# Coulomb constant

The Coulomb constant, the electric force constant, or the electrostatic constant (denoted ke, k or K) is a proportionality constant in electrodynamics equations. The value of this constant is dependent upon the medium that the charged objects are immersed in. In SI units, in the case of vacuum, it is equal to approximately 8987551787.3681764 N·m2·C−2 or 8.99×109 N·m2·C−2. 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,

$\mathbf {F} =k_{\text{e}}{\frac {Qq}{r^{2}}}\mathbf {\hat {e}} _{r}$ where êr is a unit vector in the r-direction and

$k_{\text{e}}=\alpha {\frac {\hbar c}{e^{2}}}$ ,

where α is the fine-structure constant, c is the speed of light, ħ is the reduced Planck constant, and e is elementary charge. In SI:

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

where $\varepsilon _{0}$ is the vacuum permittivity. This formula can be derived from Gauss' law, ${S}$ $\mathbf {E} \cdot {\rm {d}}\mathbf {A} ={\frac {Q}{\varepsilon _{0}}}$ Taking this integral for a sphere, radius r, around a point charge, we note that the electric field points radially outwards at all times and is normal to a differential surface element on the sphere, and is constant for all points equidistant from the point charge. ${S}$ $\mathbf {E} \cdot {\rm {d}}\mathbf {A} =|\mathbf {E} |\int _{S}dA=|\mathbf {E} |\times 4\pi r^{2}$ Noting that E = F/q for some test charge q,

{\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}} In 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/ and in SI ke = 1/ε0, where the vacuum permittivity ε0 = 1/μ0c2 8.85418782×10−12 F⋅m−1, the speed of light in vacuum c is 299792458 m/s, the vacuum permeability μ0 is 4π×107 H⋅m−1, so that

{\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}} A redefinition of SI base units is scheduled to come into force on 20 May 2019. This redefinition will make the Coulomb Constant not defined and will be subject to measurement error.

## Use

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

$k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}.$ The Coulomb constant appears in many expressions including the following:

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