Coulomb's constant

Coulomb's constant, the electric force constant, or the electrostatic constant (denoted $k e$ , $k$ or $K$ ) is a proportionality constant in electrodynamics equations. In SI units, it is exactly equal to 7009898755178736817♠8987551787.3681764 N·m²·C⁻², or roughly equaling 7009899000000000000♠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

Coulomb's constant is 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 and

k_{\text{e}}=\alpha {\frac {\hbar c}{e^{2}}}

,

where $α$ is the fine-structure constant, $c$ is the speed of light, $ħ$ is the reduced Planck constant, and $e$ is elementary charge.^[1] 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,

$\oiint$

{\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 $q$ ,

{\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 modern systems of units Coulomb's constant $k e$ is an exact constant, in Gaussian units $k e = 1$ , in Lorentz–Heaviside units (also called rationalized) $k e = 1 / 4π$ and in SI $k e = 1 / 4π ε 0$ , where the vacuum permittivity $ε 0 = 1 / μ 0 c 2 \approx$ 6988885418782000000♠8.85418782×10⁻¹² F m⁻¹, the speed of light in vacuum $c$ is 7008299792458000000♠299792458 m/s, the vacuum permeability $μ 0$ is 4 $π$ ×10⁻⁷ H m⁻¹,^[2] so that^[3]

{\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\ m} ^{-1})\\&=8.987\,551\,787\,368\,1764\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

↑ Tomilin, K. (1999). "Fine-structure constant and dimension analysis". European Journal of Physics. 20 (5): L39–L40. Bibcode:1999EJPh...20L..39T. doi:10.1088/0143-0807/20/5/404.
↑ CODATA Value: electric constant. Physics.nist.gov. Retrieved on 2010-09-28.
↑ Coulomb's constant, Hyperphysics

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[Tomilin1999-1] Tomilin, K. (1999). "Fine-structure constant and dimension analysis". European Journal of Physics. 20 (5): L39–L40. Bibcode:1999EJPh...20L..39T. doi:10.1088/0143-0807/20/5/404.

[2] CODATA Value: electric constant. Physics.nist.gov. Retrieved on 2010-09-28.

[3] Coulomb's constant, Hyperphysics

Coulomb's constant

Value of the constant

Use of Coulomb's constant

See also

References