Electron Mass Calculator

The proton-to-electron mass ratio m_p/m_e ≈ 1836 is a dimensionless constant whose value is unexplained from first principles. The proton mass comes mainly from the kinetic energy of its quarks and the gluon field (QCD binding energy), not from the quark rest masses. Understanding why this ratio is exactly 1836 (and not 100 or 10000) is an open problem in physics.

The Lorentz factor γ = 1/√(1-v²/c²) quantifies relativistic effects. At v = 0.1c, γ = 1.005 (1% correction). At v = 0.5c, γ = 1.155. At v = 0.99c, γ = 7.09. Electrons in cathode ray tubes: v ≈ 0.3c, γ ≈ 1.05. Electrons in synchrotrons: v ≈ 0.9999c, γ ≈ 100. LEP collider: v ≈ 0.999999996c, γ ≈ 100,000.

When a photon with energy exceeding 2 m_e c² = 1.022 MeV passes near a nucleus, it can convert to an electron-positron pair (gamma → e⁺ + e⁻). The minimum photon energy is twice the electron rest energy because both particles must be created. This process dominates gamma attenuation above about 5 MeV.

The classical electron radius r_e = ke × e²/(m_e c²) = 2.8179 × 10⁻¹⁵ m = 2.82 fm. It is the radius at which the classical electrostatic self-energy equals m_e c². It sets the scale for Thomson scattering cross-section (σ_T = 8πr_e²/3 = 6.65 × 10⁻²⁹ m²) and appears in radiation physics formulas.

J.J. Thomson measured the charge-to-mass ratio e/m_e = 1.759 × 10¹¹ C/kg in 1897 using cathode rays in crossed electric and magnetic fields. Millikan measured e = 1.602 × 10⁻¹⁹ C in 1913 using oil drops. Combining these gives m_e = e/(e/m_e) = 9.11 × 10⁻³¹ kg. Modern measurements use Penning traps with uncertainties of parts per trillion.

The electron g-factor (g ≈ 2.002319304) is the ratio of its magnetic moment to the quantum of magnetic moment (Bohr magneton). The deviation from exactly 2 (the anomalous magnetic moment) is ae = (g-2)/2 = 1.15965 × 10⁻³ and has been calculated to 10 significant figures in QED. Agreement with experiment to 10⁻¹² is the most precise test of any physical theory.

In quantum field theory, the vacuum is not empty but filled with virtual electron-positron pairs that pop in and out of existence for times Δt ≤ ℏ/(2 m_e c²) ≈ 6 × 10⁻²²econds. These vacuum fluctuations cause measurable effects: the Lamb shift in hydrogen (first measured 1947), the Casimir force between conductors, and contribute to the anomalous magnetic moment of the electron.

The Bohr radius a₀ = ℏ²/(m_e ke e²) is inversely proportional to m_e. If m_e were 100 times larger (like a muon, m_μ ≈ 207 m_e), atoms would be 207 times smaller. Muonic hydrogen (proton + muon) has been used to measure the proton charge radius with high precision, revealing a discrepancy (the proton radius puzzle) that stimulated much research.

Spin-orbit coupling arises from the interaction between the electron's intrinsic spin (s = 1/2, magnetic moment = g m_e μ_B) and the magnetic field it experiences due to its orbital motion around the nucleus. The energy splitting is proportional to 1/(m_e²c²) × ⟨1/r³⟩ × L·S. It causes the fine structure of atomic spectral lines and is the basis for spintronics and topological insulators.