Recoila: Abraham–Lorentz Force
Electromagnetism & Plasma arc, page 80 of 193. Back to The Pixelated Physics Universe
Who they are
Recoila accelerates, radiates, and her own light shoves her back.
Recoila is a character of the Electromagnetism & Plasma arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
An accelerating charge radiates, and the radiation carries away energy and momentum, so the charge feels a recoil. In the non-relativistic limit that radiation-reaction force is proportional to the rate of change of acceleration, the jerk.
$$ \begin{gathered} F_{\text{rad}} = \frac{\mu_0 q^{2}}{6\pi c}\,\frac{da}{dt} \\ P = \frac{\mu_0 q^{2} a^{2}}{6\pi c} \qquad (v \ll c) \\ \tau = \frac{\mu_0 e^{2}}{6\pi m_e c} \approx 6.3\times10^{-24}\ \mathrm{s} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $F_{\text{rad}}$ | radiation-reaction force | N |
| $\mu_0$ | vacuum permeability ≈ 1.256 637 06 × 10⁻⁶ N/A² | N/A² |
| $q$ | charge | C |
| $c$ | speed of light, 299 792 458 m/s (exact) | m/s |
| $a$ | acceleration | m/s² |
| $da/dt$ | rate of change of acceleration (jerk) | m/s³ |
| $P$ | radiated power (Larmor) | W |
| $e$ | elementary charge, 1.602 176 634 × 10⁻¹⁹ C (exact) | C |
| $m_e$ | electron mass, 9.109 383 7 × 10⁻³¹ kg | kg |
| $\tau$ | radiation-reaction time | s |
Honest caveat: Taken literally the equation has runaway and pre-acceleration solutions. Reduced-order (Landau–Lifshitz) forms tame it; a full classical cure is still debated.
Status: part of this topic is unsolved, unconfirmed or debated. The text above says which part; treat it as an open question, not settled fact.
Sources
- Dirac (1938) Classical theory of radiating electrons, Proc. R. Soc. Lond. A 167, 148
- Jackson (1999) Classical Electrodynamics, 3rd ed., Wiley, ch. 16
- NIST CODATA: vacuum magnetic permeability
Crossovers
Recoila meets The Spark Herald (Electromagnetism / Lightning Command). The Spark Herald's Poynting flux carries energy away from every accelerating charge. Recoila feels the other side of that bill: the radiation reaction force, which takes back energy and momentum from the charge that radiated.
Recoila meets The Vertex Smith (Quantum Electrodynamics Vertex). Recoila treats radiation reaction classically, and her equation has runaway solutions. The Vertex Smith handles the electron's interaction with its own field in QED, where the self-interaction is renormalised into the measured mass.
Wear the lore
Concept keywords: Abraham–Lorentz Force, Recoila, Electromagnetism & Plasma, physics lore, Pixelated Physics