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|Unit system||Gaussian, CGS-ESU|
|Unit of||electrical charge|
|Symbol||Fr, statC, esu|
|1 Fr in ...||... is equal to ...|
|CGS base units||1 cm3/2⋅g1/2⋅s−1|
|SI (charge)||≘ ~3.33564×10−10 C|
|SI (flux)||≘ ~2.65×10−11 C|
The franklin (Fr) or statcoulomb (statC) electrostatic unit of charge (esu) is the physical unit for electrical charge used in the centimetre–gram–second electrostatic units variant (CGS-ESU) and Gaussian systems of units. It is a derived unit given by
That is, it is defined so that the CGS-ESU quantity that corresponds to the Coulomb constant is a dimensionless quantity equal to 1.
It can be converted using
- For electric charge:1 C ≘ 2997924580 statC ≈ 3.00×109 statC⇒ 1 statC ≘ ~3.33564×10−10 C.
- For electric flux (ΦD): 1 C ≘ 4π × 2997924580 statC ≈ 3.77×1010 statC⇒ 1 statC ≘ ~2.65×10−11 C.
The symbol "≘" ('corresponds to') is used instead of "=" because the two sides are not interchangeable, as discussed below. The number 2997924580 is 10 times the numeric value of the speed of light expressed in metres/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:
Definition and relation to CGS base units
The statcoulomb is such that 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 CGS-Gaussian system states:
1 and qG
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: Substituting F = 1 dyn, qG
1 = qG
2 = 1 statC, and r = 1 cm, we get:
Dimensional relation between statcoulomb and coulomb
This section may contain material not related to the topic of the article and should be moved to Gaussian units#Major differences between Gaussian and SI units instead. (February 2013)
Since ε0, the vacuum permittivity, is not dimensionless, the coulomb 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" is 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 centimetres and metres. One can, however, find a correspondence between coulombs and statcoulombs in different contexts. As described below, "1 C corresponds to 3.00×109 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 3.77×1010 statC" when describing an electric 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 in vacuum, they will electrically repel each other with a force of 1 dyne. From this definition, it is straightforward to find an equivalent charge in 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 = qSI
1 = qSI
2, the result is q = (1/2997924580) C ≈ 3.34×10−10 C. Therefore, an object with a charge of 1 statC has a charge of 3.34×10−10 C.
This can also be expressed by the following conversion, which is fully dimensionally consistent, and often useful for switching between SI and CGS formulae:[a]
As a unit of electric displacement field or flux
- As of the 2019 redefinition of the SI base units, this equality is not exact.