Coulomb Constant Calculator

The 4π factor in Coulomb's law (in rationalized SI units) ensures that simpler formulas appear in Gauss's law: ∮E·dA = Q/ε₀ (no 4π needed). This choice (rationalized units) is standard in modern physics. Non-rationalized Gaussian units put the 4π in Gauss's law instead: ∮E·dA = 4πQ.

Coulomb's law is the electrostatic approximation valid for stationary charges. For moving charges, the full relativistic treatment (Lienard-Wiechert potentials) adds velocity-dependent and acceleration-dependent terms — radiation. At very short distances (sub-femtometer), the nuclear strong force overwhelms Coulomb repulsion between protons. In quantum field theory, Coulomb's law gets small QED corrections (vacuum polarization).

At 1 nanometer from an electron, E = ke × e/r² = 8.988 × 10⁹ × 1.602 × 10⁻¹⁹ / (10⁻⁹)² = 1.44 × 10¹¹ V/m = 144 GV/m. This is an enormous field — compare to the 3 MV/m electric field in a thunderstorm. The strong fields near DNA molecules from nearby ions are thought to contribute to radiation damage mechanisms.

In Gaussian units, F = q₁q₂/r² (no explicit constant). Charge is measured in statcoulombs (esu), where 1 statcoulomb = 3.336 × 10⁻¹⁰ C. The electron charge is 4.803 × 10⁻¹⁰ statcoulombs. Gaussian units simplify many electromagnetic formulas (Maxwell's equations become symmetric) and are favored in theoretical physics and astrophysics.

In a conductor or electrolyte, free charges redistribute to partially cancel external fields — screening. The screened Coulomb potential becomes V = ke × q × exp(-r/λD)/r, where λD is the Debye screening length. In metals, the Thomas-Fermi screening length is typically less than 1 Å; in biological saline (150 mM NaCl), λD ≈ 0.8 nm.

In ionic crystals, the potential energy per ion pair includes contributions from all other ions in the lattice, not just nearest neighbors. The Madelung constant M accounts for this geometry: U = -M × ke × z₊z₋e²/r. For NaCl (face-centered cubic), M = 1.7476. The Madelung energy makes ionic crystals significantly more stable than isolated ion pairs.

When X-rays scatter from electrons in atoms, the scattering amplitude depends on the Fourier transform of the electron density, which is determined by the atomic potential (Coulomb interaction). Bragg diffraction in X-ray crystallography maps the electron density distribution by solving the inverse problem of Coulomb scattering amplitudes measured at many angles.

In a medium with relative permittivity εr (dielectric constant), Coulomb's law becomes F = ke × q₁q₂/(εr × r²). Water has εr ≈ 78.4 at 25°C, reducing Coulomb forces by a factor of 78. This is why ionic compounds dissolve in water — the electrostatic lattice energy is reduced by 78-fold, allowing thermal fluctuations to break apart the crystal.

The classical electron radius r_e = ke × e²/(m_e c²) = 2.8179 × 10⁻¹⁵ m is the scale where the classical Coulomb self-energy of the electron equals its rest energy m_e c². It appears in Thomson scattering cross-section σ_T = (8π/3) × r_e² = 6.652 × 10⁻²⁹ m², which gives the scattering cross-section of electromagnetic radiation by free electrons.