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Energy balance and Ohmic dissipation for the constant nondispersive linear model

Physical scope

The local and fixed-volume energy balance d_t u + div(E x H) = -J_f.E with u=(E.D+B.H)/2 and the Ohmic dissipation reading E.sigma E>=0 for the constant symmetric nondispersive linear model with symmetric positive-semidefinite sigma, under AC_omega for the fixed-volume form. Restricted to this model: no stored-energy claim for dispersive, nonlocal, nonlinear or hysteretic media; boundary term retained; no conservation assertion and no empirical claim.

Statement

Assume the elementary inertial-rest branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch: constant symmetric positive-definite matrices $ε, μ \in R^{3 \times 3}$ (isotropic scalar case included) with $D = ε E$ and $B = μ H$ , and the conduction relation $J_{f} = σ E + J_{e x t}$ with $σ$ symmetric positive semidefinite and a declared external free current $J_{e x t}$ . Assume the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches with the regularity hypotheses of The macroscopic Maxwell equations in terms of D, H and the free sources, so that its macroscopic equations hold on $Ω \times I$ . Define the stored-energy density and the energy-flux vector

$u : = \frac{1}{2} (E \cdot D + B \cdot H), S : = E \times H,$

with the Euclidean inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and the cross product of The cross product in $R^{3}$ ; $u$ carries $J m^{- 3} = k g m^{- 1} s^{- 2}$ and $S$ carries $W m^{- 2}$ . Then:

(a) [Local balance and positivity] On $Ω \times I$ ,

$\partial_{t} u + div S = - J_{f} \cdot E .$

Moreover $u = \frac{1}{2} (E \cdot ε E + H \cdot μ H) \geq 0$ , with $u (x, t) > 0$ whenever $(E (x, t), H (x, t)) \neq (0, 0)$ .

(b) [Dissipation reading] With $J_{f} = σ E + J_{e x t}$ and $σ$ symmetric positive semidefinite, $E \cdot σ E \geq 0$ and the balance reads

$\partial_{t} u + div S + E \cdot σ E = - J_{e x t} \cdot E;$

the nonnegative term $E \cdot σ E$ is the Ohmic dissipation density of the model.

(c) [Fixed-volume balance] 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 $(t_{0}, t_{1}) \subseteq I$ , assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) as in that divergence theorem, and let $n$ be the outward unit normal of $\partial Ω^{'}$ . Then for every $t \in (t_{0}, t_{1})$ ,

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

(d) [Scope] The density $u = \frac{1}{2} (E \cdot D + B \cdot H)$ is a stored-energy density only for this constant symmetric nondispersive model; for general dispersive, nonlocal, nonlinear or hysteretic response no instantaneous stored-energy density is defined or claimed by this item, and the general dispersive work balance is not supplied. The identities hold only where the fields and sources have the stated regularity, the fixed-volume form keeps its boundary term, and conservation of $\int_{Ω'} u$ requires the sum of outward flux and volume work on the right of (c) to vanish, not merely zero boundary flux. No empirical claim is made.

Facts & Assumptions

Given: The laboratory region and units of the Statement, the smooth-branch data of the declared constant model with symmetric positive-definite $ε, μ$ and symmetric positive-semidefinite $σ$ of [A2], and the regularity of [A1]; the operators are the divergence, curl and partial derivatives of the named definitions, with the Euclidean inner and cross products of the Statement.

[A1]

Under the smooth branch of the vacuum postulate with the regularity hypotheses of the macroscopic theorem, the macroscopic equations hold on $Ω \times I$ : $div D = ρ_{f}$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl H = J_{f} + \partial_{t} D$ (The macroscopic Maxwell equations in terms of D, H and the free sources, clauses (a)-(b); The microscopic vacuum Maxwell equations: smooth and weak branches, smooth branch).

[A2]

The elementary inertial-rest branch: $D = ε E$ , $B = μ H$ with constant symmetric positive-definite matrices $ε, μ$ , and $J_{f} = σ E + J_{e x t}$ with $σ$ symmetric positive semidefinite (Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch, clause (b)).

[L1]

$div (E \times H) = H \cdot curl E - E \cdot curl H$ for $C^{1}$ fields (The divergence and curl of a cross product).

[L2]

For a symmetric real matrix $ε$ , $E \cdot ε E$ is differentiable in $t$ with $\frac{d}{d t} (E \cdot ε E) = 2 \partial_{t} E \cdot ε E$ when $E$ is $C^{1}$ , by the product rule (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 $g (c) \neq 0$ ); and $E \cdot ε E \geq 0$ with equality only for $E = 0$ when $ε$ is positive definite, and likewise for $μ$ and $H$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[L3]

If $K \subseteq R^{n}$ is compact, measurable and of finite Lebesgue measure, $f$ and $\partial_{t} f$ are continuous on a neighbourhood of $K \times [a, b]$ , then $\frac{d}{d t} \int_{K} f = \int_{K} \partial_{t} f$ on $[a, b]$ (Differentiation under the integral sign over a compact measurable set and over a compact $C^{1}$ surface).

[L4]

For a bounded piecewise $C^{1}$ domain, the divergence theorem holds: $\int_{Ω^{'}} div F d x = \oint_{\partial Ω^{'}} F \cdot n d S$ for the outward unit normal, under the stated finite piecewise $C^{1}$ presentation and Countable Choice (Divergence for finite piecewise C1 presentations).

[L5]

The image of a $C^{1}$ map from a lower-dimensional smooth manifold to $R^{3}$ is Lebesgue-null under Countable Choice (The image of a lower-dimensional $C^{1}$ manifold is null), and finite unions and subsets of null sets are null (Subsets and countable unions of null subsets of $R^{m}$ are null).

Proof

technique · dot the macroscopic curl equations with the complementary fields and use the cross-product divergence identity, then integrate over a fixed volume

1.1A1given

By [A1] the two curl equations $E \cdot curl H = E \cdot J_{f} + E \cdot \partial_{t} D$ and $H \cdot curl E = - H \cdot \partial_{t} B$ hold on $Ω \times I$ after taking the Euclidean inner product with the complementary field.

1.2A2L2

The identity $u = \frac{1}{2} (E \cdot ε E + H \cdot μ H)$ follows from [A2], and both quadratic forms are nonnegative by [L2], with equality in both simultaneously only when $E = H = 0$ ; hence $u \geq 0$ , and $u (x, t) > 0$ whenever $(E (x, t), H (x, t)) \neq (0, 0)$ .

1.3A2L2

By [A2], $J_{f} \cdot E = E \cdot σ E + J_{e x t} \cdot E$ , and $E \cdot σ E \geq 0$ because $σ$ is symmetric positive semidefinite; the term is the Ohmic dissipation density of the model.

2.1step 1.1A2L1L2

By [L1], $div S = div (E \times H) = H \cdot curl E - E \cdot curl H$ , so the two scalar identities of step 1.1 give $div S = - (E \cdot J_{f} + E \cdot \partial_{t} D + H \cdot \partial_{t} B)$ ; differentiating $u$ in time and using the symmetry of $ε$ and $μ$ with the product rule of [L2] gives $\partial_{t} u = \frac{1}{2} \partial_{t} (E \cdot ε E) + \frac{1}{2} \partial_{t} (H \cdot μ H) = \partial_{t} E \cdot ε E + \partial_{t} H \cdot μ H = E \cdot \partial_{t} D + H \cdot \partial_{t} B$ by [A2]; adding the two identities yields $\partial_{t} u + div S = - J_{f} \cdot E$ on $Ω \times I$ , proving (a).

3.1step 1.3step 2.1L3L4L5∎

For (c), fix $t \in (t_{0}, t_{1})$ . Each face in the finite presentation of $\partial Ω'$ lies in a regular $C^{1}$ parametrized patch and is volume-null by [L5]. The finite union $\partial Ω'$ is therefore volume-null. Hence the compact finite-measure set $K = \overline{Ω'}$ and $Ω'$ have the same integrals for the continuous functions used here. Apply [L3] to $K$ and $f = u$ : $\frac{d}{d t} \int_{Ω^{'}} u d x = \int_{Ω^{'}} \partial_{t} u d x$ , where the regularity hypotheses on the fields supply the continuous $\partial_{t} u$ on a neighbourhood of $\overline{Ω^{'}} \times [a, b]$ for a compact interval $[a, b] \subset I$ containing $t$ in its interior; substituting the local balance of step 2.1 and applying the divergence theorem [L4] to $F = S$ gives $\frac{d}{d t} \int_{Ω^{'}} u = - \int_{Ω^{'}} J_{f} \cdot E - \int_{Ω^{'}} div S = - \int_{Ω^{'}} J_{f} \cdot E - \oint_{\partial Ω^{'}} S \cdot n d S$ , under the ${A C}_{ω}$ hypothesis carried by [L4]. Thus (a)-(c) hold; the reading (b) is step 1.3 substituted into the balance, and (d) is the scope: $u$ is a stored-energy density only for this constant symmetric nondispersive model, no instantaneous stored energy is claimed for dispersive, nonlocal, nonlinear or hysteretic response, the boundary term is retained, and no conservation of $\int_{Ω^{'}} u$ or empirical statement is asserted.

Energy balance and Ohmic dissipation for the constant nondispersive linear model

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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