# Statcoulomb

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

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 ≈ ×10
^{9}statC 3.00 - ⇒ 1 statC ↔ ~64×10
^{−10}C. 3.335

- 1 C ↔ 2997924580 statC ≈ ×10

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

- 1 C ↔ 4π×2997924580 statC ≈×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}≈ ×10^{6}statC/cm^{2}3.77 - ⇒ 1 statC/cm
^{2}↔ ~×10^{−7}C/m^{2}2.65

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.

## Contents

## 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*_{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: Plugging in *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 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) |

### General incompatibility

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

- (cgs-Gaussian)
- (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* ×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 3.00*corresponds to* ×10^{10} 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

- (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 ≈ ×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

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:

- (cgs)
- (SI)

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}).