The statcoulomb (statC) or franklin (Fr) or electrostatic unit of charge (esu) is the physical unit for electrical charge used in the esu-cgs (centimetre-gram-second system of units) and Gaussian units. It is a derived unit given by
- 1 statC =dyn1/2 cm= cm3/2 g1/2 s-1.
- For electric charge:
- 1 C ↔ 2997924580 statC ≈ ×109 statC3.00
- ⇒ 1 statC ↔ ~64×10−10 C. 3.335
- For electric flux (ΦD):
- 1 C ↔ 4π×2997924580 statC ≈×1010 statC3.77
- ⇒ 1 statC ↔ ~×10−11 C. 2.65
The symbol "↔" is used instead of "=" because the two sides are not necessarily interchangeable, as discussed below. The number 2997924580 is 10 times the value of the speed of light expressed in meters/second, and the conversions are exact except where indicated. The second context implies that the SI and cgs units for an electric displacement field (D) are related by:
- 1 C/m2 ↔ 4π×2997924580×10−4 statC/cm2 ≈ ×106 statC/cm23.77
- ⇒ 1 statC/cm2 ↔ ~×10−7 C/m22.65
Definition and relation to cgs base units
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 dyn, q1 = q2 = 1 statC, and r = 1 cm, we get:
- 1 statC = g1/2 cm3/2 s−1
Dimensional relation between Statcoulomb and Coulomb
This section possibly contains original research. (February 2013)
This section may stray from the topic of the article into the topic of another article, Gaussian units #Major differences between Gaussian and SI units. (February 2013)
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, a conversion equation like "1 C = N 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. One can, however, find a correspondence between coulombs and statcoulombs in different contexts. As described below, "1 C corresponds to ×109 statC" when describing the charge of objects. In other words, if a physical object has a charge of 1 C, it also has a charge of 3.00×109 statC. Likewise, "1 C corresponds to 3.00×1010 statC" when describing an 3.77electric displacement field flux.
As a unit of charge
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. From this definition, it is straightforward to find an equivalent charge in SI coulombs. Using the SI equation
and plugging in F = 1 dyn = 10−5 N, and r = 1 cm = 10−2 m, and then solving for q = q1 = q2, the result is q = (1/2997924580)C ≈ ×10−10 C. Therefore, an object with a charge of 1 statC has a charge of 3.34×10−10 C. 3.34
This can also be expressed by the following conversion, which is fully dimensionally consistent, and often useful for switching between SI and cgs formulae:
As a unit of electric displacement field or flux
Therefore, the conversion factor for flux is 4π different from the conversion factor for charge:
- (as unit of ΦD).
The dimensionally consistent version is:
- (as unit of ΦD).