# Statcoulomb

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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

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

This conversion is exact. However, the symbol "↔" is used instead of "=" because of the dimensional-analysis complications discussed below. 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:

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:

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 equation "1 C = 2997924580 statC" can be misleading: the units on the two sides are not consistent. One cannot freely switch between Coulombs and statcoulombs within a formula or equation, as one would freely switch between centimeters and meters. A clearer statement is to say ""1 C corresponds to 2997924580 statC", instead of "1 coulomb equals 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:

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