The ideal harmonic electric-dipole radiation model and its adopted status

Let $\omega >0$, $k:=\omega \mathrm{/}c$ with $c$ the vacuum speed of Framework-local SI units, the vacuum constants and the exact/measured status of their values, ${\mu }_0>0$ the vacuum permeability of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and $p_0\in {\mathbb{C}}^3$ (the complex dipole-moment amplitude, in $\mathrm{C}\mathrm{m}$). Define, on the punctured space ${\mathbb{R}}^3\setminus \{0\}$ and all times $t\in \mathbb{R}$,

$E_{\mathrm{d}}(x,t):=\mathrm{Re}(-\frac{{\mu }_0{\omega }^2}{4\pi r}n\times (n\times p_0)e^{i(kr-\omega t)}),B_{\mathrm{d}}(x,t):=\frac{1}{c}n\times E_{\mathrm{d}}(x,t),$

where $r:=\mathrm{\mid }x\mathrm{\mid }$ and $n:=x\mathrm{/}r$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, The cross product in ${\mathbb{R}}^3$), with the real part and complex exponential of Real and imaginary parts, complex conjugation, and modulus and The complex exponential by its power series and the componentwise real-part encoding convention of Complex plane-wave fields: real parts of complex exponentials (the spatial amplitude here varies with $x$; no plane-wave instance is asserted). Call $(E_{\mathrm{d}},B_{\mathrm{d}})$ the ideal harmonic electric-dipole radiation model of frequency $\omega$ and amplitude $p_0$, and call $S_{\mathrm{d}}:=E_{\mathrm{d}}\times B_{\mathrm{d}}\mathrm{/}{\mu }_0$ its model flux (Vacuum electromagnetic energy density and energy flux (the Poynting vector)); the model fields are read as an electric and a magnetic field of The electric and magnetic fields as fixed-time spatial vector fields.

Adopted status

This is an explicitly adopted ideal radiation-law model, fixed by the following declarations.

1. It is the leading transverse far-zone field of Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source(c) evaluated at the harmonic transverse moment $V(u)=\mathrm{Re}(-{\omega }^2{p}_0{e}^{-i\omega u})$, i.e. at $p^{\mathrm{\prime }\mathrm{\prime }}$ for the waveform $p(u)=\mathrm{Re}(p_0{e}^{-i\omega u})$: by part (a) of Periodic responses of $\ddot{z}=\mathrm{Re}(w{e}^{-i\omega t})$ and the affine classification of all solutions the second derivative of that explicit $C^{\mathrm{\infty }}$ waveform is $\mathrm{Re}(-{\omega }^2{p}_0{e}^{-i\omega u})$, and substituting $u=t-r\mathrm{/}c$ into ${\mu }_{0}n\times (n\times V(u))\mathrm{/}(4\pi r)$ gives exactly $E_{\mathrm{d}}$, because $e^{i(kr-\omega t)}=e^{-i\omega (t-r\mathrm{/}c)}$.
2. The model contains no independent magnetic-dipole or higher-multipole radiation, no near-zone or static field and no other radiating component.
3. It is not asserted to solve the vacuum Maxwell equations at finite $r$, to be the field of any actual source, to satisfy any boundary condition or to have finite total energy on ${\mathbb{R}}^3\setminus \{0\}$.
4. For compact smooth conserved sources, comparison with these prescribed model fields uses the actual source's moment $p_{\mathrm{a}\mathrm{c}\mathrm{t}}(u)$ of Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source, the harmonic waveform $p_{\mathrm{h}}(u)=\mathrm{Re}(p_0{e}^{-i\omega u})$, their second-derivative mismatch

$M(u):=\mathrm{\mid }{p}_{\mathrm{a}\mathrm{c}\mathrm{t}}^{\mathrm{\prime }\mathrm{\prime }}(u)-p_{\mathrm{h}}^{\mathrm{\prime }\mathrm{\prime }}(u)\mathrm{\mid },$

and the explicit $D,\mathrm{\Gamma }$ bounds of Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source (b), (c) and (e). A bound depending only on that source's $D$ and $\mathrm{\Gamma }$ is valid when $M\equiv 0$ on the observation window, in which case the theorem's leading fields for the source's moment coincide with the model fields; for $M\equiv ̸0$ the comparison carries the additional mismatch term. The source's moment is not automatically the independently prescribed $p_0$. These actual-source comparisons assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) carried by the cited theorem. The explicit ideal fields themselves and their pointwise flux definition remain choice-free.

No measured, empirical or material-response content is adopted: $p_0$ is prescribed model data, not a derived polarizability.

Well-definedness and conventions

On the declared domain $r>0$, so $r=\mathrm{\mid }x\mathrm{\mid }$ is the positive Euclidean norm and $n=x\mathrm{/}r$ is a unit vector; $n\times (n\times p_0)$ and its real part are defined componentwise, and $E_{\mathrm{d}}$, $B_{\mathrm{d}}$ are $C^{\mathrm{\infty }}$ on $({\mathbb{R}}^3\setminus \{0\})\times \mathbb{R}$ because $r$ is $C^{\mathrm{\infty }}$ there and the real and imaginary parts of $-\frac{{\mu }_0{\omega }^2}{4\pi r}n\times (n\times p_0)e^{i(kr-\omega t)}$ are smooth combinations of the smooth functions $r^{-1}$, $x_i$, $\cos (kr-\omega t)$ and $\sin (kr-\omega t)$ (Continuity and derivatives of positive-base real powers, The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$, 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 $g(c)\mathrm{\ne }0$, $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$). The field is transverse in the sense that $E_{\mathrm{d}}\cdot n=0$ and $B_{\mathrm{d}}\cdot n=0$, because $n\times (n\times p_0)$ is orthogonal to $n$ and $n\times E_{\mathrm{d}}$ is orthogonal to $n$ by The cross product is bilinear, alternating, and orthogonal to both factors applied separately to real and imaginary parts, and $B_{\mathrm{d}}=n\times E_{\mathrm{d}}\mathrm{/}c$ holds by definition with $\mathrm{\mid }{B}_{\mathrm{d}}\mathrm{\mid }=\mathrm{\mid }{E}_{\mathrm{d}}\mathrm{\mid }\mathrm{/}c$. The model is meaningful at $p_0=0$, where both fields and the flux vanish identically. The varying radial amplitude $1\mathrm{/}r$ is not a plane-wave amplitude, and the componentwise real-part convention is that of Complex plane-wave fields: real parts of complex exponentials; the cross product, inner product and norms are those of The cross product in ${\mathbb{R}}^3$ and The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. No equation of motion of any source, no determination of $p_0$ from any charge distribution and no energy or flux reading beyond the pointwise definition of $S_{\mathrm{d}}$ in Vacuum electromagnetic energy density and energy flux (the Poynting vector) is part of this item.