Statcoulomb

statcoulomb
Unit system	Gaussian, CGS-ESU
Unit of	electric charge
Symbol	Fr, statC, esu
Derivation	dyn1/2⋅cm
Conversions
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 statcoulomb (statC), franklin (Fr), or electrostatic unit of charge (esu) is the unit of measurement for electrical charge used in the centimetre–gram–second electrostatic units variant (CGS-ESU) and Gaussian systems of units. In terms of the Gaussian base units, it is

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

That is, it is defined so that the proportionality constant in Coulomb's law using CGS-ESU quantities is a dimensionless quantity equal to 1.

It can be converted to the corresponding quantity of the International System of Quantities (ISQ) that underlies the International System of Units (SI) using

1 newton = 10⁵ dyne,

1 cm = 10⁻² m.

The SI uses the coulomb (C) as its unit of electric charge. The conversion factor between corresponding quantities with the units coulomb and statcoulomb depends on which quantity is to be converted. The most common cases are:

• For electric charge:
  1 C ≘ ~ 2997924580 statC ≈ 3.00×10⁹ statC
  ⇒ 1 statC ≘ ~ 3.33564×10⁻¹⁰ C.
• For electric flux (Φ_D):
  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 in § Dimensional relation between statcoulomb and coulomb below. The numerical part of the conversion factor of 2997924580 statC/C is very close to 10 times the numeric value of the speed of light when expressed in the unit metre/second, with a small uncertainty. In the context of electric flux, the SI and CGS units for an electric displacement field (D) are related by:

1 C/m² ≘ ~ 4π × 2997924580×10⁻⁴ statC/cm² ≈ 3.77×10⁶ statC/cm²

⇒ 1 statC/cm² ≘ ~ 2.65×10⁻⁷ C/m²

due to the relation between the metre and the centimetre.

Definition and relation to CGS base units

The statcoulomb is defined such that if two stationary spherically symmetric objects each carry a charge of 1 statC and are 1 cm apart, the force of mutual electrical repulsion will be 1 dyne, and such that Coulomb's law in the CGS-Gaussian system takes the form: $$F={\frac {q_{1}^{\text{G}}q_{2}^{\text{G}}}{r^{2}}},$$ 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 the ISQ; 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

Distinct systems of equations

Coulomb's law as expressed in the Gaussian system and the International System of Quantities that underlies the SI are respectively:

$F={\frac {q_{1}^{\text{G}}q_{2}^{\text{G}}}{r^{2}}}$ (Gaussian),

$F={\frac {q_{1}^{\text{SI}}q_{2}^{\text{SI}}}{4\pi \varepsilon _{0}r^{2}}}$ (ISQ).

Since ε₀, the vacuum permittivity, is not dimensionless, the coulomb is not dimensionally equivalent to M^1/2L^3/2T⁻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

$F={\frac {q_{1}^{\text{SI}}q_{2}^{\text{SI}}}{4\pi \varepsilon _{0}r^{2}}}$,

and setting 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

$q={\sqrt {4\pi \varepsilon _{0}\times \mathrm {10^{-9}~kg{\cdot }m^{3}{\cdot }s^{-2}} }}\approx \mathrm {3.33564\times 10^{-10}~C}$

Therefore, an object with a CGS charge of 1 statC has a charge of approximately 3.34×10⁻¹⁰ C.