Coulomb 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 ê_r is a unit vector in the r-direction.[2] 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, centred on a point charge, the electric field points radially outwards and is normal to a differential surface element on the sphere with constant magnitude for all points on the sphere.

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

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 some modern systems of units, the Coulomb constant k_e has an exact numeric value; in Gaussian units k_e = 1, in Lorentz–Heaviside units (also called rationalized) k_e = 1/4π. This was previously true in SI when the vacuum permeability was defined as μ₀ = 4π×10⁻⁷ 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, which gave an exact value of[3]

${\displaystyle {\begin{aligned}k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}={\frac {c^{2}\mu _{0}}{4\pi }}&=c^{2}\times (10^{-7}\ \mathrm {H{\cdot }m} ^{-1})\\&=8.987\ 551\ 787\ 368\ 1764\times 10^{9}~\mathrm {N{\cdot }m^{2}{\cdot }C^{-2}} .\end{aligned}}}$

Since the redefinition of SI base units,[4][5] the Coulomb constant is no longer exactly defined and is subject to the measurement error in the fine structure constant, as calculated from CODATA 2018 recommended values being[1]

${\displaystyle k_{\text{e}}=8.987\,551\,7923\,(14)\times 10^{9}\;\mathrm {kg{\cdot }m^{3}{\cdot }s^{-4}{\cdot }A^{-2}} .}$

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:

Coulomb's law:

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

Electric potential energy:

${\displaystyle U_{\text{E}}(r)=k_{\text{e}}{\frac {Qq}{r}}.}$

Electric field:

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

• Gravitational constant
• Vacuum permittivity
• Vacuum permeability