# Coulomb constant

(Redirected from Coulomb force 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 8987551788.7 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. 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 some 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/. This was previously true in SI when the vacuum permeability was defined as μ0 = 4π×107 H⋅m−1. Together with the speed of light in vacuum c, defined as 299792458 m/s, the vacuum permittivity ε0 can be written as 1/μ0c2, giving an exact value of

{\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}}

Since the redefinition of SI base units, the Coulomb constant is no longer exactly defined and is subject to the measurement error in the fine structure constant so that

{\begin{aligned}k_{\rm {e}}&=8.987\ 551\ 7887(14)\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:

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

The Coulomb constant appears in many expressions including the following:

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