# Coulomb's constant

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) who first used it in Coulomb's Law.

## Value of the constant

Coulomb's constant can be empirically derived as the constant of proportionality in Coulomb's law,

$\mathbf{F} = k_{e}\frac{Qq}{r^2}\mathbf{\hat{e}}_r$

where êr is a unit vector in the r direction. However, its theoretical value can be derived from Gauss' law,

${\scriptstyle 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.

${\scriptstyle S}$$\mathbf{E} \cdot {\rm d}\mathbf{A} = |\mathbf{E}|\mathbf{\hat{e}}_r\int_{S} dA = |\mathbf{E}|\mathbf{\hat{e}}_r \times 4\pi r^{2}$

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

$\mathbf{F} = \frac{1}{4\pi\varepsilon_0}\frac{Qq}{r^2}\mathbf{\hat{e}}_r = k_{e}\frac{Qq}{r^2}\mathbf{\hat{e}}_r$
$\therefore k_{e} = \frac{1}{4\pi\varepsilon_0}$

This 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 299792458 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_\text{e} = \frac{1}{4\pi\varepsilon_0}=\frac{c_0^2\mu_0}{4\pi}&=c_0^2\times 10^{-7}\ \mathrm{H\ m}^{-1}\\ &= 8.987\ 551\ 787\ 368\ 176\ 4\times 10^9\ \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:

$k_\text{e} = \frac{1}{4\pi\varepsilon_0}$.

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

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