# Coulomb's constant

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Coulomb's constant, the electric force constant, or the electrostatic constant (denoted ke ) is a proportionality constant in equations relating electric variables and is exactly equal to ke  = 8.9875517873681764×109 N·m2/C2 (m/F). It was named after the French physicist Charles-Augustin de Coulomb (1736–1806).

## Value of the constant

The exact value of Coulomb's constant ke  comes from three of the fundamental, invariant quantities that define free space in the SI system: the speed of light c0 , magnetic permeability μ0 , and electric permittivity ε0 , related by Maxwell as:

$\frac{1}{\mu_0\varepsilon_0}=c_0^2.$

Because of the way the SI base unit system made the natural units for electromagnetism, the speed of light in vacuum c0  is 299,792,458 m s−1, the magnetic permeability μ0  of free space is 4π·10−7 H m−1, and the electric permittivity ε0  of free space is 1 (μ0 c2
0

) ≈ 8.85418782×10−12 F m−1
,[1] so that[2]

\begin{align} k_e &= \frac{1}{4\pi\varepsilon_0}=\frac{c_0^2\mu_0}{4\pi}=c_0^2\cdot10^{-7}\mathrm{H\ m}^{-1}\\ &= 8.987\ 551\ 787\ 368\ 176\ 4\cdot10^9\mathrm{N\ m^2\ C}^{-2}. \end{align}

## Use of Coulomb's constant

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

\begin{align} k_\text{e} &= \frac{1}{4\pi\varepsilon_0} \end{align}.

Some examples of use of Coulomb's constant are the following:

$|\boldsymbol{F}|=k_\text{e}{|q_1q_2|\over r^2}$.
$U_\text{E}(r) = k_\text{e}\frac{qQ}{r}$.
$\mathbf{E} = k_\text{e} \sum_{i=1}^N \frac{Q_i}{r_i^2} \mathbf{\hat{r}}_i$.