Radiation of an Electron on a Radial Path: The Larmor Formula

Problem Statement

An electron of charge \(-e\) moves in uniform circular motion with angular frequency \(\omega\) on a circle of radius \(r\). Because the electron is constantly accelerating (centripetally), it radiates electromagnetic energy. Assuming the electron is non-relativistic and that energy losses are negligible over one orbit, calculate the total radiated power.

This is a classic problem in electrodynamics and one of the earliest questions that forced physicists to confront the limitations of classical mechanics when applied to atomic structure ↗. The tool we need is the Larmor formula, which gives the power radiated by any non-relativistic accelerating point charge.

Hints

Hint 1. The key formula is the Larmor formula for the total radiated power of a non-relativistic point charge \(q\) experiencing acceleration of magnitude \(a\):

\[P = \frac{\mu_0 \, q^2 \, a^2}{6\pi c}\]

in SI units, where \(\mu_0\) is the permeability of free space and \(c\) is the speed of light. In Gaussian (CGS) units the same result reads

\[P = \frac{2\,q^2\,a^2}{3\,c^3} \; .\]

Hint 2. For uniform circular motion, the velocity has constant magnitude \(v = r\omega\) but continuously changing direction. The acceleration is purely centripetal:

\[a = \frac{v^2}{r} = r\,\omega^2 \; .\]

Combining these two ingredients gives the answer directly.

Solution

We proceed in two short steps.

Step 1 — Centripetal acceleration. The electron moves on a circle of radius \(r\) with constant angular frequency \(\omega\). Its position can be written as

\[\mathbf{r}(t) = r\,\cos(\omega t)\,\hat{\mathbf{x}} + r\,\sin(\omega t)\,\hat{\mathbf{y}} \; .\]

Differentiating twice:

\[\mathbf{a}(t) = \ddot{\mathbf{r}}(t) = -r\,\omega^2 \left[\cos(\omega t)\,\hat{\mathbf{x}} + \sin(\omega t)\,\hat{\mathbf{y}}\right] \; .\]

The magnitude of the acceleration is constant:

\[a = |\mathbf{a}| = r\,\omega^2 \; .\]

Step 2 — Apply the Larmor formula. Substituting \(q = e\) (the elementary charge; the sign does not matter since \(q\) enters squared) and \(a = r\omega^2\) into the Larmor formula:

\[\boxed{P = \frac{\mu_0\,e^2\,r^2\,\omega^4}{6\pi\,c}} \; .\]

Equivalently, since \(v = r\omega\), we can write this as

\[P = \frac{\mu_0\,e^2\,v^4}{6\pi\,c\,r^2} \; .\]

Several features of this result are worth noting:

The radiated power scales as \(\omega^4\). Doubling the angular frequency increases the power by a factor of 16.
It scales as \(r^2\): a larger orbit at the same \(\omega\) means a higher velocity and therefore a larger acceleration.
The formula is valid only in the non-relativistic regime \(v \ll c\). For relativistic electrons the relativistic Larmor formula (Lienard's result) must be used, which introduces an additional factor of \(\gamma^4\) ↗.

Historical Background: Larmor and the Radiation of Accelerating Charges

The formula we just applied is named after the Irish physicist Joseph Larmor (1857–1942), who derived it in 1897 in his treatise Aether and Matter [^1]. Larmor was Lucasian Professor of Mathematics at Cambridge — the same chair once held by Newton and later by Dirac and Hawking. His work on radiation from accelerating charges was part of a broader effort to reconcile the electron theory of Hendrik Lorentz with Maxwell's electrodynamics.

Larmor's result was one of the first quantitative statements about radiation from a single charged particle, and it had immediate consequences. If the Larmor formula is applied to an electron orbiting an atomic nucleus — the classical Bohr-like picture — one finds that the electron should radiate away its energy and spiral into the nucleus in roughly \(10^{-11}\) seconds [^2]. This radiation catastrophe was a central puzzle of pre-quantum physics and one of the strongest motivations for the development of quantum mechanics ↗.

Beyond atomic physics, the Larmor formula and its relativistic generalizations became indispensable in understanding synchrotron radiation, bremsstrahlung, and cyclotron emission — processes that are central to accelerator physics, astrophysics, and plasma physics.

Derivation Outline of the Larmor Formula

The full derivation of the Larmor formula is a standard exercise in graduate electrodynamics courses (see Jackson [^3], Ch. 14, or Griffiths [^4], Ch. 11). Here we outline the main steps. The approach connects directly to the radiation theory developed for antennas on this site ↗ ↗.

1. Retarded potentials. The starting point is Maxwell's equations for a point charge \(q\) following a trajectory \(\mathbf{r}_s(t)\). In the Lorenz gauge, the scalar potential \(\Phi\) and vector potential \(\mathbf{A}\) satisfy wave equations whose solutions are the retarded potentials:

\[\Phi(\mathbf{r},t) = \frac{1}{4\pi\varepsilon_0}\frac{q}{\mathscr{R}\left(1 - \hat{\boldsymbol{\mathscr{R}}}\cdot\boldsymbol{\beta}\right)}\Bigg|_{t_r}\]

\[\mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4\pi}\frac{q\,\mathbf{v}}{\mathscr{R}\left(1 - \hat{\boldsymbol{\mathscr{R}}}\cdot\boldsymbol{\beta}\right)}\Bigg|_{t_r}\]

where \(\boldsymbol{\mathscr{R}} = \mathbf{r} - \mathbf{r}_s(t_r)\) is the vector from the charge's retarded position to the field point, \(\boldsymbol{\beta} = \mathbf{v}/c\), and \(t_r\) is the retarded time defined implicitly by \(c(t - t_r) = |\mathbf{r} - \mathbf{r}_s(t_r)|\). These are the Lienard–Wiechert potentials [^3].

2. Electric and magnetic fields. From the potentials, one computes the fields via \(\mathbf{E} = -\nabla\Phi - \partial\mathbf{A}/\partial t\) and \(\mathbf{B} = \nabla \times \mathbf{A}\). The result splits naturally into two terms: a velocity field (Coulomb-like, falls off as \(1/\mathscr{R}^2\)) and a radiation field (falls off as \(1/\mathscr{R}\)). Only the radiation field carries energy to infinity. In the non-relativistic limit (\(\beta \to 0\)), the radiation electric field at position \(\mathbf{r}\) is:

\[\mathbf{E}_{\text{rad}} = \frac{q}{4\pi\varepsilon_0\,c^2}\frac{\hat{\boldsymbol{\mathscr{R}}}\times\left(\hat{\boldsymbol{\mathscr{R}}}\times\mathbf{a}\right)}{\mathscr{R}}\]

where \(\mathbf{a}\) is the acceleration of the charge at the retarded time. The magnetic radiation field is \(\mathbf{B}_{\text{rad}} = \hat{\boldsymbol{\mathscr{R}}}\times\mathbf{E}_{\text{rad}}/c\).

3. Poynting vector and angular distribution. The instantaneous power radiated per unit solid angle is found from the far-field Poynting vector:

\[\frac{dP}{d\Omega} = \frac{1}{\mu_0}\,\mathscr{R}^2\,|\mathbf{E}_{\text{rad}}|^2 / c = \frac{q^2\,a^2}{16\pi^2\,\varepsilon_0\,c^3}\,\sin^2\theta\]

where \(\theta\) is the angle between the acceleration vector \(\mathbf{a}\) and the direction of observation \(\hat{\boldsymbol{\mathscr{R}}}\). This \(\sin^2\theta\) pattern is the same toroidal (doughnut-shaped) radiation pattern found for the infinitesimal dipole ↗. No radiation is emitted along the direction of acceleration; the maximum is in the plane perpendicular to it.

4. Integration over all angles. Integrating \(dP/d\Omega\) over the full sphere,

\[P = \int_0^{2\pi}d\varphi\int_0^{\pi}\frac{dP}{d\Omega}\,\sin\theta\,d\theta = \frac{q^2\,a^2}{16\pi^2\,\varepsilon_0\,c^3}\cdot 2\pi\int_0^{\pi}\sin^3\theta\,d\theta \; .\]

The angular integral evaluates to

\[\int_0^{\pi}\sin^3\theta\,d\theta = \frac{4}{3} \; ,\]

so that

\[P = \frac{q^2\,a^2}{6\pi\,\varepsilon_0\,c^3} = \frac{\mu_0\,q^2\,a^2}{6\pi\,c}\]

where we used \(\mu_0 = 1/(\varepsilon_0\,c^2)\). This is the Larmor formula in SI units. Converting to Gaussian units (setting \(4\pi\varepsilon_0 = 1\)) gives the equivalent form \(P = 2q^2a^2/(3c^3)\).

Physical Applications

Synchrotron radiation. When electrons are accelerated to relativistic speeds in circular accelerators (synchrotrons), they emit intense electromagnetic radiation. The power is vastly enhanced over the non-relativistic Larmor result by a factor of \(\gamma^4\), where \(\gamma = (1 - v^2/c^2)^{-1/2}\) is the Lorentz factor. For a relativistic electron in a magnetic field \(B\), the radiated power becomes [^3]

\[P_{\text{rel}} = \frac{\mu_0\,e^2\,c}{6\pi}\,\gamma^4\,\left(\frac{eB}{\gamma m_e}\right)^2 = \frac{\mu_0\,e^4\,B^2\,\gamma^2}{6\pi\,m_e^2\,c} \; .\]

At the Large Electron-Positron Collider (LEP) at CERN, each electron lost about 3 GeV per revolution to synchrotron radiation [^5]. This energy cost is the main reason why modern high-energy lepton colliders are designed as linear rather than circular machines. For protons, the radiation is suppressed by a factor of \((m_e/m_p)^4 \approx 10^{-13}\), which is why the proton-based Large Hadron Collider (LHC) can operate as a ring.

Synchrotron radiation is not merely a nuisance for accelerator builders. Dedicated synchrotron light sources — such as the European Synchrotron Radiation Facility (ESRF), the Advanced Photon Source (APS), and dozens of others worldwide — exploit this radiation to produce extremely bright, tunable X-ray beams used in crystallography, materials science, medical imaging, and protein structure determination [^6].

Astrophysics. Synchrotron radiation is observed across the electromagnetic spectrum from astrophysical sources. Relativistic electrons spiraling along magnetic field lines in supernova remnants, active galactic nuclei, and the Crab Nebula produce characteristic polarized radiation with a power-law spectrum [^7]. The observed spectrum and polarization allow astrophysicists to infer the magnetic field strength and the energy distribution of the radiating electrons — a direct application of the Larmor formula and its relativistic generalization.

Cyclotron radiation and electron mass measurement. In the non-relativistic limit, electrons in a magnetic field emit at the cyclotron frequency \(\omega_c = eB/m_e\). Detection of single-electron cyclotron radiation has been used in Penning trap experiments to make precision measurements of the electron magnetic moment, one of the most precisely tested predictions in all of physics [^8].

Classical instability of the atom. As noted above, the Larmor formula predicts that a classical electron orbiting a proton would radiate away all its kinetic energy in about \(1.6 \times 10^{-11}\) seconds [^2]. The resolution of this paradox — that atoms are in fact stable — required the development of quantum mechanics. The Bohr model (1913) postulated discrete stationary orbits without radiation; the full resolution came with quantum electrodynamics, which replaced the classical orbit with a probability distribution that has no net acceleration in a stationary state.

Summary

For a non-relativistic electron on a circular orbit of radius \(r\) and angular frequency \(\omega\), the radiated power follows directly from the Larmor formula:

\[P = \frac{\mu_0\,e^2\,r^2\,\omega^4}{6\pi\,c} \; .\]

The derivation route — from Maxwell's equations through retarded potentials and the far-field Poynting vector to the \(\sin^2\theta\) angular integration — is one of the most instructive calculations in classical electrodynamics. The result, compact as it is, connects to some of the deepest questions in physics: why atoms are stable, how synchrotrons produce light, and how astrophysical magnetic fields reveal themselves through radiation.

Note: This article was updated in March 2026 using Claude Opus 4.6.