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Period-averaged energy density, Poynting vector and momentum flux of a plane wave

Physical scope

Exact period-average identities for an ideal whole-space plane-wave solution of the smooth source-free vacuum Maxwell branch, in a named inertial chart with SI constants: averaged energy density, Poynting flux, momentum density and Maxwell-stress/momentum-flux tensor, with the physical balance reading additionally conditional on Lorentz continuum coupling (ii) in the explicitly defined empty-matter model. Excluded: finite beams and pulses, sources, boundaries, material media, any ensemble or detection statistics, any nonzero finite-total-energy whole-space plane wave, and any empirical claim. The admitted zero amplitude has zero total energy.

Statement

Let $k \in R^{3} ∖ {0}$ , $ω > 0$ with $ω = c ∣ k ∣$ , let $e \in C^{3}$ with $k \cdot e = 0$ and $b = k \times e / ω$ , and let $(E, B)$ be the corresponding plane-wave solution of Dispersion, transversality and the two polarization conditions for plane-wave solutions (Real plane-wave fields encoded by complex amplitudes) on $R^{3} \times R$ . Let $u$ and $S$ be the energy density and Poynting vector of Vacuum electromagnetic energy density and energy flux (the Poynting vector), and $g = S / c^{2}$ , $σ$ , $Π = - σ$ the momentum density, Maxwell stress and momentum flux of Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum. Write $T = 2 π / ω$ , $∣ e ∣^{2} = e \cdot \overline{e}$ , $\hat{k} = k / ∣ k ∣$ , and let $⟨ f ⟩ : = \frac{1}{T} \int_{0}^{T} f (x, s) d s$ be the period average at fixed $x$ . Then, for every $x \in R^{3}$ :

(a) [Averaged energy density] $⟨ u ⟩ = \frac{ε_{0}}{2} ∣ e ∣^{2}$ , independent of $x$ and of the starting time.

(b) [Averaged Poynting vector] $⟨ S ⟩ = \frac{ε_{0} c}{2} ∣ e ∣^{2} \hat{k}$ ; hence $⟨ S ⟩ = ⟨ u ⟩ c \hat{k}$ , and the period-averaged intensity is $I : = ∣ ⟨ S ⟩ ∣ = c ⟨ u ⟩ = \frac{ε_{0} c}{2} ∣ e ∣^{2}$ .

(c) [Averaged momentum density, stress and momentum flux] $⟨ g ⟩ = \frac{ε_{0}}{2 c} ∣ e ∣^{2} \hat{k}$ , and componentwise $⟨ σ_{i j} ⟩ = - ⟨ u ⟩ {\hat{k}}_{i} {\hat{k}}_{j}$ , equivalently $⟨ Π_{i j} ⟩ = ⟨ u ⟩ {\hat{k}}_{i} {\hat{k}}_{j}$ ; the period-averaged momentum flux across a plane with unit normal $n$ (with the physical balance interpretation conditional as in (d)) is therefore $⟨ u ⟩ (\hat{k} \cdot n) \hat{k}$ , a pure flux along $\hat{k}$ with signed coefficient $(I / c) (\hat{k} \cdot n)$ and magnitude $(I / c) ∣ \hat{k} \cdot n ∣$ . The tensor identity holds for every complex transverse amplitude $e$ , not only for real or circular ones.

(d) [Balance reading with continuum coupling] For this clause additionally adopt The Lorentz coupling: external force on a test charge and the continuum force and power densities(ii) for the empty-matter model on $R^{3} \times R$ : the matter energy density $w : R^{3} \times R \to R$ , energy flux $Q : R^{3} \times R \to R^{3}$ , momentum density $p : R^{3} \times R \to R^{3}$ and momentum flux $P : R^{3} \times R \to R^{3 \times 3}$ are all identically zero, as are the source fields $ρ, J$ . The adopted electromagnetic force and power densities are then $f : = ρ E + J \times B = 0$ and $J \cdot E = 0$ , respectively; the zero matter balances are satisfied. Under this additional interpretation premise the local identities of Local and fixed-volume electromagnetic energy balance and Local and fixed-volume momentum balance for smooth sources hold for this field; since $u, g, S, σ$ are $T$ -periodic in $t$ , the period averages of their time derivatives vanish (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ), so averaging the identities gives $div ⟨ S ⟩ = 0$ and $div ⟨ σ ⟩ = 0$ consistently (both averaged fields are in fact constant), and the reading of $u, S, g, Π$ as energy and momentum densities and fluxes is that of those two theorems.

(e) [Scope] These are exact identities for the ideal plane wave; $I$ is the intensity used by the radiation-pressure thought experiment of the companion page and in radiation-flux arguments. No detection statistics, no finite-beam or finite-time realisation and no empirical claim is made, and the zero amplitude gives the zero wave and zero total energy, whereas every nonzero amplitude gives infinite total energy over $R^{3}$ .

Facts & Assumptions

Given: The wavevector $k \neq 0$ , the frequency $ω = c ∣ k ∣$ , the amplitude $e \in C^{3}$ with $k \cdot e = 0$ , $b = k \times e / ω$ , the plane-wave pair $(E, B)$ on $R^{3} \times R$ with phase $θ (x, t) = k \cdot x - ω t$ , the constants of the SI conventions, and the quantities $u, S, g, σ, Π$ of the two definition items. For clause (d)'s physical balance interpretation additionally [A1].

[F1]

Period averages (Period averages of products of real parts of complex exponentials): for $ω > 0$ , $T = 2 π / ω$ and $a, d \in C$ , $\frac{1}{T} \int_{0}^{T} Re (a e^{- i ω t}) Re (d e^{- i ω t}) d t = \frac{1}{2} Re (a \overline{d})$ ; for $a, d \in C^{3}$ the same identity holds with the componentwise complex-bilinear Euclidean inner product, and $\frac{1}{T} \int_{0}^{T} Re (a e^{- i ω t}) \times Re (d e^{- i ω t}) d t = \frac{1}{2} Re (a \times \overline{d})$ componentwise.

[F2]

Energy density and Poynting vector (Vacuum electromagnetic energy density and energy flux (the Poynting vector)): $u = \frac{ε_{0}}{2} ⟨ E, E ⟩ + \frac{1}{2 μ_{0}} ⟨ B, B ⟩$ and $S = \frac{1}{μ_{0}} E \times B$ are $C^{\infty}$ and $u \geq 0$ .

[F3]

Momentum density, Maxwell stress and momentum flux (Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum): $g = S / c^{2}$ , $σ_{i j} = ε_{0} (E_{i} E_{j} - \frac{1}{2} δ_{i j} ∣ E ∣^{2}) + \frac{1}{μ_{0}} (B_{i} B_{j} - \frac{1}{2} δ_{i j} ∣ B ∣^{2})$ and $Π = - σ$ , with $σ$ symmetric.

[F4]

Balance identities (Local and fixed-volume electromagnetic energy balance(a), (c), Local and fixed-volume momentum balance for smooth sources(a), (c)): under [A1], for the jointly $C^{1}$ fields of the smooth branch with $ρ = J = 0$ one has $\partial_{t} u + div S = 0$ and $\partial_{t} g - div σ = 0$ (equivalently $\partial_{t} g + div Π = 0$ ) componentwise on the whole-space region. The momentum theorem requires clause (ii) of The Lorentz coupling: external force on a test charge and the continuum force and power densities for its force-density and balance interpretation; the Poynting identity (a) follows from Maxwell alone, while its physical matter-exchange interpretation (c) also uses that coupling.

[F5]

Plane-wave solutions (Dispersion, transversality and the two polarization conditions for plane-wave solutions): with $∣ k ∣ = ω / c$ , $k \cdot e = 0$ and $b = k \times e / ω$ the pair $(E, B)$ lies in the source-free smooth Maxwell branch on $R^{3} \times R$ ; the fields are $C^{\infty}$ , $B = \hat{k} \times E / c$ pointwise and $∣ B ∣ = ∣ E ∣ / c$ .

[F6]

Plane-wave encoding (Real plane-wave fields encoded by complex amplitudes): $E = Re (e \exp (i θ))$ , $B = Re (b \exp (i θ))$ , $θ = k \cdot x - ω t$ , and the amplitudes determine the fields uniquely.

[F7]

Fundamental theorem of calculus (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ): for $F$ of class $C^{1}$ on $[0, T]$ , $\int_{0}^{T} F^{'} (t) d t = F (T) - F (0)$ ; more generally the integral of the derivative over any interval equals the increment.

[F8]

Cross and inner product algebra (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product is bilinear, alternating, and orthogonal to both factors, The squared cross-product norm is the Gram determinant of two vectors): the cross product is bilinear and alternating and the inner product is bilinear and symmetric, so the coordinate identities $u \times (v \times w) = v ⟨ u, w ⟩ - w ⟨ u, v ⟩$ and $∣ u \times v ∣^{2} = ∣ u ∣^{2} ∣ v ∣^{2} - ⟨ u, v ⟩^{2}$ hold for real vectors, The triple-product identity extends to complex entries by the same coordinate polynomial calculation. The norm identity is used only for real vectors; complex norms mean $∣ z ∣^{2} = \sum_{j} ∣ z_{j} ∣^{2}$ and require conjugation.

[F9]

Real and imaginary parts (Real and imaginary parts, complex conjugation, and modulus): $Re$ and conjugation are $R$ -linear, $\overline{z w} = \overline{z} \overline{w}$ , $z \overline{z} = ∣ z ∣^{2}$ , and $Re (\overline{z}) = Re (z)$ .

[F10]

SI constants (Framework-local SI units, the vacuum constants and the exact/measured status of their values): $ε_{0} = (μ_{0} c^{2})^{- 1}$ , so $1 / (μ_{0} c^{2}) = ε_{0}$ and $1 / (μ_{0} c) = ε_{0} c$ ; $c > 0$ .

[F11]

Riemann integration on rectangles (Axis-parallel rectangles in $R^{m}$ and their volume, The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ , Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections, Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable): continuous functions on closed nondegenerate rectangles are Riemann integrable, the multiple integral equals the iterated integral when the sections are integrable, and the Darboux integral of a nonnegative function is monotone in the rectangle.

[A1]

Interpretation premise for clause (d) and the physical flux reading in (c): adopt The Lorentz coupling: external force on a test charge and the continuum force and power densities(ii) in the empty-matter model defined in (d), with $ρ = J = 0$ and $w, Q, p, P$ identically zero. Then its adopted force density $f = ρ E + J \times B$ (N m $^{- 3}$ ) and power density $J \cdot E$ (W m $^{- 3}$ ) vanish. This is an additional model assumption, not a consequence of the source-free Maxwell equations; the numerical average identities in (a)-(c) and the total-energy assertion in (e) do not require it.

Proof

technique · direct

1.1givenF1F5F6F9

Setup and componentwise averages. Fix $x \in R^{3}$ and put $α : = e e^{i k \cdot x}$ , $δ : = b e^{i k \cdot x} \in C^{3}$ , so that $E_{i} = Re (α_{i} e^{- i ω t})$ and $B_{i} = Re (δ_{i} e^{- i ω t})$ for each $i$ by [F6], with $θ = k \cdot x - ω t$ . Applying the scalar and vector forms of [F1] with $a = α$ , $d = α$ ; $a = δ$ , $d = δ$ ; and the cross-product form with $a = α$ , $d = δ$ , and using $α_{i} \overline{α_{j}} = e_{i} e^{i k \cdot x} \overline{e_{j}} e^{- i k \cdot x} = e_{i} \overline{e_{j}}$ and $∣ α_{i} ∣^{2} = ∣ e_{i} ∣^{2}$ , gives $⟨ E_{i} E_{j} ⟩ = \frac{1}{2} Re (e_{i} \overline{e_{j}})$ , $⟨ B_{i} B_{j} ⟩ = \frac{1}{2} Re (b_{i} \overline{b_{j}})$ and $⟨ E \times B ⟩ = \frac{1}{2} Re (e \times \overline{b})$ , and $⟨ ∣ E ∣^{2} ⟩ = \frac{1}{2} ∣ e ∣^{2}$ , $⟨ ∣ B ∣^{2} ⟩ = \frac{1}{2} ∣ b ∣^{2}$ , all independent of the starting time and of $x$ .

2.1step 1.1F2F8F10algebra

Clause (a). By [F2] and step 1.1, $⟨ u ⟩ = \frac{ε_{0}}{2} ⟨ ∣ E ∣^{2} ⟩ + \frac{1}{2 μ_{0}} ⟨ ∣ B ∣^{2} ⟩ = \frac{ε_{0}}{4} ∣ e ∣^{2} + \frac{1}{4 μ_{0}} ∣ b ∣^{2}$ . Write $e = a + i d$ with $a, d \in R^{3}$ . Since $k$ is real and $k \cdot e = 0$ , both $k \cdot a$ and $k \cdot d$ vanish. The real Gram identity of [F8] applied separately to $a$ and $d$ gives $∣ b ∣^{2} = (∣ k \times a ∣^{2} + ∣ k \times d ∣^{2}) / ω^{2} = ∣ k ∣^{2} (∣ a ∣^{2} + ∣ d ∣^{2}) / ω^{2} = ∣ e ∣^{2} / c^{2}$ because $∣ k ∣ = ω / c$ ; hence $⟨ u ⟩ = \frac{ε_{0}}{4} ∣ e ∣^{2} + \frac{ε_{0}}{4} ∣ e ∣^{2} = \frac{ε_{0}}{2} ∣ e ∣^{2}$ by $1 / (μ_{0} c^{2}) = ε_{0}$ of [F10]. The value is independent of $x$ (the phase factors cancel in step 1.1) and of the starting time: the integrand is $T$ -periodic and continuous, so its integral over any interval of length $T$ equals $\int_{0}^{T}$ by [F7] applied to the function $G (s) = \int_{s}^{s + T} f$ with $G^{'} = 0$ .

3.1step 1.1F2F8F10algebra

Clause (b). By [F2] and the cross-product average of step 1.1, $⟨ S ⟩ = \frac{1}{μ_{0}} ⟨ E \times B ⟩ = \frac{1}{2 μ_{0}} Re (e \times \overline{b})$ . Since $b = k \times e / ω$ and $k$ is real, $e \times \overline{b} = \frac{1}{ω} e \times (k \times \overline{e}) = \frac{1}{ω} [k ⟨ e, \overline{e} ⟩ - \overline{e} ⟨ e, k ⟩] = \frac{1}{ω} k ∣ e ∣^{2}$ by the triple-product identity of [F8] and $⟨ e, k ⟩ = 0$ ; hence $⟨ S ⟩ = \frac{∣ e ∣^{2}}{2 μ_{0} ω} k = \frac{∣ e ∣^{2} ∣ k ∣}{2 μ_{0} ω} \hat{k} = \frac{∣ e ∣^{2}}{2 μ_{0} c} \hat{k} = \frac{ε_{0} c}{2} ∣ e ∣^{2} \hat{k}$ by $∣ k ∣ = ω / c$ and $1 / (μ_{0} c) = ε_{0} c$ of [F10]. Comparing with step 2.1 gives $⟨ S ⟩ = ⟨ u ⟩ c \hat{k}$ and $I = ∣ ⟨ S ⟩ ∣ = \frac{ε_{0} c}{2} ∣ e ∣^{2} = c ⟨ u ⟩$ .

3.2step 1.1step 2.1F3F8F9F10algebra

Clause (c), tensor identity. By [F3] and step 1.1, $⟨ σ_{i j} ⟩ = ε_{0} (\frac{1}{2} Re (e_{i} \overline{e_{j}}) - \frac{1}{4} δ_{i j} ∣ e ∣^{2}) + \frac{1}{μ_{0}} (\frac{1}{2} Re (b_{i} \overline{b_{j}}) - \frac{1}{4} δ_{i j} ∣ b ∣^{2})$ . For the magnetic term, the coordinate identity of [F8] for the cross product applied componentwise gives $Re (b_{i} \overline{b_{j}}) = \frac{1}{ω^{2}} Re [(k \times e)_{i} \overline{(k \times e)_{j}}] = \frac{1}{ω^{2}} [∣ k ∣^{2} ∣ e ∣^{2} δ_{i j} - k_{i} k_{j} ∣ e ∣^{2} - ∣ k ∣^{2} Re (e_{i} \overline{e_{j}})]$ : multiplying the coordinate formula for $(k \times e)_{i}$ and $\overline{(k \times e)_{j}}$ and summing over the repeated indices leaves exactly the three displayed contractions, using $⟨ k, e ⟩ = 0$ and the reality of $k$ , and $Re (\overline{e_{i}} e_{j}) = Re (e_{i} \overline{e_{j}})$ by [F9]. Substituting, the two $Re (e_{i} \overline{e_{j}})$ coefficients are $\frac{ε_{0}}{2} - \frac{∣ k ∣^{2}}{2 μ_{0} ω^{2}} = \frac{ε_{0}}{2} - \frac{1}{2 μ_{0} c^{2}} = 0$ by $∣ k ∣^{2} / ω^{2} = 1 / c^{2}$ and [F10], and the total $δ_{i j}$ contribution is $- \frac{ε_{0}}{4} ∣ e ∣^{2} + \frac{∣ k ∣^{2} ∣ e ∣^{2}}{2 μ_{0} ω^{2}} - \frac{1}{4 μ_{0}} ∣ b ∣^{2} = (- \frac{ε_{0}}{4} + \frac{ε_{0}}{2} - \frac{ε_{0}}{4}) ∣ e ∣^{2} = 0$ using $∣ b ∣^{2} = ∣ e ∣^{2} / c^{2}$ from step 2.1; there remains $⟨ σ_{i j} ⟩ = - \frac{∣ k ∣^{2} ∣ e ∣^{2}}{2 μ_{0} ω^{2}} \frac{k_{i} k_{j}}{∣ k ∣^{2}} = - \frac{ε_{0}}{2} ∣ e ∣^{2} {\hat{k}}_{i} {\hat{k}}_{j} = - ⟨ u ⟩ {\hat{k}}_{i} {\hat{k}}_{j}$ by [F10] and step 2.1, so $⟨ Π_{i j} ⟩ = - ⟨ σ_{i j} ⟩ = ⟨ u ⟩ {\hat{k}}_{i} {\hat{k}}_{j}$ .

3.3step 1.1step 2.1F2F5F6F7F9F11algebra

Clause (e), zero and nonzero amplitudes. If $e = 0$ then $b = k \times e / ω = 0$ , so $E = B = 0$ by [F6] and $u = 0$ identically by [F2]; all box integrals and their supremum vanish, so the zero amplitude gives the zero wave and zero total energy. Assume now $e \neq 0$ and write $e = a + i d$ with $a, d \in R^{3}$ ; then $u = ε_{0} ∣ a \cos θ - d \sin θ ∣^{2}$ by [F2] and [F6], with $θ = k \cdot x - ω t$ . Choose a coordinate $j$ with $k_{j} \neq 0$ and put $P : = 2 π / ∣ k_{j} ∣$ ; at fixed $t$ and fixed coordinates $x_{i}$ , $i \neq j$ , the map $x_{j} \mapsto θ$ is affine with nonzero slope $k_{j}$ , so over any $x_{j}$ -interval of length $P$ the phase runs through an interval of length $2 π$ ; the average $⟨ u ⟩ = \frac{ε_{0}}{2} ∣ e ∣^{2}$ of step 2.1 is strictly positive here, so $e \neq 0$ gives a nonzero wave, and the $2 π$ -periodicity of $θ \mapsto ∣ a \cos θ - d \sin θ ∣^{2}$ (by [F9] and the periodicity of the trigonometric functions implicit in the encoding of [F6]) gives $\int_{x_{j}}^{x_{j} + P} u d x_{j} = \frac{ε_{0}}{∣ k_{j} ∣} \int_{0}^{2 π} ∣ a \cos ϕ - d \sin ϕ ∣^{2} d ϕ = \frac{ε_{0}}{∣ k_{j} ∣} [π ∣ a ∣^{2} + π ∣ d ∣^{2}] = \frac{ε_{0} P}{2} ∣ e ∣^{2} > 0$ , where the elementary integrals of $\cos^{2}$ , $\sin^{2}$ and $\sin \cos$ over a full period are computed from their derivatives by [F7] and $\int_{0}^{2 π} \sin ϕ \cos ϕ d ϕ = [\frac{1}{2} \sin^{2} ϕ]_{0}^{2 π} = 0$ by [F7]. By [F11] the continuous density is Riemann integrable on every closed box and the iterated integral equals the multiple integral, so for $R = N P$ with a positive integer $N$ , integrating first in $x_{j}$ over the $2 N$ full periods of $[- R, R]$ and then over the square $[- R, R]^{2}$ in the other coordinates gives $\int_{[- R, R]^{3}} u = (2 N) \cdot (2 R)^{2} \cdot \frac{ε_{0} P}{2} ∣ e ∣^{2} = (2 R)^{2} N ε_{0} P ∣ e ∣^{2} \to + \infty$ as $N \to \infty$ ; since $u \geq 0$ , the Darboux integrals over the boxes are monotone in $R$ by [F11], so the box-exhaustion total energy $\sup_{R > 0} \int_{[- R, R]^{3}} u$ is $+ \infty$ .

4.1step 3.1step 3.2F3algebra

Momentum density and flux reading. By [F3], $⟨ g ⟩ = ⟨ S ⟩ / c^{2} = \frac{ε_{0}}{2 c} ∣ e ∣^{2} \hat{k}$ by step 3.1, and $⟨ Π ⟩ = - ⟨ σ ⟩$ with $⟨ Π_{i j} ⟩ = ⟨ u ⟩ {\hat{k}}_{i} {\hat{k}}_{j}$ by step 3.2; the period-averaged momentum flux across a plane with unit normal $n$ is the vector with components $\sum_{j} ⟨ Π_{i j} ⟩ n_{j} = ⟨ u ⟩ {\hat{k}}_{i} (\hat{k} \cdot n)$ , that is $⟨ u ⟩ (\hat{k} \cdot n) \hat{k}$ , with signed coefficient $(I / c) (\hat{k} \cdot n)$ and magnitude $(I / c) ∣ \hat{k} \cdot n ∣$ because $∣ \hat{k} ∣ = 1$ and $I = c ⟨ u ⟩$ by step 3.1.

4.2step 2.1step 3.1step 3.2F2F3F4F5F7A1

Balance reading (clause (d)). Under [A1], the empty-matter model has zero force and power densities, and [F5] supplies smooth Maxwell fields with continuous zero sources; thus all hypotheses of both balance theorems in [F4], including the Lorentz-coupling premise, hold. By step 2.1 the function $u$ is $C^{\infty}$ on the whole-space region and $T$ -periodic in $t$ , so the mean of $\partial_{t} u$ vanishes, $\frac{1}{T} \int_{0}^{T} \partial_{t} u (x, s) d s = \frac{u (x, T) - u (x, 0)}{T} = 0$ , by [F7]; the same computation applies to each component of $g$ by [F3] and step 3.1. The averaged fields $⟨ S ⟩$ of step 3.1 and $⟨ σ ⟩$ of step 3.2 are constant, so $div ⟨ S ⟩ = 0$ and $div ⟨ σ ⟩ = 0$ componentwise; hence the local identities of [F4] with $f = 0$ hold term by term after period-averaging, and under the additional adopted premise [A1] the identification of $u, S, g, Π$ as energy and momentum densities and fluxes is exactly the interpretation stated by those two balance theorems (Vacuum electromagnetic energy density and energy flux (the Poynting vector)(e), Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum(f)).

5.1step 2.1step 3.1step 3.2step 3.3step 4.1step 4.2given∎

Conclusion. Clause (a) is step 2.1, clause (b) is step 3.1, the tensor identity of clause (c) is step 3.2 and its algebraic flux contraction is step 4.1, its physical balance reading under [A1] and clause (d) are step 4.2, and the zero- and nonzero-amplitude statements of clause (e) are step 3.3. The identities hold for every complex transverse amplitude $e$ because no step used reality or any relation between $a$ and $d$ beyond the amplitude conditions $k \cdot e = 0$ , $b = k \times e / ω$ , $∣ k ∣ = ω / c$ ; no detection statistics, finite-beam or empirical content is asserted, and $I$ is the intensity used by the radiation-pressure thought experiment and the radiation-flux arguments on the companion pages.

Source notes

The cited treatments compute the cycle averages of energy density, Poynting flux and momentum density for the monochromatic plane wave and identify $\overset{ˉ}{S} = c \overset{ˉ}{u} \hat{k}$ ; the deduction above is the componentwise period-average calculation applied to the framework's definitions, and the Maxwell-stress average is proved from the coordinate identity for the cross product. No energy or momentum content is imported from the treatments beyond the definitions and the balance identities of this framework.

Period-averaged energy density, Poynting vector and momentum flux of a plane wave

Statement

Facts & Assumptions

Proof

Source notes

Depends on

Dependency tree · 2 levels

Sources