|
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 system of 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 thereCGS remainis certainstill subfieldsprevalent wherein CGScertain is prevalentsubfields.
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 and rather trivial;: 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 moreless subtlestraightforward. 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, and "CGS units" is often intended to refer to CGS-Gaussian units.
== History ==
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. SICGS units arehave predominantlybeen useddeprecated in engineeringfavor applicationsof SI units by [[National Institute of Standards and Technology|NIST]],<ref>{{Cite report |url=https://physics.nist.gov/cuu/pdf/sp811.pdf education,|title=Guide whilefor Gaussianthe CGSUse unitsof arethe commonlyInternational usedSystem inof theoreticalUnits physics,(SI) describing|last1=Thompson microscopic|first1=Ambler systems,|last2=Taylor relativistic|first2=Barry [[electrodynamics]]N. |date=March 2008 |page=10 |access-date=March 3, and2024 }}</ref> as well as organizations such as the [[astrophysicsAmerican Physical Society]].<ref name=Jack /><ref>{{cite webCitation |last1=WeissteinWaldron |first1=Eric W.Anne |title=cgsPhysical Review Style and Notation Guide |date=February 1993 |page=15 |url=https://scienceworldcdn.wolframjournals.aps.comorg/physicsfiles/cgsstyleguide-pr.htmlpdf |websiteaccess-date=EricMarch Weisstein's3, World2024 of|publisher=American PhysicsPhysical Society |last2=Judd |first2=Peggy |last3=Miller |languagefirst3=enValerie}}</ref> CGSand unitsthe are[[International todayAstronomical noUnion]].<ref>{{Citation longer|last=Wilkins accepted|first=George byA. the|title=The houseIAU stylesStyle ofManual most|date=1989 scientific|page=20 journals,{{ref|url=https://www.iau.org/static/publications/stylemanual1989.pdf needed|access-date=FebruaryMarch 2021}} textbook publishers3,{{ref needed2024 |datepublisher=FebruaryInternational 2021Astronomical Union}}</ref> orSI standardsunits bodiesare predominantly used in engineering applications and physics education, althoughwhile theyGaussian CGS units are still commonly used in astronomicaltheoretical journalsphysics, suchdescribing asmicroscopic systems, relativistic ''[[Theelectrodynamics]], Astrophysicaland Journal[[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.
* 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 [[metre]] apart and carrying a current of 1 [[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 C = 1 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.
=== Alternative derivations of CGS units in electromagnetism ===
Electromagnetic relationships to length, time and mass may be derived by several equally appealing methods. Two of them rely on the forces observed on charges. Two fundamental laws relate (seemingly independently of each other) the electric charge or its [[derivative|rate of change]] (electric current) to a mechanical quantity such as force. They can be written<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> in system-independent form as follows:
* The first is [[Coulomb's law]], <math>F = k_{\rm C} \frac{q \, q^\prime}{d^2},</math> which describes the electrostatic force ''F'' between electric charges <math>q</math> and <math>q^\prime,</math> separated by distance ''d''. Here <math>k_{\rm C}</math> is a constant which depends on how exactly the unit of charge is derived from the base units.
* The second is [[Ampère's force law]], <math>\frac{dF}{dL} = 2 k_{\rm A}\frac{I \, I^\prime}{d},</math> which describes the magnetic force ''F'' per unit length ''L'' between currents ''I'' and ''I′'' flowing in two straight parallel wires of infinite length, separated by a distance ''d'' that is much greater than the wire diameters. Since <math>I=q/t\,</math> and <math> I^\prime=q^\prime/t,</math> the constant <math>k_{\rm A}</math> also depends on how the unit of charge is derived from the base units.
[[Maxwell's equations|Maxwell's theory of electromagnetism]] relates these two laws to each other. It states that the ratio of proportionality constants <math>k_{\rm C}</math> and <math>k_{\rm A}</math> must obey <math>k_{\rm C} / k_{\rm A} = c^2,</math> where ''c'' is the [[speed of light]] in [[vacuum]]. Therefore, if one derives the unit of charge from the Coulomb's law by setting <math>k_{\rm C}=1</math> then Ampère's force law will contain a factor <math>2/c^2.</math> Alternatively, deriving the unit of current, and therefore the unit of charge, from the Ampère's force law by setting <math> k_{\rm A} = 1</math> or <math>k_{\rm A} = 1/2,</math> will lead to a constant factor in the Coulomb's law.
Indeed, both of these mutually exclusive approaches have been practiced by the users of CGS system, leading to the two independent and mutually exclusive branches of CGS, described in the subsections below. However, the freedom of choice in deriving electromagnetic units from the units of length, mass, and time is not limited to the definition of charge. While the electric field can be related to the work performed by it on a moving electric charge, the magnetic force is always perpendicular to the velocity of the moving charge, and thus the work performed by the magnetic field on any charge is always zero. This leads to a choice between two laws of magnetism, each relating magnetic field to mechanical quantities and electric charge:
* The first law describes the [[Lorentz force]] produced by a magnetic field '''B''' on a charge '''q''' moving with velocity '''v''': <math display="block"> \mathbf{F} = \alpha_{\rm L} q\;\mathbf{v} \times \mathbf{B}\;. </math>
* The second describes the creation of a static magnetic field '''B''' by an electric current ''I'' of finite length d'''l''' at a point displaced by a vector '''r''', known as [[Biot–Savart law]]: <math display="block"> d\mathbf{B} = \alpha_{\rm B}\frac{I d\mathbf{l} \times \mathbf{\hat r}}{r^2}\;,</math> where ''r'' and <math>\mathbf{\hat r}</math> are the length and the unit vector in the direction of vector '''r''' respectively.
These two laws can be used to derive [[Ampère's force law]] above, resulting in the relationship: <math>k_{\rm A} = \alpha_{\rm L} \cdot \alpha_{\rm B}\;.</math> Therefore, if the unit of charge is based on the [[Ampère's force law]] such that <math>k_{\rm A} = 1,</math> it is natural to derive the unit of magnetic field by setting <math>\alpha_{\rm L} = \alpha_{\rm B}=1\;.</math> However, if it is not the case, a choice has to be made as to which of the two laws above is a more convenient basis for deriving the unit of magnetic field.
Furthermore, if we wish to describe the [[electric displacement field]] '''D''' and the [[magnetic field]] '''H''' in a medium other than vacuum, we need to also define the constants ''ε''<sub>0</sub> and ''μ''<sub>0</sub>, which are the [[vacuum permittivity]] and [[magnetic constant|permeability]], respectively. <!-- These two values are related by <math>\sqrt{\mu_0\epsilon_0}=\alpha_{\rm B} / c.</math> // removed this statement - seems impossible to prove! --> Then we have<ref name=Jack/> (generally) <math>\mathbf{D} = \epsilon_0 \mathbf{E} + \lambda \mathbf{P}</math> and <math>\mathbf{H} = \mathbf{B} / \mu_0 - \lambda^\prime \mathbf{M},</math> where '''P''' and '''M''' are [[polarization density]] and [[magnetization]] vectors. The units of '''P''' and '''M''' are usually so chosen that the factors ''λ'' and ''λ''′ are equal to the "rationalization constants" <math>4 \pi k_{\rm C} \epsilon_0</math> and <math>4 \pi \alpha_{\rm B} / (\mu_0 \alpha_{\rm L}),</math> respectively. If the rationalization constants are equal, then <math>c^2 = 1 / (\epsilon_0 \mu_0 \alpha_{\rm L}^2).</math> If they are equal to one, then the system is said to be "rationalized":<ref>
{{cite book
| last1 = Cardarelli |first1 = F.
| year = 2004
| title = Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins
| publisher = Springer
| edition = 2nd
| page = [https://archive.org/details/encyclopaediaofs0000card/page/20 20]
| isbn= 1-85233-682-X
| url= https://archive.org/details/encyclopaediaofs0000card
| url-access = registration
}}</ref> the laws for systems of [[spherical geometry]] contain factors of 4''π'' (for example, [[point charge]]s), those of cylindrical geometry – factors of 2π (for example, [[wire]]s), and those of planar geometry contain no factors of ''π'' (for example, parallel-plate [[capacitor]]s). However, the original CGS system used ''λ'' = ''λ''′ = 4''π'', or, equivalently, <math>k_{\rm C} \epsilon_0 = \alpha_{\rm B} / (\mu_0 \alpha_{\rm L}) = 1.</math> Therefore, Gaussian, ESU, and EMU subsystems of CGS (described below) are not rationalized.
=== Various extensions of the CGS system to electromagnetism ===
The table below shows the values of the above constants used in some common CGS subsystems:
{| class="wikitable" style="text-align: center;"
|-
! System !!width=75| <math>k_{\rm C}</math> !!width=75| <math>\alpha_{\rm B}</math> !!width=75| <math>\epsilon_0</math>!!width=75|<math>\mu_0</math>!! <math>k_{\rm A}=\frac{k_{\rm C}}{c^2}</math>!! <math>\alpha_{\rm L}=\frac{k_{\rm C}}{\alpha_{\rm B}c^2}</math>!! <math>\lambda=4\pi k_{\rm C}\epsilon_0</math>!! style="width:75px;"|<math>\lambda'=\frac{4\pi\alpha_{\rm B}}{\mu_0\alpha_{\rm L}}</math>
|-
| style="text-align:left;"| Electrostatic<ref name=Jack/> CGS<br />(ESU, esu, or stat-) || 1 || ''c''{{i sup|−2}} || 1 ||''c''{{i sup|−2}} ||''c''{{i sup|−2}} || 1 || 4''π'' || 4''π''
|-
| style="text-align:left;"| Electromagnetic<ref name=Jack/> CGS<br />(EMU, emu, or ab-) || ''c''{{i sup|2}} || 1 || ''c''{{i sup|−2}} || 1|| 1|| 1|| 4''π'' || 4''π''
|-
| style="text-align:left;"| [[Gaussian units|Gaussian]]<ref name=Jack/> CGS || 1 || ''c''{{i sup|−1}} || 1 || 1 || ''c''{{i sup|−2}} || ''c''{{i sup|−1}} || 4''π'' || 4''π''
|-
| style="text-align:left;"| [[Heaviside–Lorentz units|Heaviside–Lorentz]]<ref name=Jack/> CGS || <math>\frac{1}{4\pi}</math> || <math>\frac{1}{4\pi c}</math> || 1 || 1 ||<math>\frac{1}{4\pi c^2}</math> || ''c''{{i sup|−1}} || 1 || 1
|-
| [[SI]] || <math>\frac{1}{4\pi\epsilon_0}</math> || <math>\frac{\mu_0}{4\pi}</math> || <math>\epsilon_0</math>||<math>\mu_0</math>||<math>\frac{\mu_0}{4\pi}</math> || 1 || 1 || 1
|}
Also, note the following correspondence of the above constants to those in Jackson<ref name=Jack/> and Leung:<ref name=leu/>
<math display="block">\begin{align}
k_{\rm C} & =k_1=k_{\rm E} \\
\alpha_{\rm B} &=\alpha\cdot k_2=k_{\rm B} \\
k_{\rm A} &= k_2 =k_{\rm E}/c^2 \\
\alpha_{\rm L} &= k_3 = k_{\rm F}
\end{align}</math>
Of these variants, only in Gaussian and Heaviside–Lorentz systems <math>\alpha_{\rm L}</math> equals <math>c^{-1}</math> rather than 1. As a result, vectors <math>\vec E</math> and <math>\vec B</math> of an [[electromagnetic wave]] propagating in vacuum have the same units and are equal in [[Magnitude (mathematics)#Euclidean vector space|magnitude]] in these two variants of CGS.
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.
|-
! 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
=== 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). In the EMU CGS subsystem, this is done by setting the Ampere force constant <math>k_{\rm A} = 1</math>, so that [[Ampère's force law]] simply contains 2 as an explicit [[Proportionality (mathematics)|factor]].
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/>
=== Relations between ESU and EMU units ===
The ESU and EMU subsystems of CGS are connected by the fundamental relationship <math>k_{\rm C} / k_{\rm A} = c^2</math> (see above), where ''c'' = {{val|29979245800}} ≈ {{val|3|e=10}} is the [[speed of light]] in vacuum in centimetres per second. Therefore, the ratio of the corresponding "primary" electrical and magnetic units (e.g. current, charge, voltage, etc. – quantities proportional to those that enter directly into [[Coulomb's law]] or [[Ampère's force law]]) is equal either to ''c''<sup>−1</sup> or ''c'':<ref name=cardsgc/>
: <math>\mathrm{\frac{1\,statcoulomb}{1\,abcoulomb}}=
\mathrm{\frac{1\,statampere}{1\,abampere}}=c^{-1}</math>
and
: <math>\mathrm{\frac{1\,statvolt}{1\,abvolt}}=
\mathrm{\frac{1\,stattesla}{1\,gauss}}=c</math>.
Units derived from these may have ratios equal to higher powers of ''c'', for example:
: <math>\mathrm{\frac{1\,statohm}{1\,abohm}}=
\mathrm{\frac{1\,statvolt}{1\,abvolt}}\times\mathrm{\frac{1\,abampere}{1\,statampere}}=c^2</math>.
=== Practical CGS units ===
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=''λ''{{lambda}} = 1}} and {{nowrap|1=''λ''{{lambda}}′ = 4π4{{pi}}}}, but the above formula for ''λ''{{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
|publisher = U.S. Government Printing Office
|location = Washington, D.C.
}}</ref> which has a different unit of mass so that the formula for ''λ''{{lambda}}′ 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''π''{{pi}}/10)''μμ'''''H'''}} and {{nowrap|1='''B''' = (4''π''{{pi}}/10)''μμ''<sub>0</sub>'''H''' + ''μμ''<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 ===
== Electromagnetic units in various CGS systems ==
{| class="wikitable"
|+ Conversion of SI units in electromagnetism to ESU, EMU, and Gaussian subsystems of CGS<ref name=cardsgc/>{{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 flux]]
| style="text-align:center;"| ''Φ''<sub>E</sub> ||1 [[Volt|V]]⋅[[Metre|m]] || colspan="2" | ≘ (4π × 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<sup>−1<Fr/sup>s) || ≘ (10<sup>−1</sup>) [[BiotAbampere|abA]] (unitBi)|Bi]]
|-
! [[electric potential]] / [[voltage]]
| style="text-align:center;"|''φ'' / ''V, UE''||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]]
|-
! [[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>]] (Fr/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]]
|-
! [[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]]
|-
! [[magnetic dipole momentpermittivity]]
| style="text-align:center;"|'''μ''' {{mvar|ε}} ||1 [[AmpereFarad|AF]]⋅/[[Square metreMetre|m<sup>2</sup>]] || colspan="2" | ≘ (4{{pi}} × 10<sup>3−11</sup> ''c'') [[Statcoulomb|statC]]⋅[[square centimetre|cm<sup>2</sup>) [[Centimetre|cm]]/cm || colspan="2"≘ |(4{{pi}} ≘× (10<sup>3−11</sup>) [[AbampereSecond|Bis]]⋅[[square centimetre|cm<sup>2</sup>/[[Centimetre|cm]] = (10<sup>32</sup>) [[erg]]/[[Gauss (unit)|G]]
|-
! [[Magnetic field|magnetic B field]]
|-
! [[Magnetic field|magnetic H field]]
| style="text-align:center;"|'''H'''||1 [[Ampere|A]]/[[Metre|m]] || ≘ (4π4{{pi}} × 10<sup>−3</sup> ''c'') [[StatAmpere|statA]]/[[Centimetre|cm]] || colspan="2" | ≘ (4π4{{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]] (Bi⋅cm<sup>2</sup>)
|-
! [[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
|-
! [[electric resistance|resistance]]
|}
In this table, ''c'' = {{val|29979245800}} is the dimensionless 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 quantitiesunits are ''corresponding'' but not in general ''equal'', even between CGS variants. For example, according to the next-to-lastcapacitance row of the table, if a capacitor has a capacitance of 1 F in SI, then it has a capacitance of (10<sup>−9</sup> ''c''<sup>2</sup>) cm in ESU; ''but'' it is incorrect to replace "1 F" with "(10<sup>−9</sup> ''c''<sup>2</sup>) cm" within an equation or formula. (This warning is a special aspect of electromagnetism units in CGS. By contrast, for example, it is ''always'' correct to replace, e.g., "1 m" with "100 cm" within an equation or formula.)
One can think of the SI value of the [[Coulomb constant]] ''k''<sub>C</sub> as:
: <math>k_{\rm C}=\frac{1}{4\pi\epsilon_0}=\frac{\mu_0 (c/100)^2}{4\pi}=10^{-7}~\mathrm{N/A^2} \cdot 10^{-4} \cdot c^2 = 10^{-11}~\mathrm{N} \cdot c^2/\mathrm{A^2}.</math>
This explains why SI to ESU conversions involving factors of ''c''<sup>2</sup> lead to significant simplifications of the ESU units, such as 1 statF = 1 cm and 1 statΩ = 1 s/cm: this is the consequence of the fact that in ESU system ''k''<sub>C</sub> = 1. For example, a centimetre of capacitance is the capacitance of a sphere of radius 1 cm in vacuum. The capacitance ''C'' between two concentric spheres of radii ''R'' and ''r'' in ESU CGS system is:
: <math>\frac{1}{\frac{1}{r}-\frac{1}{R}}</math>.
By taking the limit as ''R'' goes to infinity we see ''C'' equals ''r''.
== Physical constants in CGS units ==
| [[atomic mass constant]]
| style="text-align:center;"| ''m''{{sub|u}}
| {{val|1.660539066660539069|e=-24|ul=g}}
|-
| rowspan="2"|[[Bohr magneton]]
| style="text-align:center;" rowspan="2"|''μ''<sub>B</sub>
| {{val|9.274010078274010066|e=-21|u=[[erg]]/[[Gauss (unit)|G]]}} (EMU, Gaussian)
|-
| {{val|2.780 278 00 × 780278273|e=-10<sup>−10</sup> |u=statA⋅cm<sup>2</sup>}} (ESU)
|-
| [[Bohr radius]]
| style="text-align:center;"| ''a''<sub>0</sub>
| {{val|5.2917721090291772105|e=-9|ul=cm}}
|-
| [[Boltzmann constant]]
| [[electron mass]]
| style="text-align:center;"| ''m''<sub>e</sub>
| {{val|9.1093837010938371|e=-28|ul=g}}
|-
| rowspan="2"|[[elementary charge]]
| style="text-align:center;" rowspan="2"|''e''
| {{val|4.803 204 27 × 80320471|e=-10<sup>−10</sup> |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=912|ref=no}}
|-
| [[Newtonian constant of gravitation]]
| style="text-align:center;"| ''G''
| {{val|6.674306743|e=-8|u=[[dyne|dyn]]⋅[[Centimetre|cm]]<sup>2</sup>/[[Gram|g]]<sup>2</sup>}}
|-
| [[Planck constant]]
== Advantages and disadvantages ==
While the absence of constant coefficients in the formulae expressing some relation between the quantities in some CGS subsystems simplifies some calculations, it has the disadvantage that sometimes the units in CGS are hard to define through experiment. Also, lackLack of unique unit names leads to a greatpotential confusion: thus "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]]. On the other hand, SI starts with a unit of current, the [[ampere]], that is easier to determine through experiment, but which requires extra coefficients in the electromagnetic equations. With its system of uniquely named units, the SI also removes any confusion in usage: 1 ampere is a fixed value of a specified quantity, and so are 1 [[henry (unit)|henry]], 1 [[ohm]], and 1 volt.
An advantage ofIn the [[Gaussian units|CGS-Gaussian system]] is that, electric and magnetic fields have the same units, 4''πε''{{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), but it is a "rationalized" system (as is SI) in which the charges and fields are defined in such a way that there are fewer factors of 4''π'' appearing in the formulas, and it is in Heaviside–Lorentz units that the Maxwell equations take their simplest form.
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''π''{{pi}}, those concerning coils of current and straight wires contain 2''π''{{pi}} and those dealing with charged surfaces lack ''π''{{pi}} entirely, which was the most convenient choice for applications in [[electrical engineering]]. However, modern [[calculator|hand calculator]]s and [[personalrelates computer]]sdirectly haveto eliminatedthe thisgeometric "advantage".symmetry Inof somethe fieldssystem wherebeing formulas concerning spheres are common (for example, in astrophysics), it has been argueddescribed by Einstein himself that the nonrationalized CGS system can be somewhat more convenient notationallyequation.
Specialized unit systems are used to simplify formulas even 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 electronvoltsunits of energy by inserting factors of [[speed of light|speed of light]] ''c'']] and the [[Planck constant|reduced Planck constant]] ''ħ'']]. This unit system is convenient for calculations in [[particle physics]], but it would be consideredis impractical in other contexts.
== See also ==
|