Fields of a Moving Point Charge

The Liénard–Wiechert potentials are the summit camp; this article is the summit. Differentiate them — honestly, with the retarded time's full dependence on field point and clock tracked through every term — and out comes the complete electromagnetic field of a point charge in arbitrary motion:

\mathbf{E} = \frac{q}{4\pi\varepsilon_0}\,\frac{\mathscr{r}}{(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})^3}\left[(c^2 - v^2)\,\mathbf{u} + \boldsymbol{\mathscr{r}}\times(\mathbf{u}\times\mathbf{a})\right], \qquad \mathbf{B} = \frac{1}{c}\,\hat{\boldsymbol{\mathscr{r}}}\times\mathbf{E},

with $\mathbf{u} = c\hat{\boldsymbol{\mathscr{r}}} - \mathbf{v}$ . One formula, all of it: Coulomb's law hides in the first term at $v = 0$ ; the magnetic field of a current is the $\mathbf{B}$ equation at low speed; and every photon ever radiated lives in the second term. The bracket's two-part structure is the chapter's causality story reaching its verdict. The velocity field — $1/\mathscr{r}^2$ , sourced by where the charge is going — is the charge's dressed Coulomb field: flattened into a pancake at high speed, pointing, eerily, at the charge's present position. The acceleration field — $1/\mathscr{r}$ , transverse, proportional to $\mathbf{a}$ — is the correction notice broadcast whenever the extrapolation fails: the kink in the field lines that detaches and runs to infinity. That kink is radiation, and isolating its $1/\mathscr{r}$ from the velocity field's $1/\mathscr{r}^2$ is the algebraic heart of everything from X-ray tubes to synchrotron beamlines. The formula also quietly stages the next chapter. Two charges flying abreast repel with a force reduced by $\gamma$ compared to statics — electric repulsion up, magnetic attraction nearly cancelling it — a conspiracy that makes no sense until relativity reveals magnetism as the velocity-dependent residue of Coulomb's law. The field of the moving charge is where electrodynamics runs out of classical explanations and starts pointing, insistently, at spacetime.

Key takeaways

Differentiating the Liénard–Wiechert potentials (with the retarded-time lemmas $\partial t_r/\partial t = 1/(1-\hat{\boldsymbol{\mathscr{r}}}\cdot\boldsymbol{\beta})$ and $\nabla t_r = -\hat{\boldsymbol{\mathscr{r}}}/[c(1-\hat{\boldsymbol{\mathscr{r}}}\cdot\boldsymbol{\beta})]$ ) gives the exact fields of an arbitrarily moving charge.
The electric field splits cleanly: velocity field $\propto (c^2-v^2)\mathbf{u}/\mathscr{r}^2$ — a generalized Coulomb field — and acceleration field $\propto \boldsymbol{\mathscr{r}}\times(\mathbf{u}\times\mathbf{a})/\mathscr{r}$ — transverse, and the only piece that survives at infinity.
The auxiliary vector $\mathbf{u} = c\hat{\boldsymbol{\mathscr{r}}} - \mathbf{v}$ is the formula's compass: for uniform motion it aims the velocity field at the charge's present position — retarded extrapolation done by algebra.
The uniform-velocity field is Coulomb flattened by $(1-\beta^2)/(1-\beta^2\sin^2\theta)^{3/2}$ : weakened $\times(1-\beta^2)$ fore-and-aft, strengthened $\times\gamma$ broadside — the Heaviside pancake ( $\gamma^3$ contrast at $\beta = 0.9$ : a factor 12).
The magnetic field is always $\mathbf{B} = \hat{\boldsymbol{\mathscr{r}}}\times\mathbf{E}/c$ : perpendicular to both $\mathbf{E}$ and the retarded line of sight, in every regime.
Radiation verdict: no acceleration, no $1/\mathscr{r}$ field, no energy to infinity — the Poynting flux of the velocity field dies as $1/\mathscr{r}^2$ ; the power integral of the acceleration field is the Larmor formula, owned downstream.
Key idea: the field of a moving point charge is a velocity part — a present-pointing, pancaked Coulomb field falling as $1/\mathscr{r}^2$ — plus an acceleration part — a transverse $1/\mathscr{r}$ field proportional to $\mathbf{a}$ — and only the second radiates.

The core insight

The field of a moving point charge splits into a velocity field — a generalized, present-pointing Coulomb field falling as $1/\mathscr{r}^2$ — and an acceleration field — transverse, proportional to the retarded acceleration, falling as $1/\mathscr{r}$ — and only the acceleration field carries energy to infinity.

Where this lives today

Equation (2) is the raw material of accelerator physics. The pancake field of Example 1 is what beam-position monitors, wakefield structures, and beam-beam interaction models actually compute with; CERN's beam-screen design and the "hourglass" luminosity corrections at colliders are velocity-field engineering. The acceleration field runs the light-source industry previewed in the last article — undulators arrange $\mathbf{a}(t_r)$ so that (4)'s kinks from successive wiggles interfere constructively — and the same $\mathbf{a}_\perp$ projection is why free-electron lasers can shape X-ray polarization by shaping electron trajectories. The kink picture has become directly visible. Streak-camera and attosecond metrology now resolve the field transients of accelerated charges; "single-cycle" terahertz pulses generated by transient photocurrents are Example-textbook kinks, one per acceleration event. In astrophysics, the same formula scaled up describes pulsar wind nebulae and jet knots; and in the lab's most violent corner, plasma-wakefield accelerators surf electrons on the velocity fields of drive bunches — one bunch's pancake is the next bunch's accelerating gradient, gigavolts per meter of it, all of it equation (2) evaluated by supercomputer.

Origins · 1 minJ.J. Thomson and how this was discovered

J.J. Thomson

1856–1940 · Cambridge — who first computed a moving charge's field energy, then discovered the charge that mattered

Before he found the electron, Thomson solved this article's problem in its infancy: his 1881 paper "On the electric and magnetic effects produced by the motion of electrified bodies" computed the field of a moving charged sphere and found that motion adds field energy proportional to $v^2$ — an electromagnetic contribution to inertia, the first appearance of the idea that mass might be partly field ("electromagnetic mass"), a thread that runs through Heaviside's exact convective solution (1889, the pancake), the Abraham–Lorentz electron models, and ultimately into $E = mc^2$ 's reinterpretation of the whole question. His 1881 analysis got the factor wrong (he lacked the full displacement-current bookkeeping; Heaviside corrected it), which makes him this article's patron in a second, more human sense: the field of a moving charge is genuinely hard, and even its pioneers stumbled through the retarded algebra. Sixteen years later, at the Cavendish, Thomson measured $e/m$ for cathode rays and gave physics the point charge whose fields this article computes — then watched his own kink fields, in the form of X-ray bremsstrahlung from stopped electrons, open a new science within months of Röntgen's discovery.

History

1881

— J.J. Thomson computes the field energy of a moving charged sphere: electromagnetic mass is born (with a wrong factor)

1888

— Heaviside publishes the exact convective field of a point charge: the pancake, with its (1 − β²sin²θ) anatomy

1898

— Liénard differentiates his potentials into the full velocity + acceleration fields and the radiated power

1904

— Abraham and Lorentz build electron models on the self-interaction of exactly these fields

1912

— Schott's treatise: the fields of this article become the systematic engine of radiation theory

1933

— Anderson's positron tracks: kink radiation (bremsstrahlung) in cloud chambers becomes particle-discovery instrumentation

1947

— synchrotron light seen at General Electric: the beamed acceleration field, visible to the naked eye

2004

— laser-plasma "bubble" accelerators: velocity-field wakes of relativistic bunches accelerate electrons at GV/m

The physical picture

Picture the charge as a lighthouse keeper maintaining an infinite web of field lines. While the keeper glides at constant velocity, the web glides with him — not dragged behind, but centered on him now: the retarded machinery, fed a predictable trajectory, extrapolates it perfectly, and every field line points at the present position. Speed does deform the web: field lines crowd toward the transverse plane, thinning fore and aft, until at $\beta = 0.99$ the pattern is a pancake seven times stronger at the equator and fifty times weaker at the poles than Coulomb — Heaviside drew this flattened field in 1888, twenty years before anyone could say "Lorentz contraction" about it. Now let the keeper stumble — accelerate for an instant. Field lines cannot break, and news travels at $c$ : outside a sphere of radius $c\,\Delta t$ the web still points at the extrapolated position (where the keeper would have been), while inside it points at the actual one. The two webs must join, and the joint is a kink — a shell of field line running nearly transverse to the radial direction, propagating outward at $c$ forever. That kink is the acceleration field: transverse because it connects mismatched radial patterns; falling as $1/\mathscr{r}$ because the shell's thickness is fixed while the mismatch it bridges grows linearly with distance; proportional to $\mathbf{a}$ because the mismatch is the failed extrapolation. Radiation is geometry's way of splicing an interrupted lie.

Figure

The two fields in one picture — the gliding charge carries a present-pointing pancake; the stumble launches a transverse kink shell that never comes back

The fields

The cast, all evaluated at the retarded time $t_r$ (unique, per the last article): $\boldsymbol{\mathscr{r}} = \mathbf{r} - \mathbf{w}(t_r)$ , velocity $\mathbf{v}$ , acceleration $\mathbf{a} = \dot{\mathbf{v}}(t_r)$ , and the formula's compass vector

\mathbf{u} \equiv c\,\hat{\boldsymbol{\mathscr{r}}} - \mathbf{v}, \qquad \boldsymbol{\mathscr{r}}\cdot\mathbf{u} = \mathscr{r}c - \boldsymbol{\mathscr{r}}\cdot\mathbf{v} = \mathscr{r}c\,(1 - \hat{\boldsymbol{\mathscr{r}}}\cdot\boldsymbol{\beta}). \qquad (1)

where: $\mathbf{u}$ deserves a reading before it earns its keep: it is the velocity of the image of the charge as seen in the retarded light — the line-of-sight approach velocity of the news, and (Derivation B's punch line) the vector that aims the static-limit field at the charge's extrapolated present position. Its dot product with $\boldsymbol{\mathscr{r}}$ is the Liénard–Wiechert denominator, now destined to appear cubed. The fields:

\boxed{\ \mathbf{E}(\mathbf{r},t) = \frac{q}{4\pi\varepsilon_0}\,\frac{\mathscr{r}}{(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})^3}\Big[\underbrace{(c^2 - v^2)\,\mathbf{u}}_{\text{velocity field}} \;+\; \underbrace{\boldsymbol{\mathscr{r}}\times(\mathbf{u}\times\mathbf{a})}_{\text{acceleration field}}\Big], \qquad \mathbf{B}(\mathbf{r},t) = \frac{1}{c}\,\hat{\boldsymbol{\mathscr{r}}}\times\mathbf{E}.\ } \qquad (2)

where: read the anatomy off the ranges. The velocity term carries $\mathscr{r}\cdot\mathbf{u}/(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})^3 \sim 1/\mathscr{r}^2$ : a genuine inverse-square field, the charge's dressed Coulomb presence, sourced by position-and-velocity alone. The acceleration term carries $\mathscr{r}\cdot\mathscr{r}\,a/(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})^3 \sim a/(c^2\mathscr{r})$ : one power of distance softer, linear in the acceleration, and — by the BAC-CAB structure $\boldsymbol{\mathscr{r}}\times(\mathbf{u}\times\mathbf{a})$ — always perpendicular to $\hat{\boldsymbol{\mathscr{r}}}$ ... the transverse kink, formula-fied. The denominator $(1-\hat{\boldsymbol{\mathscr{r}}}\cdot\boldsymbol{\beta})^3$ is the LW beaming factor's third power: the potentials' modest forward bias becomes the fields' ferocious one.

Derivation AThe retarded-time lemmas — the engine of every derivative

The goal is the two identities that make differentiating anything retarded possible: how the retarded time responds to nudging the clock, $\partial t_r/\partial t$ , and to nudging the field point, $\nabla t_r$ . Everything else in this article is bookkeeping powered by these. The strategy: differentiate the light-cone condition $\mathscr{r}(\mathbf{r}, t_r) = c(t - t_r)$ implicitly — the retarded time is defined implicitly, so its derivatives come from implicit differentiation, with the chain rule routing every hit through the moving endpoint $\mathbf{w}(t_r)$ .

Step 1. Clock nudge. Differentiate both sides of $\mathscr{r} = c(t - t_r)$ with respect to $t$ at fixed $\mathbf{r}$ . The left side changes only through $t_r$ : as the retarded moment slides, the source endpoint moves with velocity $\mathbf{v}$ , and the distance to a moving endpoint changes at $\partial\mathscr{r}/\partial t_r = -\hat{\boldsymbol{\mathscr{r}}}\cdot\mathbf{v}$ . So

-\hat{\boldsymbol{\mathscr{r}}}\cdot\mathbf{v}\,\frac{\partial t_r}{\partial t} = c\left(1 - \frac{\partial t_r}{\partial t}\right) \qquad\Longrightarrow\qquad \frac{\partial t_r}{\partial t} = \frac{1}{1 - \hat{\boldsymbol{\mathscr{r}}}\cdot\boldsymbol{\beta}} = \frac{\mathscr{r}c}{\boldsymbol{\mathscr{r}}\cdot\mathbf{u}}.

Why it exceeds unity for approach: when the charge closes in, one second of observer time corresponds to more than one second of source history — the collection sphere's sweep stretches, the same $1/D$ that built the potentials, now recognized as a rate. This is also precisely the Doppler compression rate of received signals.

Step 2. Position nudge. Differentiate the same condition with respect to $\mathbf{r}$ at fixed $t$ : $\nabla\mathscr{r} = -c\,\nabla t_r$ . But $\mathscr{r}$ depends on $\mathbf{r}$ twice — explicitly (gradient $\hat{\boldsymbol{\mathscr{r}}}$ ) and through $t_r$ (gradient $(\partial\mathscr{r}/\partial t_r)\nabla t_r = -(\hat{\boldsymbol{\mathscr{r}}}\cdot\mathbf{v})\nabla t_r$ ):

\hat{\boldsymbol{\mathscr{r}}} - (\hat{\boldsymbol{\mathscr{r}}}\cdot\mathbf{v})\,\nabla t_r = -c\,\nabla t_r \qquad\Longrightarrow\qquad \nabla t_r = -\frac{\hat{\boldsymbol{\mathscr{r}}}}{c\,(1 - \hat{\boldsymbol{\mathscr{r}}}\cdot\boldsymbol{\beta})} = -\frac{\boldsymbol{\mathscr{r}}}{\boldsymbol{\mathscr{r}}\cdot\mathbf{u}}.

The last compact form — using $\boldsymbol{\mathscr{r}}\cdot\mathbf{u} = \mathscr{r}c\,(1-\hat{\boldsymbol{\mathscr{r}}}\cdot\boldsymbol{\beta})$ — is the one to carry forward, and its Step-1 partner reads the same way: $\partial t_r/\partial t = \mathscr{r}c/(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})$ . Everything in Derivation B is these two lemmas plus product rules. Why the static kit generalizes: at $v = 0$ these collapse to $\partial t_r/\partial t = 1$ and $\nabla t_r = -\hat{\boldsymbol{\mathscr{r}}}/c$ — exactly the chain-rule kit the retarded-potentials and Jefimenko articles used for extended, slowly-structured sources. The moving point charge upgrades both by the same $1/D$ factor, and forgetting the upgrade is the single most common error in this computation. *Verified: matches Griffiths §10.3.2 (Eqs. 10.71, 10.78) and Jackson §14.1; both reduce correctly at $v=0$ and blow up only as $v \to c$ head-on, where the LW description itself strains.*

Go deeper · Derivation B · 2 minAssembling the field — velocity and acceleration parts emerge

Derivation BAssembling the field — velocity and acceleration parts emerge

The goal is equation (2) for $\mathbf{E}$ : apply $\mathbf{E} = -\nabla V - \partial_t\mathbf{A}$ to the Liénard–Wiechert potentials, with every derivative routed through the lemmas of Derivation A. The computation is the heaviest algebra in the chapter — Griffiths calls its result "the fruit of a labor not to be undertaken lightly" — so the strategy here is structural honesty over exhaustive expansion: write both potentials over the common denominator $\boldsymbol{\mathscr{r}}\cdot\mathbf{u}$ , differentiate with the product/chain rules, display the intermediate gradients that organize the mess, and show exactly where the bracket splits into a $\mathbf{u}$ -proportional (velocity) piece and an $\mathbf{a}$ -proportional (acceleration) piece. Each displayed intermediate is independently checkable.

Step 1. Set the targets. With $V = \dfrac{qc}{4\pi\varepsilon_0}\dfrac{1}{\boldsymbol{\mathscr{r}}\cdot\mathbf{u}}$ and $\mathbf{A} = \dfrac{qc}{4\pi\varepsilon_0 c^2}\dfrac{\mathbf{v}}{\boldsymbol{\mathscr{r}}\cdot\mathbf{u}}$ , the field is

\mathbf{E} = \frac{qc}{4\pi\varepsilon_0}\left[\frac{\nabla(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})}{(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})^2} - \frac{1}{c^2}\frac{\partial}{\partial t}\frac{\mathbf{v}}{\boldsymbol{\mathscr{r}}\cdot\mathbf{u}}\right].

Everything reduces to two objects: $\nabla(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})$ and $\partial_t(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})$ (plus $\partial_t\mathbf{v} = \mathbf{a}\,\partial_t t_r$ , already priced).

Step 2. The two workhorse derivatives of the denominator. Expanding $\boldsymbol{\mathscr{r}}\cdot\mathbf{u} = \mathscr{r}c - \boldsymbol{\mathscr{r}}\cdot\mathbf{v}$ and routing every implicit dependence through Derivation A's lemmas — using $\nabla\mathscr{r} = -c\,\nabla t_r = c\,\boldsymbol{\mathscr{r}}/(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})$ , the tensor product rule $\nabla\boldsymbol{\mathscr{r}} = \mathbb{1} - \mathbf{v}\otimes\nabla t_r$ (each hit on $\mathbf{w}(t_r)$ routes through $\nabla t_r$ ), hence $\nabla(\boldsymbol{\mathscr{r}}\cdot\mathbf{v}) = \mathbf{v} + (\boldsymbol{\mathscr{r}}\cdot\mathbf{a} - v^2)\,\nabla t_r$ ; and on the time side $\partial\boldsymbol{\mathscr{r}}/\partial t = -\mathbf{v}\,\partial t_r/\partial t$ :

\nabla(\boldsymbol{\mathscr{r}}\cdot\mathbf{u}) = -\mathbf{v} + \left(c^2 - v^2 + \boldsymbol{\mathscr{r}}\cdot\mathbf{a}\right)\frac{\boldsymbol{\mathscr{r}}}{\boldsymbol{\mathscr{r}}\cdot\mathbf{u}}, \qquad \frac{\partial(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})}{\partial t} = \left[v^2 - \boldsymbol{\mathscr{r}}\cdot\mathbf{a} - c\,(\hat{\boldsymbol{\mathscr{r}}}\cdot\mathbf{v})\right]\frac{\mathscr{r}c}{\boldsymbol{\mathscr{r}}\cdot\mathbf{u}}.

Inserting both into Step 1 (together with $\partial\mathbf{v}/\partial t = \mathbf{a}\,\mathscr{r}c/(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})$ ) and grouping every surviving term by whether it multiplies $\mathbf{u}$ or $\mathbf{a}$ :

\mathbf{E} = \frac{q}{4\pi\varepsilon_0}\,\frac{\mathscr{r}}{(\boldsymbol{\mathscr{r}}\cdot\mathbf{u})^3}\left[(c^2 - v^2)\,\mathbf{u} + \boldsymbol{\mathscr{r}}\times(\mathbf{u}\times\mathbf{a})\right],

where the identity $\boldsymbol{\mathscr{r}}\times(\mathbf{u}\times\mathbf{a}) = (\boldsymbol{\mathscr{r}}\cdot\mathbf{a})\,\mathbf{u} - (\boldsymbol{\mathscr{r}}\cdot\mathbf{u})\,\mathbf{a}$ (BAC-CAB) is what packages the two $\mathbf{a}$ -bearing terms into a single manifestly transverse triple product. Why trust the packaging: three independent audits. (i) Statics: $\mathbf{v} = \mathbf{a} = 0$ gives $\mathbf{u} = c\hat{\boldsymbol{\mathscr{r}}}$ , denominator $(\mathscr{r}c)^3$ , so $\mathbf{E} = q\hat{\boldsymbol{\mathscr{r}}}/4\pi\varepsilon_0\mathscr{r}^2$ — Coulomb, exactly. (ii) Dimensions and ranges: the $\mathbf{u}$ term scales as $1/\mathscr{r}^2$ , the triple product as $1/\mathscr{r}$ , matching Jefimenko's anatomy term for term. (iii) The transversality of the acceleration piece: $\boldsymbol{\mathscr{r}}\cdot[\boldsymbol{\mathscr{r}}\times(\mathbf{u}\times\mathbf{a})] = 0$ identically — the kink is perpendicular to the line of sight, as the field-line splice demands.

Step 3. The magnetic partner. Computing $\mathbf{B} = \nabla\times\mathbf{A}$ with the same kit — the curl routes through $\nabla t_r$ , planting $\hat{\boldsymbol{\mathscr{r}}}$ 's in cross products — collapses, after the same BAC-CAB reorganization, to the compact universal statement

\mathbf{B} = \frac{1}{c}\,\hat{\boldsymbol{\mathscr{r}}}\times\mathbf{E}.

Why the relation is exact and not just far-zone: both the velocity and acceleration parts of $\mathbf{E}$ get crossed with the same retarded line of sight. For the velocity field this reproduces, at low speed, the Biot–Savart field of a charge in flight ( $\mathbf{B} \approx \mu_0 q\,\mathbf{v}\times\hat{\boldsymbol{\mathscr{r}}}/4\pi\mathscr{r}^2$ ); for the acceleration field it delivers the mutually perpendicular, in-phase $\{\mathbf{E}, \mathbf{B}, \hat{\boldsymbol{\mathscr{r}}}\}$ triad that the radiation-zone articles will recognize as a local plane wave. One cross product, both personalities. *Verified: matches Griffiths Eq. 10.72 and Jackson Eq. 14.14 (whose covariant derivation is the E11 twin's route); audits (i)–(iii) plus the constant-velocity closed form of worked example 1 all check.*

The two personalities, examined

The velocity field generalizes Coulomb with two twists. First, the aim: for uniform motion the algebra (worked example 1) rotates every field vector to point radially from the charge's present position — the potentials' extrapolation trick surviving differentiation. Second, the shape: in present-position coordinates,

\mathbf{E}_{\text{vel}} = \frac{q}{4\pi\varepsilon_0}\,\frac{1 - \beta^2}{\left(1 - \beta^2\sin^2\theta\right)^{3/2}}\,\frac{\hat{\mathbf{R}}}{R^2}, \qquad (3)

where: $R, \theta$ are measured from the present position, $\theta$ from the velocity. The angular factor is the Heaviside pancake: fore-and-aft ( $\theta = 0$ ) the field is weakened to $(1-\beta^2) = 1/\gamma^2$ of Coulomb; broadside ( $\theta = 90°$ ) it is strengthened to $\gamma$ ; the equator-to-pole contrast is $\gamma^3$ — already a factor 12 at $\beta = 0.9$ . The flux through any surrounding sphere remains exactly $q/\varepsilon_0$ (the enhancement and deficit trade off precisely), and nothing radiates: the Poynting vector of a $1/\mathscr{r}^2$ field dies as $1/\mathscr{r}^4$ against sphere areas growing as $\mathscr{r}^2$ . A charge in uniform motion — however fast — carries its field as luggage and loses nothing. (It must be so by relativity: boost to the charge's rest frame and it is a static charge; statics does not radiate, and "radiates" is frame-invariant.)

The acceleration field is everything the velocity field is not: transverse ( $\perp\hat{\boldsymbol{\mathscr{r}}}$ , the BAC-CAB audit), linear in $\mathbf{a}$ , and falling as $1/\mathscr{r}$ — the unique falloff whose energy flux $\sim E^2\mathscr{r}^2 \sim \mathscr{r}^0$ survives to infinity. In the far zone, with $\mathbf{u} \to c\hat{\boldsymbol{\mathscr{r}}}$ (low speed for simplicity):

\mathbf{E}_{\text{rad}} \approx \frac{q}{4\pi\varepsilon_0 c^2}\,\frac{\hat{\boldsymbol{\mathscr{r}}}\times(\hat{\boldsymbol{\mathscr{r}}}\times\mathbf{a})}{\mathscr{r}} = -\frac{q\,\mathbf{a}_\perp(t_r)}{4\pi\varepsilon_0 c^2\,\mathscr{r}}, \qquad (4)

where: $\mathbf{a}_\perp$ is the acceleration's projection transverse to the line of sight — the $\sin\theta$ of the coming dipole formula in vector clothing, and the third Jefimenko correspondent ( $-\dot{\mathbf{J}}_\perp/c^2\mathscr{r}$ ) specialized to a point. Squaring and integrating (4)'s Poynting flux over the sphere yields the Larmor power — the flagship calculation owned by the Larmor-formula article downstream and deferred to it by name; this article's deliverable is the field itself and the verdict it seals: charges radiate if and only if they accelerate, with the radiated field reading the acceleration at the retarded time, projected on the sky.

Limiting cases

case	result	meaning
$v = 0$ , $a = 0$	Coulomb + $\mathbf{B} = 0$	the static ancestor, recovered exactly
$v \ll c$ , $a = 0$	Coulomb + $\mathbf{B} = \tfrac{\mu_0}{4\pi}\tfrac{q\mathbf{v}\times\hat{\boldsymbol{\mathscr{r}}}}{\mathscr{r}^2}$	the "Biot–Savart of a point charge"; the elementary current's field
uniform $v$ , any speed	present-pointing pancake (3); zero radiation	Heaviside 1888; boost-equivalent to statics
$v \ll c$ , $a \neq 0$	$\mathbf{E}_{\text{rad}} = -q\mathbf{a}_\perp(t_r)/4\pi\varepsilon_0c^2\mathscr{r}$	the radiation field; Larmor power downstream
$v \to c$ , head-on	fields carry $D^{-3} \to (2\gamma^2)^3$	ferocious forward beaming; synchrotron searchlight, priced in the relativistic-Larmor article

Common trapThe velocity field points at the present position — do not read that as faster-than-light news, and do not let the extrapolation fool you into "no radiation ever"

Two opposite misreadings of the same miracle. First: "the field knows where the charge is now, so information beat light." No — the field knows where the charge was and how it was moving, both read at $t_r$ , and the pointing at the present is an extrapolation computed by the algebra (the compass vector $\mathbf{u}$ does it: $\mathscr{r}\,\mathbf{u} = c\,[\boldsymbol{\mathscr{r}} - \mathscr{r}\mathbf{v}/c]$ is $c$ times the vector from the extrapolated present position to the field point). Let the charge swerve after $t_r$ and the field faithfully points at the now-empty extrapolated spot — no clairvoyance, just confident bookkeeping. Second, the opposite trap: since uniform motion gives such a tidy dressed field, one might imagine gentle accelerations produce only gently adjusted pancakes. They do not — any $\mathbf{a} \neq 0$ generates the $1/\mathscr{r}$ kink, and at sufficient distance the kink always dominates the pancake. There is no acceleration too small to radiate; there is only radiation too weak to notice yet.

Worked examples

1 · The pancake, quantified (confidence builder)

Setup. A charge glides at constant $\beta = 0.9$ . Compare its field to Coulomb at the same present distance $R$ , fore/aft versus broadside; repeat at $\beta = 0.99$ .

Step 1. The two extremes of (3). Along the motion ( $\theta = 0$ ): factor $\dfrac{1-\beta^2}{(1-0)^{3/2}} = 1 - 0.81 = \mathbf{0.19}$ — the field ahead of (and behind) the charge is five times weaker than Coulomb. Broadside ( $\theta = 90°$ ): $\dfrac{1-\beta^2}{(1-\beta^2)^{3/2}} = \dfrac{1}{\sqrt{1-\beta^2}} = \gamma = \mathbf{2.29}$ .

Step 2. The contrast. Equator-to-pole ratio $= \gamma^3 = \mathbf{12.1}$ . At $\beta = 0.99$ : pole factor $0.0199$ , equator factor $7.09$ , contrast $\gamma^3 = 356$ — the field is a disk. This is the field a detector actually feels when a relativistic bunch passes: a sharp transverse spike, nearly nothing fore/aft — the "flying pancake" that accelerator beam-position monitors are literally designed around.

Step 3. The flux audit. Integrating (3) over the sphere: the $\gamma$ excess on the thin equatorial band and the $1/\gamma^2$ deficit over the broad polar caps cancel to give exactly $q/\varepsilon_0$ — charge invariance, verified in the field pattern.

The lesson: speed reshapes the Coulomb field dramatically but conservatively — and entirely without radiation. Every watt a fast charge emits traces to $\mathbf{a}$ , never to $v$ .

Figure

Two charges flying abreast at beta = 0.9 — enhanced electric repulsion, magnetic attraction clawing back beta-squared of it, net force = statics over gamma

2 · Two charges flying abreast — magnetism caught impersonating relativity (exam level)

Setup. Two equal charges $q = 1$ nC travel side by side, separation $d = 1$ cm transverse to their common velocity $\beta = 0.9$ . Find the force each exerts on the other, and compare with statics. (Griffiths Ex. 10.5's physics, with numbers.)

Step 1. The electric push. Each charge sits broadside of the other, so (3) gives the transverse-enhanced field $E = \gamma\,\dfrac{q}{4\pi\varepsilon_0 d^2}$ . Static force would be $F_0 = \dfrac{q^2}{4\pi\varepsilon_0 d^2} = \dfrac{(8.99\times10^9)(10^{-18})}{10^{-4}} = 8.99\times10^{-5}$ N; the moving-charge electric repulsion is $\gamma F_0 = 2.29\times F_0 = \mathbf{2.06\times10^{-4}\ N}$ .

Step 2. The magnetic pull. Each charge is a current; $\mathbf{B} = \hat{\boldsymbol{\mathscr{r}}}\times\mathbf{E}/c$ at the partner gives $B = vE/c^2$ , and the force $q\mathbf{v}\times\mathbf{B}$ points inward (parallel currents attract): magnitude $qvB = \beta^2\,qE = 0.81\times2.06\times10^{-4} = \mathbf{1.67\times10^{-4}\ N}$ of attraction.

Step 3. The net. $F = qE(1 - \beta^2) = \gamma F_0/\gamma^2 = F_0/\gamma = \mathbf{3.92\times10^{-5}\ N}$ — the two charges repel with less than half the static force. Electric repulsion swelled by $\gamma$ ; magnetic attraction clawed back a factor $\beta^2$ of it; net: statics divided by $\gamma$ .

Step 4. The suspicious tidiness. In the charges' own rest frame there is no motion, no magnetism, and plain Coulomb repulsion $F_0$ . The lab-frame result $F_0/\gamma$ is exactly what relativistic force transformation predicts for a transverse force — meaning the magnetic attraction is precisely the bookkeeping that keeps both frames' stories consistent. Magnetism, here, is the relativistic correction to electrostatics — the thesis the relativity chapter (Magnetism as a Relativistic Effect) will prove in general. Practical corollary: space-charge repulsion inside particle beams dies as $1/\gamma^2$ per unit... (one $\gamma$ here, plus time-dilation of the dynamics) — the reason relativistic beams hold together and injector (low-energy) sections are where beams fight themselves.

The lesson: equation (2) contains pre-relativistic relativity: the field of the moving charge already knows what spacetime will later explain.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §10.3.2 · Jackson, Classical Electrodynamics, 3rd ed., §14.1–14.2 · Zangwill, Modern Electrodynamics, §23.3 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §5.6–5.7 and Appendix H (the kink construction)

Provenance. Equations follow Griffiths §10.3.2 (Eqs. 10.71–10.72) and Jackson §14.1 (Eq. 14.14). A verifier should: (1) re-derive the retarded-time identities $\partial t_r/\partial t = 1/(1-\hat{\boldsymbol{\mathscr{r}}}\cdot\boldsymbol{\beta})$ , $\nabla t_r = -\hat{\boldsymbol{\mathscr{r}}}/[c(1-\hat{\boldsymbol{\mathscr{r}}}\cdot\boldsymbol{\beta})]$ from the light-cone condition; (2) confirm the field formula's static, slow-motion, and uniform-velocity limits (the latter against the present-position closed form); (3) verify $\mathbf{B} = \hat{\boldsymbol{\mathscr{r}}}\times\mathbf{E}/c$ and the transversality $\boldsymbol{\mathscr{r}}\cdot\mathbf{E}_{\text{acc}} = 0$ ; (4) check the abreast-force reduction $F_0/\gamma$ against the relativistic transverse-force transformation. Addendum for this edition: the base's Larmor derivation is owned by Larmor Formula (EMAG.RAD.LARMR.0616) per the chapter's division of labor — this edition carries the radiation field (4) and its flux-survival verdict, deferring the power integral by name; the ultrarelativistic beaming of the power pattern belongs to Relativistic Larmor Formula (Liénard's Result); the covariant derivation of (2) via the field tensor is the E11 twin's route. Derivation B is presented with structural honesty (organizing intermediates + three audits) rather than the full two-page expansion, per the register's no-black-box/no-padding balance; every displayed intermediate is independently checkable. Worked-example numbers (pancake $0.19/2.29/12.1$ at $\beta=0.9$ , $0.0199/7.09/356$ at $0.99$ ; abreast forces $8.99\times10^{-5}$ → $2.06\times10^{-4}$ electric, $1.67\times10^{-4}$ magnetic, net $3.92\times10^{-5} = F_0/\gamma$ N) recomputed for this edition.

Builds on: Liénard–Wiechert Potentials · Retarded Potentials · Jefimenko's Equations · Lorenz Gauge Leads to: Electric Dipole Radiation · Magnetic Dipole Radiation · Larmor Formula · Relativistic Larmor Formula (Liénard's Result) · Magnetism as a Relativistic Effect

How can the velocity field point at the present position without violating causality?

Because pointing is not knowing. Every input to the field — position, velocity — is read at the retarded time; the formula then computes $\mathbf{u} = c\hat{\boldsymbol{\mathscr{r}}} - \mathbf{v}$ , and the geometry of that vector happens to aim the result at the position the charge would occupy by straight-line extrapolation from $t_r$ . For genuinely uniform motion the extrapolation coincides with reality and the field looks prescient; the moment the charge deviates after $t_r$ , distant fields keep pointing at the empty extrapolated location until the news arrives — at which point the kink (the acceleration field) sweeps through and re-aims them. You can even weaponize the lag: the field pattern between the swerve and the news's arrival is measurably "wrong," and that wrongness, integrated over the sky, is exactly the radiated pulse. Causality is not merely respected; its enforcement mechanism is the radiation itself.

Why exactly does the acceleration field fall as 1/𝓇 when every static field falls as 1/𝓇²?

The kink construction makes it geometric. The radiation shell has fixed radial thickness ( $c$ times the acceleration's duration), but the transverse mismatch it must bridge — between the old extrapolated field pattern and the new one — grows linearly with distance, because the two patterns' angular difference is fixed while arc length $= r\,\Delta\theta$ . A field line entering the shell radially must exit displaced sideways by something $\propto r$ , within a thickness that is constant: its transverse component inside the shell is therefore $\propto r \times (\text{radial component})$ , and since the radial (Coulomb-strength) piece goes as $1/r^2$ , the transverse kink goes as $1/r$ . Same conclusion from bookkeeping: energy conservation requires flux $E^2 \cdot 4\pi r^2$ to be $r$ -independent for a wave that never stops, forcing $E \propto 1/r$ exactly. Statics has no shells and no mismatches — hence no mechanism for the softer falloff.

Do the velocity and acceleration fields interfere, or can I treat them independently?

They add as vectors at every point — the split is linear — but energy is quadratic, so cross terms exist and matter in the middle distances. Near the charge the velocity field dominates utterly; far away the acceleration field does; in between, the Poynting vector contains $\mathbf{E}_{\text{vel}}\times\mathbf{B}_{\text{acc}}$ -type cross terms that shuffle energy between the bound field and the escaping wave. This is the honest reason "radiated power" is defined by the flux through a sphere at infinity: only there do the cross terms (falling faster than $1/r^2$ in flux) vanish and the accounting become unambiguous. The same subtlety underlies the radiation-reaction story later in the chapter — the bound field's energy sloshes in and out during acceleration, and only the $1/\mathscr{r}$ part's losses are irreversible. Rule of practice: fields add, then square; never add "radiated" and "bound" energies computed separately at finite distance.

Purcell derives the radiation field from the kink picture with almost no algebra. Is that derivation equivalent to this article's?

Equivalent in content, complementary in virtue. Purcell's construction (Appendix H of Purcell & Morin) takes a charge that stops abruptly, draws the field lines — old Coulomb pattern outside the news sphere, new one inside, kink in the shell — and reads off the transverse field from the geometry: it yields exactly equation (4), including the $\sin\theta$ and the $1/r$ , in half a page. What it cannot easily deliver is the general trajectory (arbitrary $\mathbf{v}$ and $\mathbf{a}$ simultaneously), the exact $D^{-3}$ beaming denominators, or the velocity field's pancake — for those, the retarded-derivative machinery of Derivations A–B is the only honest route. Ideal division of labor: internalize Purcell's picture as the why (this article's hero figure is its cousin), keep Griffiths's algebra as the what, and check each against the other — they must and do agree in the overlap.

Is bremsstrahlung just this article's acceleration field?

Yes — bremsstrahlung ("braking radiation") is the acceleration field of charges undergoing violent deceleration, and an X-ray tube is the kink picture industrialized: electrons at tens of keV slam into a tungsten anode, $\mathbf{a}$ spikes for femtoseconds, and equation (4) broadcasts a broadband transverse pulse whose spectrum extends up to the electron's full kinetic energy (the Duane–Hunt limit — a quantum statement stapled onto a classical field). The angular pattern follows $\mathbf{a}_\perp$ : maximal broadside to the deceleration at modest energies, folding forward with the $D^{-3}$ beaming as electrons go relativistic — which is why medical and industrial tubes above ~100 keV emit predominantly forward. The classical field of this article gets the pattern and scaling right; the photon-by-photon statistics and the spectrum's quantum edge belong to QED. Röntgen's 1895 discovery was, in this precise sense, the first detection of the $\boldsymbol{\mathscr{r}}\times(\mathbf{u}\times\mathbf{a})$ term.

What happens to these fields in the extreme relativistic limit — do they stay finite?

The formulas stay exact and finite for any $v < c$ , but every interesting quantity acquires violent anisotropy through the $D^{-3} = (1-\hat{\boldsymbol{\mathscr{r}}}\cdot\boldsymbol{\beta})^{-3}$ prefactor. Head-on, $D^{-1} \to 2\gamma^2$ , so fields scale like $\gamma^6$ forward while the pancake suppresses them by $\gamma^{-2}$ sideways-of-motion — the charge becomes, electromagnetically, a needle of influence. In the strict $v \to c$ limit the velocity field degenerates toward a delta-function sheet transverse to the motion (the "boosted Coulomb field" that accelerator physicists model as an impulsive kick, and field theorists as the Weizsäcker–Williams equivalent-photon cloud: a fast charge's field is operationally indistinguishable from a flash of real photons). None of this breaks the formalism — but it does break the perturbative instincts: at large $\gamma$ , "small" angular regions carry almost all the physics, and integrals must be done with the beaming denominators respected exactly, which is the entire technical content of the relativistic-Larmor and synchrotron articles ahead.

Where is the energy that the velocity field carries — does a passing charge deliver energy to me or not?

It delivers a loan, not a gift. As a uniformly moving charge approaches, its field energy density at your location rises; the Poynting vector is genuinely nonzero (fields and $\mathbf{B} = \hat{\boldsymbol{\mathscr{r}}}\times\mathbf{E}/c$ make $\mathbf{E}\times\mathbf{B}$ point along the motion) — energy convects with the charge, like the bow wave of a ship. After the charge passes, the same accounting runs in reverse and the energy leaves with it; net transfer to any fixed volume over the full flyby: zero (for uniform motion). That is the precise content of "the velocity field does not radiate": its energy is bound, transported as luggage, recoverable, falling off too fast to leave a residue at infinity. The acceleration field breaks the loan structure — its energy departs on the light cone and never returns, which is why only it costs the charge's driver anything. The Poynting bookkeeping of both behaviors, including the beautiful pathology of the static case, was settled back in the conservation-laws chapter; here it gains its most important customer.