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 equal to ke  = 8.9875517873681764×109 N·m2/C2 (i.e. 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,

${\displaystyle {\scriptstyle S}}$ ${\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} ={\frac {Q}{\varepsilon _{0}}}}$

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.

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

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

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

This exact value of Coulomb's constant, 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:

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

Because of the way the SI base unit system made the natural units for electromagnetism, the speed of light in vacuum 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]

{\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:

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

Coulomb's constant appears in many expressions including the following:

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