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Centimetre–gram–second system of units

From Wikipedia, the free encyclopedia
Physical system of measurement that uses the centimetre, gram, and second as base units
"CGS" redirects here. For other uses, seeCGS (disambiguation).
For a topical guide, seeOutline of the metric system.

Thecentimetre–gram–second system of units (CGS orcgs) is a variant of themetric system based on thecentimetre as the unit oflength, thegram as the unit ofmass, and thesecond as the unit oftime. All CGSmechanical units are unambiguously derived from these three base units, but there are several different ways in which the CGS system was extended to coverelectromagnetism.[1][2][3]

The CGS system has mainly been supplanted by theMKS system based on themetre,kilogram, and second, which was in turn extended and replaced by theInternational 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: theunit-conversion factors are allpowers of 10 as100 cm = 1 m and1000 g = 1 kg. For example, the CGS unit of force is thedyne, which is defined as1 g⋅cm/s2, so the SI unit of force, thenewton (1 kg⋅m/s2), is equal to100000 dynes.

In contrast, converting measurements of electromagnetic quantities — such aselectric charge, electric and magnetic fields, and voltage — between CGS and SI systems is considerably more complex. This is because the form of the equations governing electromagnetic phenomena, includingMaxwell's equations, depends on the system of units employed; electromagnetic quantities are defined differently in SI and in CGS. Moreover, several distinct versions of the CGS system exist, each defining electromagnetic units differently. These include the electrostatic (ESU), electromagnetic (EMU),Gaussian units, andHeaviside–Lorentz units. Gaussian units are the most widely used in modern scientific literature,[4] and the term “CGS units” is often understood to refer specifically to the CGS–Gaussian system.[5]

History

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The CGS system goes back to a proposal in 1832 by the German mathematicianCarl Friedrich Gauss to base a system of absolute units on the three fundamental units of length, mass and time.[6] Gauss chose the units of millimetre, milligram and second.[7] In 1873, a committee of theBritish Association for the Advancement of Science, including physicistsJames Clerk Maxwell andWilliam 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 ...".[8]

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 modernSI 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 favour of SI units byNIST,[9] as well as organisations such as theAmerican Physical Society[10] and theInternational Astronomical Union.[11] SI units are predominantly used inengineering applications and physics education, while Gaussian CGS units are still commonly used in theoretical physics, describing microscopic systems, relativisticelectrodynamics, andastrophysics.[12][13]

The unitsgram andcentimetre remain useful as noncoherent units within the SI system, as with any otherprefixed SI units.

Definition of CGS units in mechanics

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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 arecoherent, the definitions of all coherentderived units in terms of the base units are the same in both systems, and there is an unambiguous relationship between derived units:

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

[edit]
QuantityQuantity symbolCGS unit nameUnit symbolUnit definitionIn SI units
length,positionL,xcentimetrecm1/100 ofmetre10−2 m
massmgramg1/1000 ofkilogram10−3 kg
timetseconds1 second1 s
velocityvcentimetre per secondcm/scm/s10−2 m/s
accelerationagalGalcm/s210−2 m/s2
forceFdynedyng⋅cm/s210−5 N
energyEergergg⋅cm2/s210−7 J
powerPerg per seconderg/sg⋅cm2/s310−7 W
pressurepbaryeBag/(cm⋅s2)10−1 Pa
dynamicviscosityμpoisePg/(cm⋅s)10−1 Pa⋅s
kinematicviscosityνstokesStcm2/s10−4 m2/s
wavenumberkkaysercm−1[14] or Kcm−1100 m−1

Derivation of CGS units in electromagnetism

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CGS approach to electromagnetic units

[edit]

The conversion factors relatingelectromagnetic 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 ofelectric current, the ampere (A), was historically defined such that themagnetic force exerted by two infinitely long, thin, parallel wires 1 metre apart and carrying a current of 1 ampere is exactly2×10−7 N/m. This definition results in allSI electromagnetic units being numerically consistent (subject to factors of someinteger 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 (seeVacuum 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,electric chargeq is defined as currentI multiplied by timet,q=It,{\displaystyle q=I\,t,} resulting in the unit of electric charge, thecoulomb (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

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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 itsrate of change (electric current) to a mechanical quantity such as force. They can be written[12] in system-independent form as follows:

Maxwell's theory of electromagnetism relates these two laws to each other. It states that the ratio of the proportionality constantskC{\displaystyle k_{\rm {C}}} andkA{\displaystyle k_{\rm {A}}} must obeykC/kA=c2{\displaystyle k_{\rm {C}}/k_{\rm {A}}=c^{2}}, wherec is thespeed of light invacuum. Therefore, if one derives the unit of charge from Coulomb's law by settingkC=1{\displaystyle k_{\rm {C}}=1} then Ampère's force law will contain a factor2/c2{\displaystyle 2/c^{2}}. Alternatively, deriving the unit of current, and therefore the unit of charge, from Ampère's force law by settingkA=1{\displaystyle k_{\rm {A}}=1} orkA=1/2{\displaystyle k_{\rm {A}}=1/2}, will lead to a constant factor in Coulomb's law.

Indeed, both of these mutually exclusive approaches have been practiced by users of the 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:

These two laws can be used to deriveAmpère's force law above, resulting in the relationship:kA=αL⋅αB{\displaystyle k_{\rm {A}}=\alpha _{\rm {L}}\cdot \alpha _{\rm {B}}\;}. Therefore, if the unit of charge is based onAmpère's force law such thatkA=1{\displaystyle k_{\rm {A}}=1}, it is natural to derive the unit of magnetic field by settingαL=αB=1{\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 theelectric displacement fieldD and themagnetic fieldH in a medium other than vacuum, we need to also define the constantsε0 andμ0, which are thevacuum permittivity andpermeability, respectively. Then we have[12] (generally)D=ϵ0E+λP{\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} +\lambda \mathbf {P} } andH=B/μ0−λ′M{\displaystyle \mathbf {H} =\mathbf {B} /\mu _{0}-\lambda ^{\prime }\mathbf {M} }, whereP andM arepolarisation density andmagnetisation vectors. The units ofP andM are usually so chosen that the factors 𝜆 and 𝜆′ are equal to the "rationalisation constants"4πkCϵ0{\displaystyle 4\pi k_{\rm {C}}\epsilon _{0}} and4παB/(μ0αL){\displaystyle 4\pi \alpha _{\rm {B}}/(\mu _{0}\alpha _{\rm {L}})}, respectively. If the rationalisation constants are equal, thenc2=1/(ϵ0μ0αL2){\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 "rationalised":[15] the laws for systems ofspherical 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 ofπ (for example, parallel-platecapacitors). However, the original CGS system used 𝜆 = 𝜆′ = 4π, or, equivalently,kCϵ0=αB/(μ0αL)=1{\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 rationalised.

Various extensions of the CGS system to electromagnetism

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The table below shows the values of the above constants used in some common CGS subsystems:

SystemkC{\displaystyle k_{\rm {C}}}αB{\displaystyle \alpha _{\rm {B}}}ϵ0{\displaystyle \epsilon _{0}}μ0{\displaystyle \mu _{0}}kA=kCc2{\displaystyle k_{\rm {A}}={\frac {k_{\rm {C}}}{c^{2}}}}αL=kCαBc2{\displaystyle \alpha _{\rm {L}}={\frac {k_{\rm {C}}}{\alpha _{\rm {B}}c^{2}}}}λ=4πkCϵ0{\displaystyle \lambda =4\pi k_{\rm {C}}\epsilon _{0}}λ′=4παBμ0αL{\displaystyle \lambda '={\frac {4\pi \alpha _{\rm {B}}}{\mu _{0}\alpha _{\rm {L}}}}}
Electrostatic[12] CGS
(ESU, esu, or stat-)		1c−21c−2c−214π4π
Electromagnetic[12] CGS
(EMU, emu, or ab-)		c21c−21114π4π
Gaussian[12] CGS1c−111c−2c−14π4π
Heaviside–Lorentz[12] CGS14π{\displaystyle {\frac {1}{4\pi }}}14πc{\displaystyle {\frac {1}{4\pi c}}}1114πc2{\displaystyle {\frac {1}{4\pi c^{2}}}}c−111
SIc2b{\displaystyle {\frac {c^{2}}{b}}}1b{\displaystyle {\frac {1}{b}}}b4πc2{\displaystyle {\frac {b}{4\pi c^{2}}}}4πb{\displaystyle {\frac {4\pi }{b}}}1b{\displaystyle {\frac {1}{b}}}111

In the CGS systemsc = 2.9979 × 1010 cm/s, and in the SI systemc = 2.9979 × 108 m/s andb ≈ 107 A2/N = 107 m/H.

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:[12][16]

SystemGauss's lawAmpère–Maxwell lawGauss's law for magnetismFaraday's law
CGS-ESU∇⋅EESU=4πρESU{\displaystyle \nabla \cdot \mathbf {E} ^{\text{ESU}}=4\pi \rho ^{\text{ESU}}}∇×BESU−c−2E˙ESU=4πc−2JESU{\displaystyle \nabla \times \mathbf {B} ^{\text{ESU}}-c^{-2}{\dot {\mathbf {E} }}^{\text{ESU}}=4\pi c^{-2}\mathbf {J} ^{\text{ESU}}}∇⋅BESU=0{\displaystyle \nabla \cdot \mathbf {B} ^{\text{ESU}}=0}∇×EESU+B˙ESU=0{\displaystyle \nabla \times \mathbf {E} ^{\text{ESU}}+{\dot {\mathbf {B} }}^{\text{ESU}}=0}
CGS-EMU∇⋅EEMU=4πc2ρEMU{\displaystyle \nabla \cdot \mathbf {E} ^{\text{EMU}}=4\pi c^{2}\rho ^{\text{EMU}}}∇×BEMU−c−2E˙EMU=4πJEMU{\displaystyle \nabla \times \mathbf {B} ^{\text{EMU}}-c^{-2}{\dot {\mathbf {E} }}^{\text{EMU}}=4\pi \mathbf {J} ^{\text{EMU}}}∇⋅BEMU=0{\displaystyle \nabla \cdot \mathbf {B} ^{\text{EMU}}=0}∇×EEMU+B˙EMU=0{\displaystyle \nabla \times \mathbf {E} ^{\text{EMU}}+{\dot {\mathbf {B} }}^{\text{EMU}}=0}
CGS-Gaussian∇⋅EG=4πρG{\displaystyle \nabla \cdot \mathbf {E} ^{\text{G}}=4\pi \rho ^{\text{G}}}∇×BG−c−1E˙G=4πc−1JG{\displaystyle \nabla \times \mathbf {B} ^{\text{G}}-c^{-1}{\dot {\mathbf {E} }}^{\text{G}}=4\pi c^{-1}\mathbf {J} ^{\text{G}}}∇⋅BG=0{\displaystyle \nabla \cdot \mathbf {B} ^{\text{G}}=0}∇×EG+c−1B˙G=0{\displaystyle \nabla \times \mathbf {E} ^{\text{G}}+c^{-1}{\dot {\mathbf {B} }}^{\text{G}}=0}
CGS-Heaviside–Lorentz∇⋅ELH=ρLH{\displaystyle \nabla \cdot \mathbf {E} ^{\text{LH}}=\rho ^{\text{LH}}}∇×BLH−c−1E˙LH=c−1JLH{\displaystyle \nabla \times \mathbf {B} ^{\text{LH}}-c^{-1}{\dot {\mathbf {E} }}^{\text{LH}}=c^{-1}\mathbf {J} ^{\text{LH}}}∇⋅BLH=0{\displaystyle \nabla \cdot \mathbf {B} ^{\text{LH}}=0}∇×ELH+c−1B˙LH=0{\displaystyle \nabla \times \mathbf {E} ^{\text{LH}}+c^{-1}{\dot {\mathbf {B} }}^{\text{LH}}=0}
SI∇⋅ESI=ρSI/ϵ0{\displaystyle \nabla \cdot \mathbf {E} ^{\text{SI}}=\rho ^{\text{SI}}/\epsilon _{0}}∇×BSI−μ0ϵ0E˙SI=μ0JSI{\displaystyle \nabla \times \mathbf {B} ^{\text{SI}}-\mu _{0}\epsilon _{0}{\dot {\mathbf {E} }}^{\text{SI}}=\mu _{0}\mathbf {J} ^{\text{SI}}}∇⋅BSI=0{\displaystyle \nabla \cdot \mathbf {B} ^{\text{SI}}=0}∇×ESI+B˙SI=0{\displaystyle \nabla \times \mathbf {E} ^{\text{SI}}+{\dot {\mathbf {B} }}^{\text{SI}}=0}

Electrostatic units (ESU)

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In theelectrostatic units variant of the CGS system, (CGS-ESU), charge is defined as the quantity that obeys a form ofCoulomb's law without amultiplying constant (and current is then defined as charge per unit time):

F=q1ESUq2ESUr2.{\displaystyle F={q_{1}^{\text{ESU}}q_{2}^{\text{ESU}} \over r^{2}}.}

The ESU unit of charge,franklin (Fr), also known asstatcoulomb oresu charge, is therefore defined as follows:[17]

two equal point charges spaced 1centimetre apart are said to be of 1 franklin each if the electrostatic force between them is 1dyne.

Therefore, in CGS-ESU, a franklin is equal to a centimetre times square root of dyne:

1Fr=1statcoulomb=1esucharge=1dyne1/2⋅cm=1g1/2⋅cm3/2⋅s−1.{\displaystyle \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}} .}

The unit of current is defined as:

1Fr/s=1statampere=1esucurrent=1dyne1/2⋅cm⋅s−1=1g1/2⋅cm3/2⋅s−2.{\displaystyle \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}} .}

In the CGS-ESU system, chargeq therefore has the dimension of M1/2L3/2T−1.

Other units in the CGS-ESU system include thestatampere (1 statC/s) andstatvolt (1 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'.[18]:3

ESU notation

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All electromagnetic units in the CGS-ESU 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".[17] The franklin was introduced as a fourth ESU base unit; it is not strictly identical with the statcoulomb. (The unit of permittivity is also sometimes used as a fourth base unit.)

Electromagnetic units (EMU)

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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 ofampere as well).

The EMU unit of current,biot (Bi), also known asabampere oremu current, is therefore defined as follows:[17]

Thebiot is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed onecentimetre apart invacuum, would produce between these conductors a force equal to twodynes per centimetre of length.

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

1Bi=1abampere=1emucurrent=1dyne1/2=1g1/2⋅cm1/2⋅s−1.{\displaystyle \mathrm {1\,Bi=1\,abampere=1\,emu\;current=1\,dyne^{1/2}=1\,g^{1/2}{\cdot }cm^{1/2}{\cdot }s^{-1}} .}

The unit of charge in CGS EMU is:

1Bi⋅s=1abcoulomb=1emucharge=1dyne1/2⋅s=1g1/2⋅cm1/2.{\displaystyle \mathrm {1\,Bi{\cdot }s=1\,abcoulomb=1\,emu\,charge=1\,dyne^{1/2}{\cdot }s=1\,g^{1/2}{\cdot }cm^{1/2}} .}

Dimensionally in the CGS-EMU system, chargeq is therefore equivalent to M1/2L1/2. Hence, neither charge nor current is an independent physical quantity in the CGS-EMU system.

EMU notation

[edit]

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".[17] The biot was introduced as a fourth EMU base unit; it is not strictly identical with the abampere. (The unit of permeability is also sometimes used as a fourth base unit.) EMU magnetic unit names formed from abampere, biot, orabvolt should not be used for Gaussian units; other names should be used instead, e.g.,oersted,gilbert,erg pergauss, andmaxwell (abtesla andabweber are rarely used even with EMU).

Practical CGS units

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The practical CGS system is a hybrid system that uses thevolt and theampere 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.[19] As well as the volt and ampere, thefarad (capacitance),ohm (resistance),coulomb (electric charge), andhenry (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.[20]

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 rationalised and magnetically unrationalised; i.e.,𝜆 = 1 and𝜆′ = 4π, but the above formula for 𝜆 is invalid. A closely related system is the International System of Electric and Magnetic Units,[21] which has a different unit of mass so that the formula for 𝜆′ 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.,P =VI andF =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 rationalising the system by dividing the units of magnetic pole strength and magnetisation by 4π. 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 areB = (4π/10)μH andB = (4π/10)μ0H +μ0M.Magnetic reluctance is given a hybrid unit to ensure the validity of Ohm's law for magnetic circuits.

In all the practical systemsε0 = 8.8542 × 10−14 A⋅s/(V⋅cm),μ0 = 1 V⋅s/(A⋅cm), andc2 = 1/(4π × 10−9ε0μ0). Maxwell's equations in free space are also the same in all the systems.

Maxwell's equations in the practical systems[22]
NameAll systems
in vacuum		Gilbert systems
in matter		Ampere-turn systems
in matter
Gauss's law∇⋅E=1ε0ρ{\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{\varepsilon _{0}}}\rho }∇⋅D=ρ{\displaystyle \nabla \cdot \mathbf {D} =\rho }∇⋅D=ρ{\displaystyle \nabla \cdot \mathbf {D} =\rho }
Gauss's law for magnetism∇⋅B=0{\displaystyle \nabla \cdot \mathbf {B} =0}∇⋅B=0{\displaystyle \nabla \cdot \mathbf {B} =0}∇⋅B=0{\displaystyle \nabla \cdot \mathbf {B} =0}
Faraday's law of induction∇×E+10−8∂B∂t=0{\displaystyle \nabla \times \mathbf {E} +10^{-8}{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {0} }∇×E+10−8∂B∂t=0{\displaystyle \nabla \times \mathbf {E} +10^{-8}{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {0} }∇×E+10−8∂B∂t=0{\displaystyle \nabla \times \mathbf {E} +10^{-8}{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {0} }
Ampère–Maxwell equation10−8c2∇×B−∂E∂t=1ε0J{\displaystyle 10^{-8}c^{2}\nabla \times \mathbf {B} -{\frac {\partial \mathbf {E} }{\partial t}}={\frac {1}{\varepsilon _{0}}}\mathbf {J} }104π∇×H−∂D∂t=J{\displaystyle {\frac {10}{4\pi }}\nabla \times \mathbf {H} -{\frac {\partial \mathbf {D} }{\partial t}}=\mathbf {J} }∇×H−∂D∂t=J{\displaystyle \nabla \times \mathbf {H} -{\frac {\partial \mathbf {D} }{\partial t}}=\mathbf {J} }

In the practical systems inductance is considered to be an electrical quantity and is defined byL = 10−8NΦB/I.[23]

Other variants

[edit]

There were at various points in time about half a dozen systems of electromagnetic units in use, most based on the CGS system.[24] These include theGaussian units and theHeaviside–Lorentz units.

Electromagnetic units in various CGS systems

[edit]
Conversion of SI units in electromagnetism to ESU, EMU, and Gaussian subsystems of CGS[25][17]
QuantitySymbolSI unitESU unitGaussian unitEMU unit
electric chargeq1C≘ (10−1c)statC (Fr)≘ (10−1)abC
electric currentI1A≘ (10−1c)statA (Fr/s)≘ (10−1)abA (Bi)
electric potential /voltageφ /V, E1V≘ (108c−1)statV (erg/Fr)≘ (108)abV
electric fieldE1V/m≘ (106c−1)statV/cm (dyn/Fr)≘ (106)abV/cm
electric displacement fieldD1C/m2≘ (4π × 10−5c)statC/cm2≘ (4π × 10−5)abC/cm2
electric dipole momentp1C⋅m≘ (10c)statC⋅cm≘ (10)abC⋅cm
electric fluxΦe1C≘ (4π × 10−1c)statC≘ (4π × 10−1)abC
permittivityε1F/m≘ (4π × 10−11c2)cm/cm≘ (4π × 10−11)s2/cm2
resistanceR1Ω≘ (109c−2)statΩ (s/cm)≘ (109)abΩ
resistivityρ1Ω⋅m≘ (1011c−2)statΩ⋅cm (s)≘ (1011)abΩ⋅cm
capacitanceC1F≘ (10−9c2)statF (cm)≘ (10−9)abF
inductanceL1H≘ (109c−2) statH (s2/cm)≘ (109)abH
magnetic B fieldB1T≘ (104c−1) statT≘ (104)G
magnetic H fieldH1A/m≘ (4π × 10−3c)statA/cm≘ (4π × 10−3)Oe
magnetic dipole momentμ1A⋅m2≘ (103c)statA⋅cm2≘ (103)erg/G
magnetic fluxΦm1Wb≘ (108c−1) statWb≘ (108)Mx
permeabilityμ1H/m≘ ((4π)−1 × 107c−2)s2/cm2≘ ((4π)−1 × 107)cm/cm
magnetomotive forceF{\displaystyle {\mathcal {F}}}1A≘ (4π × 10−1c)statA≘ (4π × 10−1)Gi
magnetic reluctanceR{\displaystyle {\mathcal {R}}}1H−1≘ (4π × 10−9c2) statH−1≘ (4π × 10−9)Gi/Mx

In this table,c =29979245800 is the numeric value of thespeed of light in vacuum when expressed in units of centimetres per second. The symbol "≘" is used instead of "=" as a reminder that the units arecorresponding but notequal. For example, according to the capacitance 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 incorrect to replace "1 F" with "(10−9 c2) cm" within an equation or formula. (This warning is a special aspect of electromagnetism units. By contrast it isalways correct to replace, e.g., "1 m" with "100 cm" within an equation or formula.)

Physical constants in CGS units

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Commonly used physical constants in CGS units[26]
ConstantSymbolValue
atomic mass constantmu1.660539069×10−24 g
Bohr magnetonμB9.274010066×10−21 erg/G (EMU, Gaussian)
2.780278273×10−10 statA⋅cm2 (ESU)
Bohr radiusa05.291772105×10−9 cm
Boltzmann constantk1.380649×10−16 erg/K
electron massme9.10938371×10−28 g
elementary chargee4.80320471×10−10 Fr (ESU, Gaussian)
1.602176634×10−20 abC (EMU)
fine-structure constantα0.007297352564
Newtonian constant of gravitationG6.6743×10−8 dyn⋅cm2/g2
Planck constanth6.62607015×10−27 erg⋅s
reduced Planck constantħ1.054571817×10−27 erg⋅s
speed of lightc2.99792458×1010 cm/s

Advantages and disadvantages

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Lack of unique unit names leads to potential confusion: "15 emu" may mean either 15abvolts, or 15 emu units ofelectric dipole moment, or 15 emu units ofmagnetic susceptibility, sometimes (but not always) pergram, or permole. 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 1henry, 1 ohm, and 1 volt.

In theCGS-Gaussian system, electric and magnetic fields have the same units, 4πε0 is replaced by 1, and the only dimensional constant appearing in theMaxwell equations isc, the speed of light. TheHeaviside–Lorentz system has these properties as well (withε0 equaling 1).

In SI, and other rationalised 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 lackπ entirely, which was the most convenient choice for applications inelectrical engineering and relates directly to the geometric symmetry of the system being described by the equation.

Specialised unit systems are used to simplify formulas further than either SI or CGS do, by eliminating constants through a convention of normalising quantities with respect to some system ofnatural units. For example, inparticle physics a system is in use where every quantity is expressed by only one unit of energy, theelectronvolt, with lengths, times, and so on all converted into units of energy by inserting factors ofspeed of lightc and thereduced Planck constantħ. This unit system is convenient for calculations inparticle physics, but is impractical in other contexts.

See also

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References and notes

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  1. ^"Centimetre-gram-second system | physics".Encyclopedia Britannica. Retrieved2018-03-27.[failed verification]
  2. ^"The Centimeter-Gram-Second (CGS) System of Units – Maple Programming Help".www.maplesoft.com. Retrieved2018-03-27.
  3. ^Carron, Neal J. (21 May 2015). "Babel of units: The evolution of units systems in classical electromagnetism".arXiv:1506.01951 [physics.hist-ph].
  4. ^Jackson, J. D. Classical Electrodynamics, 3rd ed. Wiley, 1999.
  5. ^Mohr, Peter J.; Newell, David B.; Taylor, Barry N. “CODATA Recommended Values of the Fundamental Physical Constants: 2014.” Rev. Mod. Phys. 88 (3), 035009 (2016).
  6. ^Gauss, C. F. (1832), "Intensitas vis magneticae terrestris ad mensuram absolutam revocata",Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores,8:3–44.English translation
  7. ^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.
  8. ^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. Retrieved2012-04-08.
  9. ^Thompson, Ambler; Taylor, Barry N. (March 2008).Guide for the Use of the International System of Units (SI)(PDF) (Report). p. 10. RetrievedMarch 3, 2024.
  10. ^Waldron, Anne; Judd, Peggy; Miller, Valerie (February 1993),Physical Review Style and Notation Guide(PDF), American Physical Society, p. 15, retrievedMarch 3, 2024
  11. ^Wilkins, George A. (1989),The IAU Style Manual(PDF), International Astronomical Union, p. 20, retrievedMarch 3, 2024
  12. ^abcdefghJackson, John David (1999).Classical Electrodynamics (3rd ed.). New York: Wiley. pp. 775–784.ISBN 0-471-30932-X.
  13. ^Weisstein, Eric W."cgs".Eric Weisstein's World of Physics.
  14. ^"Atomic Spectroscopy".Atomic Spectroscopy. NIST. Retrieved25 October 2015.
  15. ^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.
  16. ^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.S2CID 43177051.
  17. ^abcdeCardarelli, 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.
  18. ^Fenna, Donald (2002).A Dictionary of Weights, Measures, and Units. Oxford University Press.ISBN 978-0-19-107898-9.
  19. ^Tunbridge, Paul (1992).Lord Kelvin: His Influence on Electrical Measurements and Units. IET. pp. 34–40.ISBN 0-86341-237-8.
  20. ^Knoepfel, Heinz E. (2000).Magnetic Fields: A Comprehensive Theoretical Treatise for Practical Use. Wiley. p. 543.ISBN 3-527-61742-6.
  21. ^Dellinger, John Howard (1916).International System of Electric and Magnetic Units. Washington, D.C.: U.S. Government Printing Office.
  22. ^Knoepfel, p. 542.
  23. ^Dellinger, p. 625.
  24. ^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.PMC 6752159.PMID 34565970.
  25. ^Gray, Truman S. (1954).Applied Electronics. New York: John Wiley & Sons, Inc. pp. 830–831, Appendix B.
  26. ^A.P. French; Edwind F. Taylor (1978).An Introduction to Quantum Physics. W.W. Norton & Company.

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