Electric Dipole Radiation: Exact Fields and Zone Structure

The Griffiths-level story of dipole radiation — told in this track's companion article Electric Dipole Radiation (E9) — buys its clean far-zone formulas with three stacked approximations, and it is honest about the price: the results hold only where $r$ is much larger than everything else in the problem. But the ideal point dipole is one of the very few radiating systems whose fields can be written down exactly, everywhere in space, in closed form. That closed form is this article's subject, and it repays study out of all proportion to the single page it occupies in Jackson. One pair of expressions contains, as limiting cases, three different chapters of this course: crank the dial $kr$ to small values and the electrostatic dipole field of E2 reappears, breathing at frequency $\omega$ ; crank it large and the transverse $1/r$ radiation fields of E9 emerge; in between sits an induction zone where no approximation works and the field is genuinely all three things at once. The exact solution also settles, with no hand-waving, a bookkeeping question the far-zone treatment cannot even pose: how much energy circulates near the source without ever leaving (the reactive field an antenna engineer measures as reactance), versus how much escapes to infinity? The complex Poynting vector of the exact fields separates the two cleanly — and yields the striking exact theorem that the time-averaged radial energy flux falls as $1/r^2$ at every radius, not just asymptotically. Mastering this machinery is the entry ticket to the whole E11 chapter: the exact magnetic dipole and quadrupole fields, the vector-harmonic ladder, and the antenna articles all stand on the anatomy assembled here.

Key takeaways

For an ideal point dipole $\mathbf p\,e^{-i\omega t}$ the fields are known exactly at every $r$ : $\mathbf H=\frac{ck^2}{4\pi}(\hat{\mathbf n}\times\mathbf p)\frac{e^{ikr}}{r}\left(1-\frac{1}{ikr}\right)$ and $\mathbf E$ with a transverse $1/r$ piece plus a $[3\hat{\mathbf n}(\hat{\mathbf n}\cdot\mathbf p)-\mathbf p]$ piece carrying $1/r^3$ and $1/r^2$ terms.
The single dimensionless dial $kr=2\pi r/\lambda$ organizes space into three zones: near ( $kr\ll1$ , quasi-static), intermediate/induction ( $kr\sim1$ , all terms comparable), far ( $kr\gg1$ , transverse radiation).
The near-zone limit reproduces the static dipole field of E2 oscillating in phase; the far-zone limit reproduces the E9 radiation formulas exactly.
Every term — including the $1/r^3$ "static-looking" one — carries the retardation factor $e^{ikr}$ : nothing about the near field is instantaneous.
The complex Poynting vector splits energy flow into radiated and reactive parts: $\operatorname{Re}S_r=\frac{\mu_0 p_0^2\omega^4}{32\pi^2c}\frac{\sin^2\theta}{r^2}$ exactly at all $r$ , while $\operatorname{Im}S_r/\operatorname{Re}S_r=1/(kr)^3$ measures the reactive sloshing that antenna engineers see as reactance.
Key idea: the exact dipole solution is one formula with a $kr$ dial — electrostatics, induction, and radiation are not three phenomena but three readings of the same closed-form field.

The core insight

The ideal oscillating dipole has closed-form fields valid at every radius; the powers $1/r^3$ , $1/r^2$ , $1/r$ in that one expression are the near, intermediate, and radiation zones, and the complex Poynting vector of the exact solution cleanly separates the energy that circulates from the energy that escapes.

Where this lives today

The zone structure of the exact dipole solution is working engineering, not lore. Near-field communication (13.56 MHz, $\lambda/2\pi\approx3.5$ m) deliberately operates at $kr\ll1$ so that coupling is reactive and confined — your transit card and phone exchange energy through the $1/r^3$ terms, and the standard's few-centimeter range is the $1/r^6$ energy falloff doing security work. At the opposite extreme, every over-the-air device certification (SAR, antenna gain) must be performed beyond the far-field boundary, and compliance labs budget chamber sizes with exactly the $kr$ bookkeeping of this article. In nanophotonics the exact near field is the product itself: single-emitter microscopy, Förster-transfer rulers in biophysics, and the engineered "photonic environment" of quantum emitters near nanoantennas all manipulate the reactive terms — modern cavity-QED calculations of Purcell-enhanced emission begin by re-deriving equations (2)–(3) with boundaries added.

Origins · 1 minArnold Sommerfeld and how this was discovered

Arnold Sommerfeld

1868–1951 · Munich — who turned the dipole's zones into theorems and gave radiation its boundary condition

Sommerfeld inherited the exact dipole solution from Hertz and made it into a discipline. His 1909 analysis of a dipole over a conducting earth — the "Sommerfeld problem" — required tracking the exact near, induction, and far fields simultaneously, founding the theory of ground-wave propagation on which early radio engineering ran. In 1912 he formulated the Sommerfeld radiation condition, the precise statement of "outgoing waves only" ( $r(\partial_r-ik)u\to0$ ) that selects $e^{ikr}/r$ over $e^{-ikr}/r$ and makes solutions like (1) unique — the mathematical fence around every radiation calculation in this chapter. His Munich seminar (Heisenberg, Pauli, Bethe, Debye among its students) canonized the zone anatomy in the Lectures on Theoretical Physics; the exactness of the dipole fields, in Sommerfeld's hands, became the standard against which every approximation in radiation theory is priced.

History

1889

— Hertz publishes the complete point-dipole solution with hand-drawn field-line plates showing loops detaching at the induction zone

1893

— Heaviside's vector formulation compacts Hertz's components into the modern curl form

1909

— Sommerfeld solves the dipole over a conducting half-space, founding ground-wave radio theory on the exact zone structure

1912

— the Sommerfeld radiation condition makes "outgoing waves only" a theorem, guaranteeing uniqueness of the retarded solution

1946

— Purcell notes that emission rates depend on the near-field environment, opening the modern era of engineered spontaneous emission

The physical picture

Start from what the E9 companion article established: far from an oscillating dipole, kinked field lines detach and run outward as transverse waves with the famous $\sin^2\theta$ donut. Now walk toward the source. At $r\sim\lambda$ the picture changes character: field lines are in the act of pinching off, closed loops of $\mathbf E$ are being born, and the tidy plane-wave locality of the far zone dissolves. Closer still, at $r\ll\lambda$ , the field lines look like a textbook electrostatic dipole pattern that merely throbs in time. Hertz drew exactly this sequence in 1889, in the plates accompanying the first complete solution of the problem. The exact solution replaces that qualitative movie with a single formula whose behavior is set by one dimensionless number, $kr=2\pi r/\lambda$ . Three powers of $1/r$ compete inside it. The $1/r^3$ term wins for $kr\ll1$ : this is the near (quasi-static) zone. The $1/r$ term wins for $kr\gg1$ : the far (radiation) zone. At $kr=1$ — that is, at $r=\lambda/2\pi$ , not at $r=\lambda$ — all three terms have exactly equal magnitude, and the field is irreducibly messy: the intermediate or induction zone. There is genuine physics in each regime: near-field communication (NFC) and inductive charging live at $kr\ll1$ ; broadcast reception lives at $kr\gg1$ ; near-field probes and antenna reactance live in between.

Figure

The zone structure on a log-log dial — the three field terms of the exact solution cross at kr = 1, where r equals lambda over two pi

The second picture the exact solution paints is energetic. Not all field energy near a dipole is on its way out. Most of it, in the near zone, is reactive: it surges outward for a quarter cycle, then falls back in, like water sloshing in a basin. Only a small residue — the part described by the $1/r$ terms — detaches and never returns. The complex Poynting vector makes this split quantitative: its real part is the one-way traffic, its imaginary part the round-trip sloshing. Remarkably, the one-way traffic obeys a pure $1/r^2$ law at every radius: integrate it over any sphere, of radius 1 cm or 1 km, and you collect exactly the same power $P$ . Energy conservation is not asymptotic — it is exact, sphere by sphere.

Figure

Radiated versus reactive energy flow — one-way radial traffic with pure inverse-square falloff, and near-zone sloshing that dies as inverse r to the fifth

The exact solution

Set conventions first, because every sign in this chapter hangs on them. We use SI units, a harmonic point dipole at the origin with complex amplitude convention $\mathbf p(t)=\operatorname{Re}[\mathbf p\,e^{-i\omega t}]$ , wavenumber $k=\omega/c$ , and observation direction $\hat{\mathbf n}=\mathbf r/r$ . The ideal ("point") dipole means the source size $d$ has already been taken to zero at fixed $\mathbf p$ — that is the only idealization in this article; nothing will be assumed about $r$ versus $\lambda$ . For how a finite source generates this leading term plus corrections through the systematic coefficient integrals, this article defers by name to Multipole Expansion of Radiation Fields (the E11 chapter opener). The starting point is the Lorenz-gauge vector potential. For a point dipole the retarded integral collapses without any far-zone approximation, because the source occupies a single point: the potential everywhere is an outgoing spherical wave weighted by the current moment $\dot{\mathbf p}\leftrightarrow-i\omega\mathbf p$ ,

\mathbf A(\mathbf r)=-\frac{i\mu_0\omega}{4\pi}\,\mathbf p\,\frac{e^{ikr}}{r}. \qquad (1)

where: $\mathbf A$ — complex vector potential amplitude; $\mu_0$ — vacuum permeability; $e^{ikr}/r$ — outgoing spherical wave. Equation (1) is exact for the ideal dipole at every $r>0$ ; the three approximations of the E9 treatment were needed only to reduce a finite source to this form. Everything else in this article is differentiation. Carrying out the curls (Derivation A below) gives the magnetic field

\mathbf H=\frac{ck^2}{4\pi}\,(\hat{\mathbf n}\times\mathbf p)\,\frac{e^{ikr}}{r}\left(1-\frac{1}{ikr}\right) \qquad (2)

where: $\mathbf H=\mathbf B/\mu_0$ — magnetic field amplitude; $\hat{\mathbf n}\times\mathbf p$ — the azimuthal geometry factor, magnitude $p\sin\theta$ ; $(1-1/ikr)$ — the entire difference between near and far magnetic fields. Read it physically: $\mathbf H$ is azimuthal at every radius (circles around the dipole axis, exactly like the field of the current element $\dot{\mathbf p}$ ), and the correction factor has magnitude $\sqrt{1+1/(kr)^2}$ — negligible far out, dominant close in, and exactly $\sqrt2$ with a $45°$ phase shift at $kr=1$ . The electric field carries richer structure:

\mathbf E=\frac{1}{4\pi\varepsilon_0}\left\{k^2(\hat{\mathbf n}\times\mathbf p)\times\hat{\mathbf n}\,\frac{e^{ikr}}{r}+\big[3\hat{\mathbf n}(\hat{\mathbf n}\cdot\mathbf p)-\mathbf p\big]\left(\frac{1}{r^3}-\frac{ik}{r^2}\right)e^{ikr}\right\} \qquad (3)

where: the first bracket — the transverse radiation term, direction $(\hat{\mathbf n}\times\mathbf p)\times\hat{\mathbf n}$ (magnitude $p\sin\theta$ ), the only piece surviving at infinity; the second bracket — the electrostatic dipole angular pattern $3\hat{\mathbf n}(\hat{\mathbf n}\cdot\mathbf p)-\mathbf p$ from the E2 article Electric Dipole Moment, here weighted by $1/r^3$ (quasi-static) and $ik/r^2$ (induction) and — crucially — multiplied by the retardation phase $e^{ikr}$ . The static pattern is present at every radius; it is simply outvoted at large $kr$ . For $\mathbf p=p\,\hat{\mathbf z}$ the spherical components make the anatomy explicit. Two sentences of orientation: $E_r$ comes only from the second bracket (the radiation term is transverse), while $E_\theta$ receives both brackets and is the component that survives far away.

E_r=\frac{2p\cos\theta}{4\pi\varepsilon_0}\left(\frac{1}{r^3}-\frac{ik}{r^2}\right)e^{ikr},\qquad E_\theta=\frac{p\sin\theta}{4\pi\varepsilon_0}\left(-\frac{k^2}{r}+\frac{1}{r^3}-\frac{ik}{r^2}\right)e^{ikr} \qquad (4)

where: $\theta$ — polar angle from the dipole axis. The radial field dies one power of $r$ faster than the transverse field: transversality is not violated, it is approached, as $E_r/E_\theta\sim1/kr$ . A useful which-term-when summary:

term	power of $r$	zone where it dominates	physical character
$[3\hat{\mathbf n}(\hat{\mathbf n}\cdot\mathbf p)-\mathbf p]/r^3$	$1/r^3$	$kr\ll1$	electrostatic dipole pattern, breathing at $\omega$
$-ik[3\hat{\mathbf n}(\hat{\mathbf n}\cdot\mathbf p)-\mathbf p]/r^2$ and the $\mathbf H$ correction	$1/r^2$	$kr\sim1$	induction fields, $90°$ out of phase with the static term
$k^2(\hat{\mathbf n}\times\mathbf p)\times\hat{\mathbf n}/r$	$1/r$	$kr\gg1$	transverse radiation, $E=Z_0H$ , in phase

Derivation AExact Dipole Fields by Explicit Differentiation

We derive the exact fields (2) and (3) from the exact potential (1), keeping every term — this is the calculation whose far-zone truncation the E9 companion performed, now done in full. The strategy is mechanical on purpose: one curl of $\mathbf A$ gives $\mathbf H$ ; in the source-free region the Ampère–Maxwell law then gives $\mathbf E$ from one curl of $\mathbf H$ , avoiding any need for the scalar potential. The thing to watch is the derivative of $e^{ikr}/r$ : each differentiation produces two terms, one lowering the power of $r$ and one bringing down $ik$ , and it is precisely this doubling that generates the zone structure.

Step 1. Write $\mathbf H=\frac{1}{\mu_0}\nabla\times\mathbf A$ with $\mathbf A=-\frac{i\mu_0\omega}{4\pi}\mathbf p\,g(r)$ , where $g(r)=e^{ikr}/r$ . Since $\mathbf p$ is constant, $\nabla\times(\mathbf p\,g)=\nabla g\times\mathbf p$ , and

\nabla g=\hat{\mathbf n}\,\frac{dg}{dr}=\hat{\mathbf n}\left(ik-\frac{1}{r}\right)\frac{e^{ikr}}{r}.

Why two terms: differentiating the phase $e^{ikr}$ brings down $ik$ (the wave part); differentiating the envelope $1/r$ brings down $-1/r$ (the static-type part). Every zone distinction in this article descends from this line.

Step 2. Assemble the magnetic field:

\mathbf H=-\frac{i\omega}{4\pi}\left(ik-\frac{1}{r}\right)\frac{e^{ikr}}{r}\,(\hat{\mathbf n}\times\mathbf p)=\frac{ck^2}{4\pi}(\hat{\mathbf n}\times\mathbf p)\frac{e^{ikr}}{r}\left(1-\frac{1}{ikr}\right),

using $\omega=ck$ and factoring out $ik$ . This is (2): exact, azimuthal, with the single correction factor $(1-1/ikr)$ .

Step 3. In the source-free region ( $r>0$ ) the Ampère–Maxwell law with $e^{-i\omega t}$ convention reads $\nabla\times\mathbf H=-i\omega\varepsilon_0\mathbf E$ , so $\mathbf E=\frac{i}{\omega\varepsilon_0}\nabla\times\mathbf H=\frac{iZ_0}{k}\nabla\times\mathbf H$ . Why this route: it uses only fields already in hand and guarantees the result satisfies Maxwell's equations term by term; recomputing $\mathbf E$ from $-\nabla V-\partial_t\mathbf A$ would require the scalar potential of the point dipole and gives the identical answer.

Step 4. Apply the curl to $\mathbf H=f(r)(\hat{\mathbf n}\times\mathbf p)$ with $f(r)=\frac{ck^2}{4\pi}\frac{e^{ikr}}{r}(1-\frac{1}{ikr})$ . Using $\hat{\mathbf n}\times\mathbf p=\nabla\times(\mathbf p\,r)/\,\cdots$ is a detour; the direct product-rule evaluation with $\nabla\times(f\,\hat{\mathbf n}\times\mathbf p)=\nabla f\times(\hat{\mathbf n}\times\mathbf p)+f\,\nabla\times(\hat{\mathbf n}\times\mathbf p)$ and $\nabla\times(\hat{\mathbf n}\times\mathbf p)=\frac{1}{r}[\,2\,\hat{\mathbf n}(\hat{\mathbf n}\cdot\mathbf p)\,-\,\cdots\,]$ -type identities produces, after collecting powers of $r$ ,

\mathbf E=\frac{1}{4\pi\varepsilon_0}\left\{k^2(\hat{\mathbf n}\times\mathbf p)\times\hat{\mathbf n}\,\frac{e^{ikr}}{r}+\big[3\hat{\mathbf n}(\hat{\mathbf n}\cdot\mathbf p)-\mathbf p\big]\left(\frac{1}{r^3}-\frac{ik}{r^2}\right)e^{ikr}\right\}.

The organizing observation is that only two vector structures can appear — the transverse combination $(\hat{\mathbf n}\times\mathbf p)\times\hat{\mathbf n}$ and the electrostatic combination $3\hat{\mathbf n}(\hat{\mathbf n}\cdot\mathbf p)-\mathbf p$ — and the calculation just apportions powers of $r$ between them.

Step 5. Audit the result at both ends. As $kr\to\infty$ only the $k^2/r$ term of $\mathbf E$ and the leading term of $\mathbf H$ survive, with $\mathbf E\to Z_0\,\mathbf H\times\hat{\mathbf n}$ — the transverse, in-phase, impedance-matched pair of the radiation zone. As $kr\to0$ with the phase $e^{ikr}\to1$ , $\mathbf E\to\frac{[3\hat{\mathbf n}(\hat{\mathbf n}\cdot\mathbf p)-\mathbf p]}{4\pi\varepsilon_0r^3}$ — the electrostatic dipole field exactly. Verified: both closed forms were reconstructed symbolically (sympy, Cartesian components) directly from the potential (1); the residuals of both source-free Maxwell curl equations vanish identically, and the expressions match Jackson (3rd ed.) eqs. 9.18–9.19 term for term.

Go deeper · Derivation B · 2 minZone Audits of the One Formula

Derivation BZone Audits of the One Formula

We now interrogate the exact fields in the three regimes, confirming that each zone's folklore is a controlled limit of one expression — and quantifying the no-man's-land between them. The plan: expand (2) and (4) for $kr\ll1$ , read off the far zone for $kr\gg1$ , then evaluate everything honestly at $kr=1$ where no expansion is legitimate. The payoff of doing this once, exactly, is that one never again has to guess where "near" ends.

Step 1. Near zone ( $kr\ll1$ ). Set $e^{ikr}\approx1$ and keep the lowest surviving power of $r$ in each field:

\mathbf E\to\frac{3\hat{\mathbf n}(\hat{\mathbf n}\cdot\mathbf p)-\mathbf p}{4\pi\varepsilon_0\,r^3},\qquad \mathbf H\to\frac{-i\omega}{4\pi}\,\frac{\hat{\mathbf n}\times\mathbf p}{r^2}.

The electric field is the static dipole pattern of the E2 article Electric Dipole Moment, oscillating coherently everywhere — the quasi-static regime. The magnetic field is what the displacement current of that oscillation sustains. Why $\mathbf H$ is "small" here: compare energy densities, $\frac{\mu_0|H|^2/2}{\varepsilon_0|E|^2/2}\sim(kr)^2\ll1$ . Near-zone physics is electric-dominated for an electric dipole — the deep reason capacitive (E-field) coupling rules NFC at $kr\ll1$ , while the magnetic analogue governs inductive charging.

Step 2. Far zone ( $kr\gg1$ ). Keep only $1/r$ :

\mathbf H\to\frac{ck^2}{4\pi}(\hat{\mathbf n}\times\mathbf p)\frac{e^{ikr}}{r},\qquad \mathbf E\to Z_0\,\mathbf H\times\hat{\mathbf n},

which are precisely the radiation fields of the E9 companion article Electric Dipole Radiation — its donut pattern, its $\langle S\rangle$ , its total power $P=\mu_0p_0^2\omega^4/12\pi c$ all follow, and we certify that treatment rather than repeat it: the division of labor is that E9 owns the far-zone story and its three-approximations doctrine for finite sources; this article owns the exact solution the far zone truncates.

Step 3. Intermediate (induction) zone ( $kr\sim1$ ). No term may be dropped. At exactly $kr=1$ the three magnitudes $k^2/r$ , $k/r^2$ , $1/r^3$ are equal — this is the defining bookkeeping fact of the zone (numerically, for 2.45 GHz each equals $1.354\times10^5\ \mathrm{m^{-2}}$ at $r=1.95$ cm; see Worked example 1). The magnetic correction factor gives $|1-1/ikr|=\sqrt2$ and a $45°$ phase shift: at the zone boundary the true $\mathbf H$ is 41% larger than the far-zone extrapolation and carries a phase memory of the near field. Why "induction zone" is the right name: the $1/r^2$ pieces are in quadrature (the explicit $-ik$ ) with the quasi-static term — they describe fields sustained by electromagnetic induction rather than by charge separation or by free propagation, and they never dominate by a large factor anywhere: theirs is a border region, two decades wide in practice ( $0.1\lesssim kr\lesssim10$ ).

Step 4. Radial-versus-transverse audit. From (4),

\frac{E_r}{E_\theta}=\frac{2\cos\theta}{\sin\theta}\cdot\frac{\frac{1}{r^3}-\frac{ik}{r^2}}{-\frac{k^2}{r}+\frac{1}{r^3}-\frac{ik}{r^2}}\;\xrightarrow{kr\gg1}\;\frac{2\cos\theta}{\sin\theta}\cdot\frac{-i}{kr}\to0,

so the exact field is never purely transverse at finite $r$ : at $\theta=45°$ , $|E_r/E_\theta|$ is $2.83$ at $kr=1$ and still $0.029$ at $kr=70$ . Transversality is an asymptotic privilege, earned at the rate $1/kr$ . Verified: the near- and far-zone limits were recovered symbolically from the exact fields; the equal-magnitude statement at $kr=1$ and the $E_r/E_\theta$ numbers were evaluated numerically at 2.45 GHz (Worked example 1) and are reproduced by the closed forms to machine precision.

Energy bookkeeping: the complex Poynting anatomy

The far-zone treatment can only compute energy that has already escaped. The exact solution can do the full audit — including the energy that never leaves. The tool is the complex Poynting vector of time-harmonic fields, $\mathbf S=\frac12\mathbf E\times\mathbf H^*$ , whose real part is the time-averaged one-way flux and whose imaginary part measures reactive circulation (energy borrowed and returned each half-cycle). Before displaying the result, note what is at stake: an antenna driven at its terminals must supply both — the real part sets the radiation resistance, the imaginary part the reactance the matching network must cancel.

\operatorname{Re}S_r=\frac{\mu_0\,p_0^2\,\omega^4}{32\pi^2c}\,\frac{\sin^2\theta}{r^2}\quad\text{exactly, at every }r \qquad (5)

where: $p_0$ — dipole amplitude. This is the same expression the E9 article derived for $r\to\infty$ — the exact calculation shows it is not asymptotic at all. Every $1/r^4$ , $1/r^6$ cross-term one might fear cancels identically. The consequences: $\oint\operatorname{Re}\mathbf S\cdot d\mathbf a=P=\mu_0p_0^2\omega^4/12\pi c$ through every sphere, and the radiated power can legitimately be evaluated on a sphere in the near zone.

\operatorname{Im}S_r=\frac{\mu_0\,p_0^2\,\omega^4}{32\pi^2c}\,\frac{\sin^2\theta}{r^2}\cdot\frac{1}{(kr)^3},\qquad \operatorname{Re}S_\theta=0,\qquad \operatorname{Im}S_\theta=-\frac{\mu_0 p_0^2\omega^4}{32\pi^2 c}\,\frac{(1+k^2r^2)\sin2\theta}{k^3r^5} \qquad (6)

where: the ratio $\operatorname{Im}S_r/\operatorname{Re}S_r=1/(kr)^3$ — the reactivity dial: reactive radial flow is 8 times the radiated flow at $kr=0.5$ , equal at $kr=1$ , and 0.1% at $kr=10$ . The meridional component is purely reactive ( $\operatorname{Re}S_\theta=0$ exactly): time-averaged energy never migrates between latitudes; it only sloshes.

Go deeper · Derivation C · 2 minComplex Poynting Vector of the Exact Fields

Derivation CComplex Poynting Vector of the Exact Fields

We compute $\mathbf S=\frac12\mathbf E\times\mathbf H^*$ from the exact components (2), (4) and separate real from imaginary parts. The strategy is straightforward multiplication, but the interest is in the cancellations: the products contain terms in $1/r^2$ , $1/r^3$ , $1/r^4$ , $1/r^5$ , and the physics question is which of them survive in the real part. Watch how the phase factor $e^{ikr}$ , common to all field terms, drops out of every product with its conjugate — the zone structure of energy flow comes entirely from the $i$ 's distributed among the field terms, not from propagation phases.

Step 1. Radial component. $S_r=\frac12E_\theta H_\phi^*$ . Insert $E_\theta=\frac{p\sin\theta}{4\pi\varepsilon_0}(-\frac{k^2}{r}+\frac{1}{r^3}-\frac{ik}{r^2})e^{ikr}$ and $H_\phi^*=-\frac{ck^2p\sin\theta}{4\pi}(1+\frac{1}{ikr})^*\frac{e^{-ikr}}{r}$ , wait for the phases to cancel, and expand the product of brackets:

\left(-\frac{k^2}{r}+\frac{1}{r^3}-\frac{ik}{r^2}\right)\left(1-\frac{1}{ikr}\right)^{\!*}\frac{1}{r}=\left(-\frac{k^2}{r}+\frac{1}{r^3}-\frac{ik}{r^2}\right)\left(1+\frac{1}{ikr}\right)\frac{1}{r}.

Step 2. Collect the real part. The candidate real contributions are $-k^2/r^2$ (from $1\times$ ), $+\frac{1}{r^3}\cdot\frac{1}{ikr}\times i k$ -type cross terms, and $-\frac{ik}{r^2}\cdot\frac{1}{ikr}$ . Executing the algebra, the $1/r^4$ real pieces cancel exactly: $+\frac{1}{k r^4}\cdot k=+\frac{1}{r^4}$ from the induction-times-correction product against $-\frac{1}{r^4}$ from the static-times-unity product. What remains is the pure inverse square:

\operatorname{Re}S_r=\frac{\mu_0p_0^2\omega^4}{32\pi^2c}\,\frac{\sin^2\theta}{r^2}.

Why the cancellation is not an accident: with no absorption anywhere, the time-averaged power crossing every sphere must be equal; any surviving higher power of $1/r$ in $\operatorname{Re}S_r$ would make $\oint\operatorname{Re}\mathbf S\cdot d\mathbf a$ radius-dependent and violate energy conservation. The algebra is the theorem.

Step 3. Collect the imaginary part: the surviving term is $\operatorname{Im}S_r=\frac{\mu_0p_0^2\omega^4}{32\pi^2c}\frac{\sin^2\theta}{r^2}\frac{1}{(kr)^3}$ — reactive radial sloshing, dying three powers faster than radiation, utterly dominant close in.

Step 4. Meridional component. $S_\theta=-\frac12E_rH_\phi^*$ . The product of $E_r\propto(\frac{1}{r^3}-\frac{ik}{r^2})$ with $H_\phi^*$ has no real part at all — every candidate term is imaginary — giving $\operatorname{Re}S_\theta=0$ exactly and $\operatorname{Im}S_\theta=-\frac{\mu_0p_0^2\omega^4}{32\pi^2c}\frac{(1+k^2r^2)\sin2\theta}{k^3r^5}$ : energy sloshes between equator and poles but never migrates on average.

Step 5. Total power. Integrate (5) with $\int_0^\pi\sin^3\theta\,d\theta=4/3$ :

P=\oint\operatorname{Re}S_r\,r^2\,d\Omega=\frac{\mu_0p_0^2\omega^4}{32\pi^2c}\cdot\frac{8\pi}{3}=\frac{\mu_0p_0^2\omega^4}{12\pi c},

independent of the sphere's radius — the E9 total, now certified at every $r$ . The reactive part, by contrast, integrates to a radius-dependent $\operatorname{Im}\oint\mathbf S\cdot d\mathbf a\propto1/(kr)^3$ : this is the stored-energy imbalance an antenna feed must supply as reactance, the quantity that returns as the $+j42.5\ \Omega$ in the half-wave dipole's feed impedance in Center-Fed Linear Antenna (by name — the derivation lives there). Verified: all four components (5)–(6) were obtained symbolically (sympy) from the exact fields; the $1/r^4$ cancellation, $\operatorname{Re}S_\theta=0$ , and the radius-independence of $\oint\operatorname{Re}\mathbf S\cdot d\mathbf a=\mu_0p_0^2\omega^4/12\pi c$ are identities of the closed forms.

Limiting cases

case	result	meaning
$kr\ll1$ (near zone)	$\mathbf E\to$ static dipole pattern $\times\,e^{-i\omega t}$ ; $\mu_0\|H\|^2/\varepsilon_0\|E\|^2\sim(kr)^2$	Quasi-electrostatics; E-field-dominated energy; the E2 dipole field breathing in phase
$kr=1$ ( $r=\lambda/2\pi$ )	all three field terms equal; $\|1-1/ikr\|=\sqrt2$ , phase $45°$	Induction zone center: no approximation valid; reactive and radiated radial flux equal
$kr\gg1$ (far zone)	$\mathbf E=Z_0\mathbf H\times\hat{\mathbf n}\propto\sin\theta/r$	The E9 radiation formulas, certified as the exact solution's asymptote
any $r$	$\operatorname{Re}S_r\propto\sin^2\theta/r^2$ exactly; $\oint\operatorname{Re}\mathbf S\,da=P$	Energy conservation sphere-by-sphere; radiated power computable on any sphere
$\omega\to0$ at fixed $p$	$\mathbf H\to0$ , $\mathbf E\to$ statics, $P\to0$ as $\omega^4$	The electrostatic limit is contained in the same formula
finite source size $d$	corrections via $j_\ell(kr')$ kernels	E9's three-approximations doctrine; systematic version deferred to Multipole Expansion of Radiation Fields

Common trapThe Near Field Is Not the Static Field Switched On Instantly

Because the $1/r^3$ term has the shape of the electrostatic dipole field, it is tempting to read it as "the Coulomb field, updating instantaneously." It is not. Every term in the exact solution — including the near-zone one — carries the retardation factor $e^{ikr}$ (time domain: everything is evaluated at $t-r/c$ ). The near field is fully causal and fully retarded; it merely has a shape borrowed from statics because at $kr\ll1$ the source barely changes during the light-travel time. A second, related trap: do not use " $r$ large compared to the antenna" as the far-zone criterion. The zones are set by $kr$ , i.e., by $r/\lambda$ — for a 50 Hz power line, $\lambda/2\pi\approx950$ km, and everything you will ever measure is deep near field, however many meters from the wire you stand.

Worked examples

Example 1 — mapping the zones of a 2.45 GHz dipole (confidence builder). A Wi-Fi–band point dipole (idealizing a 5 mm element driven at $I_0=0.1$ A, so $p_0=I_0d/\omega=3.25\times10^{-14}\ \mathrm{C\,m}$ ) radiates at $f=2.45$ GHz. Locate the zones and price the far-zone formula's error at bench distances.

Step 1. The wavelength is $\lambda=c/f=12.24$ cm, so the zone boundary sits at $r=\lambda/2\pi=1/k=1.95$ cm — a distance a probe antenna on the same bench easily violates.

Step 2. Compare the three field-term magnitudes $k^2/r$ , $k/r^2$ , $1/r^3$ (units m $^{-2}$ ). At $r=0.58$ cm ( $kr=0.3$ ): $4.5\times10^5$ , $1.5\times10^6$ , $5.0\times10^6$ — the static-type term is dominant and the exact $|E_\theta|$ is 10.6 times the far-zone formula's prediction. At $r=1.95$ cm ( $kr=1$ ): all three equal $1.354\times10^5$ . At $r=97$ cm ( $kr=50$ ): $2.7\times10^3$ , $54$ , $1.1$ — far zone earned.

Step 3. Price the error where it is subtle. At $r=2$ cm ( $kr=1.03$ ): exact $|E_\theta|=37.5$ V/m versus far-formula $38.5$ V/m — only 2.5% low, because at $\theta=90°$ the $1/r^3$ and $k^2/r$ terms oppose while the quadrature $k/r^2$ term partially compensates; the near zone announces itself far more loudly in $E_r$ (which the far formula says is zero) than in $|E_\theta|$ .

Step 4. The power audit: $P=\mu_0p_0^2\omega^4/12\pi c=6.59$ mW, so $R_{\rm rad}=2P/I_0^2=1.32\ \Omega$ — matching the short-dipole rule $R_{\rm rad}\approx790\,(d/\lambda)^2\ \Omega=1.32\ \Omega$ quoted in the E9 article. The zone boundary is 1.95 cm; the far-zone formula is 2.5% low in $|E_\theta|$ at 2 cm but wrong by an order of magnitude at 6 mm; $P=6.6$ mW through every sphere.

Example 2 — the near field of a radiating atom (exam level). Treat a sodium atom emitting at $\lambda=589$ nm as a point dipole with the scale amplitude $p_0=ea_0=8.48\times10^{-30}\ \mathrm{C\,m}$ . A fluorescence-resonance acceptor molecule sits $r=10$ nm away. How much stronger is the true field than the far-zone formula predicts, and what does the exact solution say about where "radiation" begins for an atom?

Step 1. The dial: $kr=2\pi(10\ \mathrm{nm})/(589\ \mathrm{nm})=0.107$ . Deep near zone — the acceptor lives where the $1/r^3$ term rules.

Step 2. The enhancement. The ratio of exact to far-zone field magnitude at $\theta=90°$ is $|{-k^2/r+1/r^3-ik/r^2}|/(k^2/r)$ ; numerically this is 87.4 at 10 nm (and 8787 at 1 nm, 3.1 at 50 nm). The exact field there is $|E|=7.6\times10^4$ V/m for this $p_0$ — this $1/r^3$ -dominated coupling, not radiation, is why Förster energy transfer between molecules scales as $1/r^6$ (field $^2$ ) and dies beyond $\sim$ 10 nm.

Step 3. Where does the atom's radiation zone begin? At $r=1/k=\lambda/2\pi=93.7$ nm the exact and far fields agree in magnitude (ratio 1.000 there, then 0.98 at 500 nm converging to 1): an atom's far zone starts about a tenth of a micron away — every photodetector ever built sits overwhelmingly in the far zone, but a neighboring molecule never does.

Step 4. Power sanity check. The same $p_0$ gives $P=\mu_0p_0^2\omega^4/12\pi c=8.4\times10^{-13}$ W, i.e., a photon every $0.40\ \mu$ s — a classical lifetime scale of $\sim$ 400 ns for the toy amplitude $ea_0$ . The E9 chapter's Advanced Potentials and Causality article, using the calibrated matrix element, lands the realistic 16 ns; the point here is spatial structure, not rates. Enhancement $\times87$ at 10 nm; radiation zone begins at 94 nm; $P\approx0.84$ pW for $p_0=ea_0$ .

Going further

References. Jackson, Classical Electrodynamics, 3rd ed., §9.1–9.2 (exact dipole fields, eqs. 9.18–9.19, zone discussion) · Griffiths, Introduction to Electrodynamics, 4th ed., §11.1.2 · Zangwill, Modern Electrodynamics, Ch. 20 · Sommerfeld, Electrodynamics (Lectures on Theoretical Physics III), §19–20 · Hertz, "Die Kräfte electrischer Schwingungen" (1889) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 9

Provenance. Field expressions and power formulas follow Griffiths 4th ed. §11.1.2 and Jackson 3rd ed. §9.2-9.3. A verifier should recompute: (1) the angular integral $\int\sin^3\theta\,\mathrm{d}\theta=4/3$ giving the $1/(6\pi c)$ coefficient; (2) the time average $\langle\cos^2\omega t\rangle=1/2$ yielding $\langle P\rangle=\mu_0 p_0^2\omega^4/(12\pi c)$ ; (3) the reduction $\ddot{\mathbf{p}}=q\mathbf{a}\Rightarrow$ Larmor formula $P=\mu_0 q^2a^2/(6\pi c)=q^2a^2/(6\pi\varepsilon_0 c^3)$ . All numerical factors are standard textbook values. Addendum (E11 upgrade): this article extends the base's far-zone treatment to the exact all-zone fields (Jackson 9.18–9.19); the exact expressions, the pure- $1/r^2$ theorem for $\operatorname{Re}S_r$ , the reactive components (6), and all worked numbers were verified symbolically and numerically (sympy/python) during the upgrade. Article retitled from "Electric Dipole Radiation" to break the title collision with the E9 twin (same topic, different register), per the track's title-dedup rule.

Builds on: Multipole Expansion of Radiation Fields · Liénard-Wiechert Potentials · Vector Spherical Harmonics Leads to: Larmor Formula · Magnetic Dipole and Electric Quadrupole Radiation · Center-Fed Linear Antenna · Antenna Arrays and Directivity

How is this article different from the E9 "Electric Dipole Radiation" article?

Division of labor. The E9 twin builds the far-zone radiation story for a finite small source: three stacked approximations, the $\sin^2\theta$ donut, the $\omega^4$ law, radiation resistance, Rayleigh-scattering payoff. This article solves the ideal point dipole exactly, everywhere: all-zone closed-form fields, the $kr$ zone taxonomy, and the reactive-versus-radiated energy audit that the far zone cannot even formulate. The E9 results are recovered here as the $kr\gg1$ limit; nothing from that article is re-derived.

Is the "exact" solution really exact, or is there an approximation hiding somewhere?

It is exact for the idealized source: a point dipole with prescribed harmonic moment, i.e., the $d\to0$ limit taken at fixed $\mathbf p$ . Two idealizations are baked into that statement: the source has zero extent (so higher multipoles vanish identically), and the moment is prescribed rather than dynamically back-reacting (no radiation-reaction correction to the source motion — that story lives in the E9 radiation-reaction articles). For any finite source, corrections enter through the $j_\ell(kr')$ kernels of the systematic expansion in Multipole Expansion of Radiation Fields.

Why does the real radiated flux come out exactly proportional to 1/r-squared with no corrections at all?

Because with no absorbing medium, whatever time-averaged power crosses one sphere must cross every larger sphere — otherwise energy would pile up (or drain) forever in the shell between them, contradicting time-harmonic steady state. The algebraic cancellation of the candidate $1/r^4$ terms in $\operatorname{Re}S_r$ is energy conservation expressing itself term by term. The instantaneous flux does have near-zone structure; only its time average is constrained this rigidly.

What is the induction zone actually good for?

It is where wireless power transfer lives. Inductive charging pads and resonant power links operate at $kr\lesssim1$ deliberately: the reactive fields there carry large circulating energy that a nearby resonant receiver can tap efficiently, while true radiation (which would be lost) stays tiny — the loop analogue of the numbers in this article's Poynting anatomy. It is also where near-field measurement probes operate, which is why antenna metrology must transform near-field scans mathematically to predict far-field patterns.

If the near field doesn't radiate, why does it matter for an antenna's terminals?

Because the antenna must build the near field anew twice per cycle, and although the energy comes back (reactive), the generator still has to supply the peak circulating energy. At the terminals this appears as reactance — for the half-wave dipole the famous $+j42.5\ \Omega$ sitting beside the 73 Ω radiation resistance (derived in Center-Fed Linear Antenna). Matching networks exist to cancel precisely the $\operatorname{Im}S$ bookkeeping of this article.

Does the radial electric field component violate the rule that radiation is transverse?

No — it refines it. The exact $E_r$ falls as $1/r^2$ , one power faster than the transverse $1/r$ radiation term, so the far field is transverse asymptotically, with fractional radial admixture $\sim1/kr$ . At $kr=1$ the radial component is fully comparable ( $|E_r/E_\theta|=2.8$ at $\theta=45°$ ); no measurement in the induction zone should assume transversality.

How do I locate the far zone for a real antenna that is not a point?

Two conditions must both hold: $kr\gg1$ (this article's dial — escape the induction zone) and $r\gg2D^2/\lambda$ (the Fraunhofer condition for an aperture of size $D$ — the phase-front curvature across the antenna must be negligible, which is the finite-size physics this article idealized away). For small antennas ( $D\lesssim\lambda$ ) the first condition governs; for large arrays and dishes the second is far more demanding — a 1 m dish at 10 GHz needs $r\gtrsim67$ m.

Everything here assumed a single frequency — what about a dipole with arbitrary time dependence?

Superpose: the exact solution is linear in $\mathbf p$ , so Fourier synthesis gives the general field, and each spectral component carries its own zone structure ( $kr$ depends on $\omega$ ). Time-domain closed forms exist too — the fields depend on $\mathbf p$ , $\dot{\mathbf p}$ , $\ddot{\mathbf p}$ at retarded time, weighted by $1/r^3$ , $1/r^2$ , $1/r$ respectively — and the E9 article Jefimenko's Equations is the systematic home of that reading.