View source for Statcoulomb

{{Short description|CGS unit of electric charge}}

{{Refimprove|date=November 2024}}
{{Infobox unit
| name         = statcoulomb
| image        = 
| caption      = 
| standard     = [[Gaussian units|Gaussian]], [[Centimetre–gram–second system of units#Electrostatic units (ESU)|CGS-ESU]]
| quantity     = [[electrical charge]]
| symbol       = Fr
| symbol2      = statC, esu
| namedafter   = 
| extralabel   = Derivation
| extradata    = dyn<sup>1/2</sup>⋅cm
| units1       = CGS base&nbsp;units
| inunits1     = 1&nbsp;cm<sup>3/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>
| units2       = [[International System of Units|SI]] (charge)
| inunits2     = ≘&nbsp;~{{val|3.33564|e=-10|ul=C}}
| units3       = [[International System of Units|SI]] (flux)
| inunits3     = ≘&nbsp;~{{val|2.65|e=-11|ul=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 system of units#Electrostatic units (ESU)|centimetre–gram–second electrostatic units variant]] (CGS-ESU) and [[Gaussian units|Gaussian]] systems of units. It is a derived unit given by
{{block indent | em = 1.5 | text = 1&nbsp;statC = 1&nbsp;dyn<sup>1/2</sup>⋅cm = 1&nbsp;cm<sup>3/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>.}}

That is, it is defined so that the CGS-ESU quantity that the proportionality constant in [[Coulomb's law]] is a dimensionless quantity equal to&nbsp;1.

It can be converted to the corresponding SI quantity using
{{block indent | em = 1.5 | text = 1&nbsp;newton = 10<sup>5</sup>&nbsp;dyne}}
{{block indent | em = 1.5 | text = 1&nbsp;cm = 10<sup>−2</sup>&nbsp;m}}

The [[International System of Units]] uses the [[coulomb]] (C) as its unit of electric charge. The conversion between the units coulomb and the statcoulomb depends on the context. The most common contexts are{{efn|As of the [[2019 revision of the SI]], the correspondence given is not exact, although it is very close.}}:
* For [[electric charge]]:<!--
-->{{block indent | em = 1.5 | text = 1&nbsp;C ≘ ~{{val|2997924580|u=statC}} ≈ {{val|3.00|e=9|u=statC}}}}<!--
-->{{block indent | em = 1.5 | text = ⇒ 1&nbsp;statC ≘ ~{{val|3.33564|e=-10|u=C}}.}}
* For [[electric flux]] (Φ<sub>D</sub>): <!--
-->{{block indent | em = 1.5 | text =1&nbsp;C ≘ ~4π × {{val|2997924580|u=statC}} ≈ {{val|3.77|e=10|u=statC}}}}<!--
-->{{block indent | em = 1.5 | text = ⇒ 1&nbsp;statC ≘ ~{{val|2.65|e=-11|u=C}}.}}

The symbol "≘" ('corresponds to') is used instead of "=" because the two sides are not interchangeable, as discussed [[#Dimensional relation between Statcoulomb and Coulomb|below]]. The numerical part of the conversion factor of {{val|2997924580|u=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 [[centimetre gram second system of units|CGS]] units for an [[electric displacement field]] ('''D''') are related by:
{{block indent | em = 1.5 | text = 1&nbsp;C/m<sup>2</sup> ≘ ~4π × {{val|2997924580|e=-4|u=statC/cm<sup>2</sup>}} ≈ {{val|3.77|e=6|u=statC/cm<sup>2</sup>}}}}
{{block indent | em = 1.5 | text = ⇒ 1&nbsp;statC/cm<sup>2</sup> ≘ ~{{val|2.65|e=-7|u=C/m<sup>2</sup>}}}}
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 such that if two stationary spherically symmetric objects each carry a charge of 1&nbsp;statC and are {{val|1|u=cm}} apart, the force of mutual electrical repulsion will be 1&nbsp;[[dyne]]. This repulsion is governed by [[Coulomb's law]], which in the [[Gaussian units|CGS-Gaussian system]] states:
<math display="block">F = \frac{q_1^\text{G} q_2^\text{G}}{r^2} ,</math>
where ''F'' is the force, ''q''{{su|b=1|p=G|lh=0.9}} and ''q''{{su|b=2|p=G|lh=0.9}} 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]<sup>1/2</sup> [length]<sup>3/2</sup> [time]<sup>−1</sup>. (This statement is not true in the [[International System of Quantities]] upon which the [[SI]] is based; see below.) We can be more specific in light of the definition above: Substituting ''F'' = 1&nbsp;dyn, ''q''{{su|b=1|p=G|lh=0.9}} = ''q''{{su|b=2|p=G|lh=0.9}} = 1&nbsp;statC, and ''r'' = 1&nbsp;cm, we get:
{{block indent | em = 1.5 | text = 1&nbsp;statC = g<sup>1/2</sup>⋅cm<sup>3/2</sup>⋅s<sup>−1</sup>}}
as expected.

== Dimensional relation between statcoulomb and coulomb ==
{{off topic|Gaussian units#Major differences between Gaussian and SI units|date=February 2013}}

=== General incompatibility ===
[[Coulomb's law]] in the [[Gaussian units|Gaussian unit system]] and the [[SI]] are respectively:
{{block indent | em = 1.5 | text = <math>F=\frac{q_1^\text{G} q_2^\text{G}}{r^2}</math> (Gaussian)}}
{{block indent | em = 1.5 | text = <math>F=\frac{q_1^\text{SI} q_2^\text{SI}}{4\pi\epsilon_0 r^2}</math> (SI)}}
Since ''ε''<sub>0</sub>, the [[vacuum permittivity]], is {{em|not}} dimensionless, the [[coulomb]] is {{em|not}} dimensionally equivalent to [mass]<sup>1/2</sup> [length]<sup>3/2</sup> [time]<sup>−1</sup>, 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&nbsp;C = ''n''&nbsp;statC" is misleading: the units on the two sides are not consistent. One {{em|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 {{em|correspondence}} between coulombs and statcoulombs in different contexts. As described below, "1&nbsp;C {{em|corresponds to}} {{Val|3.00|e=9|u=statC}}" when describing the charge of objects. In other words, if a physical object has a charge of 1&nbsp;C, it also has a charge of {{Val|3.00|e=9|u=statC}}. Likewise, "1&nbsp;C {{em|corresponds to}} {{Val|3.77|e=10|u=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&nbsp;cm apart in vacuum, they will electrically repel each other with a force of 1&nbsp;[[dyne]]. <ref>  Jan Gyllenbok, "Encyclopaedia of Historical Metrology, Weights, and Measures Volume 1", Birkhauser, 2018, ISBN  978-3-319-57598-8, page 29  </ref> From this definition, it is straightforward to find an equivalent charge in [[coulomb]]s. Using the [[SI]] equation
{{block indent | em = 1.5 | text = <math>F=\frac{q_1^\text{SI} q_2^\text{SI}}{4\pi\epsilon_0 r^2}</math>,}}
and setting {{mvar|F}} = 1&nbsp;dyn = 10<sup>−5</sup>&nbsp;N and {{mvar|r}} = 1&nbsp;cm = 10<sup>−2</sup>&nbsp;m, and then solving for {{math|1=''q'' = ''q''{{su|b=<sub>1</sub>|p=SI|lh=0.9}} = ''q''{{su|b=<sub>2</sub>|p=SI|lh=0.9}}}}, the result is
{{block indent | em = 1.5 | text = <math>q = \sqrt{4 \pi\epsilon_0 \times \mathrm{10^{-9}~kg{\cdot}m^3{\cdot}s^{-2}}} \approx \mathrm{3.33564\times 10^{-10}~C}</math>}}
Therefore, an object with a CGS charge of 1&nbsp;statC has a charge of approximately {{val|3.34|e=-10|u=C}}.

=== As a unit of electric displacement field or flux ===
An [[electric flux]] (specifically, a flux of the [[electric displacement field]] {{math|'''D'''}}) has units of charge: statC in CGS and coulombs in SI. The conversion factor can be derived from [[Gauss's law]]:
<math display="block">\Phi_\mathbf{D}^\text{G} = 4\pi Q^\text{G}</math>
<math display="block">\Phi_\mathbf{D}^\text{SI} = Q^\text{SI}</math>
where
<math display="block">\Phi_\mathbf{D} \equiv \int_S \mathbf{D}\cdot \mathrm{d}\mathbf{A}</math>
Therefore, the conversion factor for flux and the conversion factor for charge differ by a ratio of 4''π'':
<math display="block">\mathrm{1~C} ~\overset\frown=~ \mathrm{3.7673 \times 10^{10}~statC} \qquad \text{(as unit of }\Phi_\mathbf{D}\text{).}</math>

<math display="block">\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{).}</math>

== Notes ==
{{notelist}}
==References==
<references/>

[[Category:Units of electrical charge]]
[[Category:Centimetre–gram–second system of units]]