# Coulomb constant

Value of k Units
8.9875517923(14)×109 N·m2/C2
14.3996 eV·Å·e−2
10−7 (N·s2/C2)c2

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 units it is equal to 8.9875517923(14)×109 kg⋅m3⋅s−2⋅C−2.[1] It was named after the French physicist Charles-Augustin de Coulomb (1736–1806) who introduced Coulomb's law.

## Value of the constant

The Coulomb constant is the constant of proportionality in Coulomb's law,

${\displaystyle \mathbf {F} =k_{\text{e}}{\frac {Qq}{r^{2}}}\mathbf {\hat {e}} _{r}}$

where êr is a unit vector in the r-direction.[2] In SI:

${\displaystyle k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}},}$

where ${\displaystyle \varepsilon _{0}}$  is the vacuum permittivity. This formula can be derived from Gauss' law,

${\displaystyle {\scriptstyle S}}$  ${\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} ={\frac {Q}{\varepsilon _{0}}}}$

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.

${\displaystyle {\scriptstyle S}}$  ${\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} =|\mathbf {E} |\int _{S}dA=|\mathbf {E} |\times 4\pi r^{2}}$

Noting that E = F/q for some test charge q,

{\displaystyle {\begin{aligned}\mathbf {F} &={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {e}} _{r}=k_{\text{e}}{\frac {Qq}{r^{2}}}\mathbf {\hat {e}} _{r}\\[8pt]\therefore k_{\text{e}}&={\frac {1}{4\pi \varepsilon _{0}}}\end{aligned}}}

Also note that this 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.

${\displaystyle intensity={\frac {1}{d^{2}}}}$

In some modern systems of units, the Coulomb constant ke has an exact numeric value; in Gaussian units ke = 1, in Lorentz–Heaviside units (also called rationalized) ke = 1/. This was previously true in SI when the vacuum permeability was defined as μ0 = 4π×107 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[3]

{\displaystyle {\begin{aligned}k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}={\frac {c^{2}\mu _{0}}{4\pi }}&=c^{2}\times (10^{-7}\ \mathrm {H{\cdot }m} ^{-1})\\&=8.987\ 551\ 787\ 368\ 1764\times 10^{9}~\mathrm {N{\cdot }m^{2}{\cdot }C^{-2}} .\end{aligned}}}

Since the redefinition of SI base units,[4][5] the Coulomb constant is no longer exactly defined and is subject to the measurement error in the fine structure constant, as calculated from CODATA 2018 recommended values being[1]

${\displaystyle k_{\text{e}}=8.987\,551\,7923\,(14)\times 10^{9}\;\mathrm {kg{\cdot }m^{3}{\cdot }s^{-4}{\cdot }A^{-2}} .}$

## Use

The Coulomb constant is used in many electric equations, although it is sometimes expressed as the following product of the vacuum permittivity constant:

${\displaystyle k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}.}$

The Coulomb constant appears in many expressions including the following:

${\displaystyle \mathbf {F} =k_{\text{e}}{Qq \over r^{2}}\mathbf {\hat {e}} _{r}.}$
${\displaystyle U_{\text{E}}(r)=k_{\text{e}}{\frac {Qq}{r}}.}$
${\displaystyle \mathbf {E} =k_{\text{e}}\sum _{i=1}^{N}{\frac {Q_{i}}{r_{i}^{2}}}\mathbf {\hat {r}} _{i}.}$