Physics: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 8987551787.3681764 N·m²·C⁻², or roughly equaling 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,

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

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

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

where α is the fine-structure constant, c is the speed of light, ħ is the reduced Planck constant, and e is elementary charge. In SI:

$k_{\text{e}}=\frac{1}{4\pi {\epsilon }_0}$,

where ${\epsilon }_0$ is the vacuum permittivity. This formula can be derived from Gauss' law,

$S$ $\mathbf{𝐄}\cdot \mathrm{d}\mathbf{𝐀}=\frac{Q}{{\epsilon }_0}$

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.

$S$ $\mathbf{𝐄}\cdot \mathrm{d}\mathbf{𝐀}=|\mathbf{𝐄}|{\hat{\mathbf{e}}}_r\underset{S}{\int }dA=|\mathbf{𝐄}|{\hat{\mathbf{e}}}_r\times 4\pi r^2$

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

$\begin{array}{rl}\mathbf{𝐅}& =\frac{1}{4\pi {\epsilon }_0}\frac{Qq}{r^2}{\hat{\mathbf{e}}}_r=k_{\text{e}}\frac{Qq}{r^2}{\hat{\mathbf{e}}}_r\\ \therefore k_{\text{e}}& =\frac{1}{4\pi {\epsilon }_0}\end{array}$

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 ≈ 8.85418782×10⁻¹² F m⁻¹, the speed of light in vacuum c is 299792458 m/s, the vacuum permeability μ₀ is 4π×10⁻⁷ H m⁻¹, so that

$\begin{array}{rl}{k}_{\text{e}}=\frac{1}{4\pi {\epsilon }_0}=\frac{c^2{\mu }_0}{4\pi }& =c^2\times (10^{-7}{\mathrm{H}\mathrm{m}}^{-1})\\ & =8.9875517873681764\times 10^9\mathrm{N}{\mathrm{m}}^{\mathrm{2}}{\mathrm{C}}^{\mathrm{-}\mathrm{2}}.\end{array}$

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:

$k_{\text{e}}=\frac{1}{4\pi {\epsilon }_0}.$

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

Coulomb's law:

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

Electric potential energy:

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

Electric field:

$\mathbf{𝐄}=k_{\text{e}}\sum _{i=1}^N\frac{Q_i}{r_i^2}{\hat{\mathbf{r}}}_i.$

See also

• Gravitational constant
• Vacuum permittivity
• Vacuum permeability