The Cluster's Larmor Formula
A page about the Larmor formula — the power radiated by an accelerating charge.
The non-relativistic Larmor formula
The Larmor formula gives the total power radiated by an accelerating non-relativistic point charge: P = (2/3) (e^2 a^2) / (c^3) in Gaussian units. In SI units: P = (e^2 a^2) / (6 pi epsilon_0 c^3). For an electron: P = (e^2 a^2) / (6 pi epsilon_0 c^3) = (2/3) r_0 m_e c^3 (a/c)^2, where r_0 = e^2 / (m_e c^2) = 2.82 x 10^{-15} m is the classical electron radius. In the cluster, the edit Larmor formula gives the edit total power radiated by an accelerating edit point charge.
The relativistic Larmor formula
For a relativistic charge, the Liénard formula generalizes the Larmor formula: P = (2/3) (e^2 / (m_e^2 c^3)) gamma^6 [a^2 - (v x a)^2 / c^2] where gamma = (1 - v^2/c^2)^{-1/2}. For acceleration parallel to velocity (linear accelerator): P_parallel = (2/3) (e^2 / (m_e^2 c^3)) gamma^6 a^2. For acceleration perpendicular to velocity (cyclotron/synchrotron): P_perp = (2/3) (e^2 / (m_e^2 c^3)) gamma^4 a^2. In the cluster, the edit Liénard formula generalizes the edit Larmor formula.
The applications
The Larmor formula is used in:
- Synchrotron radiation: electrons bending in magnetic fields (P_perp ~ gamma^4)
- Bremsstrahlung: electrons scattering off nuclei
- Antenna theory: accelerating charges in antennas radiate EM waves
- Radiation damping: the Abraham-Lorentz force F_rad = (2/3) (e^2 / (4 pi epsilon_0 c^3)) da/dt
- Particle accelerators: energy loss from synchrotron radiation limits circular accelerator energy
In the cluster, the edit Larmor formula is used in:
- edit Synchrotron radiation
- edit Bremsstrahlung
- edit Antenna theory
- edit Radiation damping
- edit Particle accelerators
The hydrogen atom
A classical electron orbiting a proton would radiate power P = (2/3) (e^2 a^2) / (c^3). The acceleration is a = e^2 / (m_e r^2). The total power for a circular orbit is P = (2/3) (e^6) / (m_e^2 c^3 r^4). The energy decay time is tau = E / P ~ (m_e^2 c^3 r^4) / (e^4). For the Bohr radius a_0 = 0.53 x 10^{-10} m, tau ~ 1.6 x 10^{-11} s. This is the classical collapse time. In the cluster, the edit hydrogen atom would radiate edit power.
This formula
This page is about the Larmor formula. P = (2/3) (e^2 a^2) / (c^3). Relativistic: P = (2/3) (e^2 / m_e^2 c^3) gamma^6 [a^2 - (vxa)^2/c^2]. Classical H atom collapse time: tau ~ 10^{-11} s. The formula is real.