Larmor Formula Calculator
Enter charge q, acceleration a to compute the radiated power P=(2/3)·(q²a²)/(4πε₀c³). Electron a=1e15 m/s² → P≈5.7×10⁻¹⁰ W.
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At a glance:The Larmor formula (J. J. Larmor, 1897) gives the total electromagnetic power radiated by an accelerated non-relativistic point charge: in SI units P=(2/3)·(q²a²)/(4πε₀c³), where q is the charge, a the acceleration magnitude, ε₀ the vacuum permittivity, c the speed of light. cgs units: P=(2/3)·(q²a²)/c³. Physical meaning: a charge radiates only when accelerated (curved or linear acceleration); uniform-velocity motion does not radiate. The radiation is dipole radiation; the power is proportional to q² and a² (acceleration squared). History: Larmor derived it from Maxwell's equations in 1897, a classic result of classical electrodynamics; later Liénard (1898) generalized to relativistic motion (Liénard formula). Classic example: electron q=1.602e-19 C, a=1e15 m/s² → P=(2/3)·(1.602e-19)²·(1e15)²/(4π·8.854e-12·(3e8)³)=5.7×10⁻¹⁰ W. Applications: (1) synchrotron radiation (electron accelerators, ring light sources); (2) bremsstrahlung (X-ray tubes); (3) antenna radiation; (4) cyclotron/synchrotron radiation in astrophysics (pulsars); (5) classical atomic model (why electrons should collapse — a motivation for quantum mechanics).
Formula
SI: P = (2/3)·(q²·a²)/(4πε₀·c³)
cgs: P = (2/3)·(q²·a²)/c³
Relativistic (Liénard): P = (q²·γ⁶/(6πε₀·c³))·(a² − (v×a)²/c²)
Characteristic time: τ = (2/3)·q²/(4πε₀·m·c³)
Radiation reaction (Abraham-Lorentz): F = (μ₀·q²/(6π·c))·ȧ
$$P = \frac{2}{3}\frac{q^2 a^2}{4\pi\varepsilon_0 c^3}, \quad P_{\text{rel}} = \frac{q^2\gamma^6}{6\pi\varepsilon_0 c^3}\left(a^2 - \frac{(\mathbf{v}\times\mathbf{a})^2}{c^2}\right)$$How to Use
- Enter charge q (C) and acceleration a (m/s²).
- The tool computes the radiated power P=(2/3)·(q²a²)/(4πε₀c³).
- Example: electron a=1e15 → P≈5.7e-10 W.
Larmor Radiated Power Examples
| Charge | acceleration | P (W) |
|---|---|---|
| electron 1.6e-19 C | 1e15 m/s² | 5.7e-10 |
| electron 1.6e-19 C | 1e10 m/s² | 5.7e-20 |
| proton 1.6e-19 C | 1e15 m/s² | 5.7e-10 |
| 1 C | 1 m/s² | 6.0e9 |
| electron 1.6e-19 C | 1e20 m/s² | 5.7e0 |
P∝q²a². Because q is tiny, single-charge radiation is very weak; macroscopic charges (1 C) radiate enormously. Acceleration a~1e15 is typical in atomic transitions.
Case Studies
Synchrotron Radiation and Hong Kong's Light Source
In a synchrotron, electrons at v≈c in radius R have centripetal acceleration a≈c²/R. For R=10 m, a≈9e15 m/s².
Larmor power per electron (relativistic Liénard) is enormous when γ≫1; e.g. 3 GeV electrons radiate megawatts total in the ring, concentrated in the forward cone.
The Hong Kong Synchrotron Radiation Facility / SSRL-class sources use this radiation for protein crystallography and materials science. The Larmor/Liénard formula sets the brilliance limit.
Why Classical Atoms Collapse (Quantum Motivation)
An electron orbiting a nucleus has centripetal acceleration a=v²/r≈10²³ m/s² (Bohr model). Larmor predicts radiation ~10⁻⁸ W, draining orbital energy in ~10⁻¹¹ s.
Classical theory thus predicts atoms should collapse in picoseconds — contradicting stable matter. This was a key motivation for Bohr's 1913 quantum model and later quantum mechanics.
Quantum mechanically, ground-state electrons are stationary probability clouds (no acceleration), so they do not radiate — resolving the paradox.
FAQ
Why does a uniformly moving charge not radiate?
The Larmor formula depends on acceleration a. At constant velocity a=0, so P=0 — no radiation. Only a change in the velocity vector (curving, speeding up, or slowing down) produces radiation. Physically: a uniformly moving charge's field is just a Lorentz-transformed Coulomb field, carrying no net energy flux to infinity. This is consistent with relativity (no preferred frame for uniform motion).
What is the relativistic version?
The Liénard formula (1898) generalizes Larmor to relativistic speeds: P=(q²γ⁶/(6πε₀c³))·(a²−(v×a)²/c²). When v≈0 it reduces to Larmor. For synchrotron motion (v⊥a, v≈c) P≈(q²γ⁴a²)/(6πε₀c³) — enhanced by γ⁴, explaining intense synchrotron radiation. γ=1/√(1−v²/c²) is the Lorentz factor.
What is the connection to antennas?
A transmitting antenna has electrons accelerating back and forth (a~ωv). Larmor gives the radiated power; the antenna's radiation resistance R_rad converts this to P=½R_rad I². Dipole antennas radiate because charges oscillate (accelerate). The same formula underlies radio, Wi-Fi and 5G: accelerating charges in the antenna emit EM waves. Power rises with frequency squared (since a~ωv).
What is bremsstrahlung?
Bremsstrahlung ('braking radiation') is emitted when an electron decelerates in the Coulomb field of a nucleus (acceleration a during the encounter). Larmor gives the instantaneous power; integrating over the trajectory yields the X-ray spectrum (continuous up to a cutoff). X-ray tubes and medical imaging rely on it. The spectrum is broadband because the deceleration is not sinusoidal.
What is radiation reaction?
An accelerating charge loses energy to radiation, so it experiences a recoil force (Abraham-Lorentz force) F_rad=(μ₀q²/(6πc))·ȧ, proportional to the derivative of acceleration. This is the classical 'self-force' that makes the equation of motion third-order (causing runaway solutions — a known issue). In quantum electrodynamics radiation reaction is treated via photon emission probabilities. Larmor's P is the rate of energy loss feeding this force.
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References
Content reviewed by the Calculatorism editorial team. Results are for reference only; please refer to the relevant authorities for the official figures.