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Statcoulomb

## Summary

The franklin (Fr), statcoulomb (statC), or 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

statcoulomb
Unit systemGaussian, CGS-ESU
Unit ofelectrical charge
SymbolFr, statC, esu
Derivationdyn1/2⋅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
1 statC = 1 dyn1/2⋅cm = 1 cm3/2⋅g1/2⋅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

1 newton = 105 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[a]:

• For electric charge:
1 C ≘ ~2997924580 statC3.00×109 statC
⇒ 1 statC ≘ ~3.33564×10−10 C.
• For electric fluxD):
1 C ≘ ~4π × 2997924580 statC3.77×1010 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 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:

1 C/m2 ≘ 4π × 2997924580×10−4 statC/cm23.77×106 statC/cm2
⇒ 1 statC/cm2 ≘ ~2.65×10−7 C/m2

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 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:

${\displaystyle F={\frac {q_{1}^{\text{G}}q_{2}^{\text{G}}}{r^{2}}},}$

where F is the force, qG
1
and qG
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, qG
1
= qG
2
= 1 statC, and r = 1 cm, we get:
1 statC = g1/2⋅cm3/2⋅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:

${\displaystyle F={\frac {q_{1}^{\text{G}}q_{2}^{\text{G}}}{r^{2}}}}$  (Gaussian)
${\displaystyle F={\frac {q_{1}^{\text{SI}}q_{2}^{\text{SI}}}{4\pi \epsilon _{0}r^{2}}}}$  (SI)

Since ε0, 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 3.00×109 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

${\displaystyle F={\frac {q_{1}^{\text{SI}}q_{2}^{\text{SI}}}{4\pi \epsilon _{0}r^{2}}}}$  (SI),

and plugging in F = 1 dyn = 10−5 N, and r = 1 cm = 10−2 m, and then solving for q = qSI
1
= qSI
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:[b]

${\displaystyle \mathrm {1~C} \times {\sqrt {\frac {10^{9}}{4\pi \epsilon _{0}}}}=\mathrm {1~C} \times 2997924580=\mathrm {1~statC} }$

### 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:

${\displaystyle \Phi _{\mathbf {D} }^{\text{G}}=4\pi Q^{\text{G}}}$

${\displaystyle \Phi _{\mathbf {D} }^{\text{SI}}=Q^{\text{SI}}}$

where
${\displaystyle \Phi _{\mathbf {D} }\equiv \int _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} }$

Therefore, the conversion factor for flux is 4π different from the conversion factor for charge:
${\displaystyle \mathrm {1~C} ~{\overset {\frown }{=}}~\mathrm {3.7673\times 10^{10}~statC} \qquad {\text{(as unit of }}\Phi _{\mathbf {D} }{\text{).}}}$

The dimensionally consistent version is:[b]
${\displaystyle \mathrm {1~C} \times {\sqrt {\frac {4\pi \times 10^{9}}{\epsilon _{0}}}}=\mathrm {3.7673\times 10^{10}~statC} \qquad {\text{(as unit of }}\Phi _{\mathbf {D} }{\text{).}}}$

## Notes

1. ^ As of the 2019 redefinition of the SI base units, there is no exact equality possible here.
2. ^ a b As of the 2019 redefinition of the SI base units, this equality is not exact.