View source for Centimetre–gram–second system of units

{{short description|Physical system of measurement that uses the centimetre, gram, and second as base units}}
{{redirect|CGS}}
{{outline|Outline of the metric system}}

The '''centimetre–gram–second system of units''' ('''CGS''' or '''cgs''') is a variant of the [[metric system]] based on the [[centimetre]] as the unit of [[length]], the [[gram]] as the unit of [[mass]], and the [[second]] as the unit of [[time]]. All CGS [[mechanics|mechanical]] units are unambiguously derived from these three base units, but there are several different ways in which the CGS system was extended to cover [[electromagnetism]].<ref>{{Cite encyclopedia|url=https://www.britannica.com/science/centimetre-gram-second-system|title=Centimetre-gram-second system {{!}} physics|encyclopedia=Encyclopedia Britannica|access-date=2018-03-27|language=en}}{{failed verification|date=April 2018|reason=article is about viscosity, not electromagnetism}}</ref><ref>{{Cite web|url=https://www.maplesoft.com/support/help/maple/view.aspx?path=Units/CGS|title=The Centimeter-Gram-Second (CGS) System of Units – Maple Programming Help|website=www.maplesoft.com|access-date=2018-03-27}}</ref><ref>{{cite arXiv |title=Babel of units: The evolution of units systems in classical electromagnetism |last=Carron |first=Neal J. |date=21 May 2015 |class=physics.hist-ph |eprint=1506.01951 }}</ref>

The CGS system has been largely supplanted by the [[MKS units|MKS system]] based on the [[metre]], [[kilogram]], and second, which was in turn extended and replaced by the [[International System of Units]] (SI). In many fields of science and engineering, SI is the only system of units in use, but CGS is still prevalent in certain subfields.

In measurements of purely mechanical systems (involving units of length, mass, [[force]], [[energy]], [[pressure]], and so on), the differences between CGS and SI are straightforward: the [[Unit conversion|unit-conversion factors]] are all [[Exponentiation#Powers of ten|powers of 10]] as {{nowrap|1=100 cm = 1 m}} and {{nowrap|1=1000 g = 1 kg}}. For example, the CGS unit of force is the [[dyne]], which is defined as {{val|1|u=g⋅cm/s<sup>2</sup>}}, so the SI unit of force, the [[newton (unit)|newton]] ({{val|1|u=kg⋅m/s<sup>2</sup>}}), is equal to {{val|100000|u=dynes}}.

On the other hand, in measurements of electromagnetic phenomena (involving units of [[charge (physics)|charge]], electric and magnetic fields, [[voltage]], and so on), converting between CGS and SI is less straightforward. Formulas for physical laws of electromagnetism (such as [[Maxwell's equations]]) take a form that depends on which system of units is being used, because the electromagnetic quantities are defined differently in SI and in CGS. Furthermore, within CGS, there are several plausible ways to define electromagnetic quantities, leading to different "sub-systems", including [[Gaussian units]], "ESU", "EMU", and [[Heaviside–Lorentz units]]. Among these choices, Gaussian units are the most common today{{whom|date=January 2025}}, and "CGS units" is often intended to refer to CGS-Gaussian units{{citation needed|date=January 2025}}.

== History ==
The CGS system goes back to a proposal in 1832 by the German mathematician [[Carl Friedrich Gauss]] to base a system of absolute units on the three fundamental units of length, mass and time.<ref>
{{citation | first = C. F. | last = Gauss| author-link = Carl Friedrich Gauss | title = Intensitas vis magneticae terrestris ad mensuram absolutam revocata | journal = Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores | volume = 8 | year = 1832 | pages = 3–44}}. [http://www.21stcenturysciencetech.com/translations/gaussMagnetic.pdf English translation]</ref> Gauss chose the units of millimetre, milligram and second.<ref>
{{cite book
 | first1 = William | last1 = Hallock
 | first2 = Herbert Treadwell | last2 = Wade
 | page = 200
 | year = 1906
 | place = New York
 | title = Outlines of the evolution of weights and measures and the metric system
 | publisher = The Macmillan Co
 | url = https://archive.org/stream/outlinesofevolut00halluoft#page/200/mode/2up
}}</ref> In 1873, a committee of the [[British Association for the Advancement of Science]], including physicists [[James Clerk Maxwell]] and [[Lord Kelvin|William Thomson, 1st Baron Kelvin]] recommended the general adoption of centimetre, gram and second as fundamental units, and to express all derived electromagnetic units in these fundamental units, using the prefix "C.G.S. unit of ...".<ref>
{{cite conference 
 |url= https://www.biodiversitylibrary.org/item/94452#page/323/mode/1up
 |title=First Report of the Committee for the Selection and Nomenclature of Dynamical and Electrical Units 
 |date=September 1873
 |conference= Forty-third Meeting of the British Association for the Advancement of Science
 |editor-first= Professor |editor-last = Everett |editor-link = Joseph David Everett
 |first1 = Sir W |last1 = Thomson |author1-link = William Thomson, 1st Baron Kelvin
 |first2 = Professor GC |last2 = Foster |author2-link = Carey Foster
 |first3 = Professor JC |last3 = Maxwell |author3-link = James Clerk Maxwell
 |first4 = Mr GJ |last4 = Stoney |author4-link = George Johnstone Stoney
 |first5 = Professor Fleeming |last5 = Jenkin |author5-link = Fleeming Jenkin
 |first6 = Dr |last6 = Siemens |author6-link = Carl Wilhelm Siemens
 |first7 = Mr FJ |last7 = Bramwell |author7-link = Frederick Bramwell
 |publisher= John Murray
 |location= Bradford
 |page = 223
 |access-date= 2012-04-08
}}</ref>

The sizes of many CGS units turned out to be inconvenient for practical purposes. For example, many everyday objects are hundreds or thousands of centimetres long, such as humans, rooms and buildings. Thus the CGS system never gained wide use outside the field of science. Starting in the 1880s, and more significantly by the mid-20th century, CGS was gradually superseded internationally for scientific purposes by the MKS (metre–kilogram–second) system, which in turn developed into the modern [[SI]] standard.

Since the international adoption of the MKS standard in the 1940s and the SI standard in the 1960s, the technical use of CGS units has gradually declined worldwide. CGS units have been deprecated in favor of SI units by [[National Institute of Standards and Technology|NIST]],<ref>{{Cite report |url=https://physics.nist.gov/cuu/pdf/sp811.pdf |title=Guide for the Use of the International System of Units (SI) |last1=Thompson |first1=Ambler |last2=Taylor |first2=Barry N. |date=March 2008 |page=10 |access-date=March 3, 2024 }}</ref> as well as organizations such as the [[American Physical Society]]<ref>{{Citation |last1=Waldron |first1=Anne |title=Physical Review Style and Notation Guide |date=February 1993 |page=15 |url=https://cdn.journals.aps.org/files/styleguide-pr.pdf |access-date=March 3, 2024 |publisher=American Physical Society |last2=Judd |first2=Peggy |last3=Miller |first3=Valerie}}</ref> and the [[International Astronomical Union]].<ref>{{Citation |last=Wilkins |first=George A. |title=The IAU Style Manual |date=1989 |page=20 |url=https://www.iau.org/static/publications/stylemanual1989.pdf |access-date=March 3, 2024 |publisher=International Astronomical Union}}</ref> SI units are predominantly used in [[engineering]] applications and physics education, while Gaussian CGS units are still commonly used in theoretical physics, describing microscopic systems, relativistic [[electrodynamics]], and [[astrophysics]].<ref name=Jack>
{{cite book 
 | author=Jackson, John David 
 | title=Classical Electrodynamics 
 | url=https://archive.org/details/classicalelectro00jack_697
 | url-access=limited
 | edition=3rd 
 | pages=[https://archive.org/details/classicalelectro00jack_697/page/n798 775]–784
 | location=New York 
 | publisher=Wiley 
 | year=1999 
 | isbn=0-471-30932-X
}}</ref><ref>{{cite web |last1=Weisstein |first1=Eric W. |title=cgs |url=https://scienceworld.wolfram.com/physics/cgs.html |website=Eric Weisstein's World of Physics |language=en}}</ref>

The units [[gram]] and [[centimetre]] remain useful as noncoherent units within the SI system, as with any other [[Metric prefix|prefix]]ed SI units.

== Definition of CGS units in mechanics ==
In mechanics, the quantities in the CGS and SI systems are defined identically. The two systems differ only in the scale of the three base units (centimetre versus metre and gram versus kilogram, respectively), with the third unit (second) being the same in both systems.

There is a direct correspondence between the base units of mechanics in CGS and SI.  Since the formulae expressing the laws of mechanics are the same in both systems and since both systems are [[Coherence (units of measurement)|coherent]], the definitions of all coherent [[derived unit]]s in terms of the base units are the same in both systems, and there is an unambiguous relationship between derived units:
* <math>v = \frac{dx}{dt}</math>&nbsp; (definition of [[velocity]])
* <math>F = m\frac{d^2x}{dt^2}</math>&nbsp; ([[Newton's laws of motion|Newton's second law of motion]])
* <math>E = \int \vec{F}\cdot d\vec{x}</math>&nbsp; ([[energy]] defined in terms of [[Mechanical work|work]])
* <math>p = \frac{F}{L^2} </math>&nbsp; ([[pressure]] defined as force per unit area)
* <math>\eta = \tau/\frac{dv}{dx}</math>&nbsp; (dynamic [[viscosity]] defined as [[shear stress]] per unit velocity [[gradient]]).

Thus, for example, the CGS unit of pressure, [[barye]], is related to the CGS base units of length, mass, and time in the same way as the SI unit of pressure, [[pascal (unit)|pascal]], is related to the SI base units of length, mass, and time:
: 1&nbsp;unit of pressure = 1&nbsp;unit of force / (1&nbsp;unit of length)<sup>2</sup> = 1&nbsp;unit of mass / (1&nbsp;unit of length × (1&nbsp;unit of time)<sup>2</sup>)
: 1&nbsp;Ba = 1&nbsp;g/(cm⋅s<sup>2</sup>)
: 1&nbsp;Pa = 1&nbsp;kg/(m⋅s<sup>2</sup>).

Expressing a CGS derived unit in terms of the SI base units, or vice versa, requires combining the scale factors that relate the two systems:
: 1&nbsp;Ba = 1&nbsp;g/(cm⋅s<sup>2</sup>) = 10<sup>−3</sup>&nbsp;kg / (10<sup>−2&nbsp;</sup>m⋅s<sup>2</sup>) = 10<sup>−1</sup>&nbsp;kg/(m⋅s<sup>2</sup>) = 10<sup>−1</sup>&nbsp;Pa.

=== Definitions and conversion factors of CGS units in mechanics ===
{|class="wikitable" style="text-align: left;"
|-
! Quantity 
! Quantity symbol !! CGS unit name !! Unit symbol !! Unit definition !! In SI units 
|-
! [[length]], [[position (geometry)|position]]
| style="text-align:center;"| ''L'', ''x''|| [[centimetre]] || style="text-align:center;"| cm || 1/100 of [[metre]] || 10<sup>−2</sup>&nbsp;m
|-
! [[mass]] 
| style="text-align:center;"| ''m''|| [[gram]] || style="text-align:center;"|g || 1/1000 of [[kilogram]] || 10<sup>−3</sup>&nbsp;kg
|-
! [[time]]
| style="text-align:center;"| ''t''|| [[second]]|| style="text-align:center;"|s|| 1 second || 1&nbsp;s
|-
! [[velocity]] 
| style="text-align:center;"| ''v''|| centimetre per second || style="text-align:center;"|cm/s || cm/s || 10<sup>−2</sup>&nbsp;m/s
|-
! [[acceleration]] 
| style="text-align:center;"| ''a''|| [[gal (unit)|gal]] || style="text-align:center;"|Gal || cm/s<sup>2</sup> || 10<sup>−2</sup>&nbsp;m/s<sup>2</sup>
|-
! [[force (physics)|force]] 
| style="text-align:center;"| ''F''|| [[dyne]] || style="text-align:center;"|dyn || g⋅cm/s<sup>2</sup> || 10<sup>−5</sup>&nbsp;[[newton (unit)|N]]
|-
! [[energy]] 
| style="text-align:center;"| ''E''|| [[erg]] || style="text-align:center;"|erg || g⋅cm<sup>2</sup>/s<sup>2</sup> || 10<sup>−7</sup>&nbsp;[[joule|J]]
|-
! [[power (physics)|power]] 
| style="text-align:center;"| ''P''|| [[erg]] per second|| style="text-align:center;"|erg/s || g⋅cm<sup>2</sup>/s<sup>3</sup> || 10<sup>−7</sup>&nbsp;[[watt|W]]
|-
! [[pressure]] 
| style="text-align:center;"| ''p''|| [[barye]] || style="text-align:center;"| Ba|| g/(cm⋅s<sup>2</sup>) || 10<sup>−1</sup>&nbsp;[[pascal (unit)|Pa]]
|-
! dynamic [[viscosity]] 
| style="text-align:center;"| ''μ''|| [[Poise (unit)|poise]] || style="text-align:center;"|P|| g/(cm⋅s) || 10<sup>−1</sup>&nbsp;[[pascal second|Pa⋅s]]
|-
! kinematic [[viscosity]]
| style="text-align:center;"| ''ν''|| [[stokes (unit)|stokes]] || style="text-align:center;"|St|| cm<sup>2</sup>/s || 10<sup>−4</sup>&nbsp;m<sup>2</sup>/s
|-
! [[wavenumber]]
| style="text-align:center;"| ''k'' || [[kayser (unit)|kayser]] || style="text-align:center;"|cm<sup>−1</sup><ref name=K>{{cite web|title=Atomic Spectroscopy|url=http://physics.nist.gov/Pubs/AtSpec/node01.html|website=Atomic Spectroscopy|publisher=NIST|access-date=25 October 2015}}</ref> or K|| cm<sup>−1</sup> || 100&nbsp;m<sup>−1</sup>
|}

== Derivation of CGS units in electromagnetism ==

=== CGS approach to electromagnetic units ===
The conversion factors relating [[electromagnetism|electromagnetic]] units in the CGS and SI systems are made more complex by the differences in the formulas expressing physical laws of electromagnetism as assumed by each system of units, specifically in the nature of the constants that appear in these formulas. This illustrates the fundamental difference in the ways the two systems are built: 
* In SI, the unit of [[electric current]], the ampere (A), was historically defined such that the [[magnetism|magnetic]] force exerted by two infinitely long, thin, parallel wires 1&nbsp;[[metre]] apart and carrying a current of 1&nbsp;[[ampere]] is exactly {{val|2|e=-7|u=[[newton (unit)|N]]/[[metre|m]]}}. This definition results in all [[International System of Units#Derived units|SI electromagnetic units]] being numerically consistent (subject to factors of some [[integer]] powers of 10) with those of the CGS-EMU system described in further sections. The ampere is a base unit of the SI system, with the same status as the metre, kilogram, and second. Thus the relationship in the definition of the ampere with the metre and newton is disregarded, and the ampere is not treated as dimensionally equivalent to any combination of other base units. As a result, electromagnetic laws in SI require an additional constant of proportionality (see ''[[Vacuum permeability]]'') to relate electromagnetic units to kinematic units. (This constant of proportionality is derivable directly from the above definition of the ampere.) All other electric and magnetic units are derived from these four base units using the most basic common definitions: for example, [[charge (physics)|electric charge]] ''q'' is defined as current ''I'' multiplied by time ''t'', <math display="block">q = I \, t,</math> resulting in the unit of electric charge, the [[coulomb]] (C), being defined as 1&nbsp;C = 1&nbsp;A⋅s.
* The CGS system variant avoids introducing new base quantities and units, and instead defines all electromagnetic quantities by expressing the physical laws that relate electromagnetic phenomena to mechanics with only dimensionless constants, and hence all units for these quantities are directly derived from the centimetre, gram, and second.

In each of these systems the quantities called "charge" etc. may be a different quantity; they are distinguished here by a superscript. The corresponding quantities of each system are related through a proportionality constant.

[[Maxwell's equations]] can be written in each of these systems as:<ref name=Jack/><ref name=leu>
{{cite journal 
 | last = Leung | first = P. T.
 | s2cid = 43177051
 | title = A note on the 'system-free' expressions of Maxwell's equations
 | year = 2004
 | journal = European Journal of Physics
 | volume = 25
 | issue = 2
 | pages = N1–N4
 | doi = 10.1088/0143-0807/25/2/N01 | bibcode = 2004EJPh...25N...1L }}</ref>

{| class="wikitable" style="text-align: center;"
|-
 ! System
 ! width=175 | Gauss's law
 ! width=175 | Ampère–Maxwell law
 ! width=175 | Gauss's law for magnetism
 ! width=175 | Faraday's law
|-
 | style="text-align:left;"| CGS-ESU
 | <math> \nabla \cdot  \mathbf E^\text{ESU} = 4 \pi \rho^\text{ESU} </math>
 | <math> \nabla \times \mathbf B^\text{ESU} - c^{-2} \dot \mathbf E^\text{ESU} = 4 \pi c^{-2} \mathbf J^\text{ESU} </math>
 | <math> \nabla \cdot  \mathbf B^\text{ESU} = 0 </math>
 | <math> \nabla \times \mathbf E^\text{ESU} + \dot \mathbf B^\text{ESU} = 0 </math>
|-
 | style="text-align:left;"| CGS-EMU
 | <math> \nabla \cdot  \mathbf E^\text{EMU} = 4 \pi c^2 \rho^\text{EMU} </math>
 | <math> \nabla \times \mathbf B^\text{EMU} - c^{-2} \dot \mathbf E^\text{EMU} = 4 \pi \mathbf J^\text{EMU} </math>
 | <math> \nabla \cdot  \mathbf B^\text{EMU} = 0 </math>
 | <math> \nabla \times \mathbf E^\text{EMU} + \dot \mathbf B^\text{EMU} = 0 </math>
|-
 | style="text-align:left;"| CGS-[[Gaussian units|Gaussian]]
 | <math> \nabla \cdot  \mathbf E^\text{G} = 4 \pi \rho^\text{G} </math>
 | <math> \nabla \times \mathbf B^\text{G} - c^{-1} \dot \mathbf E^\text{G} = 4 \pi c^{-1} \mathbf J^\text{G} </math>
 | <math> \nabla \cdot  \mathbf B^\text{G} = 0 </math>
 | <math> \nabla \times \mathbf E^\text{G} + c^{-1} \dot \mathbf B^\text{G} = 0 </math>
|-
 | style="text-align:left;"| CGS-[[Heaviside–Lorentz units|Heaviside–Lorentz]]
 | <math> \nabla \cdot  \mathbf E^\text{LH} = \rho^\text{LH} </math>
 | <math> \nabla \times \mathbf B^\text{LH} - c^{-1} \dot \mathbf E^\text{LH} = c^{-1} \mathbf J^\text{LH} </math>
 | <math> \nabla \cdot  \mathbf B^\text{LH} = 0 </math>
 | <math> \nabla \times \mathbf E^\text{LH} + c^{-1} \dot \mathbf B^\text{LH} = 0 </math>
|-
 | style="text-align:left;"| [[SI]]
 | <math> \nabla \cdot  \mathbf E^\text{SI} = \rho^\text{SI} / \epsilon_0 </math>
 | <math> \nabla \times \mathbf B^\text{SI} - \mu_0\epsilon_0\dot \mathbf E^\text{SI} = \mu_0 \mathbf J^\text{SI} </math>
 | <math> \nabla \cdot  \mathbf B^\text{SI} = 0 </math>
 | <math> \nabla \times \mathbf E^\text{SI} + \dot \mathbf B^\text{SI} = 0 </math>
|}

=== Electrostatic units (ESU) <span class="anchor" id="ESU"></span> ===
In the '''electrostatic units''' variant of the CGS system, (CGS-ESU), charge is defined as the quantity that obeys a form of [[Coulomb's law]] without a [[Proportionality (mathematics)|multiplying constant]] (and current is then defined as charge per unit time):
: <math>F={q^\text{ESU}_1 q^\text{ESU}_2 \over r^2} .</math>

The ESU unit of charge, '''franklin''' ('''Fr'''), also known as '''[[statcoulomb]]''' or '''esu charge''', is therefore defined as follows:<ref name=cardsgc>
{{cite book
 | author = Cardarelli, F.
 | year = 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= 1-85233-682-X
 | url= https://archive.org/details/encyclopaediaofs0000card
 | url-access = registration
}}</ref> {{Blockquote|text=two equal point charges spaced 1 [[centimetre]] apart are said to be of 1 franklin each if the electrostatic force between them is 1 [[dyne]].}} Therefore, in CGS-ESU, a franklin is equal to a centimetre times square root of dyne:
: <math>\mathrm{1\,Fr = 1\,statcoulomb = 1\,esu\; charge = 1\,dyne^{1/2}{\cdot}cm=1\,g^{1/2}{\cdot}cm^{3/2}{\cdot}s^{-1}} .</math>
The unit of current is defined as:
: <math>\mathrm{1\,Fr/s = 1\,statampere = 1\,esu\; current = 1\,dyne^{1/2}{\cdot}cm{\cdot}s^{-1}=1\,g^{1/2}{\cdot}cm^{3/2}{\cdot}s^{-2}} .</math>

In the CGS-ESU system, charge ''q'' is therefore has the dimension to M<sup>1/2</sup>L<sup>3/2</sup>T<sup>−1</sup>.

Other units in the CGS-ESU system include the [[statampere]] (1&nbsp;statC/s) and [[statvolt]] (1&nbsp;[[erg]]/statC).

In CGS-ESU, all electric and magnetic quantities are dimensionally expressible in terms of length, mass, and time, and none has an independent dimension. Such a system of units of electromagnetism, in which the dimensions of all electric and magnetic quantities are expressible in terms of the mechanical dimensions of mass, length, and time, is traditionally called an 'absolute system'.<ref name="Fenna 2002">{{cite book |last1=Fenna |first1=Donald |title=A Dictionary of Weights, Measures, and Units |date=2002 |publisher=Oxford University Press |isbn=978-0-19-107898-9 |url=https://books.google.com/books?id=uBk9DAAAQBAJ |language=en}}</ref><sup>:[https://books.google.com/books?id=uBk9DAAAQBAJ&dq=%22absolute%20system%20electromagnetics%22&pg=PT49 3]</sup>

==== Unit symbols ====
All electromagnetic units in the CGS-ESU system that have not been given names of their own are named as the corresponding SI name with an attached prefix "stat" or with a separate abbreviation "esu", and similarly with the corresponding symbols.<ref name=cardsgc/>

=== Electromagnetic units (EMU) <span class="anchor" id="EMU"></span> ===
In another variant of the CGS system, '''electromagnetic units''' ('''EMU'''), current is defined via the force existing between two thin, parallel, infinitely long wires carrying it, and charge is then defined as current multiplied by time. (This approach was eventually used to define the SI unit of [[ampere]] as well).

The EMU unit of current, '''biot''' ('''Bi'''), also known as '''[[abampere]]''' or '''emu current''', is therefore defined as follows:<ref name=cardsgc/>
{{quote|text=The '''biot''' is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one [[centimetre]] apart in [[vacuum]], would produce between these conductors a force equal to two [[dyne]]s per centimetre of length.}} Therefore, in '''electromagnetic CGS units''', a biot is equal to a square root of dyne:
: <math>\mathrm{1\,Bi = 1\,abampere = 1\,emu\; current= 1\,dyne^{1/2}=1\,g^{1/2}{\cdot}cm^{1/2}{\cdot}s^{-1}}.</math>
The unit of charge in CGS EMU is:
: <math>\mathrm{1\,Bi{\cdot}s = 1\,abcoulomb = 1\,emu\, charge= 1\,dyne^{1/2}{\cdot}s=1\,g^{1/2}{\cdot}cm^{1/2}}.</math>

Dimensionally in the CGS-EMU system, charge ''q'' is therefore equivalent to M<sup>1/2</sup>L<sup>1/2</sup>. Hence, neither charge nor current is an independent physical quantity in the CGS-EMU system.

==== EMU notation ====
All electromagnetic units in the CGS-EMU system that do not have proper names are denoted by a corresponding SI name with an attached prefix "ab" or with a separate abbreviation "emu".<ref name=cardsgc/>

=== Practical CGS units ===
The practical CGS system is a hybrid system that uses the [[volt]] and the [[ampere]] as the units of voltage and current respectively. Doing this avoids the inconveniently large and small electrical units that arise in the esu and emu systems. This system was at one time widely used by electrical engineers because the volt and ampere had been adopted as international standard units by the International Electrical Congress of 1881.<ref>{{cite book |first=Paul |last=Tunbridge |title=Lord Kelvin: His Influence on Electrical Measurements and Units |pages=34–40 |publisher=IET |year=1992 |isbn=0-86341-237-8 }}</ref> As well as the volt and ampere, the [[farad]] (capacitance), [[ohm]] (resistance), [[coulomb]] (electric charge), and [[henry (unit)|henry]] (inductance) are consequently also used in the practical system and are the same as the SI units. The magnetic units are those of the emu system.<ref>{{cite book |first=Heinz E. |last=Knoepfel |title=Magnetic Fields: A Comprehensive Theoretical Treatise for Practical Use |url=https://archive.org/details/magneticfieldsco00knoe |url-access=limited |page=[https://archive.org/details/magneticfieldsco00knoe/page/n559 543] |publisher=Wiley |year=2000 |isbn=3-527-61742-6 }}</ref>

The electrical units, other than the volt and ampere, are determined by the requirement that any equation involving only electrical and kinematical quantities that is valid in SI should also be valid in the system. For example, since electric field strength is voltage per unit length, its unit is the volt per centimetre, which is one hundred times the SI unit.

The system is electrically rationalized and magnetically unrationalized; i.e., {{nowrap|1={{lambda}} = 1}} and {{nowrap|1={{lambda}}&prime; = 4{{pi}}}}, but the above formula for {{lambda}} is invalid. A closely related system is the International System of Electric and Magnetic Units,<ref>
{{cite book
 |title = International System of Electric and Magnetic Units
 |url = https://books.google.com/books?id=amGTOGpYFgwC
 |last1 = Dellinger |first1 = John Howard
 |year = 1916
 |publisher = U.S. Government Printing Office
 |location = Washington, D.C.
}}</ref> which has a different unit of mass so that the formula for {{lambda}}&prime; is invalid. The unit of mass was chosen to remove powers of ten from contexts in which they were considered to be objectionable (e.g., {{nowrap|1=''P'' = ''VI''}} and {{nowrap|1=''F'' = ''qE''}}). Inevitably, the powers of ten reappeared in other contexts, but the effect was to make the familiar joule and watt the units of work and power respectively.

The ampere-turn system is constructed in a similar way by considering magnetomotive force and magnetic field strength to be electrical quantities and rationalizing the system by dividing the units of magnetic pole strength and magnetization by 4{{pi}}. The units of the first two quantities are the ampere and the ampere per centimetre respectively. The unit of magnetic permeability is that of the emu system, and the magnetic constitutive equations are {{nowrap|1='''B''' = (4{{pi}}/10)''μ'''''H'''}} and {{nowrap|1='''B''' = (4{{pi}}/10)''μ''<sub>0</sub>'''H''' + ''μ''<sub>0</sub>'''M'''}}. [[Magnetic reluctance]] is given a hybrid unit to ensure the validity of Ohm's law for magnetic circuits.

In all the practical systems ''ε''<sub>0</sub> = 8.8542 × 10<sup>−14</sup> A⋅s/(V⋅cm), ''μ''<sub>0</sub> = 1 V⋅s/(A⋅cm), and ''c''<sup>2</sup> = 1/(4''π'' × 10<sup>−9</sup> ''ε''<sub>0</sub>''μ''<sub>0</sub>).

=== Other variants ===
There were at various points in time about half a dozen systems of electromagnetic units in use, most based on the CGS system.<ref>
{{cite journal 
 | author1=Bennett, L. H. |author2=Page, C. H. |author3=Swartzendruber, L. J.
 | title = Comments on units in magnetism
 | year = 1978
 | journal = Journal of Research of the National Bureau of Standards
 | volume = 83
 | issue = 1
 | pages = 9–12
 | doi = 10.6028/jres.083.002|pmid=34565970 |pmc=6752159 | doi-access = free
}}</ref> These include the [[Gaussian units]] and the [[Heaviside–Lorentz units]].

== Electromagnetic units in various CGS systems ==
{| class="wikitable"
|+ Conversion of SI units in electromagnetism to ESU, EMU, and Gaussian subsystems of CGS<ref>{{cite book
|title = Applied Electronics
|url = https://archive.org/details/Applied_Electronics_Truman_S._Gray_1954
|at = pp. 830–831, Appendix B
|first1 = Truman S.
|last1 = Gray
|year = 1954
|publisher = John Wiley & Sons, Inc.
|location = New York}}</ref><ref name=cardsgc/>
! Quantity
! Symbol !! SI unit !! ESU unit !! [[Gaussian units|Gaussian unit]] !! EMU unit
|-
! [[electric charge]]
| style="text-align:center;"| ''q'' ||1 [[Coulomb|C]] || colspan="2" | ≘ (10<sup>−1</sup> ''c'') [[Statcoulomb|statC]] (Fr) || ≘ (10<sup>−1</sup>) [[Abcoulomb|abC]]
|-
! [[electric current]] 
| style="text-align:center;"| ''I'' || 1 [[Ampere|A]] || colspan="2" | ≘ (10<sup>−1</sup> ''c'') [[Statampere|statA]] (Fr/s) || ≘ (10<sup>−1</sup>) [[Abampere|abA]] (Bi)
|-
! [[electric potential]] / [[voltage]] 
| style="text-align:center;"|''φ'' / ''V, E''||1 [[Volt|V]]|| colspan="2" | ≘ (10<sup>8</sup> ''c''<sup>−1</sup>) [[statvolt|statV]] (erg/Fr) || ≘ (10<sup>8</sup>) [[abvolt|abV]]
|-
! [[electric field]] 
| style="text-align:center;"|'''E'''||1 [[Volt|V]]/[[Metre|m]] || colspan="2" | ≘ (10<sup>6</sup> ''c''<sup>−1</sup>) [[statvolt|statV]]/[[Centimetre|cm]] (dyn/Fr) || ≘ (10<sup>6</sup>) [[abvolt|abV]]/[[Centimetre|cm]]
|-
! [[electric displacement field]] 
| style="text-align:center;"|'''D'''||1 [[Coulomb|C]]/[[Square metre|m<sup>2</sup>]] || colspan="2" | ≘ (4{{pi}} × 10<sup>−5</sup> ''c'') [[statcoulomb|statC]]/[[square centimetre|cm<sup>2</sup>]] || ≘ (4{{pi}} × 10<sup>−5</sup>) [[Abcoulomb|abC]]/[[square centimetre|cm<sup>2</sup>]] 
|-
! [[electric dipole moment]] 
| style="text-align:center;"|'''p'''||1 [[Coulomb|C]]⋅[[meter|m]] || colspan="2" | ≘ (10 ''c'') [[Statcoulomb|statC]]⋅[[centimeter|cm]] || ≘ (10) [[Abcoulomb|abC]]⋅[[centimeter|cm]]
|- 
! [[Statcoulomb#As a unit of flux of electric displacement field|electric flux]] 
| style="text-align:center;"| Φ<sub>e</sub> ||1 [[Coulomb|C]] || colspan="2" | ≘ (4{{pi}} × 10<sup>−1</sup> ''c'') [[Statcoulomb|statC]] || ≘ (4{{pi}} × 10<sup>−1</sup>) [[Abcoulomb|abC]] 
|-
! [[permittivity]] 
| style="text-align:center;"| {{mvar|ε}} ||1 [[Farad|F]]/[[Metre|m]] || colspan="2" | ≘ (4{{pi}} × 10<sup>−11</sup> ''c''<sup>2</sup>) [[Centimetre|cm]]/cm || ≘ (4{{pi}} × 10<sup>−11</sup>) [[Second|s]]<sup>2</sup>/[[Centimetre|cm]]<sup>2</sup>
|-
! [[Magnetic field|magnetic B field]] 
| style="text-align:center;"|'''B'''||1 [[tesla (unit)|T]] || ≘ (10<sup>4</sup> ''c''<sup>−1</sup>) statT || colspan="2" | ≘ (10<sup>4</sup>) [[Gauss (unit)|G]]
|-
! [[Magnetic field|magnetic H field]] 
| style="text-align:center;"|'''H'''||1 [[Ampere|A]]/[[Metre|m]] || ≘ (4{{pi}} × 10<sup>−3</sup> ''c'') [[StatAmpere|statA]]/[[Centimetre|cm]] || colspan="2" | ≘ (4{{pi}} × 10<sup>−3</sup>) [[oersted|Oe]]
|-
! [[magnetic dipole moment]] 
| style="text-align:center;"|'''μ'''||1 [[Ampere|A]]⋅[[Square metre|m<sup>2</sup>]] || ≘ (10<sup>3</sup> ''c'') [[Statampere|statA]]⋅[[square centimetre|cm<sup>2</sup>]] || colspan="2" | ≘ (10<sup>3</sup>) [[erg]]/[[Gauss (unit)|G]]
|-
! [[magnetic flux]]
| style="text-align:center;"|Φ<sub>m</sub>||1 [[Weber (unit)|Wb]] || ≘ (10<sup>8</sup> ''c''<sup>−1</sup>) statWb || colspan="2" | ≘ (10<sup>8</sup>) [[Maxwell (unit)|Mx]]
|-
! [[Permeability (electromagnetism)|permeability]]
| style="text-align:center;"| {{mvar|μ}} ||1 [[Henry (unit)|H]]/[[Metre|m]] || ≘ ((4{{pi}})<sup>−1</sup> × 10<sup>7</sup> ''c''<sup>−2</sup>) [[Second|s]]<sup>2</sup>/[[Centimetre|cm]]<sup>2</sup> || colspan="2" | ≘ ((4{{pi}})<sup>−1</sup> × 10<sup>7</sup>) [[Centimetre|cm]]/cm
|-
! [[magnetomotive force]] 
| style="text-align:center;"|<math>\mathcal F</math>||1 [[Ampere|A]] || ≘ (4{{pi}} × 10<sup>−1</sup> ''c'') [[StatAmpere|statA]] || colspan="2" | ≘ (4{{pi}} × 10<sup>−1</sup>) [[Gilbert (unit)|Gi]]
|-
! [[magnetic reluctance]] 
| style="text-align:center;"|<math>\mathcal R</math>||1 [[Henry (unit)|H]]<sup>−1</sup> || ≘ (4{{pi}} × 10<sup>−9</sup> ''c''<sup>2</sup>) statH<sup>−1</sup> || colspan="2" | ≘ (4{{pi}} × 10<sup>−9</sup>) [[Gilbert (unit)|Gi]]/[[Maxwell (unit)|Mx]]
|-
! [[electric resistance|resistance]]
| style="text-align:center;"|''R''||1 [[Ohm|Ω]] || colspan="2" | ≘ (10<sup>9</sup> ''c''<sup>−2</sup>) [[Statohm|statΩ]] (s/cm) || ≘ (10<sup>9</sup>) [[Abohm|abΩ]]
|-
! [[electric resistivity|resistivity]]
| style="text-align:center;"|''ρ'' ||1 [[Ohm|Ω]]⋅[[Metre|m]] || colspan="2" | ≘ (10<sup>11</sup> ''c''<sup>−2</sup>) [[Statohm|statΩ]]⋅[[Centimetre|cm]] (s) || ≘ (10<sup>11</sup>) [[Abohm|abΩ]]⋅[[Centimetre|cm]]
|-
! [[capacitance]]
| style="text-align:center;"|''C''||1 [[Farad|F]] || colspan="2" | ≘ (10<sup>−9</sup> ''c''<sup>2</sup>) [[Statfarad|statF]] (cm) || ≘ (10<sup>−9</sup>) [[Abfarad|abF]]
|-
! [[inductance]]
| style="text-align:center;"|''L''||1 [[Henry (unit)|H]] || colspan="2" | ≘ (10<sup>9</sup> ''c''<sup>−2</sup>) statH (s<sup>2</sup>/cm) || ≘ (10<sup>9</sup>) [[Abhenry|abH]]
|}

In this table, ''c'' = {{val|29979245800}} is the numeric value of the [[speed of light]] in vacuum when expressed in units of centimetres per second. The symbol "≘" is used instead of "=" as a reminder that the units are ''corresponding'' but not ''equal''. For example, according to the capacitance row of the table, if a capacitor has a capacitance of 1&nbsp;F in SI, then it has a capacitance of (10<sup>−9</sup>&nbsp;''c''<sup>2</sup>) cm in ESU; ''but'' it is incorrect to replace "1&nbsp;F" with "(10<sup>−9</sup>&nbsp;''c''<sup>2</sup>)&nbsp;cm" within an equation or formula. (This warning is a special aspect of electromagnetism units. By contrast it is ''always'' correct to replace, e.g., "1&nbsp;m" with "100&nbsp;cm" within an equation or formula.)

== Physical constants in CGS units ==
{| class="wikitable"
 |+ Commonly used physical constants in CGS units<ref name="textbook">{{Cite book | year=1978 |author1=A.P. French |author2=Edwind F. Taylor | title= An Introduction to Quantum Physics | publisher=W.W. Norton & Company}}</ref>
 ! Constant
 ! Symbol
 ! Value
|-
 | [[atomic mass constant]]
 | style="text-align:center;"| ''m''{{sub|u}}
 | {{val|1.660539069|e=-24|ul=g}}
|-
 | rowspan="2"|[[Bohr magneton]]
 | style="text-align:center;" rowspan="2"|''μ''<sub>B</sub>
 | {{val|9.274010066|e=-21|u=[[erg]]/[[Gauss (unit)|G]]}} (EMU, Gaussian)
|-
 | {{val|2.780278273|e=-10|u=statA⋅cm<sup>2</sup>}} (ESU)
|-
 | [[Bohr radius]]
 | style="text-align:center;"| ''a''<sub>0</sub>
 | {{val|5.291772105|e=-9|ul=cm}}
|-
 | [[Boltzmann constant]]
 | style="text-align:center;"| ''k''
 | {{val|1.380649|e=-16|u=[[erg]]/[[Kelvin|K]]}}
|-
 | [[electron mass]]
 | style="text-align:center;"| ''m''<sub>e</sub>
 | {{val|9.10938371|e=-28|ul=g}}
|-
 | rowspan="2"|[[elementary charge]]
 | style="text-align:center;" rowspan="2"|''e''
 | {{val|4.80320471|e=-10|u=[[Statcoulomb|Fr]]}} (ESU, Gaussian)
|-
 | {{val|1.602176634|e=-20|u=[[Abcoulomb|abC]]}} (EMU)
|-
 | [[fine-structure constant]]
 | style="text-align:center;"| ''α''
 | {{physconst|alpha|round=12|ref=no}}
|-
 | [[Newtonian constant of gravitation]]
 | style="text-align:center;"| ''G''
 | {{val|6.6743|e=-8|u=[[dyne|dyn]]⋅[[Centimetre|cm]]<sup>2</sup>/[[Gram|g]]<sup>2</sup>}}
|-
 | [[Planck constant]]
 | style="text-align:center;"| ''h''
 | {{val|6.62607015|e=-27|u=[[erg]]⋅[[Second|s]]}}
|-
 | reduced Planck constant
 | style="text-align:center;"| ''ħ''
 | {{val|1.054571817|e=-27|u=[[erg]]⋅[[Second|s]]}}
|-
 | [[speed of light]]
 | style="text-align:center;"| ''c''
 | {{val|2.99792458|e=10|u=[[Centimetre|cm]]/[[Second|s]]}}
|-
|}

== Advantages and disadvantages ==
{{Unreferenced|section|date=November 2024}}
Lack of unique unit names leads to potential confusion: "15 emu" may mean either 15 [[abvolt]]s, or 15 emu units of [[electric dipole moment]], or 15 emu units of [[magnetic susceptibility]], sometimes (but not always) per [[gram]], or per [[mole (unit)|mole]]. With its system of uniquely named units, the SI removes any confusion in usage: 1 ampere is a fixed value of a specified quantity, and so are 1 [[henry (unit)|henry]], 1&nbsp;[[ohm]], and 1&nbsp;volt.

In the [[Gaussian units|CGS-Gaussian system]], electric and magnetic fields have the same units, 4{{pi}}{{epsilon}}<sub>0</sub> is replaced by 1, and the only dimensional constant appearing in the [[Maxwell equations]] is ''c'', the speed of light. The [[Heaviside–Lorentz units|Heaviside–Lorentz system]] has these properties as well (with ''ε''<sub>0</sub> equaling 1).

In SI, and other rationalized systems (for example, [[Heaviside–Lorentz units|Heaviside–Lorentz]]), the unit of current was chosen such that electromagnetic equations concerning charged spheres contain 4{{pi}}, those concerning coils of current and straight wires contain 2{{pi}} and those dealing with charged surfaces lack {{pi}} entirely, which was the most convenient choice for applications in [[electrical engineering]] and relates directly to the geometric symmetry of the system being described by the equation.

Specialized unit systems are used to simplify formulas further than either SI or CGS do, by eliminating constants through a convention of normalizing quantities with respect to some system of [[natural units]]. For example, in [[particle physics]] a system is in use where every quantity is expressed by only one unit of energy, the [[electronvolt]], with lengths, times, and so on all converted into units of energy by inserting factors of [[speed of light]] ''c'' and the [[reduced Planck constant]] ''ħ''. This unit system is convenient for calculations in [[particle physics]], but is impractical in other contexts.

== See also ==
* [[Outline of metrology and measurement]]
{{div col begin}}
* [[International System of Units]]
* [[International System of Electrical and Magnetic Units]]
* [[List of metric units]]
* [[List of scientific units named after people]]
* [[Metre–tonne–second system of units]]
* [[United States customary units]]
* [[Foot–pound–second system of units]]
{{div col end}}

== References and notes ==
<div class="references">
<references/>
</div>

== General literature ==
* {{cite book | author=Griffiths, David J. | author-link=David Griffiths (physicist) | title=Introduction to Electrodynamics (3rd ed.) | publisher=[[Prentice Hall]] | year=1999 | isbn=0-13-805326-X | chapter=Appendix C: Units | chapter-url-access=registration | chapter-url=https://archive.org/details/introductiontoel00grif_0 | url-access=registration | url=https://archive.org/details/introductiontoel00grif_0 }}
* {{cite book | author=Jackson, John D. |author-link=John David Jackson (physicist) | title=Classical Electrodynamics (3rd ed.) | publisher=[[John Wiley & Sons|Wiley]] | year=1999 | isbn=0-471-30932-X | chapter=Appendix on Units and Dimensions }}
* {{cite book | author=Kent, William |author-link=William Kent (Consulting Engineer) | title=The Mechanical Engineer's Pocket-book (5th ed.) |publisher=[[John Wiley & Sons|Wiley]] | year=1900 | chapter=Electrical Engineering. Standards of Measurement page 1024 }}
* {{cite web | url=http://bohr.physics.berkeley.edu/classes/221/1112/notes/emunits.pdf |archive-url=https://web.archive.org/web/20151211021336/http://bohr.physics.berkeley.edu/classes/221/1112/notes/emunits.pdf |archive-date=2015-12-11 |url-status=live | 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=2017-12-15 }}

{{systems of measurement}}

{{DEFAULTSORT:Centimetre-gram-second system of units}}
[[Category:Centimetre–gram–second system of units| ]]
[[Category:Metrology]]
[[Category:Systems of units]]
[[Category:Metric system]]
[[Category:British Science Association]]