# Physics:Statcoulomb

statcoulomb | |
---|---|

Unit system | esu-cgs, Gaussian |

Unit of | electrical charge |

Symbol | statC or Fr, esu |

Derivation | dyn^{1/2}⋅cm |

Conversions | |

1 statC 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 |

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

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 "↔" 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/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:

- [math]\displaystyle{ F=\frac{q_1q_2}{r^2} }[/math]

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: Substituting *F* = 1 dyn, *q*_{1} = *q*_{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

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) |

### General incompatibility

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

- [math]\displaystyle{ F=\frac{q_1q_2}{r^2} }[/math] (cgs-Gaussian)
- [math]\displaystyle{ F=\frac{q_1q_2}{4\pi\epsilon_0 r^2} }[/math] (SI)

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, 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* 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 SI coulombs. Using the SI equation

- [math]\displaystyle{ F=\frac{q_1q_2}{4\pi\epsilon_0 r^2} }[/math] (SI),

and plugging in F = 1 dyn = 10^{−5} N, and r = 1 cm = 10^{−2} m, and then solving for *q* = *q*_{1} = *q*_{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:

- [math]\displaystyle{ 1 \; \mathrm{C}{\sqrt{ \tfrac{10^{9}}{4 \pi\epsilon_0}}} = 2997924580 \; \mathrm{statC} }[/math]

### 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:

- [math]\displaystyle{ \Phi_\mathbf{D} = 4\pi Q }[/math] (cgs)
- [math]\displaystyle{ \Phi_\mathbf{D} = Q }[/math] (SI)

where

- [math]\displaystyle{ \Phi_\mathbf{D} \equiv \int_S \mathbf{D}\cdot \mathrm{d}\mathbf{A} }[/math]

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

- [math]\displaystyle{ 1 \; \mathrm{C} \text{ corresponds to } 3.7673 \times 10^{10} \; \mathrm{statC} }[/math] (as unit of Φ
_{D}).

The dimensionally consistent version is:

- [math]\displaystyle{ 1 \; \mathrm{C}{\sqrt{\tfrac{4 \pi 10^{9}}{\epsilon_0}}} = 3.7673 \times 10^{10} \; \mathrm{statC} }[/math] (as unit of Φ
_{D})

Original source: https://en.wikipedia.org/wiki/Statcoulomb.
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