electrostatic unit

The statcoulomb (statC) or franklin (Fr) or electrostatic unit of charge (esu) is the physical unit for electrical charge used in the centimetre-gram-second (cgs) electrostatic system of units. It is a derived unit given by

1 statC = 1 g^1/2 cm^3/2 s⁻¹ = 1 erg^1/2 cm^1/2.

The SI system of units uses the coulomb (C) instead. The conversion is

1 C = 2997924580 statC.

This equation is exact (however, see below for a warning on how to properly apply it). The number on the right-hand side is 10 times the value of the speed of light expressed in meters/second. The approximate conversions in both directions are:

1 C ≈ 2.99792×10⁹ statC,

1 statC ≈ 3.33564×10⁻¹⁰ C.

The statcoulomb is defined as follows: if two stationary objects each carry a charge of 1 statC and are 1 cm apart, they will electrically repel each other with a force of 1 dyne. This repulsion is governed by Coulomb's law, which in the Gaussian-cgs system states:

where F is the force, q₁ and q₂ are the two charges, and r is the distance between the charges. Performing dimensional analysis on Coulomb's law, the dimension of electrical charge in cgs must be [mass]^1/2 [length]^3/2 [time]^-1. (This statement is not true in SI units; see below.) We can be more specific in light of the definition above: Plugging in F=1 dyne, q₁=q₂=1 statC, and r = 1 cm, we get:

1 statC = g^1/2 cm^3/2 s⁻¹

as expected.

The coulomb is an extremely large charge rarely encountered in electrostatics, while the statcoulomb is closer to everyday charges.

Dimensional relation between Statcoulomb and Coulomb

In the cgs-Gaussian unit system, as mentioned above, Coulomb's law states

To be consistent with this equation, the statcoulomb must be (and is) dimensionally equivalent to [mass]^1/2 [length]^3/2 [time]⁻¹.

On the other hand, in SI units, Coulomb's law is different:

Since ε₀, the vacuum permittivity, is not dimensionless, the coulomb (the SI unit of charge) is not dimensionally equivalent to [mass]^1/2 [length]^3/2 [time]⁻¹, unlike the statcoulomb. In fact, it is impossible to express the Coulomb in terms of mass, length, and time alone.

Consequently, the statement 1 C = 2997924580 statC must be interpreted with caution: the units on the two sides are not consistent. Given a formula, one cannot simply use this conversion factor to switch between Coulombs and statcoulombs, as one would freely switch between centimeters and meters. Rather, this statement should be understood as: "1 coulomb corresponds to 2997924580 statcoulombs"; in other words, if a physical object has a charge of 1 coulomb, it also has a charge of 2997924580 statcoulombs.

On the other hand, the following conversion is fully dimensionally consistent, and often useful for switching between SI and cgs formulae:

where ε₀ ≈ 8.85×10⁻¹² A² s⁴kg⁻¹m⁻³ = 8.85×10⁻²¹ A²s⁴g⁻¹cm⁻³.