Coulomb’s constant, kₑ

The constant in Coulomb’s law, F = kₑq₁q₂/r². It is not independent of the vacuum permittivity: kₑ = 1/(4πε₀), so it is measured, and exactly as uncertain as ε₀.

kₑ = 8.9875517862(14) × 10⁹ N·m²/C²

Derived from the CODATA 2022 ε₀. Standard uncertainty 0.0000000014 × 10⁹ N·m²/C², which is 1.6 × 10⁻¹⁰ of the value.

Not a row of the physical constants table, which leaves the electromagnetic constants out because their digits moved between the 2018 and 2022 CODATA adjustments.

In other units

Measured values are in CODATA’s concise form: the digits in brackets are the standard uncertainty in the last two places, so 6.67430(15) means 6.67430 ± 0.00015.

Coulomb’s constant in SI units and in other units
Quantity	Value
kₑ	8.9875517862(14) × 10⁹ N·m²/C²
kₑ	8.9875517862(14) × 10⁹ V·m/C
kₑe²	1.43996454687(23) eV·nm

kₑ in V·m/C

1 N·m = 1 J = 1 V·C, so N·m²/C² = V·m/C
= 8.9875517862(14) × 10⁹ V·m/C

The same number, in the units an electric field calculation lands in.

kₑe² in eV·nm

8.9875517862(14) × 10⁹ N·m²/C² × (1.602176634 × 10⁻¹⁹ C)² ÷ (1.602176634 × 10⁻¹⁹ J/eV) × 10⁹ nm/m
= 1.43996454687(23) eV·nm

Divide by a separation in nanometres for the potential energy of two elementary charges in electronvolts. Its uncertainty is carried over from the permittivity’s rounded figure, so the last bracketed digit is approximate.

Where the value comes from

Coulomb’s law can be written with kₑ or with ε₀, and they are related by kₑ = 1/(4πε₀). This site computes kₑ from the CODATA 2022 permittivity rather than storing a second number, so the Coulomb’s law calculator and the permittivity page cannot disagree. NIST does not list kₑ at all; it follows from ε₀.

Before 2019 kₑ was exact: with μ₀ fixed, it equalled c² × 10⁻⁷ in SI units, 8,987,551,787.3681764 N·m²/C². Today’s value is lower by 1.3 × 10⁻¹⁰, less than its own uncertainty of 1.6 × 10⁻¹⁰, so the old figure still agrees with it to 9 significant figures.

kₑ from ε₀

kₑ = 1 ÷ (4πε₀)
= 1 ÷ (4π × 8.8541878188(14) × 10⁻¹²)
= 8.9875517862(14) × 10⁹ N·m²/C²

Which form to use

In Coulomb’s law, use 8.988 × 10⁹ N·m²/C² with charges in coulombs and distances in metres. The rounded 9 × 10⁹ is 0.14 percent high, fine for a quick check.

For atoms and ions, kₑe² ≈ 1.44 eV·nm is quicker: two elementary charges 0.1 nm apart have a potential energy of 14.4 eV.

In Gaussian units, used in older physics texts, the constant is set to 1 by the choice of the unit of charge, which is why Coulomb’s law there reads F = q₁q₂/r².

Common mistakes

Using kₑ in a medium without dividing by the relative permittivity. In water the force between two ions is roughly 80 times smaller than kₑ predicts.
Mixing the kₑ and ε₀ forms of the same formula, and picking up or losing a factor of 4π.
Entering microcoulombs as coulombs. The force goes with the product of two charges, so the error is 10¹² rather than 10⁶.

Tools that use it

Coulomb’s Law Calculator computes the force kₑq₁q₂/r² and the field kₑq/r² with it.
Rutherford Scattering Simulator sets the distance of closest approach with it, d = 2kₑZe²/K, where kₑe² is 1.44 MeV·fm.
Electric Field Simulator sums the field kₑq/r² and the potential kₑq/r of every charge with it.

Related constants

Vacuum permittivity, ε₀ = 8.8541878188(14) × 10⁻¹² F/m
Vacuum permeability, μ₀ = 1.25663706127(20) × 10⁻⁶ N/A²
Elementary charge, e = 1.602176634 × 10⁻¹⁹ C