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Energy momentum angular balance — Examples

1 · Prerequisites

2 · Summary

Objective. This companion carries one worked ledger of the balance identities on the companion A page em-energy-momentum-and-angular-balance: the exact energy accounting of an ideal source-free plane wave in a finite box.

A crossed plane wave. The declared configuration $E = E_{0} \cos (k z - ω t) e_{1}$ , $B = (E_{0} / c) \cos (k z - ω t) e_{2}$ with $ω = c k > 0$ and $E_{0} > 0$ is verified to be a smooth solution of the smooth Maxwell branch with $ρ = 0$ and $J = 0$ (Energy flow in crossed external fields with finite-volume accounting); its energy density and flux are $u = ε_{0} E_{0}^{2} \cos^{2} (k z - ω t)$ and $S = ε_{0} c E_{0}^{2} \cos^{2} (k z - ω t) e_{3}$ , and the local identity $\partial_{t} u + \partial_{z} S_{3} = 0$ holds exactly, with no period averaging.

Finite-box ledger. For the box $B = (0, L_{1}) \times (0, L_{2}) \times (a, b)$ the example computes $\frac{d}{d t} \int_{B} u d V = L_{1} L_{2} (S_{3} (a, t) - S_{3} (b, t))$ and the same value as the negative of the closed-surface flux: the four faces with normals $\pm e_{1}, \pm e_{2}$ carry zero flux because $S$ is parallel to $e_{3}$ , the upper face contributes $L_{1} L_{2} S_{3} (b, t)$ and the lower face $- L_{1} L_{2} S_{3} (a, t)$ in the outward orientation, hence the field-energy change receives $- L_{1} L_{2} S_{3} (b, t)$ and $+ L_{1} L_{2} S_{3} (a, t)$ respectively. The example is an ideal wave on all of $R^{3} \times R$ ; it asserts no empirical realization and no statement about the (divergent) total energy over all of space.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-adaptedOpen item page →

Energy flow in crossed external fields with finite-volume accounting

Physical scope

An ideal source-free plane-wave configuration of the classical vacuum framework on all of R^3 x R: its verification against the four smooth-branch Maxwell equations, its exact energy density and Poynting vector, the pointwise energy balance, and the finite-box energy accounting with outward face fluxes. The configuration is an exact solution of the adopted equations with no source and no empirical realization; no finite whole-space energy or empirical realization is claimed.

Example

Let $k > 0$ carry $m^{- 1}$ , $ω : = c k$ carry $s^{- 1}$ , and $E_{0} > 0$ carry $V m^{- 1}$ . On $Ω \times I = R^{3} \times R$ define, with $θ (x, t) : = k z - ω t$ in the frame $(e_{1}, e_{2}, e_{3})$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and the encoding of Complex plane-wave fields: real parts of complex exponentials,

$E (x, t) : = E_{0} \cos (k z - ω t) e_{1} = Re [E_{0} e_{1} e^{i (k z - ω t)}], B (x, t) : = \frac{E_{0}}{c} \cos (k z - ω t) e_{2} = Re [\frac{E_{0}}{c} e_{2} e^{i (k z - ω t)}] .$

The electric and magnetic fields are perpendicular to one another and to the direction $e_{3}$ ("crossed" fields); $E$ oscillates along $e_{1}$ and $B$ along $e_{2}$ . Then:

(a) [Field data] $E$ and $B$ are smooth fields of The electric and magnetic fields as fixed-time spatial vector fields on $R^{3} \times R$ , and they satisfy the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches with $ρ = 0$ and $J = 0$ .

(b) [Energy density and flux] With $u$ and $S$ of Vacuum electromagnetic energy density and energy flux (the Poynting vector),

$u (x, t) = ε_{0} E_{0}^{2} \cos^{2} (k z - ω t), S (x, t) = ε_{0} c E_{0}^{2} \cos^{2} (k z - ω t) e_{3},$

and the local balance $\partial_{t} u + \partial_{z} S_{3} = 0$ , equivalently $\partial_{t} u + div S = 0$ , holds exactly on $R^{3} \times R$ .

(c) [Finite-volume accounting] For $a < b$ , $L_{1}, L_{2} > 0$ and the box $B : = (0, L_{1}) \times (0, L_{2}) \times (a, b)$ with outward normal $n$ , for every $t$ ,

$\frac{d}{d t} \int_{B} u d V = L_{1} L_{2} (S_{3} (a, t) - S_{3} (b, t)) = - \oint_{\partial B} S \cdot n d S :$

Here $\int_{B} h d V$ for each continuous box integrand denotes its Riemann integral over the closed rectangle $\overline{B}$ ; the surface integral is the sum of the six outward-oriented Riemann patch fluxes. In particular, the four faces with normals $\pm e_{1}, \pm e_{2}$ carry zero flux because $S$ is parallel to $e_{3}$ , and the upper face $z = b$ has outward flux $L_{1} L_{2} S_{3} (b, t)$ and the lower face $z = a$ has outward flux $- L_{1} L_{2} S_{3} (a, t)$ . Their respective signed contributions to the field-energy change are $- L_{1} L_{2} S_{3} (b, t)$ and $L_{1} L_{2} S_{3} (a, t)$ .

(d) [Scope] The configuration is an ideal source-free plane wave; the finite-volume accounting of (c) is the fixed-volume energy balance of this declared solution in the sense of Local and fixed-volume electromagnetic energy balance, and the example asserts no finite whole-space energy and no empirical realization. $E_{0} > 0$ , $k > 0$ and $ω = c k$ are the declared parameters of the configuration.

Facts & Assumptions

Given: $k > 0$ , $ω = c k$ with the constant $c > 0$ of the framework, $E_{0} > 0$ , the phase $θ (x, t) = k z - ω t$ , the fields $E, B$ defined above, and the box $B = (0, L_{1}) \times (0, L_{2}) \times (a, b)$ with $a < b$ and $L_{1}, L_{2} > 0$ .

[F1]

The laboratory region, its chart coordinates $(t, x)$ , frame basis $(e_{1}, e_{2}, e_{3})$ and spatial Euclidean structure are those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields; the fields of The electric and magnetic fields as fixed-time spatial vector fields are maps $Ω \times I \to R^{3}$ of class $C^{k}$ , $E$ polar and $B$ axial, with no value asserted at singular source support.

[F2]

Encoding and its calculus: with $e : = E_{0} e_{1}$ , $b : = (E_{0} / c) e_{2}$ , $k : = k e_{3}$ , the definition of Complex plane-wave fields: real parts of complex exponentials gives $E_{j} = Re (e_{j} \exp (i θ))$ , $B_{j} = Re (b_{j} \exp (i θ))$ ; the definition asserts no regularity of its own beyond well-definedness, and identities obtained from the encoded expressions by sums, real scalar multiples, partial differentiation or integration may be projected by $Re$ term by term.

[F3]

Euler's formula: $\exp (i ϑ) = \cos ϑ + i \sin ϑ$ for every real $ϑ$ (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ ).

[F4]

The complex exponential is entire with $\exp^{'} = \exp$ (The complex exponential is entire and its complex derivative is itself); a complex differentiable map is real totally differentiable with derivative equal to multiplication by its complex derivative (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations).

[F5]

Total-derivative tools: a composite of totally differentiable maps is totally differentiable with the composite derivative (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ); total differentiability gives continuity (Total differentiability gives a local $O (∥ h ∥_{2})$ increment bound and therefore continuity); if all first partial derivatives exist near a point and are continuous there, the map is totally differentiable there with Jacobian matrix its total derivative (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative); the partial derivatives of a totally differentiable map are the total derivative applied to the coordinate vectors (A total derivative computes every directional derivative, and its matrix is the Jacobian); the partial derivative is the derivative of the corresponding line map (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[F6]

$C^{k}$ conventions: a map into $R^{q}$ is of class $C^{k}$ when every component is, and a map is smooth when it is $C^{k}$ for every $k$ ( $C^{k}$ Euclidean maps and diffeomorphisms); finite sums, products, scalar multiples and composites of $C^{k}$ Euclidean maps are $C^{k}$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition); sums, scalar multiples and products of differentiable functions of one variable obey the usual rules (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 $g (c) \neq 0$ ).

[F7]

Constants and units: $ε_{0} = (μ_{0} c^{2})^{- 1}$ with $c > 0$ and $μ_{0} > 0$ ; $E$ carries $V m^{- 1}$ , $B$ carries $T$ , $u$ carries $J m^{- 3}$ and $S$ carries $W m^{- 2}$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values, Vacuum electromagnetic energy density and energy flux (the Poynting vector)).

[F8]

Inner and cross products: $⟨ \cdot, \cdot ⟩$ is the Euclidean coordinate inner product (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ); the cross product is bilinear, alternating and right-handed in the positively oriented frame basis, so $e_{1} \times e_{2} = e_{3}$ , $e_{2} \times e_{3} = e_{1}$ and $e_{3} \times e_{1} = e_{2}$ (The cross product in $R^{3}$ ).

[F9]

The energy density and Poynting vector are $u = \frac{ε_{0}}{2} ⟨ E, E ⟩ + \frac{1}{2 μ_{0}} ⟨ B, B ⟩$ and $S = μ_{0}^{- 1} E \times B$ (Vacuum electromagnetic energy density and energy flux (the Poynting vector)); the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches consists of the four equations $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ for jointly $C^{1}$ fields.

[F10]

Rectangle integration: continuous functions on a compact Jordan set are Riemann integrable (A continuous real function on a compact Jordan measurable set is Riemann integrable over that set), and the Riemann integral on a nondegenerate closed rectangle is the common lower/upper Darboux value (Lower and upper Darboux sums over a grid partition in $R^{m}$ , The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ ). For a constant integrand $C$ , the one-cell lower and upper sums both equal $C$ times the rectangle's volume, the product of its side lengths (Axis-parallel rectangles in $R^{m}$ and their volume). The rectangle Fubini theorem reduces continuous integrands to ordinary iterated integrals (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections); the integral is linear (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ ). For a continuous $h (t, z)$ with continuous $\partial_{t} h$ on a compact time/space rectangle, $\frac{d}{d t} \int_{a}^{b} h (t, z) d z = \int_{a}^{b} \partial_{t} h (t, z) d z$ at interior times (Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral). Finally $\int_{a}^{b} G^{'} = G (b) - G (a)$ for $G$ differentiable on $[a, b]$ with integrable derivative (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F11]

A face patch with parametrization $φ$ has oriented flux $\int_{D} (S \circ φ) \cdot (φ_{u} \times φ_{v})$ (Unit normal fields, orientations, and flux through a regular surface patch); reversing the normal reverses this flux. The general fixed-volume theorem Local and fixed-volume electromagnetic energy balance(b) assumes ${A C}_{ω}$ for its finite piecewise $C^{1}$ surface-measure interface (Specified finite piecewise C1 boundary presentations). Here the six explicit rectangle patch fluxes are computed directly, without invoking that theorem or its choice assumption.

Verification

technique · direct; exhibit the fields through the complex encoding, differentiate the exponential by the chain rule, verify the four Maxwell equations and the exact energy identities by substitution, and compute the finite-box ledger with the fundamental theorem, parameter differentiation and Fubini

1.1givenF2F3algebra

Explicit real form: by [F3], $Re (e_{j} \exp (i θ)) = e_{j} \cos θ$ for the real amplitudes $e = E_{0} e_{1}$ and $b = (E_{0} / c) e_{2}$ , so $E = E_{0} \cos θ e_{1}$ and $B = (E_{0} / c) \cos θ e_{2}$ on $R^{3} \times R$ ; in particular $E ⊥ B ⊥ e_{3}$ .

1.2givenF4F5algebra

Derivatives of the exponential factor: put $G : = \exp (i θ)$ and $L : = i θ$ , a real-linear map $R^{4} \to C ≅ R^{2}$ whose partial derivatives are the constants $L (e_{j}) \in {0, 0, i k, - i ω}$ . The map $L$ has continuous partial derivatives, hence is totally differentiable with $D L = L$ by [F5]; the exponential is real totally differentiable with derivative multiplication by $\exp$ by [F4]; so the chain rule [F5] makes $G = \exp \circ L$ real totally differentiable at every point with $\partial_{j} G = D G (e_{j}) = \exp (i θ) L (e_{j})$ , that is $\partial_{x} G = \partial_{y} G = 0$ , $\partial_{z} G = i k G$ and $\partial_{t} G = - i ω G$ ; being differentiable everywhere, $G$ is continuous by [F5].

1.3givenF11algebra

Box presentation: $B = (0, L_{1}) \times (0, L_{2}) \times (a, b)$ is nonempty open and bounded with closure $[0, L_{1}] \times [0, L_{2}] \times [a, b] \subseteq R^{3} = Ω$ ; its boundary consists of the six closed rectangles $S_{1}^{\pm}, \dots, S_{3}^{\pm}$ in the coordinate planes, with edge set the twelve edges, each face is a compact Borel subset of a coordinate-plane patch, faces meet only in the edge set, and the outward unit normal is $- e_{1}, + e_{1}, - e_{2}, + e_{2}, - e_{3}, + e_{3}$ on the faces $x = 0$ , $x = L_{1}$ , $y = 0$ , $y = L_{2}$ , $z = a$ , $z = b$ respectively; the surface integral in (c) is defined as the sum of their outward-oriented patch fluxes in [F11].

2.1step 1.2F6algebra

Smoothness: we show that $G$ is $C^{\infty}$ . Every partial derivative of $G$ of order at most one is a constant multiple of $G$ by step 1.2, and those partials are continuous because $G$ is. Suppose $G$ is $C^{r}$ with $r \geq 1$ and every partial derivative of order at most $r$ is a constant multiple of $G$ ; if $P = c G$ is such a partial derivative, then for each coordinate $j$ the line map $s \mapsto P (\cdot + s e_{j}) = c G (\cdot + s e_{j})$ has derivative $c \partial_{j} G = c L (e_{j}) G$ by the scalar-multiple rule [F6], so $\partial_{j} P$ exists everywhere, equals the constant $c L (e_{j})$ times $G$ , and is continuous; thus every partial derivative of order $r + 1$ is a continuous constant multiple of $G$ , and $G$ is $C^{r + 1}$ . By induction $G$ is $C^{k}$ for every $k$ , that is smooth [F6].

2.2step 1.1F7F8F9algebra

Energy density and flux: substituting step 1.1 into [F9] and using $μ_{0} c^{2} = ε_{0}^{- 1}$ and $e_{1} \times e_{2} = e_{3}$ gives $u = \frac{ε_{0}}{2} E_{0}^{2} \cos^{2} θ + \frac{1}{2 μ_{0}} \frac{E_{0}^{2}}{c^{2}} \cos^{2} θ = ε_{0} E_{0}^{2} \cos^{2} θ$ and $S = μ_{0}^{- 1} E_{0} \frac{E_{0}}{c} \cos^{2} θ e_{3} = ε_{0} c E_{0}^{2} \cos^{2} θ e_{3}$ , which is clause (b) with the units of [F7].

3.1step 1.1step 2.1F6algebra

Field data: $E_{j} = E_{0} Re (G)$ for $j = 1$ and $E_{j} = 0$ otherwise, and $B_{2} = (E_{0} / c) Re (G)$ with the other components zero; since $G$ is smooth and real parts, scalar multiples and zero components of smooth maps are smooth by [F6], the maps $E, B : R^{3} \times R \to R^{3}$ are smooth fields of [F1]. Their relevant partial derivatives are $\partial_{t} E_{1} = E_{0} Re (- i ω G) = E_{0} ω \sin θ$ and $\partial_{z} E_{1} = E_{0} Re (i k G) = - E_{0} k \sin θ$ , and likewise $\partial_{t} B_{2} = E_{0} ω c^{- 1} \sin θ$ and $\partial_{z} B_{2} = - E_{0} k c^{- 1} \sin θ$ , all read from step 1.2.

4.1step 3.1F8F9algebra

Maxwell equations with $ρ = J = 0$ : since $E$ depends only on $z$ in the $e_{1}$ direction and $B$ only on $z$ in the $e_{2}$ direction, $div E = \partial_{x} E_{1} + \partial_{y} E_{2} + \partial_{z} E_{3} = 0$ and $div B = 0$ ; the curl formulas give $curl E = (\partial_{y} E_{3} - \partial_{z} E_{2}, \partial_{z} E_{1} - \partial_{x} E_{3}, \partial_{x} E_{2} - \partial_{y} E_{1}) = (0, - E_{0} k \sin θ, 0) = - \partial_{t} B$ and $curl B = (\partial_{y} B_{3} - \partial_{z} B_{2}, \partial_{z} B_{1} - \partial_{x} B_{3}, \partial_{x} B_{2} - \partial_{y} B_{1}) = (E_{0} k c^{- 1} \sin θ, 0, 0) = c^{- 2} \partial_{t} E$ , using $ω = c k$ ; hence the four smooth-branch equations of [F9] hold with $ρ = 0$ , $J = 0$ , which is clause (a).

4.2step 3.1step 2.2algebra

Local balance: from step 3.1, $\partial_{t} u = ε_{0} E_{0}^{2} \cdot 2 \cos θ (- \sin θ) (- ω) = 2 ε_{0} E_{0}^{2} ω \cos θ \sin θ$ and $\partial_{z} S_{3} = ε_{0} c E_{0}^{2} \cdot 2 \cos θ (- \sin θ) k = - 2 ε_{0} c E_{0}^{2} k \cos θ \sin θ$ ; their sum is $2 ε_{0} E_{0}^{2} (ω - c k) \cos θ \sin θ = 0$ because $ω = c k$ , so $\partial_{t} u + \partial_{z} S_{3} = 0$ , equivalently $\partial_{t} u + div S = 0$ since $S$ is parallel to $e_{3}$ ; this completes clause (b).

5.1step 4.2F10algebra

Time derivative of the box energy: at fixed $t$ the function $u$ is continuous on the closed box and independent of $x, y$ , so [F10] gives $\int_{B} u d V = L_{1} L_{2} \int_{a}^{b} u (z, t) d z$ . For any real $t$ , choose $α < t < β$ ; $u$ and $\partial_{t} u$ are continuous on $[a, b] \times [α, β]$ , so the one-dimensional rectangle differentiation theorem of [F10], with time as the differentiation variable, gives $\frac{d}{d t} \int_{B} u d V = L_{1} L_{2} \int_{a}^{b} \partial_{t} u (z, t) d z$ . By step 4.2 this is $- L_{1} L_{2} \int_{a}^{b} \partial_{z} S_{3} (z, t) d z = - L_{1} L_{2} (S_{3} (b, t) - S_{3} (a, t))$ by the fundamental theorem of [F10]; its derivative integrand is continuous, hence integrable.

6.1step 1.3step 2.2step 5.1F8F10F11algebra

Closed-surface flux: on the four faces with normals $\pm e_{1}, \pm e_{2}$ , $S \cdot n = 0$ because $S = S_{3} e_{3}$ . The upper face has parametrization $φ_{b} (u, v) = (u, v, b)$ on $[0, L_{1}] \times [0, L_{2}]$ , with $\partial_{u} φ_{b} \times \partial_{v} φ_{b} = e_{1} \times e_{2} = e_{3}$ , so its outward patch flux is $L_{1} L_{2} S_{3} (b, t)$ by [F10]-[F11]. On the lower face use $φ_{a} (u, v) = (v, u, a)$ on $[0, L_{2}] \times [0, L_{1}]$ : its cross product is $e_{2} \times e_{1} = - e_{3}$ , giving the outward patch flux $- L_{1} L_{2} S_{3} (a, t)$ . Summing these six fluxes gives $- \oint_{\partial B} S \cdot n d S = L_{1} L_{2} (S_{3} (a, t) - S_{3} (b, t))$ , equal to the energy derivative of step 5.1. Thus (c) follows by direct Riemann computation; the individual contributions to the energy rate are the negatives of the outward fluxes, namely $- L_{1} L_{2} S_{3} (b, t)$ and $+ L_{1} L_{2} S_{3} (a, t)$ .

7.1step 2.2step 3.1step 4.1step 5.1step 6.1F10algebra∎

Scope and whole-space idealization: the configuration is a source-free smooth solution by step 4.1, and steps 5.1-6.1 establish its finite-box ledger. At fixed $t$ , choose $z_{0} = ω t / k$ , so $\cos^{2} (k z_{0} - ω t) = 1$ . Continuity from step 3.1 gives $a < b$ about $z_{0}$ with $\cos^{2} (k z - ω t) \geq 1 / 2$ for $z \in [a, b]$ . By step 2.2 and the integral order rule of [F10], $\int_{B} u d V \geq \frac{1}{2} ε_{0} E_{0}^{2} L_{1} L_{2} (b - a)$ , unbounded as $L_{1} L_{2}$ grows. Thus the improper whole-space energy, defined as the supremum of these nonnegative finite-box energies over all boxes, is infinite; no finite whole-space energy or empirical realization is claimed.

Remarks

Why no double-angle identity is needed. The local balance is proved by differentiating $\cos^{2} θ$ , equivalently by the product of $\cos θ$ and $- \sin θ$ in each term; because $ω = c k$ the two contributions cancel exactly at every point, which is the exact statement of the balance for this solution and requires no period averaging.
The face signs. The outward normal on the lower face is $- e_{3}$ ; the statement lists the flux of the field through the face in the outward orientation as $- L_{1} L_{2} S_{3} (a, t)$ , while the contribution of that face to the rate of change of the box energy is its negative, $+ L_{1} L_{2} S_{3} (a, t)$ . Both signs are kept explicitly.
No total-energy claim. The finite-box energies are unbounded over growing boxes, as proved in step 7.1; the ideal wave therefore has no finite whole-space energy. Clause (d) retains the finite-box and nonempirical scope.