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# Coulomb's constant

**Coulomb's constant**, the **electric force constant**, or the **electrostatic constant** (denoted ke ) is a proportionality constant in equations relating electric variables and is exactly equal to ke = 8.9875517873681764×10^{9} N·m^{2}/C^{2} (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 êr 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 ke comes from three of the fundamental, invariant quantities that define free space in the SI system: the speed of light c_{0} , magnetic permeability μ_{0} , and electric permittivity ε_{0} , 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 c_{0} is 299792458 m⋅s^{−1}, the magnetic permeability μ_{0} of free space is 4*π*·10^{−7} H m^{−1}, and the electric permittivity ε_{0} of free space is 1 ⁄ (μ_{0} c^{2}_{0} ) ≈ 8.85418782×10^{−12} F m^{−1},^{[1]} so that^{[2]}

\( \begin{align} k_\text{e} = \frac{1}{4\pi\varepsilon_0}=\frac{c_0^2\mu_0}{4\pi}&=c_0^2\times 10^{-7}\ \mathrm{H\ m}^{-1}\\ &= 8.987\ 551\ 787\ 368\ 176\ 4\times 10^9\ \mathrm{N\ m^2\ C}^{-2}. \end{align} \)

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:

\( k_\text{e} = \frac{1}{4\pi\varepsilon_0} . \)

Some examples of use of Coulomb's constant are 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:

\(\mathbf{E} = k_\text{e} \sum_{i=1}^N \frac{Q_i}{r_i^2} \mathbf{\hat{r}}_i. \)

See also

Vacuum permittivity

Vacuum permeability

References

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

Coulomb's constant, Hyperphysics

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