It has been suggested that this article be merged into Vacuum permittivity. (Discuss) Proposed since March 2023.
The Coulomb constant, the electric force constant, or the electrostatic constant (denoted ke, k or K) is a proportionality constant in electrostatics equations. In SI base units it is equal to 8.9875517923(14)×109 kg⋅m3⋅s−4⋅A−2. It was named after the French physicist Charles-Augustin de Coulomb (1736–1806) who introduced Coulomb's law.
Value of the constant
|Value of k||Units|
The Coulomb constant is the constant of proportionality in Coulomb's law,
where êr is a unit vector in the r-direction. In SI, 
where is the vacuum permittivity. This formula can be derived from Gauss' law,
Taking this integral for a sphere, radius r, centered on a point charge, the electric field points radially outwards and is normal to a differential surface element on the sphere with constant magnitude for all points on the sphere.
Noting that E = F/q for some test charge q,
Coulomb's law is an inverse-square law, and thereby similar to many other scientific laws ranging from gravitational pull to light attenuation. This law states that a specified physical quantity is inversely proportional to the square of the distance.
In some modern systems of units, the Coulomb constant ke has an exact numeric value; in Gaussian units ke = 1, in Heaviside–Lorentz units (also called rationalized) ke = 1/4π. This was previously true in SI when the vacuum permeability was defined as μ0 = 4π×10−7 H⋅m−1. Together with the speed of light in vacuum c, defined as 299792458 m/s, the vacuum permittivity ε0 can be written as 1/μ0c2, which gave an exact value of
Since the redefinition of SI base units, the Coulomb constant is no longer has an exactly defined value and is subject to the measurement error in the fine structure constant, as calculated from CODATA 2018 recommended values being
The Coulomb constant is used in many electric equations, although it is frequently expressed as the following product of the vacuum permittivity constant:
The Coulomb constant appears in many expressions including the following:
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Note added in proof, Feb. 26.⠀Complete agreement between Dirac's results and kinematic cosmology can be obtained, and his proportionalities M ∝ t2, γ ∝ t−1, replaced by M = const. = M0, γ ∝ t, if we take the 'electrostatic' constant γ′ in the inverse square law to be proportional to t; γ′ is, of course, usually taken as unity. This is formally the same as taking e2 ∝ t. The t-dynamics then gives the radius of the Bohr atom as proportional to t, and a material rod is 'rigid' on the τ-scale. Electrostatic forces then become to some extent analogous to gravitational forces, and might be discussed kinematically by examining the properties of an expanding sphere of mingled positive and negative particles.
- ^ "Coulomb force". Encyclopedia Britannica. Retrieved 13 March 2023.
In the metre–kilogram–second and the SI systems, the unit of force (newton), the unit of charge (coulomb), and the unit of distance (metre), are all defined independently of Coulomb's law, so the proportionality factor k is constrained to take a value consistent with these definitions, namely, k in a vacuum equals 8.98 × 109 newton square metre per square coulomb. This choice of value for k permits the practical electrical units, such as ampere and volt, to be included with the common metric mechanical units, such as metre and kilogram, in the same system.
- ^ a b Derived from ke = 1/(4πε0) – "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
- ^ "Charles-Augustin de Coulomb". Encyclopedia Britannica. Retrieved 3 March 2021.
- ^ Tomilin, K. (1999). "Fine-structure constant and dimension analysis". European Journal of Physics. 20 (5): L39–L40. Bibcode:1999EJPh...20L..39T. doi:10.1088/0143-0807/20/5/404. S2CID 250841835.
- ^ Coulomb's constant, HyperPhysics [bad link]