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,

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

where ê_r is a unit vector in the r-direction and

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

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:

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

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, 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} |\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 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 = 1/4πε₀, where the vacuum permittivity ε₀ = 1/μ₀c² ≈6988885418782000000♠8.85418782×10⁻¹²  F⋅m⁻¹, the speed of light in vacuum c is 7008299792458000000♠299792458 m/s, the vacuum permeability μ₀ is 4π×10⁻⁷ H⋅m⁻¹, ^[2] so that ^[3]

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

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:

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:

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

See also

• Gravitational constant
• Vacuum permittivity
• Vacuum permeability

References

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