# Statcoulomb

The **franklin** (**Fr**) or **statcoulomb** (**statC**) **electrostatic unit of charge** (**esu**) is the physical unit for electrical charge used in the cgs-esu and Gaussian units. It is a derived unit given by

- 1 statC = 1 dyn
^{1/2}⋅cm = 1 cm^{3/2}⋅g^{1/2}⋅s^{−1}.

statcoulomb | |
---|---|

Unit system | Gaussian, cgs-esu |

Unit of | electrical charge |

Symbol | Fr or statC, esu |

Derivation | dyn^{1/2}⋅cm |

Conversions | |

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

CGS base units | cm^{3/2}⋅g^{1/2}⋅s^{−1} |

SI (charge) | ≘ ~3.33564×10^{−10} C |

SI (flux) | ≘ ~2.65×10^{−11} C |

That is, it is defined so that the Coulomb constant becomes a dimensionless quantity equal to 1.

It can be converted using

- 1 newton = 10
^{5}dyne - 1 cm = 10
^{−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.

- 1 C ≘ 2997924580 statC ≈ 3.00×10
- For electric flux (Φ
_{D}):- 1 C ≘ 4π × 2997924580 statC ≈ 3.77×10
^{10}statC - ⇒ 1 statC ≘ ~2.65×10
^{−11}C.

- 1 C ≘ 4π × 2997924580 statC ≈ 3.77×10

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 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/m
^{2}≘ 4π × 2997924580×10^{−4}statC/cm^{2}≈ 3.77×10^{6}statC/cm^{2} - ⇒ 1 statC/cm
^{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

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, *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 statC = g
^{1/2}⋅cm^{3/2}⋅s^{−1}

as expected.

## Dimensional relation between statcoulomb and coulomb

### General incompatibility

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

- (Gaussian)
- (SI)

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 centimeters and meters. 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

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

- (SI),

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:

### As a unit of electric displacement field or flux

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:

where

Therefore, the conversion factor for flux is 4π different from the conversion factor for charge:

- (as unit of Φ
_{D}).

The dimensionally consistent version is:

- (as unit of Φ
_{D})