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Larmor formula

The Larmor formula is a fundamental equation in classical electromagnetism that quantifies the total power radiated by a non-relativistic point charge undergoing acceleration, expressing the energy loss due to electromagnetic radiation as P = \frac{q^2 a^2}{6 \pi \epsilon_0 c^3}, where q is the charge, a is the magnitude of its acceleration, \epsilon_0 is the vacuum permittivity, and c is the speed of light.^[1]^[2] Derived by the Irish physicist Sir Joseph Larmor in 1897, the formula emerged from early efforts to understand how accelerating charges produce electromagnetic waves, building on the work of James Clerk Maxwell and others in reconciling electricity, magnetism, and optics.^[3] Larmor's result, originally published in Gaussian units as P = \frac{2 q^2 a^2}{3 c^3}, provided the first precise non-relativistic expression for radiation power and laid the groundwork for later developments, including the relativistic generalization by Alfred Liénard and Emil Wiechert.^[2]^[1] The formula's significance extends across physics, serving as a cornerstone for analyzing energy dissipation in systems like particle accelerators, where it predicts substantial radiation losses for electrons (far exceeding those for protons due to the charge-to-mass ratio), and in astrophysics for modeling synchrotron radiation from charged particles spiraling in magnetic fields.^[1]^[2] It also connects to the radiation reaction force, which describes the self-force on an accelerating charge arising from its own emitted fields, influencing the motion of charged particles in high-energy environments.^[3]

Overview

Statement of the Formula

The Larmor formula provides the total instantaneous power P radiated by a single non-relativistic point charge q undergoing acceleration \mathbf{a} in vacuum.^[4]^[5] In SI units, this power is expressed as

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

where a = |\mathbf{a}| is the magnitude of the acceleration, c is the speed of light in vacuum, and \mu_0 is the vacuum permeability.^[6]^[7] An equivalent form in Gaussian (cgs) units is

P = \frac{2 q^2 a^2}{3 c^3}.

^[6]^[7] The total energy E radiated over a time interval is obtained by integrating the power with respect to time: E = \int P \, dt.^[6]^[7] This formula applies under the assumptions of non-relativistic speeds (v \ll c), a point-like charge, and no direct influence from external fields on the radiation process itself; a relativistic generalization extends it to higher velocities.^[6]^[7]

Historical Development

The development of the Larmor formula emerged within the framework of late 19th-century classical electrodynamics, where the luminiferous ether was posited as the medium for electromagnetic wave propagation, influencing interpretations of radiation from charged particles.^[8] Joseph Larmor, a proponent of ether-based theories, sought to reconcile Maxwell's equations with emerging ideas about subatomic charges, deriving the formula as part of his broader electron theory. This work addressed how accelerating charges, modeled as disturbances in the ether, produce radiation, setting the stage for later challenges from special relativity, which in 1905 eliminated the need for an absolute ether frame.^[9] Preceding Larmor's contribution, J.J. Thomson laid foundational groundwork in 1881 by analyzing the electromagnetic fields generated by moving electrified bodies, demonstrating that such motion induces magnetic effects and energy contributions that effectively increase the particle's inertial mass, hinting at radiation implications for accelerating charges.^[10] Larmor formalized the radiation power from an accelerating non-relativistic charge in his 1897 paper "On the Theory of the Magnetic Influence on Spectra; and on the Radiation from Moving Ions," published in the Philosophical Magazine, where he derived the result from classical electrodynamics applied to oscillating electrons or ions within the ether medium.^[11] This derivation quantified the energy loss due to acceleration, proportional to the square of the acceleration, as a key outcome of his dynamical ether theory.^[12] In 1903, Max Abraham extended these ideas by incorporating radiation reaction forces in models of extended electrons, refining the non-relativistic formula through considerations of electromagnetic self-interaction and damping, which built directly on Larmor's result to address energy conservation in accelerating systems.^[3] The formula also found use in nascent atomic models, like Thomson's 1904 plum pudding structure, to estimate radiation from orbiting electrons, highlighting challenges in classical stability despite qualitative consistency with spectral line broadening in the Zeeman effect.^[3]

Theoretical Foundations

Electromagnetic Radiation from Accelerating Charges

Electromagnetic radiation arises fundamentally from the principles encoded in Maxwell's equations, which describe how time-varying electric and magnetic fields propagate as waves through space. Specifically, a changing electric field induces a magnetic field, and vice versa, leading to self-sustaining electromagnetic waves that carry energy away from their source. This radiation is produced by time-varying currents or, equivalently, by accelerating electric charges, as the acceleration causes the charge's velocity to change, generating fluctuating fields that extend to infinity.^[13]^[14] Static charges, which possess no motion, produce only electrostatic fields that diminish rapidly with distance and do not radiate energy. Similarly, charges moving with uniform velocity generate fields that, in their rest frame, resemble static Coulomb fields, resulting in no radiation; any apparent motion in the lab frame is transformed via Lorentz invariance without producing propagating waves. In contrast, accelerating charges disrupt this equilibrium: the changing velocity alters the field's configuration over time, creating a displacement current that, according to Maxwell's Ampère's law with Maxwell's correction, sustains wave propagation. This distinction underscores that radiation requires non-zero acceleration, not mere motion.^[13]^[14] The propagating component of these fields, known as radiation fields, emerges in the far-field region—far from the source relative to the wavelength—where the dominant terms in the expansion of the electromagnetic potentials vary inversely with distance, unlike the nearer inductive or static terms that fall off faster. These far-field terms represent transverse waves that detach from the source and transport energy outward indefinitely, with the electric and magnetic fields perpendicular to the direction of propagation and to each other. For non-relativistic accelerating charges, the leading-order radiation typically takes the form of electric dipole radiation, where an oscillating dipole moment—arising from the charge's back-and-forth motion—dominates the emission pattern, with intensity varying as the square of the sine of the angle between the acceleration vector and the line of sight.^[14]^[13] The energy carried by these radiation fields is quantified by the Poynting vector, which points radially outward in the radiation zone and represents the directional energy flux of the electromagnetic wave, with magnitude proportional to the product of the electric and magnetic field strengths. Integrated over a spherical surface enclosing the source, this flux yields the total power radiated, a result encapsulated quantitatively by the Larmor formula for non-relativistic cases. This mechanism explains diverse phenomena, from antenna emissions to atomic transitions, highlighting the universal role of acceleration in generating observable electromagnetic radiation.^[14]^[13]

Non-Relativistic Limit

The non-relativistic regime of the Larmor formula applies when the velocity v of the charged particle is much less than the speed of light c, typically v \ll c, such that the Lorentz factor \gamma = (1 - v^2/c^2)^{-1/2} \approx 1 and relativistic effects like Lorentz contraction become negligible.^[6]^[15] In this limit, the particle's motion can be treated classically without significant time dilation or length contraction influencing the radiation process.^[16] The Larmor formula emerges as the first-order expansion in v/c of the more general relativistic radiation formulas, where higher-order terms involving v/c vanish, simplifying the power radiated to depend primarily on the particle's acceleration.^[15]^[16] Specifically, the relativistic expression reduces to the non-relativistic form when \gamma \to 1 and the cross terms with velocity perpendicular to acceleration are neglected, yielding the scalar power invariant in the particle's instantaneous rest frame.^[15] In this approximation, radiation is dominated by the electric dipole term, with magnetic dipole and higher multipole contributions neglected due to the low velocities, which suppress magnetic field effects relative to electric ones.^[16]^[6] This electric dipole dominance holds in the long-wavelength limit where the acceleration scale is much smaller than the emitted wavelength. The formula is valid for typical atomic-scale accelerations, where electron velocities satisfy v/c \lesssim 10^{-2}, as in bound atomic orbits, ensuring the non-relativistic assumptions align with observed low-speed dynamics.^[16] However, it breaks down at high energies, such as in particle accelerators where v \approx c and \gamma \gg 1, necessitating the full relativistic treatment to account for beamed forward radiation and enhanced power output.^[6]^[15]

Derivation

Poynting Vector Approach

The Poynting vector approach to deriving the Larmor formula involves computing the electromagnetic fields produced by a non-relativistic accelerating point charge in the far-field (radiation) zone and then integrating the radial component of the Poynting vector over a closed surface surrounding the charge to obtain the total radiated power.^[17] This method relies on the Lienard-Wiechert fields in the limit where the charge velocity is much less than the speed of light (β ≪ 1), focusing on the 1/r-falling radiation terms that carry energy away from the source.^[2] In the radiation zone, at large distances r from the charge and evaluated at the retarded time, the electric field of a non-relativistic accelerating point charge q with acceleration \vec{a} is transverse and given by

\vec{E}_\mathrm{rad} = \frac{q}{4\pi \epsilon_0 c^2 r} \left[ \hat{n} \times (\hat{n} \times \vec{a}) \right]_\mathrm{ret},

where \hat{n} is the unit vector from the charge's retarded position to the observation point, and the subscript "ret" denotes evaluation at the retarded time. The magnitude of this field in spherical coordinates, assuming the acceleration is along the z-axis for simplicity, is the θ-component

E_\theta = \frac{q a \sin\theta}{4\pi \epsilon_0 c^2 r},

with no radial or ϕ-components in the radiation zone, and θ the angle between \vec{a} and \hat{n}. The associated magnetic field is azimuthal and related by

B_\phi = -\frac{E_\theta}{c},

ensuring the fields are perpendicular and transverse to the propagation direction.^[17] The time-averaged energy flux is carried by the Poynting vector \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}. In the radiation zone, the fields are perpendicular with |E| = c |B|, so the radial component simplifies to

S_r = \frac{E_\theta^2}{\mu_0 c}.

Substituting the expression for E_θ yields

S_r = \frac{1}{\mu_0 c} \left( \frac{q a \sin\theta}{4\pi \epsilon_0 c^2 r} \right)^2.

This represents the power per unit area flowing radially outward.^[2] To find the total instantaneous radiated power P, integrate S_r over a spherical surface of radius r (where r ≫ wavelength, ensuring the far-field approximation holds and azimuthal symmetry about the acceleration axis):

P = \oint \vec{S} \cdot d\vec{A} = \int_0^{2\pi} d\phi \int_0^\pi S_r r^2 \sin\theta \, d\theta.

The r^2 in the area element cancels the 1/r^2 dependence in S_r, making P independent of r and confirming energy conservation in the far field. The ϕ-integral gives 2π due to azimuthal symmetry, leaving the θ-integral over sin^3 θ (from sin^2 θ in S_r and sin θ in dΩ). Evaluating \int_0^\pi \sin^3 \theta , d\theta = 4/3, the full solid angle integral is (8π)/3. Substituting all terms and simplifying using μ_0 ε_0 = 1/c^2 produces the Larmor formula:

P = \frac{\mu_0 q^2 a^2}{6\pi c}.

This result assumes observation at large r, use of retarded times for field evaluation, and non-relativistic motion with no higher-order velocity corrections. The derivation highlights the sin^2 θ angular dependence of the instantaneous power per unit solid angle, though the total power averages over directions.^[17]

Energy Conservation Argument

One heuristic derivation of the Larmor formula relies on energy conservation, considering the work done by the electromagnetic self-fields on the accelerating charge to account for the radiated energy. This approach avoids the explicit computation of radiation fields, focusing instead on the balance between mechanical power input and electromagnetic energy loss. In an early heuristic argument, J. J. Thomson considered the energy flux through a large spherical surface surrounding the accelerating charge. He posited that the radiation electric field falls off as E \sim \frac{q a}{c^2 r}, where q is the charge, a is the acceleration, c is the speed of light, and r is the distance from the charge. The Poynting vector magnitude is then proportional to \frac{E^2}{c}, and integrating the flux over the sphere's surface—accounting for the angular dependence \sin^2 \theta—yields a total radiated power of P = \frac{2 q^2 a^2}{3 c^3} in Gaussian units. This simple flux balance matches the result from detailed field calculations but provides a quicker insight into the scaling of radiation with acceleration.^[18] Building on such ideas, Max Abraham provided a more formal treatment in 1903, using the Lienard-Wiechert potentials and conservation of energy and momentum to derive the power radiated by an accelerating charge in the non-relativistic limit. The radiated power equals the negative rate of change of the particle's mechanical energy minus the rate of change of the electromagnetic field's energy, leading to the Larmor formula P = \frac{2 q^2 a^2}{3 c^3} (Gaussian units). For time-varying accelerations, such as in periodic motion, the average power loss matches this expression through the work done against the radiation reaction force, as detailed in the radiation reaction section. This self-consistent energy balance confirms the formula without direct field integration at infinity.^[19]

Relativistic Extension

Covariant Formulation

The covariant formulation of the Larmor formula expresses the radiated power in a Lorentz-invariant manner, applicable to charged particles with arbitrary relativistic velocities. This generalization, known as Liénard's formula, ensures that the expression transforms correctly under Lorentz boosts and is derived within the framework of special relativity. The power P radiated by a point charge q is given in covariant form as

P = \frac{\mu_0 q^2}{6 \pi m^2 c} \left( -\frac{d p^\mu}{d \tau} \frac{d p_\mu}{d \tau} \right),

where p^\mu = m u^\mu is the four-momentum, u^\mu is the four-velocity, \tau is the proper time, m is the particle's rest mass, and the summation uses the Minkowski metric with signature (+,-,-,-) such that

\frac{d p^\mu}{d \tau} \frac{d p_\mu}{d \tau} &lt; 0

for the spacelike four-acceleration, yielding positive power with the explicit minus sign.^[20]^[21] This scalar expression represents the total power measured in the laboratory frame and reduces to the non-relativistic Larmor formula in the particle's instantaneous rest frame, where \gamma = 1, \beta = 0, and the three-acceleration \mathbf{a} satisfies a^\mu a_\mu = -a^2, giving P = \frac{\mu_0 q^2 a^2}{6 \pi c}.^[17] In the lab frame, the four-acceleration magnitude relates to the Lorentz factor \gamma = (1 - \beta^2)^{-1/2} and velocity components, leading to an explicit three-vector form:

P = \frac{\mu_0 q^2 \gamma^6}{6 \pi c} \left[ |\mathbf{a}|^2 - \left| \boldsymbol{\beta} \times \mathbf{a} \right|^2 \right],

with \boldsymbol{\beta} = \mathbf{v}/c and \mathbf{a} = d\mathbf{v}/dt. Decomposing the acceleration into components parallel (a_\parallel = (\mathbf{a} \cdot \boldsymbol{\beta}) \boldsymbol{\beta}) and perpendicular (\mathbf{a}_\perp = \mathbf{a} - \mathbf{a}_\parallel) to the velocity yields

P = \frac{\mu_0 q^2 \gamma^6}{6 \pi c} \left( a_\parallel^2 + (1 - \beta^2) a_\perp^2 \right).

For acceleration parallel to the velocity (a_\perp = 0), the power scales as \gamma^6 a_\parallel^2; for perpendicular acceleration (a_\parallel = 0), it scales as \gamma^4 a_\perp^2 since $1 - \beta^2 = \gamma^{-2}.^[17]^[21] The derivation proceeds from the Liénard-Wiechert potentials, which give the electromagnetic fields of a relativistic point charge. The scalar and vector potentials at a field point are evaluated at retarded proper time, incorporating the denominator factor \kappa = 1 - \boldsymbol{\beta} \cdot \hat{\mathbf{n}}, where \hat{\mathbf{n}} is the unit vector from the retarded position to the observation point. The electric and magnetic fields separate into near-field (velocity) and radiation (acceleration) terms, with the radiation fields scaling as $1/R. The time-averaged power is found by integrating the radial component of the Poynting vector \mathbf{S} = (1/\mu_0) \mathbf{E} \times \mathbf{B} over a large sphere, which simplifies to an angular average of the radiation field's squared magnitude. Relativistic effects, including \gamma enhancements and the \kappa factor, emerge naturally from this integration, confirming the \gamma^6 and \gamma^4 scalings.^[21]^[1] In highly relativistic regimes (\gamma \gg 1), the perpendicular case dominates applications like circular accelerators, where the \gamma^4 enhancement (combined with a_\perp \propto \gamma^2 for fixed radius of curvature) leads to substantial energy loss, further amplified by relativistic beaming that concentrates the radiation forward despite the total power being integrated over all angles.^[17]

Angular Distribution in Relativistic Case

In the relativistic case, the angular distribution of radiated power from an accelerating charge deviates significantly from the non-relativistic dipole pattern, exhibiting a strong forward beaming effect due to Lorentz contraction of the fields and retardation. The radiation is predominantly confined to a narrow cone of opening angle approximately $1/\gamma along the direction of the particle's velocity, where \gamma = 1/\sqrt{1 - \beta^2} and \beta = v/c. This concentration arises from the relativistic transformation, making the observed power orders of magnitude higher in the forward direction compared to backward emission for \gamma \gg 1.^[17] The general relativistic angular distribution for the instantaneous power is

\frac{dP}{d\Omega} = \frac{\mu_0 q^2 a^2}{16 \pi^2 c} \frac{ |\mathbf{n} \times [(\mathbf{n} - \boldsymbol{\beta}) \times \dot{\boldsymbol{\beta}} ] |^2 }{(1 - \mathbf{n} \cdot \boldsymbol{\beta})^5 },

where \mathbf{n} is the unit vector in the direction of observation, \boldsymbol{\beta} = \mathbf{v}/c, and \dot{\boldsymbol{\beta}} = \mathbf{a}/c, evaluated at retarded time. For acceleration perpendicular to velocity, this simplifies to a form with strong dependence on the angle \theta from the velocity direction, typically involving terms like \sin^2 \phi / (1 - \beta \cos \theta)^5, where \phi is the azimuthal angle relative to the acceleration plane, reflecting the \gamma^4 enhancement in total power and forward peaking.^[22]^[23] In the case of circular motion, relevant to synchrotron radiation, the instantaneous differential power follows a similar form, proportional to \sin^2 \psi / (1 - \beta \cos\theta)^5, where \psi is the angle between the instantaneous acceleration (radial, perpendicular to velocity) and the line of sight, and \theta is the angle from the instantaneous velocity direction. The time-averaged distribution over one orbital period yields an azimuthally symmetric pattern around the average velocity axis, with a critical angle \theta_c \approx 1/\gamma beyond which the intensity falls rapidly due to the beaming. This averaged pattern maintains the forward concentration but broadens slightly compared to the instantaneous case, emphasizing the role of trajectory curvature in shaping the observable emission.^[17]

Applications

Classical Electron Orbit

In the classical Rutherford model of the atom, electrons are envisioned as orbiting the positively charged nucleus in stable circular paths, much like planets around the sun, with the centripetal acceleration a = v^2 / r maintained by the electrostatic attraction, where v is the orbital speed and r is the orbital radius. This acceleration implies that the electron, as a charged particle, continuously radiates electromagnetic energy according to the Larmor formula. The instantaneous power radiated is given by

P = \frac{\mu_0 e^2 a^2}{6 \pi c} = \frac{\mu_0 e^2 v^4}{6 \pi c r^2},

where e is the elementary charge, \mu_0 is the vacuum permeability, and c is the speed of light. As a result, the electron loses kinetic energy, causing the orbit to shrink in a spiraling decay toward the nucleus. The timescale for this orbital collapse can be estimated by considering the rate of energy loss relative to the initial orbital energy. For atomic-scale radii on the order of the Bohr radius (r \approx 5.3 \times 10^{-11} m), the characteristic time \tau is approximately

\tau \sim \frac{3 m_e^2 c^3 r^3}{4 \mu_0 e^4} \approx 10^{-11} \, \text{s},

where m_e is the electron mass; this rapid decay, on the order of $1.6 \times 10^{-11} s for hydrogen-like atoms, demonstrates the inherent instability of the classical model, as the electron would quickly spiral into the nucleus. Furthermore, the continuous acceleration in the orbit leads to radiation across a continuous spectrum of frequencies, rather than the discrete lines observed experimentally, exacerbating the classical ultraviolet catastrophe by predicting unbounded energy emission at high frequencies in thermal equilibrium contexts. Quantum mechanics resolves this classical instability through the concept of stationary states, where bound electrons do not radiate energy during orbital motion, as postulated in Bohr's 1913 model.

Synchrotron Radiation

Synchrotron radiation arises when a relativistic charged particle moves in a circular path due to a perpendicular magnetic field, as in particle accelerators. Consider a particle with charge q, rest mass m, velocity v = \beta c (where \beta < 1), and Lorentz factor \gamma = (1 - \beta^2)^{-1/2}. The magnetic field B deflects the particle into circular motion with radius \rho = \gamma m v / (q B). The resulting centripetal acceleration is a = v^2 / \rho = \beta^2 c^2 / \rho. This setup applies the relativistic extension of the Larmor formula, where the acceleration is perpendicular to the velocity. The total power radiated in this configuration is

P = \frac{\mu_0 q^2 \gamma^4 \beta^4 c^3}{6 \pi \rho^2},

expressed in terms of the orbit radius for perpendicular incidence of the magnetic field.^[24] This formula highlights the strong dependence on \gamma^4, making synchrotron radiation dominant at high energies. The emitted radiation forms a continuous spectrum extending from infrared to X-ray wavelengths, determined by the particle's energy and the bending radius. A key feature is the critical frequency \omega_c = \frac{3}{2} \gamma^3 \frac{c}{\rho}, above which the spectral power decreases exponentially; approximately 75% of the total energy is radiated below \omega_c. This broadband nature arises from the non-uniform acceleration during the orbit. In practice, synchrotron radiation is prominent in cyclotrons and storage rings. For instance, in the Large Electron-Positron Collider (LEP) at CERN, it imposed an energy limit of about 100 GeV per beam due to significant losses, while in the Large Hadron Collider (LHC), it aids beam cooling and serves as a tool for monitoring beam properties in proton operations. Dedicated synchrotron light sources exploit this radiation for applications in materials science and biology.^[25]^[26] The energy loss per turn \Delta E = \frac{\mu_0 q^2 c^2 \gamma^4}{3 \rho} requires compensation via radiofrequency cavities to sustain beam circulation, with losses scaling rapidly as \gamma^4 / \rho.^[24] In LEP, this loss reached megawatts at peak energies, necessitating advanced RF systems.^[27]

Radiation Reaction

Origin of the Self-Force

The self-force, also known as the radiation reaction force, arises from the back-reaction of an accelerating charged particle on its own electromagnetic field. When a charge accelerates, it generates electromagnetic disturbances that propagate outward at the finite speed of light c. As a result, the self-field at the particle's location reflects its past positions and accelerations rather than the instantaneous state, creating a lag that leads to a net force opposing the motion.^[28] This concept was first elucidated through the use of retarded potentials by Alfred Liénard in 1898, who described the fields of an arbitrarily moving charge.^[3] A key aspect of this self-interaction involves the Schott energy, which represents electromagnetic energy stored temporarily in the near field surrounding the particle. This energy, proportional to the product of velocity and acceleration, acts as an intermediate reservoir: it accumulates during changes in acceleration and is later released as radiated energy in the far field or returned to the particle's kinetic energy.^[29] Introduced by George A. Schott in 1912 and further characterized as "acceleration energy" in his 1915 work, the Schott energy helps reconcile the local energy balance in the presence of radiation reaction.^[3] In the non-relativistic regime, the time-averaged self-force over an acceleration cycle takes the form

\mathbf{F}_\text{self} \approx \frac{q^2}{6\pi \epsilon_0 c^3} \dot{\mathbf{a}},

where q is the charge, \epsilon_0 is the vacuum permittivity, c is the speed of light, and \dot{\mathbf{a}} is the jerk (time derivative of acceleration).^[30] This force performs work on the particle at a rate that, on average, equals the Larmor power radiated, embodying an equivalence principle where the energy loss to radiation is balanced by the mechanical work done against the self-force.^[30] The development of the self-force concept sparked significant historical debate, particularly over "runaway solutions" in which the particle would exhibit unphysical, exponentially growing acceleration without external input, due to the higher-order time derivatives in the equations of motion. Paul Dirac resolved this in 1938 by reformulating the theory using a symmetric combination of retarded and advanced potentials, defining the radiation field as their difference to eliminate pre-acceleration and runaway behaviors while preserving causal, realistic solutions.^[31] This approach, now central to the Lorentz-Dirac equation, ensured consistency with classical electrodynamics for point charges.^[3]

Abraham-Lorentz Formula

The Abraham-Lorentz formula, first derived by Max Abraham in 1903 and independently by Hendrik Lorentz in 1904, describes the radiation reaction force on a non-relativistic accelerating point charge, derived by considering the back-reaction of the charge's own electromagnetic field. This self-field contribution to the electric field at the charge's position, after accounting for the singular Coulomb term and velocity-dependent parts, yields an effective field \mathbf{E}_\text{self} \approx \frac{q}{6\pi \epsilon_0 c^3} \dot{\mathbf{a}}, where q is the charge, c is the speed of light, \epsilon_0 is the vacuum permittivity, and \dot{\mathbf{a}} = d\mathbf{a}/dt is the jerk (time derivative of acceleration \mathbf{a}). The resulting force is then \mathbf{F}_\text{rad} = q \mathbf{E}_\text{self} = \frac{\mu_0 q^2}{6\pi c} \dot{\mathbf{a}}, with \mu_0 the vacuum permeability, incorporating the non-relativistic limit of the Lorentz force law.^[3] This third-order differential equation, when added to Newton's second law, leads to the Abraham-Lorentz equation of motion: m \mathbf{a} = \mathbf{F}_\text{ext} + \frac{\mu_0 q^2}{6\pi c} \dot{\mathbf{a}}, where m is the mass and \mathbf{F}_\text{ext} is the external force. However, naive solutions exhibit unphysical behaviors, including pre-acceleration—where the charge begins accelerating before the external force is applied—and runaway solutions, characterized by exponential growth in velocity even after the external force ceases, such as \mathbf{v}(t) \propto e^{t/\tau} with characteristic time \tau = \frac{\mu_0 q^2 m}{6\pi c^2}. These pathologies arise from the higher-order nature of the equation and the idealized point-charge assumption.^[3] To mitigate these issues, the Landau-Lifshitz approximation provides a stable, reduced-order form valid when accelerations are not extreme: \mathbf{F}_\text{rad} \approx \frac{\mu_0 q^2}{6\pi c} \left( \dot{\mathbf{a}} - \frac{\mathbf{a}^2}{c^2} \mathbf{a} \right), effectively truncating higher derivatives while preserving energy conservation in perturbative regimes. The relativistic generalization, known as the Abraham-Lorentz-Dirac formula, expresses the four-force as K^\mu = \frac{2 q^2}{3 c^3} \left( \frac{d^2 u^\mu}{d\tau^2} + u^\mu \frac{(du^\nu / d\tau)(du_\nu / d\tau)}{c^2} \right), where u^\mu is the four-velocity, \tau is proper time, and indices follow the Minkowski metric; this covariant form extends the non-relativistic case but retains similar interpretive challenges.^[3] In classical applications, the formula models damping in driven oscillators, where the radiation reaction leads to energy loss and amplitude decay proportional to \tau, as seen in analyses of bound electron motion. However, due to its classical limitations and the absence of quantum effects like spontaneous emission, the Abraham-Lorentz formula is generally avoided in quantum electrodynamics, where self-interactions are handled perturbatively via renormalization and Feynman diagrams.^[3]

References

[1]
66 Radiation by an Accelerating Charge - Galileo and Einstein
Purcell's Derivation of the Larmor Formula. (Actually we gave Landau's derivation in lecture 57, but Purcell's gives much more insight into the field behavior.).
[2]
Larmor Formula - AstroBaki - CASPER
Aug 18, 2021 · Larmor Formula · 1 Electric Field of an Accelerated Charge · 2 Derivation of Power Pattern (Cyclotron) · 3 Another derivation of Larmor.
[3]
[PDF] On the History of the Radiation Reaction1 - Kirk T. McDonald
Jan 9, 2020 · 25While Hertz' result (9) agrees with the Larmor formula [46], the latter was deduced only in 1897, after ... Planck's derivation can be ...
[4]
IX. A dynamical theory of the electric and luminiferous medium.
A dynamical theory of the electric and luminiferous medium.— Part III. relations with material media. Joseph Larmor.Missing: formula | Show results with:formula
[5]
[PDF] Classical Electrodynamics - Duke Physics
20.1 Larmor's Formula . ... At last we have made it to Jackson's equation 9.11, but look how elegant our approach was. Instead of a form that is only valid ...
[6]
[PDF] Chapter 4 Radiation By Moving Charges
Larmor's formula. The non-relativistic expressions for P and dPldCl, are often referred to as the "dipole approximation" because they are exactly what is ...
[7]
[PDF] Classical Electrodynamics
Problems, 462. chapter 14. Radiation by Moving Charges. 14.1 Liénard-Wiechert potentials and fields, 464. 14.2 Larmor's radiated power formula and its ...
[8]
Joseph Larmor - Biography - MacTutor - University of St Andrews
He was the first to calculate the rate of energy radiation from an accelerating electron and for this he gave Larmor's formula which gives the power radiated in ...
[9]
[PDF] Ether and electrons in relativity theory (1900–1911) - PhilArchive
He supposed (following. Larmor) that electron radiation was due entirely to acceleration, and he calculated thereby the energy of an electron in uniform motion.
[10]
[PDF] J.J. Thomson and “Hidden” Momentum 1 Radiation Pressure and ...
(18). 5 Field Momentum of a Pair of Moving Charged. Particles. Thomson ... 30In the author's view, Thomson's 1881 paper [56] marks the beginning of elementary- ...
[11]
http://kirkmcd.princeton.edu/examples/EM/larmor_pm_44_503_97.pdf
[12]
[physics/0205046] A Hundred Years of Larmor Formula - arXiv
May 16, 2002 · Sir Joseph LARMOR showed in 1897 that an oscillating electric charge emits radiation energy proportional to (acceleration)^2. At first sight,the ...
[13]
Contributions of John Henry Poynting to the understanding of ...
Mar 28, 2012 · This paper reviews and assesses his radiation-pressure work, with a level of coverage aimed at the reader familiar with the Maxwell electromagnetic theory.
[14]
28 Electromagnetic Radiation - Feynman Lectures
In order to do so, we need an accelerating charge. It should be a single charge, but if we can make a great many charges move together, all the same way, we ...
[15]
[PDF] Classical Electromagnetism - Richard Fitzpatrick
Now, charges moving with a constant velocity constitute a steady current, so a nonsteady current is associated with accelerating charges. ... far field region ( ...
[16]
The Larmor formula - Richard Fitzpatrick
It follows, after a little algebra, that the relativistic generalization of Larmor's formula takes the form. \begin{displaymath} P= \frac{e^2}{6, (1649). next ...
[17]
[PDF] Unit 7-4: Relativistic Larmor's Formula
The non-relativistic Larmor's formula was based on the electric dipole approximation in the long wavelength limit, kd 1, which we saw was equivalent to the non ...
[18]
[PDF] Chapter Fourteen Radiation by Moving Charges
Sep 17, 2001 · This is the relativistic generalization of the Larmor formula. It is instructive to look at some simple examples. 2.1.1. Example: Synchrotron.
[19]
Electricity and matter : Thomson, J. J. (Joseph John), Sir, 1856-1940
Nov 28, 2006 · 1904. Topics: Electricity, Matter, Radio-activity ... DOWNLOAD OPTIONS. download 1 file · ABBYY GZ download · download 1 file · B/W PDF download.
[20]
Larmor's Formula
This has a covariant generalization that is valid for any velocity of charge. First we factor out an $m^2$ and convert this to momentum coordinates. Then we ...
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[PDF] 6. Electromagnetic Radiation - DAMTP
In Section 6.2.2, we derived the Larmor formula for the emitted power in the electric dipole approximation to radiation. In this section, we present the ...Missing: heuristic | Show results with:heuristic
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[PDF] RADIATION BY AN ACCELERATING CHANGE - UT Physics
By contrast, an accelerating particle produces EM radiation, in which the electric and the magnetic fields decrease with distance as 1/r rather than 1/r2, and ...Missing: E_θ SI
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[PDF] Today in Physics 218: relativistic accelerating charges
Mar 22, 2004 · Relativistic charges and the generalized Larmor formula ... 1 cos . β θ. −. dP dΩ . v c. 1, β → cos β θ. (. ) 5. 1 cos. 1 if. 1,. 0. β.
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[PDF] Synchrotron Radiation - PHYS 473 - CERN Indico
Aug 24, 2023 · This formula tells us how much energy is radiated by the particle due to the change in its momentum. Very week radiation for low momentum ...
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[PDF] The large hadron collider (LHC) in the LEP tunnel
synchrotron radiation losses for an average circulating current of 8.4 mA at 100 GeV. It. Adequate RF power is available from the LEP RF system to compensate ...
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[PDF] Synchrotron radiation in LHC: spectrum and dynamics - CERN Indico
In LHC at 7 TeV, the critical photon energy is about 7 keV, energy loss per turn is 42 eV, and damping times are on the order of 1 day.
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[PDF] The LEP collider - Comptes Rendus de l'Académie des Sciences
the energy loss by synchrotron radiation is collinear to the trajectory while the energy gain is entirely longitudinal. The vertical amplitudes are strongly ...<|control11|><|separator|>
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[PDF] Tracking the radiation reaction energy when charged bodies ...
Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York,. 1998). 18A. M. Steane, Relativity Made Relatively Easy (Oxford U.P. ...
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[PDF] The significance of the Schott energy for energy-momentum ... - arXiv
Eq. (48) shows that the radiation reaction force is a force that retards the motion, acting like friction in a fluid. The “push” in the direction of the motion ...
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[PDF] Self-Force and Radiation Reaction - UCSB Physics
Derivation of the Abraham-Lorentz Force. The Larmor formula gives the power radiated by an oscillating charge: P = µ0q2. 6πc a2 = µ0q2. 6πc. (a · a). (2.1) ...<|control11|><|separator|>
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### Summary of Dirac's 1938 Resolution of Runaway Solutions in Classical Radiation Reaction Self-Force