# Coulomb's constant

**Coulomb's constant**, the **electric force constant**, or the **electrostatic constant** (denoted Template:SubSup) is a proportionality constant in equations relating electric variables and is exactly equal to Template:SubSup = Template:GapsTemplate:E N·m^{2}/C^{2} (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,

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

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.

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

This exact value of Coulomb's constant Template:SubSup comes from three of the fundamental, invariant quantities that define free space in the SI system: the speed of light Template:SubSup, magnetic permeability Template:SubSup, and electric permittivity Template:SubSup, related by Maxwell as:

Because of the way the SI base unit system made the natural units for electromagnetism, the speed of light in vacuum Template:SubSup is Template:Gaps, the magnetic permeability Template:SubSup of free space is 4*π*·10^{−7} H m^{−1}, and the electric permittivity Template:SubSup of free space is 1 Template:Frac (Template:SubSupTemplate:SubSup) ≈ Template:Gaps,^{[1]}
so that^{[2]}

## 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:

Some examples of use of Coulomb's constant are the following:

## See also

## References

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