The statcoulomb (statC) or franklin (Fr) or electrostatic unit of charge (esu) is the physical unit for electrical charge used in the centimetregramsecond (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 dimensionalanalysis complications discussed below. The number on the righthand 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 Gaussiancgs system states:
where F is the force, q_{1} and q_{2} 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, q_{1}=q_{2}=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 cgsGaussian 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 ε_{0}, 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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