Coulomb's constant

Coulomb's constant, the electric force constant, or the electrostatic constant (denoted $k e$ ) is a proportionality constant in equations relating electric variables and is equal to $k e$ = 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,

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

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

$\oiint$

{\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.

$\oiint$

{\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_{{\text{e}}}{\frac {Qq}{r^{2}}}{\mathbf {{\hat {e}}}}_{r}

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

This exact value of Coulomb's constant, $k e$ , 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 7008299792458000000♠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{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:

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

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

Coulomb's law:

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

Electric potential energy:

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

Electric field:

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

References

↑ CODATA Value: electric constant. Physics.nist.gov. Retrieved on 2010-09-28.
↑ Coulomb's constant, Hyperphysics

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Coulomb's constant

Value of the constant

Use of Coulomb's constant

See also

References