View source for Gaussian units

{{short description|Variant of the centimetre–gram–second unit system}}
{{Use British English |date=September 2023}}

[[File:Carl Friedrich Gauss.jpg|right|thumb|250px|[[Carl Friedrich Gauss]]]]
'''Gaussian units''' constitute a [[metric system]] of [[units of measurement]]. This system is the most common of the several electromagnetic unit systems based on the [[centimetre–gram–second system of units]] (CGS). It is also called the '''Gaussian unit system''', '''Gaussian-cgs units''', or often just '''cgs units'''.{{efn|One of many examples of using the term "cgs units" to refer to Gaussian units is: [http://nlpc.stanford.edu/nleht/Science/reference/conversion.pdf Lecture notes from Stanford University]}} The term "cgs units" is ambiguous and therefore to be avoided if possible: there are several variants of CGS, which have conflicting definitions of electromagnetic quantities and units.

[[International System of Units|SI units]] predominate in most fields, and continue to increase in popularity at the expense of Gaussian units.<ref name=Rowlett/>{{efn|name=JacksonEditions}} Alternative unit systems also exist. Conversions between quantities in the Gaussian and SI systems are {{em|not}} direct unit conversions, because the quantities themselves are defined differently in each system. This means that the equations that express physical laws of electromagnetism—such as [[Maxwell's equations]]—will change depending on the system of quantities that is employed. As an example, quantities that are [[dimensionless]] in one system may have dimension in the other.

== Alternative unit systems ==
{{main|Centimetre–gram–second system of units#Alternate derivations of CGS units in electromagnetism|l1=Alternative CGS units in electromagnetism}}
The Gaussian unit system is just one of several electromagnetic unit systems within CGS. Others include "[[Centimetre–gram–second system of units#Electrostatic units (ESU)|electrostatic units]]", "[[Centimetre–gram–second system of units#Electromagnetic units (EMU)|electromagnetic units]]", and [[Heaviside–Lorentz units]].

Some other unit systems are called "[[natural units]]", a category that includes [[atomic units]], [[Planck units]], and others.

The [[International System of Units]] (SI), with the associated [[International System of Quantities]] (ISQ), is by far the most common system of units today. In [[engineering]] and practical areas, SI is nearly universal and has been for decades.<ref name=Rowlett>[https://web.archive.org/web/20130120021655/hhttp://www.unc.edu/~rowlett/units/cgsmks.html "CGS"], in ''How Many? A Dictionary of Units of Measurement'', by Russ Rowlett and the [[University of North Carolina at Chapel Hill]]</ref> In technical, scientific literature (such as [[theoretical physics]] and [[astronomy]]), Gaussian units were predominant until recent decades, but are now getting progressively less so.<ref name=Rowlett/>{{efn|name=JacksonEditions|For example, one widely used graduate electromagnetism textbook is ''[[Classical Electrodynamics (book)|Classical Electrodynamics]]'' by [[John David Jackson (physicist)|J. D. Jackson]]. The second edition, published in 1975, used Gaussian units exclusively, but the third edition, published in 1998, uses mostly SI units. Similarly, ''Electricity and Magnetism'' by Edward Purcell is a popular undergraduate textbook. The second edition, published in 1984, used Gaussian units, while the third edition, published in 2013, switched to SI units.}} The 8th SI Brochure mentions the CGS-Gaussian unit system,<ref>{{SIbrochure8th}}, p. 128</ref> but the 9th SI Brochure makes no mention of CGS systems.

Natural units may be used in more theoretical and abstract fields of physics, particularly [[particle physics]] and [[string theory]].

== Major differences between Gaussian and SI systems ==

=== "Rationalized" unit systems ===

One difference between the Gaussian and SI systems is in the factor {{math|4''π''}} in various formulas that relate the quantities that they define. With SI electromagnetic units, called ''rationalized'',<ref name=Littlejohn>{{cite web | url=http://bohr.physics.berkeley.edu/classes/221/1112/notes/emunits.pdf | title=Gaussian, SI and Other Systems of Units in Electromagnetic Theory | work=Physics 221A, University of California, Berkeley lecture notes|author-link1=Robert Grayson Littlejohn | author=Littlejohn, Robert | date=Fall 2017 | access-date=2018-04-18 }}</ref><ref name=Kowalski>Kowalski, Ludwik, 1986, [http://alpha.montclair.edu/~kowalskiL/SI/SI_PAGE.HTML "A Short History of the SI Units in Electricity"], {{webarchive |url=https://web.archive.org/web/20090429035624/http://alpha.montclair.edu/~kowalskiL/SI/SI_PAGE.HTML |date=2009-04-29 }}  ''The Physics Teacher'' 24(2): 97–99. [https://dx.doi.org/10.1119/1.2341955 Alternate web link (subscription required)]</ref> [[Maxwell's equations]] have no explicit factors of {{math|4''π''}} in the formulae, whereas the [[inverse-square law|inverse-square]] force laws &ndash; [[Coulomb's law]] and the [[Biot–Savart law]] &ndash; {{em|do}} have a factor of {{math|4''π''}} attached to the {{math|''r''{{i sup|2}}}}. With Gaussian units, called ''unrationalized'' (and unlike [[Heaviside–Lorentz units]]), the situation is reversed: two of Maxwell's equations have factors of {{math|4''π''}} in the formulas, while both of the inverse-square force laws, Coulomb's law and the Biot–Savart law, have no factor of {{math|4''π''}} attached to {{math|''r''{{i sup|2}}}} in the denominator.

(The quantity {{math|4''π''}} appears because {{math|4''πr''{{i sup|2}}}} is the [[sphere#Surface area|surface area of the sphere]] of radius {{mvar|r}}, which reflects the geometry of the configuration. For details, see the articles ''[[Gauss's law#Relation to Coulomb's law|Relation between Gauss's law and Coulomb's law]]'' and ''[[Inverse-square law]]''.)

=== Unit of charge ===

A major difference between the Gaussian system and the ISQ is in the respective definitions of the quantity charge. In the ISQ, a separate base dimension, electric current, with the associated SI unit, the [[ampere]], is associated with electromagnetic phenomena, with the consequence that a unit of electrical charge (1&nbsp;[[coulomb]]&nbsp;= 1&nbsp;ampere&nbsp;× 1&nbsp;second) is a physical quantity that cannot be expressed purely in terms of the mechanical units (kilogram, metre, second). On the other hand, in the Gaussian system, the unit of electric charge (the [[statcoulomb]], statC) {{em|can}} be written entirely as a dimensional combination of the non-electrical base units (gram, centimetre, second), as:

{{block indent|1={{val|1|u=statC}} = {{val|1|u=g<sup>1/2</sup>⋅cm<sup>3/2</sup>⋅s<sup>−1</sup>}}.}}

For example, [[Coulomb's law]] in Gaussian units has no constant:
<math display="block">F = \frac{Q^{_\mathrm{G}}_1 Q^{_\mathrm{G}}_2}{r^2} ,</math>
where {{mvar|F}} is the repulsive force between two electrical charges, {{math|''Q''{{su|p={{small|G}}|b=1|lh=0.8em}}}} and {{math|''Q''{{su|p={{small|G}}|b=2|lh=0.8em}}}} are the two charges in question, and {{mvar|r}} is the distance separating them. If {{math|''Q''{{su|p={{small|G}}|b=1|lh=0.8em}}}} and {{math|''Q''{{su|p={{small|G}}|b=2|lh=0.8em}}}} are expressed in [[statC]] and {{mvar|r}} in [[centimetre]]s, then the unit of {{mvar|F}} that is coherent with these units is the [[dyne]].

The same law in the ISQ is:
<math display="block">F = \frac{1}{4\pi\varepsilon_0} \frac{Q^{_\mathrm{I}}_1 Q^{_\mathrm{I}}_2}{r^2}</math>
where {{math|''ε''{{sub|0}}}} is the [[vacuum permittivity]], a quantity that is not dimensionless: it has dimension ([[electric charge|charge]])<sup>2</sup> ([[time]])<sup>2</sup> ([[mass]])<sup>−1</sup> ([[length]])<sup>−3</sup>. Without {{math|''ε''{{sub|0}}}}, the equation would be dimensionally inconsistent with the quantities as defined in the ISQ, whereas the quantity {{math|''ε''{{sub|0}}}} does not appear in Gaussian equations.  This is an example of how some dimensional [[physical constant]]s can be eliminated from the expressions of [[physical law]] by the choice of definition of quantities.  In the ISQ, {{math|1/''ε''<sub>0</sub>}} converts or scales [[Electric displacement field|electric flux density]], {{math|'''D'''}}, to the corresponding [[electric field]], {{math|'''E'''}} (the latter has dimension of [[force]] per [[electric charge|charge]]), while in the Gaussian system, electric flux density is the same quantity as electric field strength in [[free space]] aside from a dimensionless constant factor.

In the Gaussian system, the [[speed of light]] {{mvar|c}} appears directly in electromagnetic formulas like [[Maxwell's equations]] (see below), whereas in the ISQ it appears via the product {{math|1=''μ''<sub>0</sub>''ε''<sub>0</sub> = 1/''c''<sup>2</sup>}}.

=== Units for magnetism ===

In the Gaussian system, unlike the ISQ, the electric field {{math|'''E'''{{ssup|G}}}} and the [[magnetic field]] {{math|'''B'''{{ssup|G}}}} have the same dimension. This amounts to a factor of [[speed of light|{{mvar|c}}]] between how {{math|'''B'''}} is defined in the two unit systems, on top of the other differences.<ref name=Littlejohn/> (The same factor applies to other magnetic quantities such as the [[magnetic field]], {{math|'''H'''}}, and [[magnetization]], {{math|'''M'''}}.) For example, in a [[Sinusoidal plane-wave solutions of the electromagnetic wave equation|planar light wave in vacuum]], {{math|1={{abs|'''E'''{{ssup|G}}('''r''', ''t'')}} = {{abs|'''B'''{{ssup|G}}('''r''', ''t'')}}}} in Gaussian units, while {{math|1={{abs|'''E'''{{ssup|I}}('''r''', ''t'')}} = ''c''&nbsp;{{abs|'''B'''{{ssup|I}}('''r''', ''t'')}}}} in the ISQ.

=== Polarization, magnetization ===

There are further differences between Gaussian system and the ISQ in how quantities related to polarization and magnetization are defined. For one thing, in the Gaussian system, ''all'' of the following quantities have the same dimension: {{math|'''[[electric field|E]]'''{{ssup|G}}}}, {{math|'''[[electric displacement field|D]]'''{{ssup|G}}}}, {{math|'''[[polarization density|P]]'''{{ssup|G}}}}, {{math|[[magnetic field|'''B''']]{{ssup|G}}}}, {{math|[[magnetic field|'''H''']]{{ssup|G}}}}, and {{math|'''[[magnetization|M]]'''{{ssup|G}}}}. A further point is that the [[electric susceptibility|electric]] and [[magnetic susceptibility]] of a material is dimensionless in both the Gaussian system and the ISQ, but a given material will have a different numerical susceptibility in the two systems. (Equation is given below.)

== List of equations ==

This section has a list of the basic formulae of electromagnetism, given in both the Gaussian system and the [[International System of Quantities|International System of Quantities (ISQ)]]. Most symbol names are not given; for complete explanations and definitions, please click to the appropriate dedicated article for each equation. A simple conversion scheme for use when tables are not available may be found in Garg (2012).<ref name=Garg>A. Garg, 2012, "Classical Electrodynamics in a Nutshell" (Princeton University Press).</ref>
All formulas except otherwise noted are from Ref.<ref name=Littlejohn/>

=== Maxwell's equations ===
{{main|Maxwell's equations}}

Here are Maxwell's equations, both in macroscopic and microscopic forms. Only the "differential form" of the equations is given, not the "integral form"; to get the integral forms apply the [[divergence theorem]] or [[Stokes' theorem|Kelvin–Stokes theorem]].

{| class="wikitable plainrowheaders"
|+ Maxwell's equations in Gaussian system and ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[Gauss's law]]{{br}}(macroscopic)
| <math>\nabla \cdot \mathbf{D}^{_\mathrm{G}} = 4\pi\rho_\mathrm{f}^{_\mathrm{G}}</math>
| <math>\nabla \cdot \mathbf{D}^{_\mathrm{I}} = \rho_\mathrm{f}^{_\mathrm{I}}</math>
|-
! scope="row" | [[Gauss's law]]<br />(microscopic)
| <math>\nabla \cdot \mathbf{E}^{_\mathrm{G}} = 4\pi\rho^{_\mathrm{G}}</math>
| <math>\nabla \cdot \mathbf{E}^{_\mathrm{I}} = \frac{1}{\varepsilon_0} \rho^{_\mathrm{I}}</math>
|-
! scope="row" | [[Gauss's law for magnetism]]
|<math>\nabla \cdot \mathbf{B}^{_\mathrm{G}} = 0</math>
|<math>\nabla \cdot \mathbf{B}^{_\mathrm{I}} = 0</math>
|-
! scope="row" | Maxwell–Faraday equation<br />([[Faraday's law of induction]])
| <math>\nabla \times \mathbf{E}^{_\mathrm{G}} + \frac{1}{c}\frac{\partial \mathbf{B}^{_\mathrm{G}}} {\partial t} = 0</math>
| <math>\nabla \times \mathbf{E}^{_\mathrm{I}} + \frac{\partial \mathbf{B}^{_\mathrm{I}}} {\partial t} = 0</math>
|-
! scope="row" | [[Ampère–Maxwell equation]]<br /> (macroscopic)
| <math>\nabla \times \mathbf{H}^{_\mathrm{G}} - \frac{1}{c} \frac{\partial \mathbf{D}^{_\mathrm{G}}} {\partial t} = \frac{4\pi}{c}\mathbf{J}_\mathrm{f}^{_\mathrm{G}}</math>
| <math>\nabla \times \mathbf{H}^{_\mathrm{I}} - \frac{\partial \mathbf{D}^{_\mathrm{I}}} {\partial t}= \mathbf{J}_\mathrm{f}^{_\mathrm{I}}</math>
|-
! scope="row" | [[Ampère–Maxwell equation]]<br /> (microscopic)
| <math>\nabla \times \mathbf{B}^{_\mathrm{G}} - \frac{1}{c}\frac{\partial \mathbf{E}^{_\mathrm{G}}} {\partial t} = \frac{4\pi}{c}\mathbf{J}^{_\mathrm{G}}</math>
| <math>\nabla \times \mathbf{B}^{_\mathrm{I}} - \frac{1}{c^2}\frac{\partial \mathbf{E}^{_\mathrm{I}}} {\partial t} = \mu_0\mathbf{J}^{_\mathrm{I}}</math>
|}

=== Other basic laws ===

{| class="wikitable plainrowheaders"
|+ Other electromagnetic laws in Gaussian system and ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[Lorentz force]]
| <math>\mathbf{F} = q^{_\mathrm{G}}\,\left(\mathbf{E}^{_\mathrm{G}}+\tfrac{1}{c}\,\mathbf{v}\times\mathbf{B}^{_\mathrm{G}}\right)</math>
| <math>\mathbf{F} = q^{_\mathrm{I}}\,\left(\mathbf{E}^{_\mathrm{I}}+\mathbf{v}\times\mathbf{B}^{_\mathrm{I}}\right)</math>
|-
! scope="row" | [[Coulomb's law]]
| <math>\mathbf{F} = \frac{q^{_\mathrm{G}}_1 q^{_\mathrm{G}}_2}{r^2}\,\mathbf{\hat r}</math>
| <math>\mathbf{F} = \frac{1}{4\pi\varepsilon_0}\,\frac{q^{_\mathrm{I}}_1 q^{_\mathrm{I}}_2}{r^2}\, \mathbf{\hat r}</math> <br />
|-
! scope="row" | Electric field of<br />[[Coulomb's law|stationary point charge]]
| <math>\mathbf{E}^{_\mathrm{G}} = \frac{q^{_\mathrm{G}}}{r^2}\,\mathbf{\hat r}</math>
| <math>\mathbf{E}^{_\mathrm{I}} = \frac{1}{4\pi\varepsilon_0}\,\frac{q^{_\mathrm{I}}}{r^2}\,\mathbf{\hat r}</math>
|-
! scope="row" | [[Biot–Savart law]]<ref>[https://books.google.com/books?id=RvZFuqw6mXEC&pg=PA180 Introduction to Electrodynamics by Capri and Panat, p180]</ref>
| <math> \mathbf{B}^{_\mathrm{G}} = \frac{1}{c}\!\oint\frac{I^{_\mathrm{G}} \times \mathbf{\hat r}}{r^2}\,\operatorname{d}\!\mathbf{\boldsymbol{\ell}}</math>
| <math> \mathbf{B}^{_\mathrm{I}} = \frac{\mu_0}{4\pi}\!\oint\frac{I^{_\mathrm{I}} \times \mathbf{\hat r}}{r^2}\,\operatorname{d}\!\mathbf{\boldsymbol{\ell}}</math>
|-
! scope="row" | [[Poynting vector]]<br />(microscopic)
| <math>\mathbf{S} = \frac{c}{4\pi}\,\mathbf{E}^{_\mathrm{G}} \times \mathbf{B}^{_\mathrm{G}}</math>
| <math>\mathbf{S} = \frac{1}{\mu_0}\,\mathbf{E}^{_\mathrm{I}} \times \mathbf{B}^{_\mathrm{I}}</math>
|}

=== Dielectric and magnetic materials ===

Below are the expressions for the various fields in a dielectric medium. It is assumed here for simplicity that the medium is homogeneous, linear, isotropic, and nondispersive, so that the [[permittivity]] is a simple constant.

{| class="wikitable"
|+ Expressions for fields in dielectric media
|-
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
| <math>\mathbf{D}^{_\mathrm{G}} = \mathbf{E}^{_\mathrm{G}}+4\pi\mathbf{P}^{_\mathrm{G}}</math>
| <math>\mathbf{D}^{_\mathrm{I}} = \varepsilon_0 \mathbf{E}^{_\mathrm{I}}+\mathbf{P}^{_\mathrm{I}}</math>
|-
| <math>\mathbf{P}^{_\mathrm{G}} = \chi^{_\mathrm{G}}_\mathrm{e}\mathbf{E}^{_\mathrm{G}}</math>
| <math>\mathbf{P}^{_\mathrm{I}} = \chi^{_\mathrm{I}}_\mathrm{e}\varepsilon_0\mathbf{E}^{_\mathrm{I}}</math>
|-
| <math>\mathbf{D}^{_\mathrm{G}} = \varepsilon^{_\mathrm{G}}\mathbf{E}^{_\mathrm{G}}</math>
| <math>\mathbf{D}^{_\mathrm{I}} = \varepsilon^{_\mathrm{I}}\mathbf{E}^{_\mathrm{I}}</math>
|-
| <math>\varepsilon^{_\mathrm{G}} = 1+4\pi\chi^{_\mathrm{G}}_\mathrm{e}</math>
| <math>\varepsilon^{_\mathrm{I}}/\varepsilon_0 = 1+\chi^{_\mathrm{I}}_\mathrm{e}</math>
|}
where
* {{math|'''E'''}} and {{math|'''D'''}} are the [[electric field]] and [[Electric displacement field|displacement field]], respectively;
* {{math|'''P'''}} is the [[polarization density]];
* <math>\varepsilon</math> is the [[permittivity]];
* <math>\varepsilon_0</math> is the [[permittivity of vacuum]] (used in the SI system, but meaningless in Gaussian units); and
* <math>\chi_\mathrm{e}</math> is the [[electric susceptibility]].

The quantities <math>\varepsilon^{_\mathrm{G}}</math> and <math>\varepsilon^{_\mathrm{I}}/\varepsilon_0</math> are both dimensionless, and they have the same numeric value. By contrast, the [[electric susceptibility]] <math>\chi_\mathrm{e}^{_\mathrm{G}}</math> and <math>\chi_\mathrm{e}^{_\mathrm{I}}</math> are both unitless, but have {{em|different numeric values}} for the same material:
<math display="block">4\pi \chi_\mathrm{e}^{_\mathrm{G}} = \chi_\mathrm{e}^{_\mathrm{I}}\,.</math>

Next, here are the expressions for the various fields in a magnetic medium. Again, it is assumed that the medium is homogeneous, linear, isotropic, and nondispersive, so that the [[Permeability (electromagnetism)|permeability]] is a simple constant.

{| class="wikitable"
|+ Expressions for fields in magnetic media
|-
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
| <math>\mathbf{B}^{_\mathrm{G}} = \mathbf{H}^{_\mathrm{G}}+4\pi\mathbf{M}^{_\mathrm{G}}</math>
| <math>\mathbf{B}^{_\mathrm{I}} = \mu_0 (\mathbf{H}^{_\mathrm{I}}+\mathbf{M}^{_\mathrm{I}})</math>
|-
| <math>\mathbf{M}^{_\mathrm{G}} = \chi^{_\mathrm{G}}_\mathrm{m}\mathbf{H}^{_\mathrm{G}}</math>
| <math>\mathbf{M}^{_\mathrm{I}} = \chi^{_\mathrm{I}}_\mathrm{m}\mathbf{H}^{_\mathrm{I}}</math>
|-
| <math>\mathbf{B}^{_\mathrm{G}} = \mu^{_\mathrm{G}}\mathbf{H}^{_\mathrm{G}}</math>
| <math>\mathbf{B}^{_\mathrm{I}} = \mu^{_\mathrm{I}}\mathbf{H}^{_\mathrm{I}}</math>
|-
| <math>\mu^{_\mathrm{G}} = 1+4\pi\chi^{_\mathrm{G}}_\mathrm{m}</math>
| <math>\mu^{_\mathrm{I}}/\mu_0 = 1+\chi^{_\mathrm{I}}_\mathrm{m}</math>
|}
where
* {{math|'''B'''}} and {{math|'''H'''}} are the [[magnetic field]]s;
* {{math|'''M'''}} is [[magnetization]];
* <math>\mu</math> is [[magnetic permeability]];
* <math>\mu_0</math> is the [[permeability of vacuum]] (used in the SI system, but meaningless in Gaussian units); and
* <math>\chi_\mathrm{m}</math> is the [[magnetic susceptibility]].

The quantities <math>\mu^{_\mathrm{G}}</math> and <math>\mu^{_\mathrm{I}}/\mu_0</math> are both dimensionless, and they have the same numeric value. By contrast, the [[magnetic susceptibility]] <math>\chi_\mathrm{m}^{_\mathrm{G}}</math> and <math>\chi_\mathrm{m}^{_\mathrm{I}}</math> are both unitless, but has {{em|different numeric values}} in the two systems for the same material:
<math display="block">4\pi \chi_\mathrm{m}^{_\mathrm{G}} = \chi_\mathrm{m}^{_\mathrm{I}}</math>

=== Vector and scalar potentials ===
{{main|Magnetic vector potential|Electric potential}}

The electric and magnetic fields can be written in terms of a vector potential {{math|'''A'''}} and a scalar potential {{mvar|ϕ}}:

{| class="wikitable plainrowheaders"
|+ Electromagnetic fields in Gaussian system and ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[Electric field]]
| <math>\mathbf{E}^{_\mathrm{G}} = -\nabla\phi^{_\mathrm{G}}-\frac{1}{c}\frac{\partial \mathbf{A}^{_\mathrm{G}}}{\partial t}</math>
| <math>\mathbf{E}^{_\mathrm{I}} = -\nabla\phi^{_\mathrm{I}}-\frac{\partial \mathbf{A}^{_\mathrm{I}}}{\partial t}</math>
|-
! scope="row" | [[Magnetic field|Magnetic '''B''' field]]
| <math>\mathbf{B}^{_\mathrm{G}} = \nabla \times \mathbf{A}^{_\mathrm{G}}</math>
| <math>\mathbf{B}^{_\mathrm{I}} = \nabla \times \mathbf{A}^{_\mathrm{I}}</math>
|}

=== Electrical circuit ===

{| class="wikitable plainrowheaders"
|+ Electrical circuit values in Gaussian system and ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[Charge conservation]]
| <math>I^{_\mathrm{G}} = \frac{\mathrm{d}Q^{_\mathrm{G}}}{\mathrm{d}t}</math>
| <math>I^{_\mathrm{I}} = \frac{\mathrm{d}Q^{_\mathrm{I}}}{\mathrm{d}t}</math>
|-
! scope="row" | [[Lenz's law]]
| <math>V^{_\mathrm{G}} = \frac{1}{c}\frac{\mathrm{d}\mathrm{\Phi}^{_\mathrm{G}}}{\mathrm{d}t}</math>
| <math>V^{_\mathrm{I}} = -\frac{\mathrm{d}\mathrm{\Phi}^{_\mathrm{I}}}{\mathrm{d}t}</math>
|-
! scope="row" | [[Ohm's law]]
| <math>V^{_\mathrm{G}} = R^{_\mathrm{G}} I^{_\mathrm{G}}</math>
| <math>V^{_\mathrm{I}} = R^{_\mathrm{I}} I^{_\mathrm{I}}</math>
|-
! scope="row" | [[Capacitance]]
| <math>Q^{_\mathrm{G}} = C^{_\mathrm{G}} V^{_\mathrm{G}}</math>
| <math>Q^{_\mathrm{I}} = C^{_\mathrm{I}} V^{_\mathrm{I}}</math>
|-
! scope="row" | [[Inductance]]
| <math>\mathrm{\Phi}^{_\mathrm{G}} = cL^{_\mathrm{G}} I^{_\mathrm{G}}</math>
| <math>\mathrm{\Phi}^{_\mathrm{I}} = L^{_\mathrm{I}} I^{_\mathrm{I}}</math>
|-
|}
where
* {{mvar|Q}} is the [[electric charge]]
* {{mvar|I}} is the [[electric current]]
* {{mvar|V}} is the [[electric potential]]
* {{math|&Phi;}} is the [[magnetic flux]]
* {{mvar|R}} is the [[electrical resistance]]
* {{mvar|C}} is the [[capacitance]]
* {{mvar|L}} is the [[inductance]]

=== Fundamental constants ===

{| class="wikitable plainrowheaders"
|+ Fundamental constants in Gaussian system and ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[Impedance of free space]]
| <math>Z_0^{_\mathrm{G}} = \frac{4\pi}{c}</math>
| <math>Z_0^{_\mathrm{I}} = \sqrt{\frac{\mu_0}{\varepsilon_0}}</math>
|-
! scope="row" | [[Electric constant]]
| <math>1 = \frac{4\pi}{Z_0^{_\mathrm{G}}c}</math>
| <math>\varepsilon_0 = \frac{1}{Z_0^{_\mathrm{I}}c}</math>
|-
! scope="row" | [[Magnetic constant]]
| <math>1 = \frac{Z_0^{_\mathrm{G}}c}{4\pi}</math>
| <math>\mu_0 = \frac{Z_0^{_\mathrm{I}}}{c}</math>
|-
! scope="row" | [[Fine-structure constant]]
| <math>\alpha = \frac{(e^{_\mathrm{G}})^2}{\hbar c}</math>
| <math>\alpha = \frac{1}{4\pi\varepsilon_0} \frac{(e^{_\mathrm{I}})^2}{\hbar c}</math>
|-
! scope="row" | [[Magnetic flux quantum]]
| <math>\phi_0^{_\mathrm{G}} = \frac{hc}{2e^{_\mathrm{G}}}</math>
| <math>\phi_0^{_\mathrm{I}} = \frac{h}{2e^{_\mathrm{I}}}</math>
|-
! scope="row" | [[Conductance quantum]]
| <math>G_0^{_\mathrm{G}} = \frac{2(e^{_\mathrm{G}})^2}{h}</math>
| <math>G_0^{_\mathrm{I}} = \frac{2(e^{_\mathrm{I}})^2}{h}</math>
|-
! scope="row" | [[Bohr radius]]
| <math>a_\mathrm{B} =\frac{\hbar^2}{m_\mathrm{e}(e^{_\mathrm{G}})^2}</math>
| <math>a_\mathrm{B} =\frac{4\pi\varepsilon_0\hbar^2}{m_\mathrm{e}(e^{_\mathrm{I}})^2}</math>
|-
! scope="row" | [[Bohr magneton]]
| <math>\mu_\mathrm{B}^{_\mathrm{G}} =\frac{e^{_\mathrm{G}}\hbar}{2m_\mathrm{e}c}</math>
| <math>\mu_\mathrm{B}^{_\mathrm{I}} =\frac{e^{_\mathrm{I}}\hbar}{2m_\mathrm{e}}</math>
|}

== Electromagnetic unit names ==
{{for|non-electromagnetic units|Centimetre–gram–second system of units}}

{{Table alignment}}
{| class="wikitable plainrowheaders defaultcenter col5left"
|+ Table 1: Common electromagnetism units in SI vs Gaussian<ref name=cardsgc>
{{cite book
 | last = Cardarelli | first = F.
 | date = 2004
 | title = Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins
 | publisher = Springer
 | edition = 2nd
 | pages = [https://archive.org/details/encyclopaediaofs0000card/page/20 20–25]
 | isbn= 978-1-85233-682-0
 | url= https://archive.org/details/encyclopaediaofs0000card
 | url-access = registration
}}</ref>
! scope="col" | Quantity
! scope="col" | Symbol
! scope="col" | SI unit
! scope="col" | Gaussian unit{{br}}(in base units)
! scope="col" | Conversion factor
|-
! scope="row" | [[Electric charge]] 
| {{mvar|q}}
| [[Coulomb|C]]
| [[Statcoulomb|Fr]]{{br}}(cm<sup>3/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>)
| <math>\frac{q^{_\mathrm{G}}}{q^{_\mathrm{I}}} = \frac{1}{\sqrt{4\pi\varepsilon_0}} \approx \frac{2.998 \times 10^9 \, \mathrm{Fr}}{1\, \mathrm{C}}</math>
|-
! scope="row" | [[Electric current]] 
| {{mvar|I}}
| [[Ampere|A]]
| [[Statampere|statA]]{{br}}(cm<sup>3/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−2</sup>)
| <math>\frac{I^{_\mathrm{G}}}{I^{_\mathrm{I}}} = \frac{1}{\sqrt{4\pi\varepsilon_0}} \approx \frac{2.998 \times 10^9 \, \mathrm{statA}}{1\, \mathrm{A}}</math>
|-
! scope="row" | [[Electric potential]],{{br}}[[Voltage]] 
| {{mvar|φ}}{{br}}{{mvar|V}}
| [[Volt|V]]
| [[statvolt|statV]]{{br}}(cm<sup>1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>)
| <math>\frac{V^{_\mathrm{G}}}{V^{_\mathrm{I}}} = \sqrt{4\pi\varepsilon_0} \approx \frac{1\, \mathrm{statV}}{2.998 \times 10^2 \, \mathrm{V}}</math>
|-
! scope="row" | [[Electric field]] 
| {{math|'''E'''}}
| [[Volt|V]]/[[Metre|m]]
| [[statvolt|statV]]/[[Centimetre|cm]]{{br}}(cm<sup>−1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>)
| <math>\frac{\mathbf{E}^{_\mathrm{G}}}{\mathbf{E}^{_\mathrm{I}}} = \sqrt{4\pi\varepsilon_0} \approx \frac{1 \, \mathrm{statV/cm}}{2.998 \times 10^4 \, \mathrm{V/m}}</math>
|-
! scope="row" | [[Electric displacement field]] 
| {{math|'''D'''}}
| [[Coulomb|C]]/[[Square metre|m<sup>2</sup>]]
| [[statcoulomb|Fr]]/[[Centimetre|cm]]<sup>2</sup>{{br}}(cm<sup>−1/2</sup>g<sup>1/2</sup>s<sup>−1</sup>)
| <math>\frac{\mathbf{D}^{_\mathrm{G}}}{\mathbf{D}^{_\mathrm{I}}} = \sqrt{\frac{4\pi}{\varepsilon_0}} \approx \frac{4\pi\times 2.998 \times 10^5 \, \mathrm{Fr/cm}^2}{ 1 \, \mathrm{C/m}^2}</math>
|-
! scope="row" | [[Electric dipole moment]] 
| {{math|'''p'''}}
| [[Coulomb|C]]⋅[[Metre|m]]
| [[Statcoulomb|Fr]]⋅[[Centimetre|cm]]{{br}}(cm<sup>5/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>)
| <math>\frac{\mathbf{p}^{_\mathrm{G}}}{\mathbf{p}^{_\mathrm{I}}} = \frac{1}{\sqrt{4\pi\varepsilon_0}} \approx \frac{2.998 \times 10^{11} \, \mathrm{Fr}{\cdot}\mathrm{cm}}{1 \, \mathrm{C}{\cdot}\mathrm{m}}</math>
|-
! scope="row" | [[Electric flux]]{{efn|The quantity here is the flux of the {{em|displacement field}} ({{math|'''D'''}}), not the electric field ({{math|'''E'''}}).}}
| {{math|Φ<sub>D</sub>}}
| [[Coulomb|C]]
| [[Statcoulomb|Fr]]{{br}}(cm<sup>3/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>)
| <math>\frac{\Phi^{_\mathrm{G}}_{\mathrm{e}}}{\Phi^{_\mathrm{I}}_{\mathrm{e}}} = \sqrt{\frac{4\pi}{\varepsilon_0}} \approx \frac{4\pi\times 2.998 \times 10^9 \, \mathrm{Fr}}{1 \, \mathrm{C}}</math> 
|-
! scope="row" | [[Permittivity]]
| {{mvar|ε}}
| [[Farad|F]]/[[Metre|m]]
| [[Centimetre|cm]]/cm
| <math>\frac{\varepsilon^{_\mathrm{G}}}{\varepsilon^{_\mathrm{I}}} = \frac{1}{\varepsilon_0} \approx \frac{4\pi \times 2.998^2 \times 10^{9} \, \mathrm{cm/cm}}{1 \, \mathrm{F/m}}</math>
|-
! scope="row" | [[Magnetic field|Magnetic '''B''' field]] 
| {{math|'''B'''}}
| [[tesla (unit)|T]]
| [[Gauss (unit)|G]]{{br}}(cm<sup>−1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>)
| <math>\frac{\mathbf{B}^{_\mathrm{G}}}{\mathbf{B}^{_\mathrm{I}}} = \sqrt{\frac{4\pi}{\mu_0}} \approx \frac{10^4 \, \mathrm{G}}{1 \, \mathrm{T}}</math>
|-
! scope="row" | [[magnetic field|Magnetic '''H''' field]] 
| {{math|'''H'''}}
| [[Ampere|A]]/[[Metre|m]]
| [[oersted|Oe]]{{br}}(cm<sup>−1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>)
| <math>\frac{\mathbf{H}^{_\mathrm{G}}}{\mathbf{H}^{_\mathrm{I}}} = \sqrt{4\pi\mu_0} \approx \frac{4\pi \times 10^{-3} \, \mathrm{Oe}}{1 \, \mathrm{A/m}}</math>
|-
! scope="row" | [[Magnetic dipole moment]] 
| {{math|'''m'''}}
| [[Ampere|A]]⋅[[Square metre|m<sup>2</sup>]]
| [[erg]]/[[Gauss (unit)|G]]{{br}}(cm<sup>5/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>)
| <math>\frac{\mathbf{m}^{_\mathrm{G}}}{\mathbf{m}^{_\mathrm{I}}} = \sqrt{\frac{\mu_0}{4\pi}} \approx \frac{10^3 \, \mathrm{erg/G}}{1 \, \mathrm{A}{\cdot}\mathrm{m}^2}</math>
|-
! scope="row" | [[Magnetic flux]]
| {{math|Φ<sub>m</sub>}}
| [[Weber (unit)|Wb]]
| [[Maxwell (unit)|Mx]]{{br}}(cm<sup>3/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>)
| <math>\frac{\Phi^{_\mathrm{G}}_{\mathrm{m}}}{\Phi^{_\mathrm{I}}_{\mathrm{m}}} = \sqrt{\frac{4\pi}{\mu_0}} \approx \frac{10^8 \, \mathrm{Mx}}{1 \, \mathrm{Wb}}</math> 
|-
! scope="row" | [[Permeability (electromagnetism)|Permeability]]
| {{mvar|μ}}
| [[Henry (unit)|H]]/[[Metre|m]]
| [[Centimetre|cm]]/cm
| <math>\frac{\mu^{_\mathrm{G}}}{\mu^{_\mathrm{I}}} = \frac{1}{\mu_0} \approx \frac{1 \, \mathrm{cm/cm}}{4\pi \times 10^{-7} \, \mathrm{H/m}}</math> 
|-
! scope="row" | [[Magnetomotive force]] 
| <math>\mathcal F</math>
| [[Ampere|A]]
| [[Gilbert (unit)|Gi]]{{br}}(cm<sup>1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>)
| <math>\frac{\mathcal F^{_\mathrm{G}}}{\mathcal F^{_\mathrm{I}}} = \sqrt{4\pi\mu_0} \approx \frac{4\pi \times 10^{-1} \, \mathrm{Gi}}{1 \, \mathrm{A}}</math>
|-
! scope="row" | [[Magnetic reluctance]] 
| <math>\mathcal R</math>
| [[Henry (unit)|H]]<sup>−1</sup>
| [[Gilbert (unit)|Gi]]/[[Maxwell (unit)|Mx]]{{br}}(cm<sup>−1</sup>)
| <math>\frac{\mathcal R^{_\mathrm{G}}}{\mathcal R^{_\mathrm{I}}} = \mu_0 \approx \frac{4\pi \times 10^{-9} \, \mathrm{Gi/Mx}}{1 \, \mathrm{H}^{-1}}</math>
|-
! scope="row" | [[Electrical resistance|Resistance]]
| {{mvar|R}}
| [[Ohm|Ω]]
| [[Second|s]]/[[Centimetre|cm]]
| <math>\frac{R^{_\mathrm{G}}}{R^{_\mathrm{I}}} = 4\pi\varepsilon_0 \approx \frac{1 \, \mathrm{s/cm}}{2.998^2 \times 10^{11}  \, \Omega}</math>
|-
! scope="row" | [[Electrical resistivity|Resistivity]]
| {{mvar|ρ}}
| [[Ohm|Ω]]⋅[[Metre|m]]
| [[Second|s]]
| <math>\frac{\rho^{_\mathrm{G}}}{\rho^{_\mathrm{I}}} = 4\pi\varepsilon_0 \approx \frac{1 \, \mathrm{s}}{2.998^2 \times 10^{9}  \, \Omega{\cdot}\mathrm{m}}</math>
|-
! scope="row" | [[Capacitance]]
| {{mvar|C}}
| [[Farad|F]]
| [[Centimetre|cm]]
| <math>\frac{C^{_\mathrm{G}}}{C^{_\mathrm{I}}} = \frac{1}{4\pi\varepsilon_0} \approx \frac{2.998^2 \times 10^{11} \, \mathrm{cm}}{1 \, \mathrm{F}}</math>
|-
! scope="row" | [[Inductance]]
| {{mvar|L}}
| [[Henry (unit)|H]]
| [[Second|s]]<sup>2</sup>/[[Centimetre|cm]]
| <math>\frac{L^{_\mathrm{G}}}{L^{_\mathrm{I}}} = 4\pi\varepsilon_0 \approx \frac{1 \, \mathrm{s}^2/\mathrm{cm}}{2.998^2 \times 10^{11} \, \mathrm{H}}</math> 
|}
'''Note''': The SI quantities <math>\varepsilon_0</math> and <math>\mu_0</math> satisfy {{tmath|1= \varepsilon_0\mu_0 = 1/c^2 }}.

The conversion factors are written both symbolically and numerically. The numerical conversion factors can be derived from the symbolic conversion factors by [[dimensional analysis]]. For example, the top row says {{nowrap|<math>{1} \,/\, {\sqrt{4\pi\varepsilon_0}} \approx {2.998 \times 10^9 \,\mathrm{Fr}} \,/\, {1\,\mathrm{C}}</math>,}} a relation which can be verified with dimensional analysis, by expanding <math>\varepsilon_0</math> and [[coulomb]]s (C) in [[SI base units]], and expanding [[statcoulomb]]s (or franklins, Fr) in Gaussian base units.

It is surprising to think of measuring capacitance in centimetres. One useful example is that a centimetre of capacitance is the capacitance between a sphere of radius 1&nbsp;cm in vacuum and infinity.

Another surprising unit is measuring [[resistivity]] in units of seconds. A physical example is: Take a [[parallel-plate capacitor]], which has a "leaky" dielectric with permittivity 1 but a finite resistivity. After charging it up, the capacitor will discharge itself over time, due to current leaking through the dielectric. If the resistivity of the dielectric is {{mvar|t}} seconds, the half-life of the discharge is {{math|~0.05 ''t''}} seconds.<!-- 0.05 ~ ln(2)/(4pi) --> This result is independent of the size, shape, and charge of the capacitor, and therefore this example illuminates the fundamental connection between resistivity and time units.

=== Dimensionally equivalent units ===

A number of the units defined by the table have different names but are in fact dimensionally equivalent – i.e., they have the same expression in terms of the base units cm, g, s. (This is analogous to the distinction in SI between [[newton-metre]] and [[joule]].) The different names help avoid ambiguities and misunderstandings as to what physical quantity is being measured. In particular, {{em|all}} of the following quantities are dimensionally equivalent in Gaussian units, but they are nevertheless given different unit names as follows:<ref>{{cite book|url=https://books.google.com/books?id=CQMsK5xW1DcC&pg=PA155 |title=Demystifying Electromagnetic Equations|page=155|access-date=2012-12-25|isbn=9780819442345|last1=Cohen|first1=Douglas L.|year=2001|publisher=SPIE Press }}</ref>

{| class="wikitable plainrowheaders" style="text-align: center;"
|+ Dimensionally equivalent units
|-
! scope="col" | Quantity
! scope="col" | Gaussian symbol
! scope="col" | In Gaussian<br />base units
! scope="col" | Gaussian unit<br />of measure
|-
! scope="row" | [[Electric field]]
| {{math|'''E'''{{ssup|G}}}}
| cm<sup>−1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>
| [[statV]]/cm
|-
! scope="row" | [[Electric displacement field]]
| {{math|'''D'''{{ssup|G}}}}
| cm<sup>−1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>
| [[statC]]/cm<sup>2</sup>
|-
! scope="row" | [[Polarization density]]
| {{math|'''P'''{{ssup|G}}}}
| cm<sup>−1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>
| [[statC]]/cm<sup>2</sup>
|-
! scope="row" | [[Magnetic field#The B-field|Magnetic flux density]]
| {{math|'''B'''{{ssup|G}}}}
| cm<sup>−1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>
| [[Gauss (unit)|G]]
|-
! scope="row" | [[Magnetic field#The H-field|Magnetizing field]]
| {{math|'''H'''{{ssup|G}}}}
| cm<sup>−1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>
| [[Oersted|Oe]]
|-
! scope="row" | [[Magnetization]]
| {{math|'''M'''{{ssup|G}}}}
| cm<sup>−1/2</sup>⋅g<sup>1/2</sup>⋅s<sup>−1</sup>
| [[Dyne|dyn]]/[[Maxwell (unit)|Mx]]
|}

== General rules to translate a formula ==

Any formula can be converted between Gaussian and SI units by using the symbolic conversion factors from Table 1 above.

For example, the [[Coulomb's law|electric field of a stationary point charge]] has the ISQ formula
<math display="block">\mathbf{E}^{_\mathrm{I}} = \frac{q^{_\mathrm{I}}}{4\pi \varepsilon_0 r^2} \hat{\mathbf{r}} ,</math>
where {{mvar|r}} is distance, and the "{{smaller|{{math|I}}}}" superscript indicates that the electric field and charge are defined as in the ISQ. If we want the formula to instead use the Gaussian definitions of electric field and charge, we look up how these are related using Table 1, which says:
<math display="block">\begin{align}
\frac{\mathbf{E}^{_\mathrm{G}}}{\mathbf{E}^{_\mathrm{I}}} &= \sqrt{4\pi\varepsilon_0}\,, \\
\frac{q^{_\mathrm{G}}}{q^{_\mathrm{I}}} &= \frac{1}{\sqrt{4\pi\varepsilon_0}}\,.
\end{align}</math>

Therefore, after substituting and simplifying, we get the Gaussian-system formula:
<math display="block">\mathbf{E}^{_\mathrm{G}} = \frac{q^{_\mathrm{G}}}{r^2}\hat{\mathbf{r}}\,,</math>
which is the correct Gaussian-system formula, as mentioned in a previous section.

For convenience, the table below has a compilation of the symbolic conversion factors from Table 1. To convert any formula from the Gaussian system to the ISQ using this table, replace each symbol in the Gaussian column by the corresponding expression in the SI column (vice versa to convert the other way). Replace <math>1/c^2</math> by <math>\varepsilon_0 \mu_0</math> (or vice versa). This will reproduce any of the specific formulas given in the list above, such as Maxwell's equations, as well as any other formula not listed.<ref>{{cite book |last1= Бредов |first1= М. М. |last2= Румянцев |first2= В. В. |last3= Топтыгин |first3= И. Н. |date= 1985 |title= Классическая электродинамика |trans-title= Classical Electrodynamics |lang= ru |publisher=[[Nauka (publisher)|Nauka]] |chapter= Appendix 5: Units transform |page= 385}}</ref><ref>{{cite web |last1=Simpson |first1=David |title=SI / Gaussian Formula Conversion Table |url=http://www.pgccphy.net/ref/gaussian-conv.pdf |website=Prince George's Community College |access-date=23 February 2024}}</ref><ref name="Jackson">{{cite book |last1=Jackson |first1=John |title=Classical Electrodynamics |date=14 August 1998 |publisher=John Wiley & Sons, Inc. |isbn=0-471-30932-X |page=782 |edition=3}}</ref>{{efn|For some examples of how to use this table, see: [http://www.qsl.net/g4cnn/units/units.htm Units in Electricity and Magnetism]. See the section "Conversion of Gaussian formulae into SI" and the subsequent text.}}

{| class="wikitable plainrowheaders"
|+ Table 2A: Replacement rules for translating formulas from Gaussian to ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[electric field]], [[electric potential]], [[electromotive force]]
| <math> \left(\mathbf{E}^{_\mathrm{G}}, \varphi^{_\mathrm{G}}, \mathcal E^{_\mathrm{G}}\right) </math>
| <math> \sqrt{4\pi\varepsilon_0}\left(\mathbf{E}^{_\mathrm{I}}, \varphi^{_\mathrm{I}}, \mathcal E^{_\mathrm{I}}\right) </math>
|-
! scope="row" | [[electric displacement field]]
| <math> \mathbf{D}^{_\mathrm{G}} </math>
| <math> \sqrt{\frac{4\pi}{\varepsilon_0}}\mathbf{D}^{_\mathrm{I}} </math>
|-
! scope="row" | [[charge (physics)|charge]], [[charge density]], [[Electric current|current]],<br />[[current density]], [[polarization density]],<br />[[electric dipole moment]]
| <math> \left(q^{_\mathrm{G}}, \rho^{_\mathrm{G}}, I^{_\mathrm{G}}, \mathbf{J}^{_\mathrm{G}},\mathbf{P}^{_\mathrm{G}}, \mathbf{p}^{_\mathrm{G}}\right) </math>
| <math> \frac{1}{\sqrt{4\pi\varepsilon_0}}\left(q^{_\mathrm{I}}, \rho^{_\mathrm{I}}, I^{_\mathrm{I}}, \mathbf{J}^{_\mathrm{I}},\mathbf{P}^{_\mathrm{I}},\mathbf{p}^{_\mathrm{I}}\right) </math>
|-
! scope="row" | [[Magnetic field|magnetic {{math|'''B'''}} field]], [[magnetic flux]],<br />[[magnetic vector potential]]
| <math> \left(\mathbf{B}^{_\mathrm{G}}, \Phi_\mathrm{m}^{_\mathrm{G}},\mathbf{A}^{_\mathrm{G}}\right) </math>
| <math> \sqrt{\frac{4\pi}{\mu_0}}\left(\mathbf{B}^{_\mathrm{I}}, \Phi_\mathrm{m}^{_\mathrm{I}},\mathbf{A}^{_\mathrm{I}}\right) </math>
|-
! scope="row" | [[Magnetic field|magnetic {{math|'''H'''}} field]], [[magnetic scalar potential]], [[magnetomotive force]]
| <math> \left(\mathbf{H}^{_\mathrm{G}}, \psi^{_\mathrm{G}}, \mathcal F^{_\mathrm{G}}\right) </math>
| <math> \sqrt{4\pi\mu_0}\left(\mathbf{H}^{_\mathrm{I}}, \psi^{_\mathrm{I}}, \mathcal F^{_\mathrm{I}}\right) </math>
|-
! scope="row" | [[magnetic moment]], [[magnetization]], [[magnetic pole strength]]
| <math> \left(\mathbf{m}^{_\mathrm{G}}, \mathbf{M}^{_\mathrm{G}}, p^{_\mathrm{G}}\right) </math>
| <math> \sqrt{\frac{\mu_0}{4\pi}}\left(\mathbf{m}^{_\mathrm{I}}, \mathbf{M}^{_\mathrm{I}}, p^{_\mathrm{I}}\right) </math>
|-
! scope="row" | [[permittivity]],<br />[[permeability (electromagnetism)|permeability]]
| <math> \left(\varepsilon^{_\mathrm{G}}, \mu^{_\mathrm{G}}\right) </math>
| <math> \left(\frac{\varepsilon^{_\mathrm{I}}}{\varepsilon_0}, \frac{\mu^{_\mathrm{I}}}{\mu_0}\right) </math>
|-
! scope="row" | [[electric susceptibility]],<br />[[magnetic susceptibility]]
| <math> \left(\chi_\mathrm{e}^{_\mathrm{G}}, \chi_\mathrm{m}^{_\mathrm{G}}\right) </math>
| <math> \frac{1}{4\pi}\left(\chi_\mathrm{e}^{_\mathrm{I}}, \chi_\mathrm{m}^{_\mathrm{I}}\right) </math>
|-
! scope="row" | [[Electrical conductivity|conductivity]], [[Electrical conductance|conductance]], [[capacitance]]
| <math> \left(\sigma^{_\mathrm{G}}, S^{_\mathrm{G}}, C^{_\mathrm{G}}\right) </math>
| <math> \frac{1}{4\pi\varepsilon_0}\left(\sigma^{_\mathrm{I}},S^{_\mathrm{I}},C^{_\mathrm{I}}\right) </math>
|-
! scope="row" | [[resistivity]], [[electrical resistance|resistance]], [[inductance]], [[memristance]], [[electrical impedance|impedance]]
| <math> \left(\rho^{_\mathrm{G}},R^{_\mathrm{G}},L^{_\mathrm{G}},M^{_\mathrm{G}},Z^{_\mathrm{G}}\right) </math>
| <math> 4\pi\varepsilon_0\left(\rho^{_\mathrm{I}},R^{_\mathrm{I}},L^{_\mathrm{I}},M^{_\mathrm{I}},Z^{_\mathrm{I}}\right) </math>
|-
! scope="row" | [[magnetic reluctance]]
| <math> \mathcal{R}^{_\mathrm{G}}</math>
| <math> \mu_0\mathcal{R}^{_\mathrm{I}} </math>
|}

{| class="wikitable plainrowheaders"
|+ Table 2B: Replacement rules for translating formulas from ISQ to Gaussian
|-
! scope="col" | Name
! scope="col" | {{abbr|ISQ|International System of Quantities}}
! scope="col" | Gaussian system
|-
! scope="row" | [[electric field]], [[electric potential]], [[electromotive force]]
| <math> \left(\mathbf{E}^{_\mathrm{I}}, \varphi^{_\mathrm{I}}, \mathcal E^{_\mathrm{I}}\right) </math>
| <math> \frac{1}{\sqrt{4\pi\varepsilon_0}}\left(\mathbf{E}^{_\mathrm{G}}, \varphi^{_\mathrm{G}}, \mathcal E^{_\mathrm{G}}\right) </math>
|-
! scope="row" | [[electric displacement field]]
| <math> \mathbf{D}^{_\mathrm{I}} </math>
| <math> \sqrt{\frac{\varepsilon_0}{4\pi}}\mathbf{D}^{_\mathrm{G}} </math>
|-
! scope="row" | [[charge (physics)|charge]], [[charge density]], [[Electric current|current]],<br />[[current density]], [[polarization density]],<br />[[electric dipole moment]]
| <math> \left(q^{_\mathrm{I}}, \rho^{_\mathrm{I}}, I^{_\mathrm{I}}, \mathbf{J}^{_\mathrm{I}},\mathbf{P}^{_\mathrm{I}}, \mathbf{p}^{_\mathrm{I}}\right) </math>
| <math> \sqrt{4\pi\varepsilon_0}\left(q^{_\mathrm{G}}, \rho^{_\mathrm{G}}, I^{_\mathrm{G}}, \mathbf{J}^{_\mathrm{G}},\mathbf{P}^{_\mathrm{G}},\mathbf{p}^{_\mathrm{G}}\right) </math>
|-
! scope="row" | [[Magnetic field|magnetic {{math|'''B'''}} field]], [[magnetic flux]],<br />[[magnetic vector potential]]
| <math> \left(\mathbf{B}^{_\mathrm{I}}, \Phi_\mathrm{m}^{_\mathrm{I}},\mathbf{A}^{_\mathrm{I}}\right) </math>
| <math> \sqrt{\frac{\mu_0}{4\pi}}\left(\mathbf{B}^{_\mathrm{G}}, \Phi_\mathrm{m}^{_\mathrm{G}},\mathbf{A}^{_\mathrm{G}}\right) </math>
|-
! scope="row" | [[Magnetic field|magnetic {{math|'''H'''}} field]], [[magnetic scalar potential]], [[magnetomotive force]]
| <math> \left(\mathbf{H}^{_\mathrm{I}}, \psi^{_\mathrm{I}}, \mathcal F^{_\mathrm{I}}\right) </math>
| <math> \frac{1}{\sqrt{4\pi\mu_0}}\left(\mathbf{H}^{_\mathrm{G}}, \psi^{_\mathrm{G}}, \mathcal F^{_\mathrm{G}}\right) </math>
|-
! scope="row" | [[magnetic moment]], [[magnetization]], [[magnetic pole strength]]
| <math> \left(\mathbf{m}^{_\mathrm{I}}, \mathbf{M}^{_\mathrm{I}}, p^{_\mathrm{I}}\right) </math>
| <math> \sqrt{\frac{4\pi}{\mu_0}}\left(\mathbf{m}^{_\mathrm{G}}, \mathbf{M}^{_\mathrm{G}}, p^{_\mathrm{G}}\right) </math>
|-
! scope="row" | [[permittivity]],<br />[[permeability (electromagnetism)|permeability]]
| <math> \left(\varepsilon^{_\mathrm{I}}, \mu^{_\mathrm{I}}\right) </math>
| <math> \left(\varepsilon_0\varepsilon^{_\mathrm{G}}, \mu_0\mu^{_\mathrm{G}}\right) </math>
|-
! scope="row" | [[electric susceptibility]],<br />[[magnetic susceptibility]]
| <math> \left(\chi_\mathrm{e}^{_\mathrm{I}}, \chi_\mathrm{m}^{_\mathrm{I}}\right) </math>
| <math> 4\pi \left(\chi_\mathrm{e}^{_\mathrm{G}}, \chi_\mathrm{m}^{_\mathrm{G}}\right) </math>
|-
! scope="row" | [[Electrical conductivity|conductivity]], [[Electrical conductance|conductance]], [[capacitance]]
| <math> \left(\sigma^{_\mathrm{I}}, S^{_\mathrm{I}}, C^{_\mathrm{I}}\right) </math>
| <math> 4\pi\varepsilon_0\left(\sigma^{_\mathrm{G}},S^{_\mathrm{G}},C^{_\mathrm{G}}\right) </math>
|-
! scope="row" | [[resistivity]], [[electrical resistance|resistance]], [[inductance]], [[memristance]], [[electrical impedance|impedance]]
| <math> \left(\rho^{_\mathrm{I}},R^{_\mathrm{I}},L^{_\mathrm{I}},M^{_\mathrm{I}},Z^{_\mathrm{I}}\right) </math>
| <math> \frac{1}{4\pi\varepsilon_0}\left(\rho^{_\mathrm{G}},R^{_\mathrm{G}},L^{_\mathrm{G}},M^{_\mathrm{G}},Z^{_\mathrm{G}}\right) </math>
|-
! scope="row" | [[magnetic reluctance]]
| <math> \mathcal{R}^{_\mathrm{I}}</math>
| <math> \frac{1}{\mu_0}\mathcal{R}^{_\mathrm{G}} </math>
|}
After the rules of the table have been applied and the resulting formula has been simplified, replace all combinations <math>\varepsilon_0 \mu_0</math> by <math>1/c^2</math>.

== Notes ==

{{notelist}}

== References ==

{{reflist|30em}}

== External links ==

* [http://www.pgccphy.net/1030/gaussian.html Comprehensive list of Gaussian unit names, and their expressions in base units]
* [http://www.gsjournal.net/old/science/danescu.pdf The evolution of the Gaussian Units] {{Webarchive|url=https://web.archive.org/web/20160109073004/http://www.gsjournal.net/old/science/danescu.pdf |date=2016-01-09 }} by Dan Petru Danescu

{{Carl Friedrich Gauss}}

[[Category:Centimetre–gram–second system of units]]
[[Category:Systems of units]]
[[Category:Carl Friedrich Gauss]]