# Statcoulomb

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statcoulomb | |
---|---|

Unit system | Gaussian, CGS-ESU |

Unit of | electrical charge |

Symbol | Fr, statC, esu |

Derivation | dyn^{1/2}⋅cm |

Conversions | |

1 Fr in ... | ... is equal to ... |

CGS base units | 1 cm^{3/2}⋅g^{1/2}⋅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

^{1/2}⋅cm = 1 cm

^{3/2}⋅g

^{1/2}⋅s

^{−1}.

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

^{5}dyne

^{−2}m

The SI system of units uses the coulomb (C) instead. The conversion between C and statC is different in different contexts. The most common contexts are:

- For electric charge:1 C ≘ 2997924580 statC ≈ 3.00×10
^{9}statC⇒ 1 statC ≘ ~3.33564×10^{−10}C. - For electric flux (Φ
_{D}):1 C ≘ 4π × 2997924580 statC ≈ 3.77×10^{10}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:

^{2}≘ 4π × 2997924580×10

^{−4}statC/cm

^{2}≈ 3.77×10

^{6}statC/cm

^{2}

^{2}≘ ~2.65×10

^{−7}C/m

^{2}

due to the relation between the metre and the centimetre. The coulomb is an extremely large charge rarely encountered in electrostatics, while the statcoulomb is closer to everyday charges.

## Definition and relation to CGS base units[edit]

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:

*F*is the force,

*q*

^{G}

_{1}and

*q*

^{G}

_{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,

*q*

^{G}

_{1}=

*q*

^{G}

_{2}= 1 statC, and

*r*= 1 cm, we get:

^{1/2}⋅cm

^{3/2}⋅s

^{−1}

as expected.

## Dimensional relation between statcoulomb and coulomb[edit]

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) |

### General incompatibility[edit]

Coulomb's law in the Gaussian unit system and the SI are respectively:

Since *ε*_{0}, 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* 3.00×10^{9} 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 3.00×10^{9} statC. Likewise, "1 C *corresponds to* 3.77×10^{10} statC" when describing an electric displacement field flux.

### As a unit of charge[edit]

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* = *q*^{SI}_{1} = *q*^{SI}_{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[edit]

An electric flux (specifically, a flux of the electric displacement field **D**) has units of charge: statC in CGS and coulombs in SI. The conversion factor can be derived from Gauss's law:

^{[a]}

## Notes[edit]

- ^
^{a}^{b}As of the 2019 redefinition of the SI base units, this equality is not exact.