The Larmor Radiation Formula

The formula, symbol by symbol

The Larmor formula is the total power (energy per unit time) carried away as electromagnetic radiation by a single point charge whose speed is small compared to light:

P = q² a² / (6 π ε₀ c³)

• P — total radiated power, watts (W = J/s).
• q — the charge, coulombs (C). For an electron, q = −1.602 × 10⁻¹⁹ C; only q² appears, so sign is irrelevant.
• a — magnitude of the instantaneous acceleration, m/s². It is the square of this quantity that governs the output.
• ε₀ — permittivity of free space, 8.854 × 10⁻¹² F/m (equivalently C²·N⁻¹·m⁻²).
• c — speed of light in vacuum, 2.998 × 10⁸ m/s. Because it enters as c³, radiation is a weak effect at everyday accelerations.

In Gaussian (CGS) units, where the awkward 4πε₀ disappears, the same physics reads P = 2q²a²/(3c³). The two forms are identical once you substitute the unit conventions; the SI version simply makes ε₀ explicit.

Why the power goes as acceleration squared

Start from the fields of a moving charge. Far from the charge, the radiation field (also called the acceleration field) dominates. For v ≪ c it has magnitude

E_rad = q·a·sinθ / (4πε₀ c² r)

Two features are decisive. First, E_rad is proportional to the acceleration a — no acceleration, no radiation field. Second, it falls off only as 1/r, not the 1/r² of a static Coulomb field. The Poynting vector — the energy flux — is proportional to E², so it goes as a²/r². When you integrate that flux over a sphere of radius r, whose area grows as r², the r-dependence cancels exactly, leaving a finite total power that depends on r not at all and on a² directly. That surviving 1/r radiation tail is precisely what allows energy to escape to infinity.

Doing the angular integral of the sin²θ pattern (below) supplies the numerical factor 6π, and the result is the Larmor formula. The physical takeaway: uniform motion is silent, acceleration is loud, and the loudness grows as the square of how hard you shake the charge.

The sin²θ dipole radiation pattern

Radiation is not emitted equally in all directions. The power per unit solid angle is

dP/dΩ = q² a² sin²θ / (16 π² ε₀ c³)

where θ is measured from the acceleration vector. The consequences:

• At θ = 90° (perpendicular to the acceleration) the emission is maximal.
• At θ = 0° and 180° (straight along the acceleration) the emission is exactly zero — a charge does not radiate in the direction it is being pushed.
• The three-dimensional pattern is a torus (doughnut) wrapped around the acceleration axis — the very same shape produced by a short dipole antenna, which is just a charge sloshing back and forth.

Integrating sin²θ over the full sphere gives ∫sin²θ dΩ = 8π/3, which converts dP/dΩ into the total P = q²a²/(6πε₀c³). This is why the effect is called dipole radiation: the leading term in the multipole expansion of any accelerating charge distribution is the electric dipole, and its angular signature is sin²θ.

The classical atom should collapse

Here is the historically explosive consequence. Model the hydrogen atom classically: an electron in a circular orbit of radius r₀ = 5.29 × 10⁻¹¹ m (the Bohr radius) around a proton. Circular motion means the electron is always accelerating toward the center with

a = v²/r = e² / (4πε₀ m_e r²) ≈ 9 × 10²² m/s²

Plug that into Larmor. The orbiting electron continuously radiates power on the order of 10⁻⁸ W, bleeding away its kinetic and potential energy. As it loses energy it spirals inward, accelerating harder, radiating even more — a runaway collapse. Integrating the energy loss gives a lifetime of only about

t_collapse ≈ 1.6 × 10⁻¹¹ s

In other words, a classical atom would emit a burst of ultraviolet light and self-destruct in about 16 picoseconds. Matter could not exist. The blatant contradiction with a stable, structured universe was one of the sharpest failures of classical physics and a direct spur toward Bohr's 1913 quantized orbits and, ultimately, quantum mechanics — where stationary states have definite energy and simply do not radiate.

Worked example: an electron in an antenna

Suppose an electron oscillates sinusoidally, x(t) = x₀ cos(ωt), with amplitude x₀ = 1 nm at frequency f = 10 GHz (ω = 2πf ≈ 6.28 × 10¹⁰ rad/s). The peak acceleration is a₀ = ω²x₀ ≈ (6.28 × 10¹⁰)² × 10⁻⁹ ≈ 3.9 × 10¹² m/s². The time-averaged radiated power (using ⟨a²⟩ = a₀²/2) is

⟨P⟩ = q² a₀² / (12 π ε₀ c³)
    = (1.6×10⁻¹⁹)² (3.9×10¹²)² / (12π · 8.854×10⁻¹² · (3×10⁸)³)
    ≈ 4 × 10⁻²⁹ W

Tiny for one electron — but multiply by the ~10²³ electrons oscillating coherently in a real antenna and the numbers become the milliwatts and watts of everyday radio. This is the microscopic root of all antenna emission: coherent Larmor radiation from many charges.

Synchrotron radiation and bremsstrahlung

Any acceleration radiates, but the direction of the acceleration relative to the velocity names the phenomenon:

Mechanism	Source of acceleration	Geometry	Typical setting
Synchrotron / cyclotron radiation	Magnetic Lorentz force (transverse)	a ⟂ v (centripetal)	Storage rings, pulsars, radio galaxies
Bremsstrahlung ("braking radiation")	Coulomb field of a nucleus	a mostly ∥ v (deceleration)	X-ray tubes, electron-target collisions
Dipole / antenna radiation	Applied oscillating field	a oscillating along wire	Radio, TV, cellular transmitters
Thomson / Compton scattering	Incident EM wave drives the charge	a set by wave polarization	Light scattering, plasma diagnostics

Both synchrotron and bremsstrahlung are computed from the relativistic Liénard generalization of Larmor,

P = q² γ⁶ [ a² − (v × a / c)² ] / (6 π ε₀ c³)

where γ = 1/√(1 − v²/c²) is the Lorentz factor. For a particle on a circular path (a ⟂ v) this reduces to a γ⁴ enhancement; for straight-line acceleration it reduces to γ⁶. At the γ ≈ 10⁴ of a modern electron storage ring, this factor is what makes synchrotron light sources billions of times brighter than a laboratory X-ray tube, and it is why light electrons radiate far more than heavy protons at the same energy — the acceleration for a given force scales as 1/mass, so P scales as 1/mass².

Larmor radiation in numbers

Situation	Acceleration a (m/s²)	Order-of-magnitude radiated power
Electron in Bohr ground-state orbit	≈ 9 × 10²²	~10⁻⁸ W (→ 16 ps collapse)
Electron, 10 GHz, 1 nm amplitude	≈ 3.9 × 10¹²	~4 × 10⁻²⁹ W (per electron, time-avg)
Proton vs electron, same force	a_p ≈ a_e / 1836	P_p ≈ P_e / 1836² (~3×10⁻⁷)
Charge at constant velocity	0	0 W (no radiation)
Electron in a synchrotron (γ≈10⁴)	huge transverse a	γ⁴-enhanced — kilowatts per beam

JavaScript — Larmor power calculations

const q_e   = 1.602176634e-19;   // elementary charge, C
const eps0  = 8.8541878128e-12;  // vacuum permittivity, F/m
const c     = 2.99792458e8;      // speed of light, m/s
const m_e   = 9.1093837015e-31;  // electron mass, kg

// Larmor formula (non-relativistic): total radiated power
function larmorPower(q, a) {
  return (q * q * a * a) / (6 * Math.PI * eps0 * c * c * c);
}

// Bohr ground-state orbit: centripetal acceleration a = v^2 / r
const a0 = 5.29177e-11;                       // Bohr radius, m
const v_orbit = q_e / Math.sqrt(4 * Math.PI * eps0 * m_e * a0); // ~2.19e6 m/s
const a_centripetal = v_orbit * v_orbit / a0; // ~9e22 m/s^2
console.log(`Orbit acceleration: ${a_centripetal.toExponential(2)} m/s^2`);      // ~9.0e22
console.log(`Radiated power:     ${larmorPower(q_e, a_centripetal).toExponential(2)} W`); // ~4.6e-8

// Angular distribution: power per unit solid angle at angle theta from a-vector
function dPdOmega(q, a, thetaRad) {
  const s = Math.sin(thetaRad);
  return (q * q * a * a * s * s) / (16 * Math.PI * Math.PI * eps0 * c * c * c);
}
console.log(`dP/dOmega at 90 deg: ${dPdOmega(q_e, a_centripetal, Math.PI/2).toExponential(2)}`); // max
console.log(`dP/dOmega at 0 deg:  ${dPdOmega(q_e, a_centripetal, 0).toExponential(2)}`);          // 0

// Relativistic (Lienard) generalization for circular motion (a perpendicular v)
function lienardCircular(q, a, gamma) {
  return larmorPower(q, a) * Math.pow(gamma, 4);
}
console.log(`gamma=1e4 circular: ${lienardCircular(q_e, 1e18, 1e4).toExponential(2)} W`);

// Time-averaged power of a sinusoidal oscillator, x = x0 cos(w t)
function oscillatorAvgPower(q, x0, omega) {
  const aPeak = omega * omega * x0;   // peak acceleration
  return larmorPower(q, aPeak) / 2;   //  = aPeak^2 / 2
}
const omega = 2 * Math.PI * 10e9;     // 10 GHz
console.log(`10 GHz e- osc: ${oscillatorAvgPower(q_e, 1e-9, omega).toExponential(2)} W`); // ~4e-29

Common misconceptions

• "A charge radiates because it is moving." No — only accelerating charges radiate. A charge in uniform motion (constant velocity) has a = 0 and emits nothing; its field is just a Lorentz-boosted Coulomb field carrying no energy to infinity.
• "The radiation is strongest along the direction of acceleration." Exactly backwards. The sin²θ pattern is zero along the acceleration axis and maximal perpendicular to it.
• "Larmor works at any speed." Only for v ≪ c. At relativistic speeds you must use the Liénard formula with its γ⁶ (or γ⁴ for circular motion) factors; ignoring them underestimates synchrotron output by many orders of magnitude.
• "Heavy and light particles radiate the same for the same force." For a given force, a = F/m, so radiated power scales as 1/m². An electron radiates ~1836² ≈ 3.4 million times more than a proton pushed by the same force — the reason electron machines, not proton machines, dominate synchrotron light production.
• "Larmor forbids stable atoms, so it must be wrong." It is not wrong classically; it correctly shows the classical atom is unstable. The fix is quantum mechanics, not a repair of Larmor. The formula remains exact for classical, non-relativistic point charges.
• "P depends on velocity." In the non-relativistic Larmor formula it does not appear at all — only a² does. Velocity re-enters only through the relativistic γ factors.