Larmor Formula from the Liénard–Wiechert Fields

The strong-UG article Larmor Formula presented the bill: integrate the Poynting flux of the acceleration field over a distant sphere and $P=\mu_0q^2a^2/6\pi c$ falls out, with the $\sin^2\theta$ donut and the sixteen-picosecond hydrogen catastrophe as its famous corollaries. This article — the Jackson-register twin — does not re-present the bill. It audits the books behind it, because every advanced application of radiation theory lives or dies on four pieces of bookkeeping the flagship derivation could leave implicit. First, whose clock? The energy crossing a sphere per second of observer time and the energy leaving the charge per second of source time differ by a retardation Jacobian, and we promote that Jacobian to the formal definition of emitted power — the definition every relativistic calculation downstream will spend. Second, which axis? We state and prove the general-orientation form: the pattern is $\sin^2\Theta$ with $\Theta$ measured from the acceleration, and to leading order the velocity does not appear anywhere — a theorem, not an accident of choosing $v=0$ . Third, how many charges? Radiation superposes as fields, not powers: $N$ charges shaken in phase radiate $N^2$ times one charge, and incoherent charges only $N$ times — the single distinction that separates an antenna from a hot gas, priced here exactly. Fourth, which frequencies? Parseval's theorem converts the time-integrated Larmor energy into a spectrum, the master tool the synchrotron and bremsstrahlung articles will run. Four questions, four ledgers, one formula — Larmor's, now with the bookkeeping that makes it load-bearing at the graduate level.

Key takeaways

Emitted power is a definition: $dP(t_r)/d\Omega\equiv R^2(\mathbf{S}\cdot\hat{\mathbf{n}})\,(\partial t/\partial t_r)$ , with the retardation Jacobian $\kappa=1-\hat{\mathbf{n}}\cdot\boldsymbol\beta$ converting detector-clock flux into source-clock energy loss.
In the $v\to0$ limit the distribution is $dP/d\Omega=\dfrac{q^2a^2}{16\pi^2\varepsilon_0c^3}\sin^2\Theta$ with $\Theta$ measured from the acceleration — no velocity anywhere, at any orientation of $\mathbf{a}$ relative to $\mathbf{v}$ .
Velocity-independence is a theorem: a constant-velocity charge is a boosted static charge (zero radiation for every $v$ ), so $P$ must be velocity-blind at leading order; corrections are $O(\beta)$ in the pattern.
Coherence theorem: fields add, then square — $P=\mu_0\big|\sum_iq_i\mathbf{a}_i\big|^2/6\pi c$ for a sub-wavelength cluster: in-phase charges give $N^2$ , random phases give $N$ ; equivalently $P=\mu_0\ddot p^2/6\pi c$ with $\ddot{\mathbf{p}}=\sum q_i\mathbf{a}_i$ .
Spectral ledger: Parseval converts $W=\frac{\mu_0q^2}{6\pi c}\int a^2dt$ into $dW/d\omega=\frac{\mu_0q^2}{6\pi^2 c}|\tilde a(\omega)|^2$ — the radiated energy is the acceleration's power spectrum.
Momentum form $P=\dfrac{q^2}{6\pi\varepsilon_0m^2c^3}\left(\dfrac{d\mathbf{p}}{dt}\right)^2$ is the nonrelativistic bridge to the covariant generalization next door.
Key idea: the Larmor formula becomes a graduate tool once four ledgers are kept explicitly — source clock versus observer clock, angle from the acceleration, coherent versus incoherent summation, and the Parseval spectrum of the acceleration.

The core insight

Radiated power is bookkeeping on the Liénard–Wiechert acceleration field: define emitted power with the retardation Jacobian, measure angles from the acceleration, sum fields (not powers) over charges, and read the spectrum from the Fourier transform of the acceleration.

Where this lives today

The bookkeeping ledgers of this article are the daily working tools of accelerator light sources. Storage-ring designers track exactly when their bunches cross the coherence threshold: at wavelengths below the bunch length, emission is incoherent ( $N$ -scaling, the normal synchrotron-light business); above it the $N^2$ term switches on as coherent synchrotron radiation — prized in dedicated THz sources, feared elsewhere as a beam-destabilizing drain, in both roles priced by Derivation B. Free-electron lasers push the theorem to hard X-rays by force: the undulator plus the beam's own radiation microbunches the electrons at the optical wavelength, converting a $10^9$ -electron bunch from $N$ to $N^2$ at ångström scales — the four-decade brightness jump of LCLS-class machines is the cross terms of Step 3, turned on deliberately. The spectral ledger runs the diagnostic side: pulse-duration measurements at FELs and plasma accelerators invert Derivation C — measure $dW/d\omega$ , infer the emitting current's time structure, with $\Delta\omega\,\tau_d\sim1$ as the resolution limit. And the two-clocks definition of Derivation A is the convention every synchrotron-radiation code and pulsar-emission model is built on — the $\kappa$ bookkeeping formalized here is spent five powers at a time in the angular-distribution article.

Origins · 1 minJoseph John Larmor and how this was discovered

Joseph John Larmor

1857–1942 · the 1897 paper that priced radiation — written to explain why spectral lines are sharp

The formula entered the world in Larmor's 1897 Philosophical Magazine paper "On the theory of the magnetic influence on spectra; and on the radiation from moving ions" — a title worth reading twice, because neither half says "radiated power." Larmor's quarry was spectroscopy: the newly measured Zeeman splitting, which his precession theorem (orbits of charged "ions" in a weak magnetic field precess at $\omega_L=eB/2m$ , superposing unchanged) explained so economically it survives in every magnetic-resonance textbook. The power formula appears as supporting analysis — how fast do orbiting ions drain energy into the aether? — with the damping of spectral lines, this article's Lorentzian business, as its intended application: would radiation damping smear the sharp lines his theorem organized? His answer — slowly, since the loss per period is tiny — is the first appearance of the small parameter the radiation-reaction articles formalize. He remained a man of the dynamical aether, declining to follow his own tool into relativity and the quantum; the formula outgrew its author's universe within a decade.

History

1889

— Rayleigh states the energy theorem for Fourier pairs — the Parseval machinery radiation spectra will run on

1897

— Larmor's "magnetic influence on spectra" paper: precession theorem, and the radiated-power formula as its damping estimate

1898

— Liénard extends the power result to arbitrary velocity, κ factors included, seven years before relativity

1912

— Schott's Electromagnetic Radiation systematizes the angular and spectral bookkeeping of accelerated charges

1954

— Nodvick and Saxon work out coherent synchrotron radiation from bunched electrons: the N² term becomes accelerator physics

1971

— Madey's free-electron laser principle: engineered microbunching converts N to N² at optical wavelengths

2009

— LCLS lases at 1.5 Å: coherence bookkeeping delivers a four-decade brightness jump in hard X-rays

The physical picture

Start with the two clocks. A distant detector integrates Poynting flux against its wall time $t$ . But the energy it collects between $t$ and $t+dt$ was emitted between the corresponding retarded times $t_r$ and $t_r+dt_r$ — and if the charge is moving, those intervals differ: news emitted while the charge advances toward you arrives compressed. The compression factor is $\kappa=1-\hat{\mathbf{n}}\cdot\boldsymbol\beta$ , the same retardation Jacobian the Liénard–Wiechert construction carries in its denominator. "Power radiated by the charge" — the number that debits the charge's energy budget — must be energy per unit of the source's time, and that bookkeeping choice, invisible when $v\to0$ , is the hinge on which every relativistic radiation formula turns. We make it the definition from the start, so that the nonrelativistic Larmor formula and its fierce relativistic completions are one framework with one convention, rather than a formula and a patch. The second picture is the donut's true axis. The radiation field is the acceleration projected on the sky — the transverse part of $\mathbf{a}$ , whatever the velocity is doing. A charge drifting along $x$ while shoved along $y$ wears its $\sin^2\Theta$ donut around $y$ : to leading order the velocity leaves no fingerprint at all, because a charge with only velocity is a boosted electrostatics problem and radiates nothing in any frame. Velocity-blindness is not an approximation artifact; it is relativity policing the books. The third picture is a crowd of charges. Each contributes a field linear in $q_i\mathbf{a}_i$ ; the detector squares the sum. If a bunch much smaller than a wavelength moves as one, the amplitudes add before squaring: $N$ electrons radiate $N^2$ single-electron powers — milliwatts from a centimeter of antenna wire whose current is nothing but $10^{20}$ electrons politely agreeing. Random phases instead kill the cross terms on average, leaving $N$ . One theorem, and it is the entire difference between a transmitter and a toaster.

Figure

The two clocks — flux lands on the detector clock, but the charge pays on its own clock; the Jacobian between them is the retardation factor

Figure

The general-orientation donut — the pattern reads the acceleration and ignores the velocity to leading order

The math: one field, four ledgers

Everything below runs on the acceleration part of the point-charge field, quoted by name — its systematic covariant construction is Fields of a Relativistic Moving Charge in this chapter, and its three-dimensional anatomy is the strong-UG Fields of a Moving Point Charge. In the radiation zone,

\mathbf{E}_{\text{rad}}=\frac{q}{4\pi\varepsilon_0c}\,\frac{\hat{\mathbf{n}}\times\big[(\hat{\mathbf{n}}-\boldsymbol\beta)\times\dot{\boldsymbol\beta}\big]}{\kappa^3\,R}\Bigg|_{t_r},\qquad \mathbf{B}_{\text{rad}}=\frac{\hat{\mathbf{n}}}{c}\times\mathbf{E}_{\text{rad}}, \qquad (1)

where: $R$ and $\hat{\mathbf{n}}$ run from the retarded position to the field point, $\boldsymbol\beta=\mathbf{v}/c$ and $\dot{\boldsymbol\beta}$ are evaluated at the retarded time $t_r$ , and $\kappa=1-\hat{\mathbf{n}}\cdot\boldsymbol\beta$ is the retardation factor those articles derived. Locally the triad is a plane wave headed outward, so the Poynting flux is $|\mathbf{E}_{\text{rad}}|^2/\mu_0c$ along $\hat{\mathbf{n}}$ . This article never re-derives (1); it spends it.

Derivation AEmitted power as a definition and the general-orientation Larmor distribution

The goal is twofold: first, to define what "power radiated by the charge" means when the charge moves — the source-clock bookkeeping that the strong-UG twin could bypass at $v\to0$ — and second, to extract the leading-order angular distribution and show it is $\sin^2\Theta$ about the acceleration with the velocity absent entirely. The strategy: write the energy ledger connecting detector time to retarded time, promote the Jacobian to the definition of emitted power, then take $\beta\to0$ after the definitions are in place, so that nothing about the result depends on having pointed $\mathbf{a}$ along $\mathbf{v}$ or set $\mathbf{v}=0$ by hand. The subtlety to watch: the Jacobian enters the definition once, not the fields — double-counting it is the classic relativistic bookkeeping error, flagged in the strong-UG relativistic article and formalized here.

Step 1. The energy ledger. A detector patch at distance $R$ subtending $d\Omega$ collects energy $R^2(\mathbf{S}\cdot\hat{\mathbf{n}})\,d\Omega\,dt$ during detector interval $dt$ . That energy left the charge during the retarded interval $dt_r$ , related by the retarded-time derivative the Liénard–Wiechert construction supplies:

\frac{\partial t}{\partial t_r}=\kappa=1-\hat{\mathbf{n}}\cdot\boldsymbol\beta .

Why this factor exists: the sphere's news deadline moves as the charge moves — emission stretched over $dt_r$ of the charge's life arrives compressed (approach) or dilated (recession) into $dt=\kappa\,dt_r$ of detector time. It is the same factor sitting cubed inside the field (1); here it appears once more, as raw time bookkeeping.

Step 2. The definition. Power emitted, per solid angle, per unit source coordinate time:

\frac{dP(t_r)}{d\Omega}\;\equiv\;R^2\big(\mathbf{S}\cdot\hat{\mathbf{n}}\big)\,\frac{\partial t}{\partial t_r}=R^2\big(\mathbf{S}\cdot\hat{\mathbf{n}}\big)\,\kappa .

This is the quantity whose angular integral debits the particle's energy budget, and it is the definition the relativistic articles of this chapter run on: the strong-UG Relativistic Larmor Formula (Liénard's Result) introduced the doctrine, and Angular Distribution of Radiation from Accelerated Charges will integrate its $\kappa^{-5}$ consequences. Received flux — energy per detector time, the raw $R^2(\mathbf{S}\cdot\hat{\mathbf{n}})$ — remains a legitimate quantity answering a different question, and conflating the two is the warning box below.

Step 3. Substitute the field and keep the orientation general. With (1),

\frac{dP(t_r)}{d\Omega}=\frac{q^2}{16\pi^2\varepsilon_0c}\,\frac{\big|\hat{\mathbf{n}}\times[(\hat{\mathbf{n}}-\boldsymbol\beta)\times\dot{\boldsymbol\beta}]\big|^2}{\kappa^5},

six powers of $\kappa$ from the squared field minus one returned by the Jacobian: five net. Nothing has been assumed about the angle between $\boldsymbol\beta$ and $\dot{\boldsymbol\beta}$ .

Step 4. The nonrelativistic limit, orientation intact. Set $\beta\to0$ with $\dot{\boldsymbol\beta}=\mathbf{a}/c$ finite: $\kappa\to1$ and $(\hat{\mathbf{n}}-\boldsymbol\beta)\to\hat{\mathbf{n}}$ , so the numerator becomes $|\hat{\mathbf{n}}\times(\hat{\mathbf{n}}\times\mathbf{a})|^2/c^2=a^2\sin^2\Theta/c^2$ with $\Theta$ measured from $\mathbf{a}$ — the velocity direction has vanished from the formula, not been chosen away:

\frac{dP}{d\Omega}=\frac{q^2a^2}{16\pi^2\varepsilon_0c^3}\,\sin^2\Theta .

Why velocity-independence had to happen: a charge with $\dot{\boldsymbol\beta}=0$ and any constant $\boldsymbol\beta$ is a static charge in boosted clothing — zero radiation in every frame, exactly. So the distribution can contain $\boldsymbol\beta$ only in products with $\dot{\boldsymbol\beta}$ , which are higher order; the leading term is built from $\mathbf{a}$ alone. The first surviving correction is the $O(\beta)$ forward tipping hiding in $\kappa^{-5}\approx1+5\hat{\mathbf{n}}\cdot\boldsymbol\beta$ .

Step 5. Total power, by citation. The angular integral $\oint\sin^2\Theta\,d\Omega=8\pi/3$ is executed in the strong-UG Larmor Formula, whose flagship derivation owns it; applying it here gives

P=\frac{q^2a^2}{6\pi\varepsilon_0c^3}=\frac{\mu_0q^2a^2}{6\pi c},

Larmor's result, now carried by a definition that survives to any velocity. Verified: Step 3 matches Jackson Eq. 14.38 (the source-time distribution with $\kappa^{-5}$ ); Step 4 matches Jackson Eq. 14.21 with $\Theta$ from the acceleration; Step 5 reproduces Jackson Eq. 14.22 and Griffiths Eq. 11.70. Dimensional audit: $q^2a^2/\varepsilon_0c^3$ carries watts.

The result of Step 5 has a spelling worth its own equation card. Using $\mathbf{F}=d\mathbf{p}/dt=m\mathbf{a}$ at nonrelativistic speeds,

P=\frac{q^2}{6\pi\varepsilon_0m^2c^3}\left(\frac{d\mathbf{p}}{dt}\right)^{\!2} \qquad (2)

where: $m$ is the particle's mass and $d\mathbf{p}/dt$ the applied force. Written this way the formula reads "radiation prices the force, weighted by charge-to-mass squared" — electrons out-radiate protons by $(m_p/m_e)^2\approx3.4\times10^6$ under the same force. This is the exact form that survives relativistically for collinear motion, and replacing $d/dt$ by the proper-time derivative $d/d\tau$ of both energy and momentum is precisely the covariant completion built next door in Relativistic Generalization of Larmor's Formula — equation (2) is the bridge, crossed in one step from here.

Figure

The coherence ledger — one kicked electron, then a hundred million of them: fields add before the square

Go deeper · Derivation B · 2 minThe coherence theorem — N squared versus N

Derivation BThe coherence theorem — N squared versus N

The goal is the rule for many charges: when does a cluster radiate like $N$ independent Larmor emitters, and when like a single super-charge worth $N^2$ of them? The strategy is disarmingly short — superpose the fields of (1) at $\beta\to0$ , square, and inspect the cross terms — but the consequences run from antenna engineering to free-electron lasers, and the derivation makes precise the one geometric condition (cluster small compared to the radiated wavelength) that the folklore usually leaves vague.

Step 1. Superpose fields. For charges $q_i$ at positions $\mathbf{r}_i$ with accelerations $\mathbf{a}_i$ , the radiation fields add. If the cluster's extent $d$ satisfies $d\ll\lambda$ for every relevant Fourier component of the motion (and $d\ll R$ trivially), the retarded times and unit vectors of all charges coincide to leading order — the phase spread $kd$ across the cluster is negligible — and

\mathbf{E}_{\text{rad}}^{\text{tot}}=\frac{1}{4\pi\varepsilon_0c^2R}\,\hat{\mathbf{n}}\times\Big(\hat{\mathbf{n}}\times\sum_iq_i\mathbf{a}_i\Big)\Big|_{t_r}.

Why the condition is $d\ll\lambda$ and not just $d\ll R$ : distance only guarantees the $1/R$ envelopes agree; it is the comparison of $d$ with the wavelength that decides whether emission phases agree. The antenna-array article prices exactly what happens when $d\sim\lambda$ and the phases are engineered instead of ignored.

Step 2. Square and integrate. Every step of Derivation A goes through with $q\mathbf{a}\to\sum_iq_i\mathbf{a}_i$ :

P=\frac{\mu_0}{6\pi c}\,\Big|\sum_iq_i\mathbf{a}_i\Big|^2=\frac{\mu_0\,\ddot p^2}{6\pi c},\qquad \ddot{\mathbf{p}}=\frac{d^2}{dt^2}\sum_iq_i\mathbf{r}_i ,

the second spelling identifying the sum as the second derivative of the cluster's electric dipole moment — Larmor for a crowd is dipole radiation, the identification the strong-UG Electric Dipole Radiation made from the other direction.

Step 3. The two regimes. Identical motions, $\mathbf{a}_i=\mathbf{a}$ for all $i$ : the sum is $Nq\mathbf{a}$ and $P=N^2P_1$ — coherent, the cross terms all constructive. Random relative phases (thermal motion, uncorrelated scattering): writing $\mathbf{a}_i=\mathbf{a}\,e^{i\varphi_i}$ for a narrowband motion, $\big\langle|\sum_ie^{i\varphi_i}|^2\big\rangle=N+\sum_{i\ne j}\langle e^{i(\varphi_i-\varphi_j)}\rangle=N$ — the $N(N-1)$ cross terms average to zero and $\langle P\rangle=NP_1$ , incoherent. Why no energy law is violated by $N^2$ : the radiated energy is paid by whatever agent drives the charges, and driving $N$ charges coherently against their own collective radiation field costs $N^2$ worth of work — each charge moves against the field of all the others. The enhancement is real, and so is the bill presented to the transmitter.

Step 4. Read it as engineering. A macroscopic current element is the coherent limit industrialized: for a wire element of length $dz$ carrying $I(t)$ , the sum $\sum q_i\mathbf{a}_i=\frac{d}{dt}\sum q_i\mathbf{v}_i=\dot I\,dz\,\hat{\mathbf{z}}$ — some $10^{20}$ conduction electrons acting as one charge, which is why centimeter antennas radiate milliwatts while single electrons radiate $10^{-27}$ joules. Antenna Arrays and Directivity spends this theorem at full scale, arranging coherence in chosen directions; coherent synchrotron radiation and free-electron lasers arrange it in frequency, microbunching beams at the radiated wavelength to convert $N$ into $N^2$ . Verified: Step 3's incoherent average is the standard random-phase computation (Jackson §14, introductory remarks to bremsstrahlung; Zangwill §23); Step 4's current-element identification reproduces the short-element radiation resistance $\approx789\,(dz/\lambda)^2\ \Omega$ — recomputed independently in worked example 2 against the strong-UG value quoted in Electric Dipole Radiation.

Figure

The spectral ledger — an exponentially damped kick and its Lorentzian spectrum, computed

Go deeper · Derivation C · 2 minThe spectral resolution of radiated energy

Derivation CThe spectral resolution of radiated energy

The goal is the frequency ledger: convert the time-integrated Larmor energy into an integral over frequencies, obtaining the master formula $dW/d\omega\propto|\tilde a(\omega)|^2$ that the synchrotron and bremsstrahlung articles will feed with their own accelerations. The strategy is Parseval's theorem applied to the acceleration history — honest about its one prerequisite (finite total energy: the motion must be a completed event, not an eternal sinusoid) — followed by a fully worked mini-application, the exponentially damped kick, chosen because its Lorentzian answer is the ancestral shape of every collision-broadened line.

Step 1. The energy of an event. For an acceleration history $\mathbf{a}(t)$ that begins and ends (a collision, a kick, a decaying oscillation), the total radiated energy is

W=\int_{-\infty}^{\infty}P(t)\,dt=\frac{\mu_0q^2}{6\pi c}\int_{-\infty}^{\infty}a^2(t)\,dt .

Step 2. Parseval. With the convention $\tilde{\mathbf{a}}(\omega)=\int\mathbf{a}(t)e^{i\omega t}dt$ , Parseval's theorem states $\int a^2dt=\frac{1}{2\pi}\int|\tilde a(\omega)|^2d\omega$ ; reality of $\mathbf{a}(t)$ gives $|\tilde a(-\omega)|=|\tilde a(\omega)|$ , so the two half-axes carry equal energy and fold onto physical frequencies $\omega\ge0$ :

W=\frac{\mu_0q^2}{6\pi^2c}\int_0^{\infty}\big|\tilde{\mathbf{a}}(\omega)\big|^2\,d\omega \quad\Longrightarrow\quad \frac{dW}{d\omega}=\frac{\mu_0q^2}{6\pi^2c}\,\big|\tilde{\mathbf{a}}(\omega)\big|^2 .

Why this is a theorem about events, not instants: a spectrum is a property of a whole history — "the frequency content at time $t$ " is not a well-posed classical quantity, and the honest objects are $P(t)$ (time ledger) and $dW/d\omega$ (frequency ledger), two marginals of one event. Windowed spectra interpolate, at the usual time-bandwidth price.

Step 3. The damped kick, end to end. Take $a(t)=a_0e^{-t/\tau_d}$ for $t\ge0$ , zero before — a particle struck and relaxing. The transform is $\tilde a(\omega)=a_0/(1/\tau_d-i\omega)$ , so

\frac{dW}{d\omega}=\frac{\mu_0q^2}{6\pi^2c}\,\frac{a_0^2}{\omega^2+1/\tau_d^2}

— a Lorentzian of full width $\Delta\omega=2/\tau_d$ at half maximum: the shorter the kick, the broader the spectrum, hyperbolically.

Step 4. Close the books both ways. Time route: $W=\frac{\mu_0q^2}{6\pi c}\int_0^\infty a_0^2e^{-2t/\tau_d}dt=\frac{\mu_0q^2a_0^2\tau_d}{12\pi c}$ . Frequency route: $\int_0^\infty\frac{d\omega}{\omega^2+1/\tau_d^2}=\frac{\pi\tau_d}{2}$ , giving $\frac{\mu_0q^2}{6\pi^2c}\cdot\frac{\pi a_0^2\tau_d}{2}=\frac{\mu_0q^2a_0^2\tau_d}{12\pi c}$ — identical. The ledgers balance. Why the Lorentzian matters beyond the exercise: an exponentially damped oscillation $a_0e^{-t/\tau_d}\cos\omega_0t$ just translates this peak to $\omega_0$ — the classical line shape of a radiatively damped oscillator, whose width the radiation-reaction articles connect to the damping constant. The same machinery, fed the swept acceleration of a relativistic orbit, produces the synchrotron spectrum — by name, Synchrotron Radiation's opening move. *Verified: Parseval closure checked numerically for this edition (time integral 0.500000 vs frequency integral 0.4999, grid-limited, of the normalized $a_0^2\tau_d/2$ ); the $dW/d\omega$ master formula matches Jackson §14.5 (Eq. 14.51 specialized to the total over angles at $\beta\to0$ ); the Lorentzian width–duration reciprocity is the standard time-bandwidth statement.*

Limiting cases

case	result	meaning
$\beta\to0$ , any orientation of $\mathbf{a}$	$dP/d\Omega\propto\sin^2\Theta$ from $\mathbf{a}$	velocity absent: the general-orientation theorem
constant $\mathbf{v}$ , $\mathbf{a}=0$	$P=0$ exactly, every $v$	boosted statics; the theorem's anchor
$O(\beta)$ correction	pattern tips forward by $\sim5\beta\cos$ terms	seed of the $\kappa^{-5}$ lobes — Angular Distribution of Radiation from Accelerated Charges
$N$ charges, in phase, $d\ll\lambda$	$P=N^2P_1$	coherent: fields add; antennas, FELs
$N$ charges, random phases	$\langle P\rangle=NP_1$	incoherent: cross terms cancel; thermal bremsstrahlung
macroscopic current element	$P=\frac{\mu_0(\dot I\,dz)^2}{6\pi c}$	$\sum q_ia_i=\dot I\,dz$ : Larmor industrialized
harmonic event $a_0e^{-t/\tau_d}\cos\omega_0t$	Lorentzian at $\omega_0$ , width $2/\tau_d$	ancestral line shape; radiative damping's fingerprint
force spelling	$P=\frac{q^2}{6\pi\varepsilon_0m^2c^3}(dp/dt)^2$	the bridge crossed by Relativistic Generalization of Larmor's Formula

Common trapTwo ledgers, two classic frauds — mixing the clocks, and summing powers when nature sums fields

First fraud: quoting received flux as emitted power. They differ by the Jacobian $\kappa=1-\hat{\mathbf{n}}\cdot\boldsymbol\beta$ — invisible at $v\ll c$ , a factor $\sim2\gamma^2$ ahead of a relativistic charge — and only the $\kappa$ -corrected, source-clock quantity belongs in the particle's energy budget. Declare the clock before integrating anything; the habit is free here and mandatory from the next article on. Second fraud: computing a bunch's radiation by multiplying one particle's Larmor power by $N$ . That is correct only for random phases. If the charges move in step within a sub-wavelength region — an antenna current, a microbunched beam — the fields add first and the power is $N^2P_1$ : for the picosecond kick of worked example 1 the two prescriptions disagree by eight orders of magnitude. Neither error announces itself dimensionally; both are pure bookkeeping, which is why this article exists.

Worked examples

1 · The picosecond mirror — one electron, then a hundred million in step (confidence builder)

Setup. An electron with kinetic energy $1$ keV hits a field region that reverses its velocity in $\tau=1$ ps. Find the radiated energy — then repeat for a bunch of $N=10^8$ electrons reflected in phase, the bunch much smaller than the radiated wavelength.

Step 1. Kinematics of the kick. $v=\sqrt{2K/m}=\sqrt{2(1.602\times10^{-16})/9.109\times10^{-31}}=1.88\times10^{7}$ m/s ( $\beta=0.063$ — comfortably nonrelativistic). Reversal means $\Delta v=2v$ in $\tau$ :

a=\frac{2v}{\tau}=\frac{3.75\times10^{7}}{10^{-12}}=3.75\times10^{19}\ \text{m/s}^2 .

Step 2. One electron. $P=\dfrac{\mu_0e^2a^2}{6\pi c}=8.03\times10^{-15}$ W during the kick, so the radiated energy is $W_1=P\tau=8.0\times10^{-27}$ J $=5.0\times10^{-8}$ eV — a fraction $5\times10^{-11}$ of the electron's kinetic energy. Single slow electrons are abysmal radiators; this is why cathode-ray tubes didn't glow from deceleration alone.

Step 3. The coherent bunch. The radiated wavelength scale is $c\tau\approx0.3$ mm, so a micron-scale bunch satisfies $d\ll\lambda$ easily and Derivation B applies with all phases equal:

W_{\text{coh}}=N^2W_1=(10^{8})^2(8.0\times10^{-27})=\mathbf{8.0\times10^{-11}\ J}=0.50\ \text{GeV}

— a fraction $W_{\text{coh}}/(NK)=5\times10^{-3}$ of the bunch's total kinetic energy: the fractional loss grew by the factor $N$ . The incoherent prescription would have predicted $NW_1=8.0\times10^{-19}$ J — wrong by $10^{8}$ .

Step 4. Read the moral. Coherence converts a hopeless emitter into a serious one: bunch charges below the emission wavelength and radiation costs stop being negligible. This is the operating principle of coherent synchrotron radiation sources (and their beam-dynamics headaches), and — with the bunching maintained actively at optical wavelengths — of the free-electron laser.

The lesson: always ask whether the phases are locked before choosing $N$ or $N^2$ — eight decades ride on the answer.

2 · The centimeter antenna element — Larmor's crowd theorem meets radio engineering (exam level)

Setup. A wire element of length $dz=1$ cm carries $I(t)=I_0\cos\omega t$ with $I_0=1$ A at $f=100$ MHz ( $\lambda=3.0$ m, so $dz\ll\lambda$ ). Compute the time-averaged radiated power from the coherence theorem, and cross-check against the short-element radiation resistance.

Step 1. The collective acceleration. From Derivation B Step 4, $\big|\sum q_i\mathbf{a}_i\big|=|\dot I|\,dz=\omega I_0\,dz\,|\sin\omega t|$ , amplitude $\omega I_0dz=(2\pi\times10^{8})(1)(0.01)=6.28\times10^{6}$ A·m/s.

Step 2. Time-averaged power. $\langle\dot I^2\rangle=\tfrac12\omega^2I_0^2$ :

\langle P\rangle=\frac{\mu_0(\omega I_0\,dz)^2}{12\pi c}=\frac{(1.257\times10^{-6})(6.28\times10^{6})^2}{12\pi(2.998\times10^{8})}=\mathbf{4.39\times10^{-3}\ W} .

Step 3. Cross-check via radiation resistance. Writing $\langle P\rangle=\tfrac12I_0^2R_{\text{rad}}$ defines $R_{\text{rad}}=\frac{2\pi Z_0}{3}(dz/\lambda)^2=789\,(dz/\lambda)^2\ \Omega=789\times(0.01/3)^2=8.77\times10^{-3}\ \Omega$ , giving $\langle P\rangle=4.39$ mW — identical. (The textbook " $80\pi^2\approx790$ " coefficient is this $2\pi Z_0/3$ with the engineering convention $Z_0=120\pi$ ; the strong-UG Electric Dipole Radiation quotes exactly this resistance, and Center-Fed Linear Antenna builds the full-size version where the current distribution, not just its element, must be honored.)

Step 4. Count the crowd. The element carries $\sim10^{20}$ conduction electrons; one electron oscillating with the same velocity amplitude would radiate $\sim10^{-40}$ -class power. Milliwatts from a centimeter of wire is the coherence theorem in commercial operation — every transmitting antenna is a standing monument to $N^2$ .

The lesson: $\sum q_ia_i=\dot I\,dz$ turns statistical mechanics' worth of electrons into one tractable source term, and the radiation-resistance bookkeeping is just Larmor's coefficient wearing an engineer's units.

Going further

References. Jackson, Classical Electrodynamics, 3rd ed., §14.1–14.2, §14.5 · Griffiths, Introduction to Electrodynamics, 4th ed., §11.2.1 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §10.6 · Zangwill, Modern Electrodynamics, §20.5, §23 · Landau & Lifshitz, Classical Theory of Fields, §66–67

Provenance. All equations standard. A verifier should: (1) reproduce the Larmor derivation by integrating $S=\varepsilon_0 c E_{\mathrm{rad}}^2$ over a sphere using $\int\sin^2\theta\,\mathrm{d}\Omega=8\pi/3$ to obtain $P=\mu_0 q^2 a^2/(6\pi c)$ (Griffiths §11.2.1 Eq. 11.70; Jackson §14.1 Eq. 14.22, Gaussian $\tfrac{2}{3}q^2a^2/c^3$ ); (2) confirm the Gaussian/SI equivalence via $\mu_0=1/(\varepsilon_0 c^2)$ ; (3) recompute the hydrogen collapse time $\tau=4\pi^2\varepsilon_0^2 m^2 c^3 a_0^3/(3e^4)\approx1.6\times10^{-11}\,\mathrm{s}$ using $a_0=5.29\times10^{-11}\,\mathrm{m}$ (Griffiths Problem 11.20). Constants: $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ , $c=2.998\times10^8\,\mathrm{m/s}$ . Cross-checks in Purcell §10.6 and Zangwill §20.5. Addendum for this edition (division of labor and corrections): this E11 edition is retitled "Larmor Formula from the Liénard–Wiechert Fields" to break the title duplication with the landed strong-UG twin Larmor Formula, which owns the flagship Poynting integration (including the $8\pi/3$ integral, quoted here by citation), the Thomson kink construction, the hydrogen-collapse and cyclotron-cooling numbers; none are re-derived here. This edition owns the source-clock definition $dP(t_r)/d\Omega=R^2(\mathbf{S}\cdot\hat{\mathbf{n}})\kappa$ and the general-orientation $\sin^2\Theta$ theorem (Derivation A), the coherence theorem with the $d\ll\lambda$ condition (Derivation B), and the Parseval spectral ledger with the Lorentzian worked closure (Derivation C). Two base defects corrected: the base's eqCard (8) "characteristic radiation time scale" was garbled (malformed exponent and inverted constant) — the intended object is $\tau_c=q^2/6\pi\varepsilon_0mc^3=6.27\times10^{-24}$ s for the electron, stated correctly in the FAQ below and owned by the radiation-reaction articles; and the base provenance's collapse closed form $4\pi^2\varepsilon_0^2m^2c^3a_0^3/(3e^4)$ carries a stray factor $1/3$ (the correct closed form is $4\pi^2\varepsilon_0^2m^2c^3a_0^3/e^4=a_0^3/4r_e^2c\approx1.6\times10^{-11}$ s, consistent with the number quoted; the calculation itself is the strong-UG twin's). Worked numbers for this edition recomputed and machine-verified: kick $a=3.75\times10^{19}$ m/s², $W_1=8.0\times10^{-27}$ J, coherent bunch $8.0\times10^{-11}$ J $=0.50$ GeV; current element $4.39$ mW both routes, $R_{\text{rad}}=2\pi Z_0/3\,(dz/\lambda)^2=789\,(dz/\lambda)^2\ \Omega$ ; Parseval closure $\mu_0q^2a_0^2\tau_d/12\pi c$ by both ledgers.

Builds on: Liénard–Wiechert Potentials: Covariant Formulation · Fields of a Relativistic Moving Charge · Electric Dipole Radiation: Exact Fields and Zone Structure Leads to: Relativistic Generalization of Larmor's Formula · Angular Distribution of Radiation from Accelerated Charges · Synchrotron Radiation · Bremsstrahlung

How is this article different from the strong-UG Larmor Formula article — which one should I read first?

Read the strong-UG Larmor Formula first: it owns the flagship derivation (Poynting flux of the acceleration field, the $8\pi/3$ integral), the Thomson kink construction, and the celebrity applications — hydrogen's sixteen-picosecond collapse, cyclotron cooling. This E11 edition assumes all of that and supplies the graduate bookkeeping layered on top: the source-clock definition of emitted power (the $\kappa$ Jacobian, formalized before it becomes unavoidable relativistically), the proof that the pattern reads the acceleration at any orientation with velocity absent, the coherence theorem ( $N^2$ versus $N$ ) with its precise sub-wavelength condition, and the Parseval spectral ledger. The two articles share one formula and zero derivations.

Why formalize the κ Jacobian here at all, if it reduces to 1 in every formula this article actually closes?

Because definitions adopted in easy regimes are the ones that survive hard ones. Every relativistic radiation quantity downstream — Liénard's $\gamma^6$ power, the $\kappa^{-5}$ angular lobes, synchrotron pulse compression — hinges on which clock the power refers to, and retrofitting the convention after the $\gamma$ factors arrive is how factors of $2\gamma^2$ go missing. Adopting $dP(t_r)/d\Omega=R^2(\mathbf{S}\cdot\hat{\mathbf{n}})\kappa$ while $\kappa\approx1$ costs nothing and means the strong-UG relativistic article's doctrine, this chapter's angular-distribution machinery, and the nonrelativistic Larmor formula are one continuous framework. It is the same pedagogy as defining proper time before you need it: the bookkeeping is the physics, installed early.

Doesn't the N-squared coherent enhancement violate energy conservation?

No — it presents a bigger bill to the driver, which pays it. Radiated energy comes from the work done by whatever agent accelerates the charges, and when $N$ charges move coherently, each one is pushed not only against its own radiation field but against the fields of all the others (this inter-charge back-force is the collective version of radiation reaction). The work integral then scales as $N^2$ , exactly matching the radiated energy. In accelerators this is not academic: coherent synchrotron radiation drains bunch energy fast enough to distort and destabilize short bunches, and the "microbunching instability" is the machine feeling the $N^2$ bill arrive. Nothing is free; coherence just reorganizes who pays how much.

When exactly does the coherent N-squared break down as the bunch gets bigger?

Continuously, frequency by frequency. The exact statement keeps the phases: the power at wavenumber $k$ carries a form factor $\big|\sum_ie^{-ik\hat{\mathbf{n}}\cdot\mathbf{r}_i}\big|^2$ — $\approx N^2$ for $kd\ll1$ , decaying to $\approx N$ once $kd\gtrsim1$ for a smooth bunch, with the crossover set by the bunch's density Fourier spectrum. A bunch of length $\sigma$ therefore radiates coherently at $\lambda\gtrsim2\pi\sigma$ and incoherently below — why storage-ring THz programs fight for short bunches, and why an FEL must create density structure at the optical wavelength to buy coherence there. Derivation B's clean $N^2/N$ dichotomy is the two ends of this form-factor interpolation.

The spectral formula integrates only over positive frequencies. Where did the negative frequencies go?

Folded in, honestly. A real acceleration history satisfies $\tilde a(-\omega)=\tilde a(\omega)^*$ , so the energy integrand is even and the two half-axes carry identical halves of the total; since detectors register only $|\omega|$ , the convention here doubles the positive-frequency integrand and integrates $\omega\ge0$ . When comparing references, check two conventions before trusting factors of 2 and $\pi$ : the placement of $2\pi$ in the transform pair, and one-sided versus two-sided spectra — the classic reason two correct-looking bremsstrahlung spectra disagree by exactly $2\pi$ .

If the pattern is velocity-blind at leading order, at what speed does the velocity actually start to show?

At first order in β, and in the pattern before the total. Expanding the exact $\kappa^{-5}$ distribution for small β gives $dP/d\Omega\propto\sin^2\Theta\,(1+5\,\hat{\mathbf{n}}\cdot\boldsymbol\beta+\ldots)$ — a forward tipping of the donut proportional to β, easily visible at $\beta\sim0.1$ , while the total power picks up velocity corrections only at $O(\beta^2)$ (Liénard's $\gamma$ factors expanded). So a 10 keV electron beam ( $\beta\approx0.2$ ) already radiates measurably more forward than backward even though its total emission is still Larmor to a percent. The full story — lobes, the $1/2\gamma$ peak angle, the received-versus-emitted compounding — is Angular Distribution of Radiation from Accelerated Charges, which grows exactly this $O(\beta)$ seed.

What is the characteristic time built from the Larmor coefficient, and why does it keep appearing?

Divide the Larmor coefficient by $mc^2$ 's natural scales and one time emerges: $\tau_c=q^2/6\pi\varepsilon_0mc^3=6.27\times10^{-24}$ s for the electron (the base article's garbled "characteristic radiation time scale" card intended this object — restored here). Physically, $\tau_c$ is the time over which radiation losses compete with the particle's own inertia: radiation reaction matters when the motion has structure on scales approaching $\tau_c$ , spectral lines acquire the classical width $\gamma_{\text{cl}}=\omega_0^2\tau_c$ , and the electromagnetic self-energy problem hides at the length $c\tau_c=2r_e/3$ . It is the single knob of the radiation-reaction articles (Radiation Reaction Force, Abraham–Lorentz Equation and Its Pathologies), which own everything about it beyond this definition.

Larmor Formula from the Liénard–Wiechert Fields

Where this lives today

The physical picture

The math: one field, four ledgers

Limiting cases

Worked examples

1 · The picosecond mirror — one electron, then a hundred million in step (confidence builder)

2 · The centimeter antenna element — Larmor's crowd theorem meets radio engineering (exam level)

Going further

Before this

Where this leads