Larmor formula
The classical expression P = q²a²/(6πε₀c³) for the total electromagnetic power radiated by a non-relativistic point charge of acceleration a. Derived by Joseph Larmor in 1897; the foundational result of classical radiation theory.
Definition
The Larmor formula gives the total electromagnetic power radiated by a non-relativistic point charge q undergoing instantaneous acceleration a as P = q²a²/(6πε₀c³). It is the foundational result of classical radiation theory: every antenna (§10.3), every synchrotron (§10.4), every bremsstrahlung X-ray tube (§10.5), and every radiating star's luminosity traces back to this one equation. The derivation proceeds from the retarded-potential expression for the far-field electric radiation field E_rad ∝ q·a_⊥/(rc²), computes the Poynting flux S = |E_rad|²/(μ₀c), integrates over a sphere at infinity, and uses the angular integral ∫sin²θ dΩ = 8π/3. The factor of 1/(6π) is a direct geometric consequence of integrating the sin²θ radiation pattern (a doughnut-shaped lobe perpendicular to a) over the full solid angle.
The formula has two critical limitations. It is non-relativistic — for speeds approaching c, the formula generalises to the Liénard result P = (q²γ⁶/6πε₀c³)[a² − (v×a)²/c²], which can exceed the non-relativistic value by a factor of γ⁶ when acceleration is parallel to velocity, or γ⁴ when perpendicular (as in circular motion, where it yields synchrotron radiation). And it is classical — quantum radiation (spontaneous emission, bremsstrahlung at high energies) obeys different spectrum shapes, though the total power agrees with Larmor in the correspondence limit ℏω ≪ mc². Dimensional check: q² has units of C² = A²·s²; ε₀ is F/m = A²·s⁴/(kg·m³); a² is m²/s⁴; c³ is m³/s³. Combining: A²·s² · m²/s⁴ · kg·m³/(A²·s⁴) · 1/m³ = kg·m²/s³ = W. Correct.
History
Joseph Larmor derived the formula in 1897 at Cambridge, in a paper on the magnetic influence on spectra. Henri Liénard independently extended it to the relativistic case in 1898.