# Coulomb's constant

Coulomb's constant, the electric force constant, or the electrostatic constant (denoted Template:SubSup) is a proportionality constant in equations relating electric variables and is exactly equal to Template:SubSup = Template:GapsTemplate:E 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,

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

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

Template:Oiint

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.

Template:Oiint

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

${\displaystyle \mathbf {F} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Qq}{r^{2}}}\mathbf {\hat {e}} _{r}=k_{\text{e}}{\frac {Qq}{r^{2}}}\mathbf {\hat {e}} _{r}}$
${\displaystyle \therefore k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}}$

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

${\displaystyle {\frac {1}{\mu _{0}\varepsilon _{0}}}=c_{0}^{2}.}$

Because of the way the SI base unit system made the natural units for electromagnetism, the speed of light in vacuum Template:SubSup is Template:Gaps, the magnetic permeability Template:SubSup of free space is 4π·10−7 H m−1, and the electric permittivity Template:SubSup of free space is ,[1] so that[2]

{\displaystyle {\begin{aligned}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{aligned}}}

## 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:

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

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

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