# Centimetre–gram–second system of units

The centimetre–gram–second system of units (abbreviated 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 mechanical units are unambiguously derived from these three base units, but there are several different ways of extending the CGS system to cover electromagnetism.[1][2][3]

The CGS system has been largely supplanted by the 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 there remain certain subfields where CGS is prevalent.

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 factors are all powers of 10 as 100 cm = 1 m and 1000 g = 1 kg. For example, the CGS unit of force is the dyne which is defined as 1 g⋅cm/s2, so the SI unit of force, the newton (1 kg⋅m/s2), is equal to 100,000 dynes.

On the other hand, in measurements of electromagnetic phenomena (involving units of charge, electric and magnetic fields, voltage, and so on), converting between CGS and SI is more subtle. Formulas for physical laws of electromagnetism (such as Maxwell's equations) need to be adjusted depending on which system of units one uses. This is because there is no one-to-one correspondence between electromagnetic units in SI and those in CGS, as is the case for mechanical units. Furthermore, within CGS, there are several plausible choices of electromagnetic units, leading to different unit "sub-systems", including Gaussian units, "ESU", "EMU", and Lorentz–Heaviside units. Among these choices, Gaussian units are the most common today, and "CGS units" often used specifically refers to CGS-Gaussian units.

## 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.[4] Gauss chose the units of millimetre, milligram and second.[5] In 1873, a committee of the British Association for the Advancement of Science, including British physicists James Clerk Maxwell and William Thomson 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 ...".[6]

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 general 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, in the United States more slowly than elsewhere. CGS units are today no longer accepted by the house styles of most scientific journals, textbook publishers, or standards bodies, although they are commonly used in astronomical journals such as The Astrophysical Journal. CGS units are still occasionally encountered in technical literature, especially in the United States in the fields of material science, electrodynamics and astronomy. The continued usage of CGS units is most prevalent in magnetism and related fields because the B and H fields have the same units in free space and there is a lot of potential for confusion when converting published measurements from cgs to MKS.[7]

The units gram and centimetre remain useful as prefixed units within the SI system, especially for instructional physics and chemistry experiments, where they match the small scale of table-top setups. However, where derived units are needed, the SI ones are generally used and taught instead of the CGS ones today. For example, a physics lab course might ask students to record lengths in centimetres, and masses in grams, but force (a derived unit) in newtons, a usage consistent with the SI system.

## Definition of CGS units in mechanics

In mechanics, the CGS and SI systems of units are built in an identical way. The two systems differ only in the scale of two out of the three base units (centimetre versus metre and gram versus kilogram, respectively), while the third unit (second as the unit of time) is the same in both systems.

There is a one-to-one correspondence between the base units of mechanics in CGS and SI, and the laws of mechanics are not affected by the choice of units. The definitions of all derived units in terms of the three base units are therefore the same in both systems, and there is an unambiguous one-to-one correspondence of derived units:

${\displaystyle v={\frac {dx}{dt}}}$  (definition of velocity)
${\displaystyle F=m{\frac {d^{2}x}{dt^{2}}}}$  (Newton's second law of motion)
${\displaystyle E=\int {\vec {F}}\cdot \mathrm {d\,} {\vec {x}}}$  (energy defined in terms of work)
${\displaystyle p={\frac {F}{L^{2}}}}$  (pressure defined as force per unit area)
${\displaystyle \eta =\tau /{\frac {dv}{dx}}}$  (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, is related to the SI base units of length, mass, and time:

1 unit of pressure = 1 unit of force/(1 unit of length)2 = 1 unit of mass/(1 unit of length⋅(1 unit of time)2)
1 Ba = 1 g/(cm⋅s2)
1 Pa = 1 kg/(m⋅s2).

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 Ba = 1 g/(cm⋅s2) = 10−3 kg/(10−2 m⋅s2) = 10−1 kg/(m⋅s2) = 10−1 Pa.

### Definitions and conversion factors of CGS units in mechanics

Quantity Quantity symbol CGS unit name Unit
symbol
Unit definition Equivalent
in SI units
length, position L, x centimetre cm 1/100 of metre = 10−2 m
mass m gram g 1/1000 of kilogram = 10−3 kg
time t second s 1 second = 1 s
velocity v centimetre per second cm/s cm/s = 10−2 m/s
acceleration a gal Gal cm/s2 = 10−2 m/s2
force F dyne dyn g⋅cm/s2 = 10−5 N
energy E erg erg g⋅cm2/s2 = 10−7 J
power P erg per second erg/s g⋅cm2/s3 = 10−7 W
pressure p barye Ba g/(cm⋅s2) = 10−1 Pa
dynamic viscosity μ poise P g/(cm⋅s) = 10−1 Pa⋅s
kinematic viscosity ν stokes St cm2/s = 10−4 m2/s
wavenumber k kayser (K) cm−1[8] cm−1 = 100 m−1

## Derivation of CGS units in electromagnetism

### CGS approach to electromagnetic units

The conversion factors relating electromagnetic units in the CGS and SI systems are made more complex by the differences in the formulae expressing physical laws of electromagnetism as assumed by each system of units, specifically in the nature of the constants that appear in these formulae. 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 magnetic force exerted by two infinitely long, thin, parallel wires 1 metre apart and carrying a current of 1 ampere is exactly 2×10−7 N/m. This definition results in all SI electromagnetic units being consistent (subject to factors of some integer powers of 10) with the EMU CGS 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 permittivity) 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, electric charge q is defined as current I multiplied by time t,
${\displaystyle q=I\cdot t}$,
therefore the unit of electric charge, the coulomb (C), is defined as 1 C = 1 A⋅s.
• The CGS system avoids introducing new base quantities and units, and instead derives all electric and magnetic units directly from the centimetre, gram, and second by specifying the form of the expression of physical laws that relate electromagnetic phenomena to mechanics.

### Alternate 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 rate of change (electric current) to a mechanical quantity such as force. They can be written[9] in system-independent form as follows:

• The first is Coulomb's law, ${\displaystyle F=k_{\rm {C}}{\frac {q\cdot q^{\prime }}{d^{2}}}}$, which describes the electrostatic force F between electric charges ${\displaystyle q}$ and ${\displaystyle q^{\prime }}$, separated by distance d. Here ${\displaystyle k_{\rm {C}}}$ 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, ${\displaystyle {\frac {dF}{dL}}=2k_{\rm {A}}{\frac {I\,I^{\prime }}{d}}}$, 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 ${\displaystyle I=q/t\,}$ and ${\displaystyle I^{\prime }=q^{\prime }/t}$, the constant ${\displaystyle k_{\rm {A}}}$ also depends on how the unit of charge is derived from the base units.

Maxwell's theory of electromagnetism relates these two laws to each other. It states that the ratio of proportionality constants ${\displaystyle k_{\rm {C}}}$ and ${\displaystyle k_{\rm {A}}}$ must obey ${\displaystyle k_{\rm {C}}/k_{\rm {A}}=c^{2}}$, where c is the speed of light in vacuum. Therefore, if one derives the unit of charge from the Coulomb's law by setting ${\displaystyle k_{\rm {C}}=1}$ then Ampère's force law will contain a prefactor ${\displaystyle 2/c^{2}}$. Alternatively, deriving the unit of current, and therefore the unit of charge, from the Ampère's force law by setting ${\displaystyle k_{\rm {A}}=1}$ or ${\displaystyle k_{\rm {A}}=1/2}$, will lead to a constant prefactor 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:
${\displaystyle \mathbf {F} =\alpha _{\rm {L}}q\;\mathbf {v} \times \mathbf {B} \;.}$
• The second describes the creation of a static magnetic field B by an electric current I of finite length dl at a point displaced by a vector r, known as Biot–Savart law:
${\displaystyle d\mathbf {B} =\alpha _{\rm {B}}{\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}\;,}$ where r and ${\displaystyle \mathbf {\hat {r}} }$ 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: ${\displaystyle k_{\rm {A}}=\alpha _{\rm {L}}\cdot \alpha _{\rm {B}}\;}$. Therefore, if the unit of charge is based on the Ampère's force law such that ${\displaystyle k_{\rm {A}}=1}$, it is natural to derive the unit of magnetic field by setting ${\displaystyle \alpha _{\rm {L}}=\alpha _{\rm {B}}=1\;}$. 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 ε0 and μ0, which are the vacuum permittivity and permeability, respectively. Then we have[9] (generally) ${\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} +\lambda \mathbf {P} }$ and ${\displaystyle \mathbf {H} =\mathbf {B} /\mu _{0}-\lambda ^{\prime }\mathbf {M} }$, 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" ${\displaystyle 4\pi k_{\rm {C}}\epsilon _{0}}$ and ${\displaystyle 4\pi \alpha _{\rm {B}}/(\mu _{0}\alpha _{\rm {L}})}$, respectively. If the rationalization constants are equal, then ${\displaystyle c^{2}=1/(\epsilon _{0}\mu _{0}\alpha _{\rm {L}}^{2})}$. If they are equal to one, then the system is said to be "rationalized":[10] the laws for systems of spherical geometry contain factors of 4π (for example, point charges), those of cylindrical geometry – factors of 2π (for example, wires), and those of planar geometry contain no factors of π (for example, parallel-plate capacitors). However, the original CGS system used λ = λ′ = 4π, or, equivalently, ${\displaystyle k_{\rm {C}}\epsilon _{0}=\alpha _{\rm {B}}/(\mu _{0}\alpha _{\rm {L}})=1}$. 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:

system ${\displaystyle k_{\rm {C}}}$ ${\displaystyle \alpha _{\rm {B}}}$ ${\displaystyle \epsilon _{0}}$ ${\displaystyle \mu _{0}}$ ${\displaystyle k_{\rm {A}}={\frac {k_{\rm {C}}}{c^{2}}}}$ ${\displaystyle \alpha _{\rm {L}}={\frac {k_{\rm {C}}}{\alpha _{\rm {B}}c^{2}}}}$ ${\displaystyle \lambda =4\pi k_{\rm {C}}\epsilon _{0}}$ ${\displaystyle \lambda '={\frac {4\pi \alpha _{\rm {B}}}{\mu _{0}\alpha _{\rm {L}}}}}$
Electrostatic[9] CGS
(ESU, esu, or stat-)
1 c−2 1 c−2 c−2 1
Electromagnetic[9] CGS
(EMU, emu, or ab-)
c2 1 c−2 1 1 1
Gaussian[9] CGS 1 c−1 1 1 c−2 c−1
Lorentz–Heaviside[9] CGS ${\displaystyle {\frac {1}{4\pi }}}$ ${\displaystyle {\frac {1}{4\pi c}}}$ 1 1 ${\displaystyle {\frac {1}{4\pi c^{2}}}}$ c−1 1 1
SI ${\displaystyle {\frac {c^{2}}{b}}}$ ${\displaystyle {\frac {1}{b}}}$ ${\displaystyle {\frac {b}{4\pi c^{2}}}}$ ${\displaystyle {\frac {4\pi }{b}}}$ ${\displaystyle {\frac {1}{b}}}$ 1 1 1

The constant b in SI system is a unit-based scaling factor defined as: ${\displaystyle b=10^{7}\,\mathrm {A} ^{2}/\mathrm {N} =10^{7}\,\mathrm {m/H} =4\pi /\mu _{0}=4\pi \epsilon _{0}c^{2}=c^{2}/k_{\rm {C}}\;}$.

Also, note the following correspondence of the above constants to those in Jackson[9] and Leung:[11]

${\displaystyle k_{\rm {C}}=k_{1}=k_{\rm {E}}}$
${\displaystyle \alpha _{\rm {B}}=\alpha \cdot k_{2}=k_{\rm {B}}}$
${\displaystyle k_{\rm {A}}=k_{2}=k_{\rm {E}}/c^{2}}$
${\displaystyle \alpha _{\rm {L}}=k_{3}=k_{\rm {F}}}$

In system-independent form, Maxwell's equations can be written as:[9][11]

${\displaystyle {\begin{array}{ccl}{\vec {\nabla }}\cdot {\vec {E}}&=&4\pi k_{\rm {C}}\rho \\{\vec {\nabla }}\cdot {\vec {B}}&=&0\\{\vec {\nabla }}\times {\vec {E}}&=&\displaystyle {-\alpha _{\rm {L}}{\frac {\partial {\vec {B}}}{\partial t}}}\\{\vec {\nabla }}\times {\vec {B}}&=&\displaystyle {4\pi \alpha _{\rm {B}}{\vec {J}}+{\frac {\alpha _{\rm {B}}}{k_{\rm {C}}}}{\frac {\partial {\vec {E}}}{\partial t}}}\end{array}}}$

Note that of all these variants, only in Gaussian and Heaviside–Lorentz systems ${\displaystyle \alpha _{\rm {L}}}$ equals ${\displaystyle c^{-1}}$ rather than 1. As a result, vectors ${\displaystyle {\vec {E}}}$ and ${\displaystyle {\vec {B}}}$ of an electromagnetic wave propagating in vacuum have the same units and are equal in magnitude in these two variants of CGS.

### Electrostatic units (ESU)

In one variant of the CGS system, Electrostatic units (ESU), charge is defined via the force it exerts on other charges, and current is then defined as charge per time. It is done by setting the Coulomb force constant ${\displaystyle k_{\rm {C}}=1}$, so that Coulomb's law does not contain an explicit prefactor.

The ESU unit of charge, franklin (Fr), also known as statcoulomb or esu charge, is therefore defined as follows:[12]

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 electrostatic CGS units, a franklin is equal to a centimetre times square root of dyne:

${\displaystyle \mathrm {1\,Fr=1\,statcoulomb=1\,esu\;charge=1\,cm{\sqrt {dyne}}=1\,g^{1/2}\cdot cm^{3/2}\cdot s^{-1}} }$.

The unit of current is defined as:

${\displaystyle \mathrm {1\,Fr/s=1\,statampere=1\,esu\;current=1\,(cm/s){\sqrt {dyne}}=1\,g^{1/2}\cdot cm^{3/2}\cdot s^{-2}} }$.

Dimensionally in the ESU CGS system, charge q is therefore equivalent to m1/2L3/2t−1. Hence, neither charge nor current is an independent physical quantity in ESU CGS. This reduction of units is the consequence of the Buckingham π theorem.

#### ESU notation

All electromagnetic units in ESU CGS system that do not have proper names are denoted by a corresponding SI name with an attached prefix "stat" or with a separate abbreviation "esu".[12]

### Electromagnetic units (EMU)

In another variant of the CGS system, electromagnetic units (EMUs), 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 ${\displaystyle k_{\rm {A}}=1}$, so that Ampère's force law simply contains 2 as an explicit prefactor (this prefactor 2 is itself a result of integrating a more general formulation of Ampère's law over the length of the infinite wire).

The EMU unit of current, biot (Bi), also known as abampere or emu current, is therefore defined as follows:[12]

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 dynes per centimetre of length.

Therefore, in electromagnetic CGS units, a biot is equal to a square root of dyne:

${\displaystyle \mathrm {1\,Bi=1\,abampere=1\,emu\;current=1\,{\sqrt {dyne}}=1\,g^{1/2}\cdot cm^{1/2}\cdot s^{-1}} }$.

The unit of charge in CGS EMU is:

${\displaystyle \mathrm {1\,Bi\cdot s=1\,abcoulomb=1\,emu\,charge=1\,s\cdot {\sqrt {dyne}}=1\,g^{1/2}\cdot cm^{1/2}} }$.

Dimensionally in the EMU CGS system, charge q is therefore equivalent to m1/2L1/2. Hence, neither charge nor current is an independent physical quantity in EMU CGS.

#### EMU notation

All electromagnetic units in EMU CGS 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".[12]

### Relations between ESU and EMU units

The ESU and EMU subsystems of CGS are connected by the fundamental relationship ${\displaystyle k_{\rm {C}}/k_{\rm {A}}=c^{2}}$ (see above), where c = 29,979,245,800 ≈ 3⋅1010 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−1 or c:[12]

${\displaystyle \mathrm {\frac {1\,statcoulomb}{1\,abcoulomb}} =\mathrm {\frac {1\,statampere}{1\,abampere}} =c^{-1}}$

and

${\displaystyle \mathrm {\frac {1\,statvolt}{1\,abvolt}} =\mathrm {\frac {1\,stattesla}{1\,gauss}} =c}$.

Units derived from these may have ratios equal to higher powers of c, for example:

${\displaystyle \mathrm {\frac {1\,statohm}{1\,abohm}} =\mathrm {\frac {1\,statvolt}{1\,abvolt}} \times \mathrm {\frac {1\,abampere}{1\,statampere}} =c^{2}}$.

### Practical cgs units

The practical cgs system is a hybrid system that uses the volt and the ampere as the unit of voltage and current respectively. Doing this avoids the inconveniently large and small quantities that arise for electromagnetic units 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.[13] As well as the volt and amp, the farad (capacitance), ohm (resistance), coulomb (electric charge), and henry are consequently also used in the practical system and are the same as the SI units. However, intensive properties (that is, anything that is per unit length, area, or volume) will not be the same as SI since the cgs unit of distance is the centimetre. For instance electric field strength is in units of volts per centimetre, magnetic field strength is in oersteds and resistivity is in ohm-cm.[14]

Some physicists and electrical engineers in North America still use these hybrid units.[15]

### 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.[16] These also include the Gaussian units and the Heaviside–Lorentz units.

## Electromagnetic units in various CGS systems

Conversion of SI units in electromagnetism to ESU, EMU, and Gaussian subsystems of CGS[12]
c = 29,979,245,800
Quantity Symbol SI unit ESU unit EMU unit Gaussian unit
electric charge q 1 C ↔ (10−1 c) statC ↔ (10−1) abC ↔ (10−1 c) Fr
electric flux ΦE 1 Vm ↔ (4π×10−1 c) statC ↔ (10−1) abC ↔ (4π×10−1 c) Fr
electric current I 1 A ↔ (10−1 c) statA ↔ (10−1) abA ↔ (10−1 c) Fr⋅s−1
electric potential / voltage φ / V 1 V ↔ (108 c−1) statV ↔ (108) abV ↔ (108 c−1) statV
electric field E 1 V/m ↔ (106 c−1) statV/cm ↔ (106) abV/cm ↔ (106 c−1) statV/cm
electric displacement field D 1 C/m2 ↔ (10−5 c) statC/cm2 ↔ (10−5) abC/cm2 ↔ (10−5 c) Fr/cm2
electric dipole moment p 1 Cm ↔ (10 c) statCcm ↔ (10) abCcm ↔ (1019 c) D
magnetic dipole moment μ 1 Am2 ↔ (103 c) statAcm2 ↔ (103) abAcm2 ↔ (103) erg/G
magnetic B field B 1 T ↔ (104 c−1) statT ↔ (104) G ↔ (104) G
magnetic H field H 1 A/m ↔ (4π×10−3 c) statA/cm ↔ (4π×10−3) Oe ↔ (4π×10−3) Oe
magnetic flux Φm 1 Wb ↔ (108 c−1) statWb ↔ (108) Mx ↔ (108) Mx
resistance R 1 Ω ↔ (109 c−2) s/cm ↔ (109) abΩ ↔ (109 c−2) s/cm
resistivity ρ 1 Ωm ↔ (1011 c−2) s ↔ (1011) abΩcm ↔ (1011 c−2) s
capacitance C 1 F ↔ (10−9 c2) cm ↔ (10−9) abF ↔ (10−9 c2) cm
inductance L 1 H ↔ (109 c−2) cm−1s2 ↔ (109) abH ↔ (109 c−2) cm−1s2

In this table, c = 29,979,245,800 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 SI and CGS units are corresponding but not equal because they have incompatible dimensions. For example, according to the next-to-last row of the table, if a capacitor has a capacitance of 1 F in SI, then it has a capacitance of (10−9 c2) cm in ESU; but it is usually incorrect to replace "1 F" with "(10−9 c2) 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 "1 m" with "100 cm" within an equation or formula.)

One can think of the SI value of the Coulomb constant kC as:

${\displaystyle k_{\rm {C}}={\frac {1}{4\pi \epsilon _{0}}}={\frac {\mu _{0}(c/100)^{2}}{4\pi }}=10^{-7}{\rm {N}}/{\rm {A}}^{2}\cdot 10^{-4}\cdot c^{2}=10^{-11}{\rm {N}}\cdot c^{2}/{\rm {A}}^{2}.}$

This explains why SI to ESU conversions involving factors of c2 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 kC = 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:

${\displaystyle {\frac {1}{{\frac {1}{r}}-{\frac {1}{R}}}}}$.

By taking the limit as R goes to infinity we see C equals r.

## Physical constants in CGS units

Commonly used physical constants in CGS units[17]
Constant Symbol Value
Atomic mass unit u 1.660 538 782 × 10−24 g
Bohr magneton μB 9.274 009 15 × 10−21 erg/G (EMU, Gaussian)
2.780 278 00 × 10−10 statA⋅cm2 (ESU)
Bohr radius a0 5.291 772 0859 × 10−9 cm
Boltzmann constant k 1.380 6504 × 10−16 erg/K
Electron mass me 9.109 382 15 × 10−28 g
Elementary charge e 4.803 204 27 × 10−10 Fr (ESU, Gaussian)
1.602 176 487 × 10−20 abC (EMU)
Fine-structure constant α ≈ 1/137 7.297 352 570 × 10−3
Gravitational constant G 6.674 28 × 10−8

Dyncm2/(g2)

Planck constant h 6.626 068 85 × 10−27 ergs
ħ 1.054 5716 × 10−27 ergs
Speed of light in vacuum c ≡ 2.997 924 58 × 1010 cm/s

While the absence of explicit prefactors in some CGS subsystems simplifies some theoretical calculations, it has the disadvantage that sometimes the units in CGS are hard to define through experiment. Also, lack of unique unit names leads to a great confusion: thus "15 emu" may mean either 15 abvolts, or 15 emu units of electric dipole moment, or 15 emu units of magnetic susceptibility, sometimes (but not always) per gram, or per 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 multiplicative factors in the electromagnetic equations. With its system of uniquely named units, the SI also removes any confusion in usage: 1.0 ampere is a fixed value of a specified quantity, and so are 1.0 henry, 1.0 ohm, and 1.0 volt.

A key virtue of the Gaussian CGS system is that electric and magnetic fields have the same units, 4πε0 is replaced by 1, and the only dimensional constant appearing in the Maxwell equations is c, the speed of light. The Heaviside–Lorentz system has these desirable properties as well (with ε0 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 many fewer factors of 4π 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), the unit of current was chosen such that electromagnetic equations concerning charged spheres contain 4π, those concerning coils of current and straight wires contain 2π and those dealing with charged surfaces lack π entirely, which was the most convenient choice for applications in electrical engineering. However, modern hand calculators and personal computers have eliminated this "advantage". In some fields where formulas concerning spheres are common (for example, in astrophysics), it has been argued that the nonrationalized CGS system can be somewhat more convenient notationally.

Specialized unit systems are used to simplify formulas even further than either SI or CGS, by eliminating constants through 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 electronvolts by inserting factors of speed of light c and the Planck constant ħ. This unit system is very convenient for calculations in particle physics, but it would be considered impractical in other contexts.

## References and notes

1. ^ "Centimetre-gram-second system | physics". Encyclopedia Britannica. Retrieved 2018-03-27.
2. ^ "The Centimeter-Gram-Second (CGS) System of Units - Maple Programming Help". www.maplesoft.com. Retrieved 2018-03-27.
3. ^ Carron, Neal J. (21 May 2015). "Babel of units: The evolution of units systems in classical electromagnetism" (PDF). Retrieved 31 March 2018.
4. ^ Gauss, C. F. (1832), "Intensitas vis magneticae terrestris ad mensuram absolutam revocata", Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores, 8: 3–44. English translation.
5. ^ Hallock, William; Wade, Herbert Treadwell (1906). Outlines of the evolution of weights and measures and the metric system. New York: The Macmillan Co. p. 200.
6. ^ Thomson, Sir W; Foster, Professor GC; Maxwell, Professor JC; Stoney, Mr GJ; Jenkin, Professor Fleeming; Siemens, Dr; Bramwell, Mr FJ (September 1873). Everett, Professor (ed.). First Report of the Committee for the Selection and Nomenclature of Dynamical and Electrical Units. Forty-third Meeting of the British Association for the Advancement of Science. Bradford: John Murray. p. 223. Retrieved 2012-04-08.
7. ^ Bennett, L. H.; Page, C. H.; Swartzendruber, L. J. (January–February 1978). "Comments on units in magnetism" (PDF). Journal of Research of the National Bureau of Standards. 83 (1): 9–12. Retrieved 15 January 2018.
8. ^ "Atomic Spectroscopy". Atomic Spectroscopy. NIST. Retrieved 25 October 2015.
9. Jackson, John David (1999). Classical Electrodynamics (3rd ed.). New York: Wiley. pp. 775–784. ISBN 0-471-30932-X.
10. ^ Cardarelli, F. (2004). Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins (2nd ed.). Springer. p. 20. ISBN 1-85233-682-X.
11. ^ a b Leung, P. T. (2004). "A note on the 'system-free' expressions of Maxwell's equations". European Journal of Physics. 25 (2): N1–N4. Bibcode:2004EJPh...25N...1L. doi:10.1088/0143-0807/25/2/N01.
12. Cardarelli, F. (2004). Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins (2nd ed.). Springer. pp. 20–25. ISBN 1-85233-682-X.
13. ^ Tunbridge, Paul (1992). Lord Kelvin: His Influence on Electrical Measurements and Units. IET. pp. 34–40. ISBN 0-86341-237-8.
14. ^ Knoepfel, Heinz E. (2000). Magnetic Fields: A Comprehensive Theoretical Treatise for Practical Use. Wiley. p. 543. ISBN 3-527-61742-6.
15. ^ Knoepfel, p. xx
16. ^ Bennett, L. H.; Page, C. H.; Swartzendruber, L. J. (1978). "Comments on units in magnetism". Journal of Research of the National Bureau of Standards. 83 (1): 9–12. doi:10.6028/jres.083.002.
17. ^ A.P. French; Edwind F. Taylor (1978). An Introduction to Quantum Physics. W.W. Norton & Company.

## General literature

Abampere

The abampere (abA), also called the biot (Bi) after Jean-Baptiste Biot, is the derived electromagnetic unit of electric current in the emu-cgs system of units (electromagnetic cgs). One abampere is equal to ten amperes in the SI system of units. An abampere of current in a circular path of one centimeter radius produces a magnetic field of 2π oersteds at the center of the circle.

The name abampere was introduced by Kennelly in 1903 as a short name for the long name (absolute) electromagnetic cgs unit of current that was in use since the adoption of the cgs system in 1875. The abampere was coherent with the emu-cgs system, in contrast to the ampere, the practical unit of current that had been adopted too in 1875.

The emu-cgs (or "electromagnetic cgs") units are one of several systems of electromagnetic units within the centimetre-gram-second system of units; others include esu-cgs, Gaussian units, and Lorentz–Heaviside units. In these other systems, the abampere is not one of the units; the "statcoulomb per second" or statampere is used instead.

The other units in this system related to the abampere are:

abcoulomb – the charge that passes in one second through any cross section of a conductor carrying a steady current of one abampere

abhenry – the self-inductance of a circuit or the mutual inductance of two circuits in which the variation of current at the rate of one abampere per second results in an induced electromotive force of one abvolt

abohm – the resistance of a conductor that, with a constant current of one abampere through it, maintains between its terminals a potential difference of one abvolt

Abcoulomb

The abcoulomb (abC or aC) or electromagnetic unit of charge (emu of charge) is the derived physical unit of electric charge in the cgs-emu system of units. One abcoulomb is equal to ten coulombs.

The name abcoulomb was introduced by Kennelly in 1903 as a short name for the long name (absolute) electromagnetic cgs unit of charge that was in use since the adoption of the cgs system in 1875. The abcoulomb was coherent with the cgs-emu system, in contrast to the coulomb, the practical unit of charge that had been adopted too in 1875.

CGS-emu (or "electromagnetic cgs") units are one of several systems of electromagnetic units within the centimetre gram second system of units; others include CGS-esu, Gaussian units, and Lorentz–Heaviside units. In these other systems, the abcoulomb is not used; CGS-esu and Gaussian units use the statcoulomb is instead, while the Lorentz-Heaviside unit of charge has no specific name.

In the electromagnetic cgs system, electric current is a fundamental quantity defined via Ampère's law and takes the permeability as a dimensionless quantity (relative permeability) whose value in a vacuum is unity. As a consequence, the square of the speed of light appears explicitly in some of the equations interrelating quantities in this system.

The definition of the abcoulomb follows from that of the abampere: given two parallel currents of one abampere separated by one centimetre, the force per distance of wire is 2 dyn/cm. The abcoulomb is the charge flowing in 1 second given a current of 1 abampere.

Abhenry

Abhenry is the CGS (centimeter-gram-second) electromagnetic unit of inductance, equal to one billionth of a henry.

Barye

The barye (symbol: Ba), or sometimes barad, barrie, bary, baryd, baryed, or barie, is the centimetre–gram–second (CGS) unit of pressure. It is equal to 1 dyne per square centimetre.

1 Ba = 0.1 Pa = 1×10−6 bar = 1×10−4 pieze = 0.1 N/m2 = 1 g⋅cm−1⋅s−2

Bril (unit)

The bril is an old, non-SI, unit of luminance. The SI unit of luminance is the candela per square metre.

Centimetre

A centimetre (international spelling as used by the International Bureau of Weights and Measures; symbol cm) or centimeter (American spelling) is a unit of length in the metric system, equal to one hundredth of a metre, centi being the SI prefix for a factor of 1/100. The centimetre was the base unit of length in the now deprecated centimetre–gram–second (CGS) system of units.

Though for many physical quantities, SI prefixes for factors of 103—like milli- and kilo-—are often preferred by technicians, the centimetre remains a practical unit of length for many everyday measurements. A centimetre is approximately the width of the fingernail of an average adult person.

Cubic centimetre

A cubic centimetre (or cubic centimeter in US English) (SI unit symbol: cm3; non-SI abbreviations: cc and ccm) is a commonly used unit of volume that extends the derived SI-unit cubic metre, and corresponds to the volume of a cube that measures 1 cm × 1 cm × 1 cm. One cubic centimetre corresponds to a volume of 1/1,000,000 of a cubic metre, or 1/1,000 of a litre, or one millilitre; thus, 1 cm3 ≡ 1 mL. The mass of one cubic centimetre of water at 3.98 °C (the temperature at which it attains its maximum density) is closely equal to one gram. SI supports only the use of symbols and deprecates the use of any abbreviations for units. Hence cm3 is preferred to cc or ccm.

Many scientific disciplines have replaced cubic centimeter measurements with milliliters, but the medical and automotive fields in the United States still use the term cubic centimetre. Much of the automotive industry outside the U.S. has switched to litres. The United Kingdom uses millilitres in preference to cubic centimetres in the medical field, but not the automotive. Most other English-speaking countries, as well as the Netherlands, follow the UK example.[citation needed]

There is currently a movement within the medical field to discontinue the use of cc in prescriptions and on medical documents, as it can be misread as "00". This could cause a hundredfold overdose of medication, which could be dangerous or even lethal. In the United States, such confusion accounts for 12.6% of all errors associated with medical abbreviations.

In automobile engines, "cc" refers to the total volume of its engine displacement in cubic centimetres. The displacement can be calculated using the formula

${\displaystyle d={\pi \over 4}\times b^{2}\times s\times n}$

where d is engine displacement, b is the bore of the cylinders, s is length of the stroke and n is the number of cylinders.

Conversions

Dyne

The dyne (symbol dyn, from Greek δύναμις, dynamis, meaning power, force) is a derived unit of force specified in the centimetre–gram–second (CGS) system of units, a predecessor of the modern SI.

Electrostatic units

The electrostatic system of units (ESU) is a system of units used to measure quantities of electric charge, electric current, and voltage within the centimeter-gram-second (or "CGS") system of metric units. In electrostatic units, electrical charge is defined by the force that it exerts on other charges.Although the CGS units have mostly been supplanted by the MKSA (meter-kilogram-second-ampere) or International System of Units (SI) units, the electrostatic units are still in occasional use in some applications, most notably in certain fields of physics such as in particle physics and astrophysics.

The main electrostatic units are:

The statcoulomb, called the Franklin or the "esu" for electric charge.

The statvolt for voltage.

The gauss for magnetic induction.

Erg

The erg is a unit of energy and work equal to 10−7 joules. It originated in the centimetre–gram–second (CGS) system of units. It has the symbol erg. The erg is not an SI unit. Its name is derived from ergon (ἔργον), a Greek word meaning work or task.An erg is the amount of work done by a force of one dyne exerted for a distance of one centimeter. In the CGS base units, it is equal to one gram centimeter-squared per second-squared (g⋅cm2/s2). It is thus equal to 10−7 joules or 100 nanojoules (nJ) in SI units. An erg is approximately the amount of work done (or energy consumed) by one common house fly performing one "push up", the leg-bending dip that brings its mouth to the surface on which it stands and back up.

1 erg = 10−7 J = 100 nJ

1 erg = 10−10sn⋅m = 100 psn⋅m = 100 picosthène-metres

1 erg = 624.15 GeV = 6.2415×1011 eV

1 erg = 1 dyn⋅cm = 1 g⋅cm2/s2

Gauss (unit)

The gauss, abbreviated as G or Gs, is the cgs unit of measurement of magnetic flux density (or "magnetic induction") (B). It is named after German mathematician and physicist Carl Friedrich Gauss. One gauss is defined as one maxwell per square centimeter. The cgs system has been superseded by the International System of Units (SI), which uses the tesla (symbol T) as the unit of magnetic flux density. One gauss equals 1×10−4 tesla (100 μT), so 1 tesla = 10,000 gauss.

Gram

The gram (alternative spelling: gramme; SI unit symbol: g; Latin: gramma, from Greek γράμμα, grámma) is a metric system unit of mass.

Originally defined as "the absolute weight of a volume of pure water equal to the cube of the hundredth part of a metre [1 cm3], and at the temperature of melting ice" (later at 4 °C, the temperature of maximum density of water). However, in a reversal of reference and defined units, a gram is now defined as one thousandth of the SI base unit, the kilogram, or 1×10−3 kg, which itself is now defined by the International Bureau of Weights and Measures, not in terms of grams, but by "the amount of electricity needed to counteract its force"

Gram per cubic centimetre

Gram per cubic centimetre is a unit of density in the CGS system, commonly used in chemistry, defined as mass in grams divided by volume in cubic centimetres. The official SI symbols are g/cm3, g·cm−3, or g cm−3. It is equivalent to the units gram per millilitre (g/mL) and kilogram per litre (kg/L). The density of water is about 1 g/cm3, since the gram was originally defined as the mass of one cubic centimetre of water at its maximum density at 4 °C.

Maxwell (unit)

The maxwell (symbol: Mx) is the CGS (centimetre-gram-second) unit of magnetic flux (Φ).

Oersted

The oersted (symbol Oe) is the unit of the auxiliary magnetic field H in the centimetre–gram–second system of units (CGS). It is equivalent to 1 dyne per maxwell.

Phot

A phot (ph) is a photometric unit of illuminance, or luminous flux through an area. It is not an SI unit, but rather is associated with the older centimetre gram second system of units. The name was coined by André Blondel in 1921.

Metric equivalence:

${\displaystyle 1\ \mathrm {phot} =1\ {\frac {\mathrm {lumen} }{\mathrm {centimeter} ^{2}}}=10,000\ {\frac {\mathrm {lumens} }{\mathrm {meter} ^{2}}}=10,000\ \mathrm {lux} =10\ \mathrm {kilolux} }$

Metric dimensions:

illuminance = luminous intensity × solid angle / length2
Poise (unit)

The poise (symbol P; /pɔɪz, pwɑːz/) is the unit of dynamic viscosity (absolute viscosity) in the centimetre–gram–second system of units. It is named after Jean Léonard Marie Poiseuille (see Hagen–Poiseuille equation).

${\displaystyle 1~{\text{P}}=0.1~{\text{kg}}{\cdot }{\text{m}}^{-1}{\cdot }{\text{s}}^{-1}=1~{\text{g}}{\cdot }{\text{cm}}^{-1}{\cdot }{\text{s}}^{-1}=1~{\text{dyne}}{\cdot }{\text{s}}{\cdot }{\text{cm}}^{-2}.}$

The analogous unit in the International System of Units is the pascal-second (Pa⋅s):

${\displaystyle 1~{\text{Pa}}{\cdot }{\text{s}}=1~{\text{N}}{\cdot }{\text{s}}{\cdot }{\text{m}}^{-2}=1~{\text{kg}}{\cdot }{\text{m}}^{-1}{\cdot }{\text{s}}^{-1}=10~{\text{P}}.}$

The poise is often used with the metric prefix centi- because the viscosity of water at 20 °C (NTP) is almost exactly 1 centipoise. A centipoise is one hundredth of a poise, or one millipascal-second (mPa⋅s) in SI units (1 cP = 10−3 Pa⋅s = 1 mPa⋅s).

The CGS symbol for the centipoise is cP. The abbreviations cps, cp, and cPs are sometimes seen.

Liquid water has a viscosity of 0.00890 P at 25 °C at a pressure of 1 atmosphere (0.00890 P = 0.890 cP = 0.890 mPa⋅s).

Statcoulomb

The statcoulomb (statC) or franklin (Fr) or electrostatic unit of charge (esu) is the physical unit for electrical charge used in the esu-cgs (centimetre–gram–second system of units) and Gaussian units. It is a derived unit given by

1 statC = dyn1/2 cm = cm3/2 g1/2 s−1.It can be converted using

1 newton = 105 dyne

1 cm = 10−2 mThe SI system of units uses the coulomb (C) instead. The conversion between C and statC is different in different contexts. The most common contexts are:

For electric charge:

1 C ↔ 2997924580 statC ≈ 3.00×109 statC

⇒ 1 statC ↔ ~3.33564×10−10 C.

For electric flux (ΦD):

1 C ↔ 4π × 2997924580 statC ≈ 3.77×1010 statC

⇒ 1 statC ↔ ~2.65×10−11 C.The symbol "↔" is used instead of "=" because the two sides are not necessarily interchangeable, as discussed below. The number 2997924580 is 10 times the value of the speed of light expressed in meters/second, and the conversions are exact except where indicated. The second context implies that the SI and cgs units for an electric displacement field (D) are related by:

1 C/m2 ↔ 4π × 2997924580×10−4 statC/cm2 ≈ 3.77×106 statC/cm2

⇒ 1 statC/cm2 ↔ ~2.65×10−7 C/m2due to the relation between the metre and the centimetre. The coulomb is an extremely large charge rarely encountered in electrostatics, while the statcoulomb is closer to everyday charges.

Statmho

The statmho is the unit of electrical conductance in the electrostatic system of units (ESU), an extension of the centimeter-gram-second (CGS) system to cover electrical units.

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