroscopic-maxwell), no constitutive relation (the item post-em-material-closure), no conservation of the bound pair (the item pthm-em-macroscopic-maxwell, clause (a)) and no identification of $P,M$ with microscopic dipole sums is asserted here.