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Local and fixed-volume electromagnetic energy balance

Physical scope

The conditional energy balance of the classical vacuum electromagnetic field: the local identity linking the time derivative of the energy density and the divergence of the Poynting vector to the work density J.E, and its fixed-volume form with the boundary energy flux retained, both deduced from the smooth branch of the adopted vacuum Maxwell equations and the adopted continuum coupling. The statement is conditional on those adopted postulates, excludes the weak branch, singular source support and non-integrable products, and asserts no conservation without the boundary term. No empirical claim is made.

Statement

Assume the smooth branch of the vacuum Maxwell postulate The microscopic vacuum Maxwell equations: smooth and weak branches on the laboratory region $U ≅ I \times Ω$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields: the fields $E, B : Ω \times I \to R^{3}$ are jointly $C^{1}$ of The electric and magnetic fields as fixed-time spatial vector fields and satisfy, with the continuous source fields $ρ, J$ of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current and the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values,

$div E = ρ / ε_{0}, div B = 0, curl E = - \partial_{t} B, curl B = μ_{0} J + c^{- 2} \partial_{t} E$

pointwise on $Ω \times I$ , with $div$ and $curl$ of Divergence and curl of a $C^{1}$ vector field applied at fixed times and $\partial_{t}$ the partial derivative at fixed position of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ . Let $u$ and $S$ be the energy density and Poynting vector of Vacuum electromagnetic energy density and energy flux (the Poynting vector). Then:

(a) [Local identity] On $Ω \times I$ ,

$\partial_{t} u + div S = - J \cdot E,$

where $div S$ is taken at fixed time and $J \cdot E$ is the Euclidean inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ .

(b) [Fixed-volume identity] Let $Ω^{'} \subseteq Ω$ be open with $\overline{Ω^{'}} \subseteq Ω$ carrying the finite piecewise $C^{1}$ presentation of Divergence for finite piecewise C1 presentations, let $t_{0} < t_{1}$ with $(t_{0}, t_{1}) \subseteq I$ , and let $n$ be the outward unit normal of $\partial Ω^{'}$ . Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the hypothesis under which that divergence theorem is stated. Then for every $t \in (t_{0}, t_{1})$ ,

$\frac{d}{d t} \int_{Ω^{'}} u (x, t) d x = - \int_{Ω^{'}} J \cdot E d x - \oint_{\partial Ω^{'}} S \cdot n d S,$

the closed-surface integral being the finite face sum of that divergence theorem and the derivative being the derivative of the one-variable function $t \mapsto \int_{Ω^{'}} u (x, t) d x$ on $(t_{0}, t_{1})$ .

(c) [Interpretation and scope] Under clause (ii) of The Lorentz coupling: external force on a test charge and the continuum force and power densities, $J \cdot E$ is the power density supplied to the matter model of that coupling, so (a) and (b) read as a local, respectively fixed-volume, energy balance of the field and that matter model. Nothing is claimed for a prescribed current without such a matter model, for the weak branch of the Maxwell postulate, for singular sources, or at field points where the products defining $u$ or $S$ are not locally integrable; the fixed-volume identity retains both the work term and the boundary flux. The field-energy rate can vanish by cancellation of these two terms; vanishing boundary flux alone does not eliminate exchange with matter. A conservation claim for field plus matter additionally requires an explicit matter balance. This is a conditional consequence of the adopted postulates, not an empirical claim.

Facts & Assumptions

Given: The laboratory region $U ≅ I \times Ω$ ; the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches with the jointly $C^{1}$ fields of The electric and magnetic fields as fixed-time spatial vector fields and the continuous sources of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current; the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values; and the energy density $u$ and Poynting vector $S$ of Vacuum electromagnetic energy density and energy flux (the Poynting vector). In clause (b) also ${A C}_{ω}$ , the admissible $Ω^{'}$ and the interval $(t_{0}, t_{1}) \subseteq I$ .

[F1]

The four smooth-branch equations hold pointwise on $Ω \times I$ as displayed in the Statement, with $ε_{0} = (μ_{0} c^{2})^{- 1}$ , $c > 0$ and $μ_{0} > 0$ (The microscopic vacuum Maxwell equations: smooth and weak branches, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F2]

The fields are jointly $C^{1}$ on the open set $Ω \times I$ ; at fixed time they are $C^{1}$ spatial fields of The electric and magnetic fields as fixed-time spatial vector fields, and for fixed position the maps $s \mapsto E_{i} (x, s)$ and $s \mapsto B_{i} (x, s)$ are defined on the open interval $I$ and have derivative $\partial_{t} E_{i} (x, s)$ , respectively $\partial_{t} B_{i} (x, s)$ , by the definition of the partial derivative (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ). For differentiable real functions on an interval, sums and scalar multiples have the expected derivatives and the product rule $(f g)^{'} = f^{'} g + f g^{'}$ holds (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F3]

The energy density and Poynting vector are $u = \frac{ε_{0}}{2} ⟨ E, E ⟩ + \frac{1}{2 μ_{0}} ⟨ B, B ⟩$ and $S = μ_{0}^{- 1} E \times B$ , of class $C^{1}$ on $Ω \times I$ ; the Euclidean inner product $⟨ \cdot, \cdot ⟩$ is bilinear and symmetric; and $div$ of a $C^{1}$ field is the sum of the coordinate partial derivatives (Vacuum electromagnetic energy density and energy flux (the Poynting vector), The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Divergence and curl of a $C^{1}$ vector field).

[F4]

For $C^{1}$ fields $F, G$ on an open set of $R^{3}$ , $div (F \times G) = ⟨ curl F, G ⟩ - ⟨ F, curl G ⟩$ pointwise (The divergence and curl of a cross product).

[F5]

Fixed-volume tools: if $K \subseteq R^{n}$ is compact and measurable with $λ_{n} (K) < \infty$ and $f, \partial_{t} f$ are continuous on an open neighbourhood of $K \times [a, b]$ , then $t \mapsto \int_{K} f (x, t) d λ_{n} (x)$ has derivative $\int_{K} \partial_{t} f (x, t) d λ_{n} (x)$ at every interior $t \in (a, b)$ (Differentiation under the integral sign over a compact measurable set and over a compact $C^{1}$ surface); if $Ω$ carries the finite piecewise $C^{1}$ presentation and $F \in C^{1} (\overline{Ω}; R^{n})$ , then $\int_{Ω} div F = \sum_{j} \int_{S_{j}} F \cdot ν_{j} d S$ with faces counted once off the edge set, under ${A C}_{ω}$ (Divergence for finite piecewise C1 presentations).

[F6]

Boundary nullity: the graph of a continuous function on a closed nondegenerate rectangle in $R^{m}$ has content zero in $R^{m + 1}$ (The graph of a continuous function on a closed nondegenerate rectangle in $R^{m}$ has content zero in $R^{m + 1}$ ), and a nonnegative measurable integral over a Lebesgue-null set vanishes (A nonnegative integral over a null set vanishes); in a finite piecewise $C^{1}$ presentation each face is a compact Borel subset of a regular $C^{1}$ hypersurface patch. Proof step 4.1 of Divergence for finite piecewise C1 presentations establishes ambient Lebesgue-nullity of each such face by graph-coordinate Fubini and rigid change of variables. The finitely many faces cover $\partial Ω^{'}$ , so $λ_{3} (\partial Ω^{'}) \leq \sum_{j} λ_{3} (S_{j}) = 0$ .

[F7]

The source and coupling conventions: $ρ, J$ are the continuous frame densities of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current on $Ω \times I$ , so $J \cdot E$ is continuous; and clause (ii) of The Lorentz coupling: external force on a test charge and the continuum force and power densities adopts $J \cdot E$ as the power density of an explicitly stated matter model on the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields.

Proof

technique · direct; differentiate the quadratic energy density along time, expand the divergence of the Poynting vector by the cross-product identity, cancel the magnetic terms with the two curl equations, and integrate the local identity over an admissible fixed volume with the divergence theorem and compact-parameter differentiation

1.1givenF2F3algebra

Time derivative of the energy density: fix $(x, t) \in Ω \times I$ . The maps $s \mapsto E_{i} (x, s)$ and $s \mapsto B_{i} (x, s)$ are differentiable at $t$ with derivatives $\partial_{t} E_{i} (x, t)$ and $\partial_{t} B_{i} (x, t)$ by [F2], so the product rule gives $\partial_{t} ⟨ E, E ⟩ = 2 \sum_{i} E_{i} \partial_{t} E_{i} = 2 ⟨ E, \partial_{t} E ⟩$ and likewise for $B$ ; hence $\partial_{t} u = ε_{0} ⟨ E, \partial_{t} E ⟩ + μ_{0}^{- 1} ⟨ B, \partial_{t} B ⟩$ at $(x, t)$ , and the right-hand side is continuous in $(x, t)$ .

1.2givenF3F4algebra

Divergence of the Poynting vector: at each fixed time the fields $E (\cdot, t), B (\cdot, t)$ are $C^{1}$ on the open set $Ω$ , so [F4] applies and $div S = μ_{0}^{- 1} div (E \times B) = μ_{0}^{- 1} (⟨ curl E, B ⟩ - ⟨ E, curl B ⟩)$ pointwise on $Ω \times I$ .

1.3givenF3F5F6algebra

Fixed-volume setup: let $Ω^{'}$ , $t_{0} < t_{1}$ be as in clause (b) and fix $t \in (t_{0}, t_{1})$ ; choose reals $a, b$ with $t_{0} < a < t < b < t_{1}$ . The set $\overline{Ω^{'}}$ is compact and contained in the open $Ω$ , and $u$ and its partial derivative $\partial_{t} u$ are continuous on a neighbourhood of $\overline{Ω^{'}} \times [a, b]$ by [F3]; the boundary $\partial Ω^{'}$ is Lebesgue-null by the finite piecewise $C^{1}$ presentation and [F6], so the integrals of $u$ and of $\partial_{t} u$ over $Ω^{'}$ and over $\overline{Ω^{'}}$ agree, the signed case following by applying the vanishing statement to the positive and negative parts. By [F5] the function $t \mapsto \int_{Ω^{'}} u (x, t) d x$ is therefore differentiable at $t$ with derivative $\int_{Ω^{'}} \partial_{t} u (x, t) d x$ .

2.1step 1.1step 1.2F1algebra

Local identity: substitute [F1] into steps 1.1 and 1.2. Since $curl E = - \partial_{t} B$ and $⟨ \cdot, \cdot ⟩$ is symmetric, $μ_{0}^{- 1} ⟨ B, \partial_{t} B ⟩ + μ_{0}^{- 1} ⟨ curl E, B ⟩ = μ_{0}^{- 1} ⟨ B, \partial_{t} B ⟩ - μ_{0}^{- 1} ⟨ \partial_{t} B, B ⟩ = 0$ ; hence $\partial_{t} u + div S = ε_{0} ⟨ E, \partial_{t} E ⟩ - μ_{0}^{- 1} ⟨ E, μ_{0} J + c^{- 2} \partial_{t} E ⟩ = - J \cdot E + (ε_{0} - (μ_{0} c^{2})^{- 1}) ⟨ E, \partial_{t} E ⟩ = - J \cdot E$ , which is clause (a).

3.1step 1.2step 1.3step 2.1F5algebra

Fixed-volume identity: by step 1.3 and clause (a) just proved, $\frac{d}{d t} \int_{Ω^{'}} u d x = \int_{Ω^{'}} \partial_{t} u d x = - \int_{Ω^{'}} J \cdot E d x - \int_{Ω^{'}} div S (\cdot, t) d x$ . The field $S (\cdot, t)$ is $C^{1}$ on a neighbourhood of $\overline{Ω^{'}}$ by [F3] and step 1.1, so the divergence theorem of [F5] applies to it and $\int_{Ω^{'}} div S (\cdot, t) d x = \oint_{\partial Ω^{'}} S \cdot n d S$ as a finite face sum; substituting gives clause (b), under the stated ${A C}_{ω}$ .

4.1step 2.1step 3.1F1F3F7∎

Clause (c) records the interpretation of $J \cdot E$ through clause (ii) of [F7], the exclusions already visible in [F1] and [F3] (weak branch, singular support, non-integrable products), and the retention of the boundary term in clause (b); the argument used only the two adopted postulates, the definitions of the field quantities, the cross-product divergence identity and the integration interface, and made no empirical claim.

Remarks

Where the two canceling terms come from. The magnetic pair $μ_{0}^{- 1} ⟨ B, \partial_{t} B ⟩$ and $μ_{0}^{- 1} ⟨ curl E, B ⟩$ cancels exactly because Faraday's law carries the minus sign; the electric residual vanishes because $ε_{0} μ_{0} c^{2} = 1$ is the defining relation of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Neither cancellation assumes $\partial_{t} u$ or $div S$ to be separately conserved.
The boundary term is not optional. On a fixed volume the balance reads $d E_{Ω^{'}} / d t = - \int_{Ω^{'}} J \cdot E - \oint_{\partial Ω^{'}} S \cdot n$ ; dropping the surface term requires either a vanishing normal flux or its inclusion in an enlarged system, which is exactly the qualification recorded in clause (c).
Regularity. The argument uses only first derivatives of $E$ and $B$ ; the additional smoothness demanded by other branches of the framework is not needed for this balance, and no assumption is made at points of singular source support.

Local and fixed-volume electromagnetic energy balance

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources