Classical Electron Radius Calculator

The classical electron radius, r_e ≈ 2.818 femtometres, is a length scale defined by setting a point charge's electrostatic self-energy equal to its rest mass energy (via E=mc²). It is a CODATA-listed fundamental physical constant, though it does not represent the electron's true physical size.

r = q²/(4πε₀mc²), where q is the particle's charge, ε₀ is the vacuum permittivity, m is its rest mass, and c is the speed of light. This calculator generalizes the formula to any charged particle, not just the electron.

No, quantum electrodynamics (QED), the modern and experimentally verified theory of electrons, treats the electron as a genuine point particle with no measurable internal size, down to the smallest scales probed by particle accelerators. The classical radius is a useful length scale from pre-quantum classical electromagnetism, not a measured physical dimension.

It appears directly in real, experimentally verified formulas, most notably the Thomson scattering cross-section σ = (8π/3)r_e², which describes how electromagnetic radiation scatters off free charges at low photon energies. It is a genuinely useful length scale in classical electrodynamics and scattering theory.

By modeling the electron as a uniformly charged sphere and setting its electrostatic self-energy, q²/(4πε₀r) (roughly, up to a numerical factor depending on the charge distribution assumed), equal to its total rest mass energy mc², then solving for the radius r.

Because r is inversely proportional to mass (r ∝ 1/m), a heavier particle needs to pack the same electrostatic self-energy into a smaller radius to match its (larger) rest mass energy. This is why the classical proton radius is about 1836 times smaller than the classical electron radius, matching the proton-to-electron mass ratio exactly.

It is a CODATA-recommended fundamental constant known to extremely high precision, 2.8179403262(13) femtometres as of the most recent adjustment, since it is derived directly from the well-measured elementary charge, electron mass, and fundamental constants ε₀ and c.

You can compute the analogous classical radius for the proton or muon (both included as presets), or for any custom particle mass, useful for comparing how this classical length scale varies across different charged particles in physics coursework and problem sets.

They are both fundamental atomic-scale lengths but represent very different physics: the Bohr radius (~52,900 fm) is the quantum mechanical size of a hydrogen atom's ground-state electron orbital, while the classical electron radius (~2.818 fm) is a purely classical electromagnetic length scale, more than 18,000 times smaller.

Results are given in femtometres (fm, 10⁻¹⁵ m), the standard unit for nuclear and particle-scale lengths, alongside the raw value in metres for reference.