Jefimenko's Equations

Two articles ago a warning was posted and left unpaid: you may not retard the fields. The retarded potentials are exact, but differentiating them — the step that produces $\mathbf{E}$ and $\mathbf{B}$ — acts on the retarded time hiding inside every integrand, and the naive "Coulomb's law with stale news" misses whole terms. This article does the differentiation honestly, once, in full, and writes down what actually comes out: Jefimenko's equations, the exact, causal, gauge-free field formulas of classical electrodynamics. Every field that has ever existed — around every antenna, atom, and lightning stroke — is an instance of these two integrals. Their anatomy repays study. The electric field carries three terms: the retarded Coulomb field ( $\rho/\mathscr{r}^2$ — statics with stale news, after all), a middle term driven by the rate of change of charge density ( $\dot\rho/c\mathscr{r}$ ), and a term driven by the rate of change of current ( $\dot{\mathbf{J}}/c^2\mathscr{r}$ ). The magnetic field carries two: retarded Biot–Savart plus a $\dot{\mathbf{J}}$ correction. The $1/\mathscr{r}$ terms are the ones the naive guess forgot — and they are the radiation: only they survive at infinity, and only changing currents generate them. Two chapter debts are settled on the spot. The Coulomb-gauge article's causality acquittal gets its receipts: here are the fields, built from past sources only, no gauge anywhere in sight. And a deeper puzzle is resolved — why did quasistatics ever work? The answer is a small miracle of cancellation: for slowly varying currents, the first-order retardation error in Biot–Savart is exactly repaid by the $\dot{\mathbf{J}}$ term, making the static law good to second order. Your intuition survived four chapters of time-dependent physics because Jefimenko's equations were quietly covering for it.

Key takeaways

Differentiating the retarded potentials honestly gives the exact fields of any localized source: \mathbf{E}(\mathbf{r},t) = \frac{1}{4\pi\varepsilon_0}\int\left[\frac{\rho(t_r)}{\mathscr{r}^2}\hat{\boldsymbol{\mathscr{r}}} + \frac{\dot\rho(t_r)}{c\,\mathscr{r}}\hat{\boldsymbol{\mathscr{r}}} - \frac{\dot{\mathbf{J}}(t_r)}{c^2\mathscr{r}}\right]d^3r' \mathbf{B}(\mathbf{r},t) = \frac{\mu_0}{4\pi}\int\left[\frac{\mathbf{J}(t_r)}{\mathscr{r}^2} + \frac{\dot{\mathbf{J}}(t_r)}{c\,\mathscr{r}}\right]\times\hat{\boldsymbol{\mathscr{r}}}\;d^3r'.
These are the time-dependent generalizations of Coulomb and Biot–Savart: the $1/\mathscr{r}^2$ terms are the static laws with retarded sources; the $1/\mathscr{r}$ terms — sourced by $\dot\rho$ and $\dot{\mathbf{J}}$ — are new, and they are the radiation.
The equations are manifestly causal and manifestly gauge-free: fields at $(\mathbf{r},t)$ are built from charge and current on the past light cone, period — the receipts promised by the Coulomb-gauge causality acquittal.
Quasistatic vindication: expanding in the retardation, the first-order error of naive Biot–Savart is exactly cancelled by the $\dot{\mathbf{J}}$ term — the static magnetic law with the present current is correct through first order, which is why circuit-era magnetostatics never failed.
The electric analog holds too: Coulomb's law with the present $\rho$ is also good to second order — but the $\dot{\mathbf{J}}$ term has no static partner and survives as genuine radiation.
Radiation requires changing currents: steady $\rho$ and $\mathbf{J}$ , however arranged, produce only $1/\mathscr{r}^2$ fields; only $\dot{\mathbf{J}} \neq 0$ (accelerating charge) reaches infinity.
Key idea: Jefimenko's equations are the honest derivatives of the retarded potentials — retarded Coulomb and Biot–Savart plus $\dot\rho$ and $\dot{\mathbf{J}}$ terms of $1/\mathscr{r}$ range that carry all the radiation and enforce causality with no reference to any gauge.

The core insight

Jefimenko's equations — obtained by honestly differentiating the retarded potentials — express the exact fields as retarded Coulomb and Biot–Savart integrals plus $\dot\rho$ and $\dot{\mathbf{J}}$ terms of $1/\mathscr{r}$ range that carry all the radiation.

Where this lives today

Jefimenko's equations are the conscience of computational electromagnetics. Time-domain integral-equation solvers evaluate exactly these convolutions over meshed source histories; near-field-to-far-field transformations in FDTD codes are (1)–(2) applied on a bounding surface; and the "causality audits" of metamaterial and superluminal-media claims consist of checking reported fields against retarded-source integrals of precisely this form. In plasma and beam physics, the equations are the standard for validating particle-in-cell field solvers, because they give the exact fields of a prescribed current history with no gauge or grid artifacts to argue about. They also settled a pedagogical war. The 1991 Griffiths–Heald paper ("Time-dependent generalizations of the Biot–Savart and Coulomb laws") brought the equations into mainstream teaching and pinned down exactly which naive retarded guesses work (potentials; B with present current, to second order) and which fail (retarded Coulomb for $\mathbf{E}$ ) — the structure this article inherited. And the vindication theorem quietly underwrites entire industries: magnetic resonance imaging computes its gradient and RF coils with Biot–Savart at kilohertz-to-megahertz frequencies, transformer design lives at 50–60 Hz, and both are licensed not by "retardation is small" hand-waving but by the exact first-order cancellation — engineering trust, secured by a conspiracy between staleness and change.

Origins · 1 minOliver Heaviside and how this was discovered

Oliver Heaviside

1850–1925 · Camden Town and Devon — the self-taught engineer who put the fields first

Heaviside left school at sixteen, learned electromagnetism as a telegraph operator, and then — jobless, half-deaf, living with his parents — rewrote Maxwell. The twenty quaternion-laden equations of the Treatise became, in Heaviside's hands (1884–85), the four vector equations every student now learns; the curl, divergence, and the modern $\mathbf{E}$ , $\mathbf{B}$ notation are substantially his. He championed exactly this article's viewpoint: the fields are the physics, the potentials mere scaffolding (he campaigned, with characteristic venom, to "murder $\psi$ and $\mathbf{A}$ "), and he computed the fields of moving charges directly — anticipating pieces of the Liénard–Wiechert story, including the deformed field of a fast charge, before either namesake. His operational calculus, invented to solve telegraph-line transients and scorned by mathematicians ("Shall I refuse my dinner because I do not fully understand the process of digestion?"), was vindicated decades later as Laplace-transform theory. The exact causal field equations he never quite assembled were finally organized by Panofsky–Phillips (1962) and named for Jefimenko (1966) — but their fields-first, source-history-in, fields-out architecture is Heaviside's program, completed.

History

1867

— Lorenz's retarded potentials contain the fields implicitly; nobody differentiates them in print

1884

— Heaviside recasts Maxwell into the four modern field equations; begins computing moving-charge fields directly

1888

— Heaviside publishes the distorted field of a moving point charge — fields-first electrodynamics in action

1912

— Schott's treatise systematizes radiation theory, still potential-first

1962

— Panofsky and Phillips's textbook presents the explicit retarded field integrals

1966

— Jefimenko's Electricity and Magnetism makes the equations a pedagogical centerpiece; the name attaches

1991

— Griffiths and Heald's paper fixes the modern teaching canon: which retarded guesses work, which fail

2000

s — time-domain integral-equation solvers make the equations industrial machinery

The physical picture

Think of the field at your location as a report compiled from the past light cone, with three species of correspondent filing to the electric desk. The first correspondent reports where the charge was: the retarded Coulomb term, $1/\mathscr{r}^2$ , pointing along the stale line of sight — the only reporter statics ever employed. The second reports how fast the charge count was changing: the $\dot\rho$ term, also radial but falling only as $1/\mathscr{r}$ — a longer-range stringer whose job, it will turn out, is mostly to correct the first one's stale bookkeeping. The third reports how the currents were accelerating: the $\dot{\mathbf{J}}$ term, $1/\mathscr{r}$ , pointing along the (negative) current acceleration rather than the line of sight — the only correspondent whose dispatches survive to infinity, because a $1/\mathscr{r}$ field carries a $1/\mathscr{r}^2$ Poynting flux through spheres of area $\mathscr{r}^2$ : finite power forever outward. Radiation is the third correspondent, and nothing else. The magnetic desk runs leaner: retarded Biot–Savart ( $\mathbf{J}\times\hat{\boldsymbol{\mathscr{r}}}/\mathscr{r}^2$ ) plus one $\dot{\mathbf{J}}\times\hat{\boldsymbol{\mathscr{r}}}/c\mathscr{r}$ stringer. No charge-density reporter at all — magnetic fields never cared where charge sits, only where it flows.

Figure

Anatomy of Jefimenko's electric field — three terms, three ranges, three jobs; only the current-acceleration term survives to infinity

Two doctrinal points before the algebra. First, gauge-freedom: these formulas contain $\rho$ , $\mathbf{J}$ , and geometry — no potentials, no gauge condition, nothing to choose. They are what the Coulomb-gauge causality acquittal appealed to as "the receipts": however weird a gauge's bookkeeping (instantaneous $V$ , nonlocal transverse currents), the fields it computes must equal these integrals, which manifestly read only the past light cone. Second, historical irony: the exact field equations of electrodynamics were not written in a textbook until 1962 (Panofsky & Phillips) and not named until Oleg Jefimenko's 1966 text made them a teaching centerpiece — nine decades after Maxwell. Nobody needed them: potentials were easier for every practical calculation, and the quasistatic cancellation below meant the static laws quietly kept working far beyond their warranty.

The equations, earned

Everything flows from the retarded potentials of two articles back, plus their differentiation kit: $\nabla t_r = -\hat{\boldsymbol{\mathscr{r}}}/c$ , so any retarded source obeys $\nabla f(t_r) = -\dot f\,\hat{\boldsymbol{\mathscr{r}}}/c$ .

\mathbf{E}(\mathbf{r},t) = \frac{1}{4\pi\varepsilon_0}\int\left[\frac{\rho(\mathbf{r}',t_r)}{\mathscr{r}^2}\,\hat{\boldsymbol{\mathscr{r}}} + \frac{\dot\rho(\mathbf{r}',t_r)}{c\,\mathscr{r}}\,\hat{\boldsymbol{\mathscr{r}}} - \frac{\dot{\mathbf{J}}(\mathbf{r}',t_r)}{c^2\,\mathscr{r}}\right]d^3r', \qquad (1)

\mathbf{B}(\mathbf{r},t) = \frac{\mu_0}{4\pi}\int\left[\frac{\mathbf{J}(\mathbf{r}',t_r)}{\mathscr{r}^2} + \frac{\dot{\mathbf{J}}(\mathbf{r}',t_r)}{c\,\mathscr{r}}\right]\times\hat{\boldsymbol{\mathscr{r}}}\;d^3r'. \qquad (2)

where: $\mathscr{r} = |\mathbf{r}-\mathbf{r}'|$ , $t_r = t - \mathscr{r}/c$ , dots are time derivatives evaluated at $t_r$ . Read the structure before deriving it: set all dots to zero and (1)–(2) collapse to Coulomb and Biot–Savart exactly — statics is the no-news-is-new-news limit. The dotted terms carry the dynamics, and their $1/\mathscr{r}$ range divides the world: near a source ( $\mathscr{r} \ll c\tau$ , with $\tau$ the source timescale) the static-shaped terms dominate; far away ( $\mathscr{r} \gg c\tau$ ) only the $\dot{\mathbf{J}}$ term (and $\dot\rho$ term, though it is radial and — after the transversality bookkeeping of the coming radiation articles — drops out of the far-zone transverse field) matters. Note also what is absent: no $\ddot\rho$ , no memory integrals, no fields-sourcing-fields. Given the source history, two integrals, done.

Derivation AThe electric field — differentiating the retarded potentials honestly

The goal is equation (1): $\mathbf{E} = -\nabla V - \partial_t\mathbf{A}$ evaluated on the retarded potentials with every retarded-time dependence tracked. The strategy is pure bookkeeping discipline, and half the work is already banked: the gradient of the retarded $V$ was computed in the verification derivation of Retarded Potentials (its Step 1), and is recalled rather than recomputed. What remains is the time derivative of $\mathbf{A}$ — trivial, since $\partial t_r/\partial t = 1$ — and assembly. Watching which term comes from which parent is the point: the naive "retarded Coulomb" guess corresponds to keeping only the kernel-gradient piece and forgetting that moving the field point also re-dates every dispatch.

Step 1. Recall the gradient of the retarded scalar potential (banked two articles ago):

\nabla V = \frac{1}{4\pi\varepsilon_0}\int\left[-\frac{\dot\rho(t_r)}{c}\frac{\hat{\boldsymbol{\mathscr{r}}}}{\mathscr{r}} - \rho(t_r)\frac{\hat{\boldsymbol{\mathscr{r}}}}{\mathscr{r}^2}\right]d^3r'.

Why two pieces: the second is the static gradient acting on the $1/\mathscr{r}$ kernel; the first exists because stepping the field point closer re-dates the news fresher — the derivative of staleness. This first piece is precisely what the naive guess forgets.

Step 2. Differentiate the vector potential in time. Since $t_r = t - \mathscr{r}/c$ has $\partial t_r/\partial t = 1$ at fixed positions, the derivative slides through the integral onto the source:

\frac{\partial\mathbf{A}}{\partial t} = \frac{\mu_0}{4\pi}\int\frac{\dot{\mathbf{J}}(t_r)}{\mathscr{r}}\,d^3r'.

Step 3. Assemble $\mathbf{E} = -\nabla V - \partial_t\mathbf{A}$ , using $\mu_0 = 1/\varepsilon_0 c^2$ to put both contributions over the common constant:

\mathbf{E} = \frac{1}{4\pi\varepsilon_0}\int\left[\frac{\rho(t_r)}{\mathscr{r}^2}\hat{\boldsymbol{\mathscr{r}}} + \frac{\dot\rho(t_r)}{c\mathscr{r}}\hat{\boldsymbol{\mathscr{r}}} - \frac{\dot{\mathbf{J}}(t_r)}{c^2\mathscr{r}}\right]d^3r'.

Why each parent matters: the Coulomb term and the $\dot\rho$ term both descend from $V$ — they are radial, tied to the line of sight; the $\dot{\mathbf{J}}$ term descends from $\mathbf{A}$ and points along the current's acceleration, with no obligation to the line of sight. That directional freedom is what will let it be transverse in the far zone — the geometric prerequisite for radiation.

Step 4. The magnetic companion. $\mathbf{B} = \nabla\times\mathbf{A}$ requires the curl of $\mathbf{J}(t_r)/\mathscr{r}$ ; the product rule gives a kernel piece and a retardation piece, $\nabla\times[\mathbf{J}(t_r)/\mathscr{r}] = \mathbf{J}(t_r)\times\hat{\boldsymbol{\mathscr{r}}}/\mathscr{r}^2 + \dot{\mathbf{J}}(t_r)\times\hat{\boldsymbol{\mathscr{r}}}/(c\mathscr{r})$ — the second term from $\nabla\times\mathbf{J}(t_r) = -\tfrac{1}{c}\nabla\mathscr{r}\times\dot{\mathbf{J}} = \dot{\mathbf{J}}\times\hat{\boldsymbol{\mathscr{r}}}/c$ — and integrating yields (2). Verified: matches Griffiths Eqs. 10.36–10.38 and Jackson §6.5 (Eqs. 6.55–6.56); setting all time derivatives to zero recovers Coulomb and Biot–Savart exactly; the pair reproduces the wire fields of the retarded-potentials article when applied to its switch-on problem.

The quasistatic vindication

Here is the puzzle four chapters of magnetostatics quietly posed. Biot–Savart was derived for steady currents; engineers apply it to 50 Hz transformers, audio circuits, even MHz coils, with the present current $I(t)$ pasted in — and it works superbly. Why? The naive answer — "retardation is small" — is only half the story, because retardation enters at first order in $\mathscr{r}/c\tau$ , and first-order errors are not usually invisible. The true answer is a conspiracy.

Figure

The quasistatic conspiracy — the staleness error of retarded Biot–Savart and the J-dot term cancel at first order, leaving the present-current static law accurate to second order

Go deeper · Derivation B · 2 minWhy Biot–Savart never failed you — the first-order cancellation

Derivation BWhy Biot–Savart never failed you — the first-order cancellation

The goal is the quasistatic expansion of Jefimenko's $\mathbf{B}$ , equation (2), for sources varying slowly on the light-crossing time — and the punch line that the naive Biot–Savart law with the present current is correct not to zeroth but to second order in the retardation. The strategy: Taylor-expand both retarded terms about the present time $t$ and watch the first-order pieces annihilate. This cancellation is the magnetic sibling of the middle-term cancellation in the retarded-potential verification, and it is the honest license behind every quasistatic magnetic calculation in the earlier chapters.

Step 1. Expand the sources about the present. With $t_r - t = -\mathscr{r}/c$ :

\mathbf{J}(t_r) = \mathbf{J}(t) - \frac{\mathscr{r}}{c}\dot{\mathbf{J}}(t) + \frac{\mathscr{r}^2}{2c^2}\ddot{\mathbf{J}}(t) - \cdots, \qquad \dot{\mathbf{J}}(t_r) = \dot{\mathbf{J}}(t) - \frac{\mathscr{r}}{c}\ddot{\mathbf{J}}(t) + \cdots

Why the expansion is legitimate: it requires the source to change smoothly over the light-crossing time to the field point — the quasistatic condition $\mathscr{r} \ll c\tau$ , i.e. field points well inside a source wavelength.

Step 2. Insert into the integrand of (2) and collect orders:

\frac{\mathbf{J}(t_r)}{\mathscr{r}^2} + \frac{\dot{\mathbf{J}}(t_r)}{c\,\mathscr{r}} = \frac{\mathbf{J}(t)}{\mathscr{r}^2} \underbrace{- \frac{\dot{\mathbf{J}}(t)}{c\,\mathscr{r}} + \frac{\dot{\mathbf{J}}(t)}{c\,\mathscr{r}}}_{=\ 0} + \underbrace{\frac{\ddot{\mathbf{J}}(t)}{2c^2} - \frac{\ddot{\mathbf{J}}(t)}{c^2}}_{-\,\ddot{\mathbf{J}}(t)/2c^2} + \cdots

The first-order terms cancel exactly: the staleness correction to the Biot–Savart term is repaid, to the penny, by the leading piece of the $\dot{\mathbf{J}}$ term. Why this is a conspiracy and not luck: the two terms have entirely different parentage — one is the Taylor error of sampling the past, the other a genuine new term of the dynamical theory — yet their coefficients are locked by the same structure that made the retarded potentials solve the wave equation. Nature arranged for the static law to be better than its derivation.

Step 3. Read off the theorem:

\mathbf{B}(\mathbf{r},t) = \frac{\mu_0}{4\pi}\int \frac{\mathbf{J}(\mathbf{r}',t)\times\hat{\boldsymbol{\mathscr{r}}}}{\mathscr{r}^2}\,d^3r' \times\left[1 + \mathcal{O}\!\left(\left(\frac{\mathscr{r}}{c\tau}\right)^{\!2}\right)\right]

— Biot–Savart with the present current, with no first-order error. At 50 Hz and one meter, the neglected terms are of order $(2\pi \mathscr{r}/\lambda)^2 \sim 10^{-12}$ : magnetostatics was never really an approximation in the lab; it was an identity with invisible corrections.

Step 4. The electric side, and where the vindication ends. The same expansion applied to (1)'s first two terms gives Coulomb's law with the present $\rho$ , also good through first order — but the third term, $-\dot{\mathbf{J}}/c^2\mathscr{r}$ , has no static partner to hide in. In the quasistatic zone it is small ( $\mathcal{O}(v/c)^2$ relative); in the far zone it is everything. The vindication is a near-zone theorem; radiation is precisely the part of the field that no present-time static law can mimic. Verified: Griffiths §10.2.2 (Eq. 10.39 and the surrounding discussion); the exact wire computation of worked example 2 exhibits the cancellation non-perturbatively — the two integrand pieces merge into the static form exactly.

Limiting cases

case	result	meaning
statics (all dots zero)	(1)–(2) → Coulomb + Biot–Savart	the parent laws, recovered exactly — with retarded sampling now irrelevant
quasistatic zone ( $\mathscr{r}\ll c\tau$ )	static laws with present sources, errors $\mathcal{O}(\mathscr{r}/c\tau)^2$	the vindication theorem: first-order retardation cancels
far zone ( $\mathscr{r}\to\infty$ )	only $1/\mathscr{r}$ terms survive; transverse part of $-\dot{\mathbf{J}}$ dominates $\mathbf{E}$	radiation: finite Poynting flux to infinity, dipole formula three articles ahead
point charge on a trajectory	delta-function sources; integrals collapse with the LW Jacobian	next two articles — Jefimenko is correct but the collapse is subtle, so the potential route is used instead
no time-varying current anywhere	$\mathbf{E}, \mathbf{B} \sim 1/\mathscr{r}^2$ only	no radiation without $\dot{\mathbf{J}}$ : accelerating charge is the only transmitter nature sells

Common trapTwo readings of Jefimenko are wrong — the equations are exact, but they are not a causal metaphysics or a fields-free shortcut

First error: treating (1)–(2) as proving that charges alone cause fields and that fields never cause fields ("Faraday's law is a myth"). The equations express the fields of given source histories with retarded data; they are silent about self-consistency (sources respond to fields, which respond to sources — the coupled problem), and a free incoming wave, allowed by different boundary data, appears in neither integral. Jefimenko's own strong interpretive claims go beyond what the mathematics warrants; use the equations as exact bookkeeping, not ontology. Second error: reaching for (1)–(2) as the default calculational tool. For almost every real problem — point charges, antennas, multipoles — differentiating after specializing the potentials is vastly cleaner than integrating the five-term Jefimenko integrands; that is why the next two articles return to the potential route for the moving point charge. Jefimenko's equations are the conceptual master formulas: consult them for structure, causality audits, and quasistatic error budgets; compute with potentials.

Worked examples

1 · The correction dial — when do the new terms matter? (confidence builder)

Setup. The dotted terms enter relative to the static ones at the ratio $\left(\dot\rho/c\mathscr{r}\right)/\left(\rho/\mathscr{r}^2\right) = \mathscr{r}\dot\rho/c\rho \sim \omega\mathscr{r}/c = 2\pi\mathscr{r}/\lambda$ for oscillation at $\omega$ . Evaluate the dial across three regimes.

Step 1. Power engineering: $f = 50$ Hz at $\mathscr{r} = 1$ m: $\omega\mathscr{r}/c = (314)(1)/(3\times10^8) = 1.0\times10^{-6}$ . The Jefimenko corrections are parts per million of the field — and by the vindication theorem, the surviving error is this number squared: $10^{-12}$ . Circuit theory is safe by twelve orders of magnitude.

Step 2. RF on a bench: $f = 1$ GHz at $\mathscr{r} = 10$ cm: $\omega\mathscr{r}/c = 2.1$ . Order unity — the "correction" terms are the field. There is no quasistatic reading of a microwave circuit at arm's length; this is why RF engineering is transmission-line and antenna theory rather than circuit theory.

Step 3. Broadcast reception: $f = 1$ MHz at $\mathscr{r} = 100$ m: $\omega\mathscr{r}/c = 2.1$ again — same dial, same verdict. An AM receiver 100 m from the tower already lives in Jefimenko's $1/\mathscr{r}$ terms; the $1/\mathscr{r}^2$ Coulomb–Biot–Savart pieces of the tower's field died within a wavelength (300 m) of the mast.

The lesson: one dimensionless dial, $2\pi\mathscr{r}/\lambda$ , decides which terms of (1)–(2) run the show — the same $d/\lambda$ logic as the retardation budget, now applied to field structure rather than source sampling.

2 · The steadily ramping wire — exact fields, and the cancellation caught in the act (exam level)

Setup. An infinite straight wire carries $I(t) = Kt$ for $t > 0$ (zero before), with $K = 100$ A/s. Compute $\mathbf{B}$ and $\mathbf{E}$ at distance $s = 0.5$ m exactly from Jefimenko's equations, and evaluate at $t = 1\ \mu$ s. (Griffiths Prob. 10.12 territory — done here as a Jefimenko showcase.)

Step 1. Set up the $\mathbf{B}$ integral. For a line current, (2) becomes $B_\varphi = \frac{\mu_0}{4\pi}\int dz\, s\left[\frac{I(t_r)}{\mathscr{r}^3} + \frac{\dot I(t_r)}{c\,\mathscr{r}^2}\right]$ with $\mathscr{r} = \sqrt{s^2+z^2}$ , the factor $s/\mathscr{r}$ coming from $|\hat{\mathbf{z}}\times\hat{\boldsymbol{\mathscr{r}}}|$ . Only elements with $t_r > 0$ , i.e. $|z| < \sqrt{c^2t^2 - s^2}$ , have reported in.

Step 2. Watch the integrand collapse. With $I(t_r) = K(t - \mathscr{r}/c)$ and $\dot I = K$ :

\frac{K(t - \mathscr{r}/c)}{\mathscr{r}^3} + \frac{K}{c\,\mathscr{r}^2} = \frac{Kt}{\mathscr{r}^3} - \frac{K}{c\,\mathscr{r}^2} + \frac{K}{c\,\mathscr{r}^2} = \frac{Kt}{\mathscr{r}^3}.

The retardation of the Biot–Savart term and the entire $\dot I$ term cancel exactly — Derivation B's conspiracy, exhibited non-perturbatively (a linear ramp has $\ddot I = 0$ , so the cancellation has no remainder from the expansion; only the finite reporting window survives as a correction).

Step 3. Integrate: $B_\varphi = \frac{\mu_0 K t\, s}{4\pi}\int_{-z_0}^{z_0}\frac{dz}{(s^2+z^2)^{3/2}} = \frac{\mu_0 K t}{2\pi s}\sqrt{1 - \frac{s^2}{c^2t^2}}$ , with $z_0 = \sqrt{c^2t^2 - s^2}$ . Numbers at $t = 1\ \mu$ s ( $I = 0.1$ mA, $ct = 300$ m): the square root is $1 - 1.4\times10^{-6}$ , so $B = \frac{\mu_0(10^{-4})}{2\pi(0.5)} = \mathbf{40.0\ pT}$ — Biot–Savart with the present current, to a part per million, three hundred meters of causal horizon notwithstanding.

Step 4. The electric field has no such luck. From (1), only the $\dot{\mathbf{J}}$ term contributes ( $\rho = 0$ on a neutral wire):

E_z = -\frac{\mu_0 K}{2\pi}\,\ln\!\left(\frac{ct + \sqrt{c^2t^2 - s^2}}{s}\right) = -\frac{\mu_0(100)}{2\pi}\ln(1200) = -\mathbf{0.142\ mV/m},

anti-parallel to the growing current (Lenz would approve) and growing logarithmically forever as the reporting window widens — a term with no static partner and no cancellation, the germ of the inductive electric field.

The lesson: the magnetic field forgets the dynamics (cancellation), the electric field remembers them (the lonely $\dot{\mathbf{J}}$ term). One source, both behaviors, straight from the anatomy of (1)–(2).

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §10.2.2 · Jackson, Classical Electrodynamics, 3rd ed., §6.5 · Zangwill, Modern Electrodynamics, §20.4 · Jefimenko, Electricity and Magnetism, 2nd ed. (1989) · Griffiths & Heald, Am. J. Phys. 59, 111 (1991)

Provenance. Equations follow Griffiths §10.2.2 (Eqs. 10.36–10.39) and Jackson §6.5. A verifier should: (1) re-derive $\mathbf{E}$ from $-\nabla V - \partial_t\mathbf{A}$ on the retarded potentials, confirming the three-term structure; (2) re-derive $\mathbf{B} = \nabla\times\mathbf{A}$ confirming the two-term structure with the $\dot{\mathbf{J}}\times\hat{\boldsymbol{\mathscr{r}}}/c\mathscr{r}$ term; (3) check the static limits (Coulomb, Biot–Savart); (4) verify the quasistatic expansion showing first-order cancellation (Eq. 10.39's discussion). Constants: $\mu_0\varepsilon_0 = 1/c^2$ exact. Addendum for this edition: the gradient of the retarded $V$ is recalled from Retarded Potentials (its verification derivation, Step 1) rather than recomputed — division of labor within the causality spine; this edition owns the field assembly (Derivation A), the quasistatic vindication with its exact cancellation (Derivation B, delivering the "why statics worked" payoff), the term-anatomy/range audit, and the Coulomb-gauge receipts connection promised by Coulomb Gauge's Derivation B. The point-charge specialization is deferred by name to Liénard–Wiechert Potentials and Fields of a Moving Point Charge (the potential route being calculationally superior — see the warning). Worked-example numbers ( $\omega\mathscr{r}/c$ dial $10^{-6}$ /2.1/2.1; wire $B = 40.0$ pT with $1-1.4\times10^{-6}$ horizon factor, $E = -0.142$ mV/m at $\ln 1200 = 7.09$ ) verified symbolically and numerically for this edition.

Builds on: Retarded Potentials · Inhomogeneous Wave Equations for Potentials · Lorenz Gauge · Coulomb Gauge (causality receipts) Leads to: Liénard–Wiechert Potentials · Fields of a Moving Point Charge · Electric Dipole Radiation · Magnetic Dipole Radiation

Why are the exact field equations of electrodynamics named after a 20th-century textbook author?

Because for ninety years nobody needed to write them down. The potential route (Lorenz 1867 onward) was calculationally superior for every practical problem — multipole expansions, point charges, antennas all flow more easily from $V$ and $\mathbf{A}$ — and the quasistatic cancellation meant experimenters saw no anomalies that demanded exact field formulas. The integrals appeared in scattered forms (Heaviside's moving-charge fields, Schott's treatise, research literature), reached textbook print in Panofsky & Phillips (1962), and became a named teaching centerpiece with Oleg Jefimenko's 1966 Electricity and Magnetism, whose insistent causal reading — fields determined by past sources, full stop — struck a chord. Griffiths & Heald (1991) then fixed the modern canon. The naming is genuinely anachronistic, a fact worth knowing: priority in physics tracks use, and the community only reached for these equations when conceptual questions (causality, what radiates, what quasistatics really assumes) became teaching priorities.

Do Jefimenko's equations prove that electric fields are never produced by changing magnetic fields?

They prove something narrower and more useful: that the fields of a given source history can be written without fields appearing as sources — $\mathbf{E}$ and $\mathbf{B}$ each expressed directly in $\rho$ and $\mathbf{J}$ . Jefimenko himself read this ontologically ("Faraday induction" as correlation, not causation, both fields being co-effects of the sources), and the reading has real pedagogical value against loose "the changing B creates E, which creates B..." talk. But the strong metaphysical claim overreaches: the formulas presuppose retarded boundary data (a free incoming wave appears in neither integral yet is perfectly physical), and in the self-consistent problem — sources responding to fields responding to sources — the causal graph is genuinely entangled. The defensible summary: given the source history and quiescent past, sources suffice to determine the fields; whether that licenses "only sources cause fields" is philosophy the equations neither need nor settle.

Why is there a ρ̇ term in E but no ρ term at all in B?

Trace the parentage in Derivation A. $\mathbf{B} = \nabla\times\mathbf{A}$ , and $\mathbf{A}$ is built from $\mathbf{J}$ alone — charge density never enters the vector potential, so no manipulation can plant a $\rho$ in $\mathbf{B}$ . (Deeper reading: static charge produces no magnetism, and where charge sits is exactly the information $\rho$ carries.) The electric field descends from both potentials: the $\rho$ and $\dot\rho$ terms from $-\nabla V$ , the $\dot{\mathbf{J}}$ term from $-\partial_t\mathbf{A}$ . The $\dot\rho$ term is the interesting middle child — radial like Coulomb, $1/\mathscr{r}$ -ranged like radiation. Its main structural role is revealed by the quasistatic expansion: it is the term that repairs the staleness of the retarded Coulomb piece (and via continuity, $\dot\rho = -\nabla'\cdot\mathbf{J}$ , it is secretly current bookkeeping too — some authors rewrite (1) to eliminate it entirely in favor of $\mathbf{J}$ terms).

If these equations are exact, why does the chapter now go back to potentials for the moving point charge?

Because exactness and convenience are different virtues. For a point charge, $\rho$ and $\mathbf{J}$ are delta functions of a moving argument, and every term of (1)–(2) requires integrating deltas whose arguments contain the integration variable through $t_r$ — the collapse produces Jacobian factors, and doing this correctly for five terms, then combining, is an error-factory (published wrong answers exist). The potential route runs the treacherous delta collapse once, at the level of two scalar integrals (the Liénard–Wiechert derivation, next article), and then differentiates — heavy algebra but honest algebra. The two routes provably agree; the chapter's division is: Jefimenko for structure and doctrine, Liénard–Wiechert for point-charge computation. This is also the general lesson of the pair of articles: the fields-first formulation is conceptually supreme and calculationally third-best.

The J̇ term in E points along the current's acceleration, not along the line of sight. Why is that so important?

Because radial fields cannot radiate. In the far zone, energy transport needs $\mathbf{E}$ and $\mathbf{B}$ transverse to the propagation direction (the Poynting flux must point outward along $\hat{\boldsymbol{\mathscr{r}}}$ ), and the two radial terms of (1) — Coulomb and $\dot\rho$ — can never supply a transverse electric field. The $\dot{\mathbf{J}}$ term is the only one free of the line of sight: far away, its component perpendicular to $\hat{\boldsymbol{\mathscr{r}}}$ , which is $-\dot{\mathbf{J}}_\perp/c^2\mathscr{r}$ , is the entire radiation field. This is also the Jefimenko-level origin of the $\sin\theta$ in dipole radiation: the transverse projection of a linear $\dot{\mathbf{J}}$ scales as $\sin\theta$ — the same projection the Coulomb-gauge article extracted from the transverse-current structure, and the pattern the dipole article will integrate into its power formula. One term, projected: that is what every antenna in the world is emitting.

How do Jefimenko's equations relate to Maxwell's equations — are they equivalent?

One-way traffic with a toll. Every Jefimenko field solves Maxwell's equations with the given $\rho$ , $\mathbf{J}$ (they descend from the retarded potentials, which were verified against the wave equations, which encode Maxwell in Lorenz gauge). But Maxwell's equations have more solutions: add any source-free field (an incoming wave from infinity, a cavity mode set ringing before your data began) and Maxwell is still satisfied while Jefimenko's integrals know nothing of it. The precise statement: Jefimenko = Maxwell + localized sources + quiescent past (no incoming radiation). That toll is exactly the retarded boundary condition prosecuted in the advanced-potentials article, and it is why Jefimenko's equations cannot be "the fundamental equations" — they are the fundamental solution of the fundamental equations, for the class of data nature usually supplies.

Can I see energy conservation directly in Jefimenko's equations?

Not easily, and the difficulty is instructive. Poynting's theorem is a local identity among fields, so it holds automatically for Jefimenko fields — but verifying it directly from (1)–(2) means multiplying retarded integrals against retarded integrals, producing double integrals over pairs of source elements with different retarded times: correlations between what one piece of the source did and what another piece did earlier. That structure is real physics (radiation is precisely the part of the field energy that escapes the mutual bookkeeping of the source's parts), but as an exercise it is brutal, and nobody audits energy that way. Practice: compute fields via potentials, form $\mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0$ , integrate over a far sphere — the route the dipole-radiation article takes to its $\omega^4$ power formula. The moral repeats: Jefimenko for structure, potentials for computation, Poynting for the ledger.