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The Coulomb constant, the electric force constant, or the electrostatic constant (denoted , or ) is a
proportionality constant In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
in
electrostatics Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for am ...
equations. In SI base units it is equal to .Derived from ''k''e = 1/(4''πε''0) – It was named after the French physicist
Charles-Augustin de Coulomb Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of attra ...
(1736–1806) who introduced
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...
.

# Value of the constant

The Coulomb constant is the constant of proportionality in
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...
, :$\mathbf = k_\text\frac\mathbf_r$ where is a unit vector in the -direction. In SI: :$k_\text = \frac,$ where $\varepsilon_0$ is the vacuum permittivity. This formula can be derived from Gauss' law, : Taking this integral for a sphere, radius , 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 for some test charge , : 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.$\text = \frac$In some modern systems of units, the Coulomb constant has an exact numeric value; in
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...
, in Heaviside–Lorentz units (also called ''rationalized'') . This was previously true in SI when the vacuum permeability was defined as . Together with the speed of light in vacuum , defined as , the vacuum permittivity can be written as , which gave an exact value of :$\begin k_\text = \frac=\frac&=c^2\times \left(10^\ \mathrm^\right)\\ &= 8.987\ 551\ 787\ 368\ 1764\times 10^9~\mathrm. \end$ Since the redefinition of
SI base unit The SI base units are the standard units of measurement defined by the International System of Units (SI) for the seven base quantities of what is now known as the International System of Quantities: they are notably a basic set from which al ...
s, The day is the 144th anniversary of the Metre Convention. the Coulomb constant is no longer exactly defined and is subject to the measurement error in the
fine structure constant In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
, as calculated from CODATA 2018 recommended values being :$k_\text = 8.987\,551\,7923\,\left(14\right)\times 10^9\;\mathrm .$

# Use

The Coulomb constant is used in many electric equations, although it is sometimes expressed as the following product of the vacuum permittivity constant: :$k_\text = \frac.$ The Coulomb constant appears in many expressions including the following: ;
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...
:$\mathbf=k_\text\mathbf_r.$ ; Electric potential energy :$U_\text\left(r\right) = k_\text\frac.$ ; Electric field :$\mathbf = k_\text \sum_^N \frac \mathbf_i.$

* Gravitational constant * Vacuum permittivity * Vacuum permeability * Inverse-square law

# References

{{Scientists whose names are used in physical constants Electricity Electromagnetism Physical constants