# Coulomb constant

(Redirected from Coulomb's constant)

The Coulomb constant, the electric force constant, or the electrostatic constant (denoted ke, k or K) is a proportionality constant in electrodynamics equations. In SI units, it is exactly equal to 8987551787.3681764 N·m2·C−2, or approximately 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,

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

where êr is a unit vector in the r-direction and

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

where α is the fine-structure constant, c is the speed of light, ħ is the reduced Planck constant, and e is elementary charge.[1] 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, around a point charge, we note that the electric field points radially outwards at all times and is normal to a differential surface element[disambiguation needed] on the sphere, and is constant for all points equidistant from the point charge.

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

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}}}

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,[2] so that[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}}}

## 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:

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