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Centimetre–gram–second system of units

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(Redirected fromCGS unit)
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 been largely 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.

On the other hand, in measurements of electromagnetic phenomena (involving units ofcharge, electric and magnetic fields,voltage, and so on), converting between CGS and SI is less straightforward. Formulas for physical laws of electromagnetism (such asMaxwell'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", includingGaussian units, "ESU", "EMU", andHeaviside–Lorentz units. Among these choices, Gaussian units are the most common today[according to whom?], and "CGS units" is often intended to refer to CGS-Gaussian units[citation needed].

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.[4] Gauss chose the units of millimetre, milligram and second.[5] 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 ...".[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 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 favor of SI units byNIST,[7] as well as organizations such as theAmerican Physical Society[8] and theInternational Astronomical Union.[9] 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.[10][11]

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

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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[12] or Kcm−1100 m−1

Derivation of CGS units in electromagnetism

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CGS approach to electromagnetic units

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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.

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:[10][13]

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:[14]

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 is therefore has the dimension to 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'.[15]:3

Unit symbols

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All electromagnetic units in the CGS-ESU system that have not been given names of their own are named as the corresponding SI name with an attached prefix "stat" or with a separate abbreviation "esu", and similarly with the corresponding symbols.[14]

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:[14]

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

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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".[14]

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.[16] 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.[17]

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.,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,[18] 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 rationalizing the system by dividing the units of magnetic pole strength and magnetization 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).

Other variants

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There were at various points in time about half a dozen systems of electromagnetic units in use, most based on the CGS system.[19] 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[20][14]
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
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
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

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

[edit]
Commonly used physical constants in CGS units[21]
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 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 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.

Specialized unit systems are used to simplify formulas further than either SI or CGS do, by eliminating constants through a convention of normalizing quantities with respect to some system 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

[edit]
  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. ^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. Retrieved2012-04-08.
  7. ^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.
  8. ^Waldron, Anne; Judd, Peggy; Miller, Valerie (February 1993),Physical Review Style and Notation Guide(PDF), American Physical Society, p. 15, retrievedMarch 3, 2024
  9. ^Wilkins, George A. (1989),The IAU Style Manual(PDF), International Astronomical Union, p. 20, retrievedMarch 3, 2024
  10. ^abJackson, John David (1999).Classical Electrodynamics (3rd ed.). New York: Wiley. pp. 775–784.ISBN 0-471-30932-X.
  11. ^Weisstein, Eric W."cgs".Eric Weisstein's World of Physics.
  12. ^"Atomic Spectroscopy".Atomic Spectroscopy. NIST. Retrieved25 October 2015.
  13. ^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.
  14. ^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.
  15. ^Fenna, Donald (2002).A Dictionary of Weights, Measures, and Units. Oxford University Press.ISBN 978-0-19-107898-9.
  16. ^Tunbridge, Paul (1992).Lord Kelvin: His Influence on Electrical Measurements and Units. IET. pp. 34–40.ISBN 0-86341-237-8.
  17. ^Knoepfel, Heinz E. (2000).Magnetic Fields: A Comprehensive Theoretical Treatise for Practical Use. Wiley. p. 543.ISBN 3-527-61742-6.
  18. ^Dellinger, John Howard (1916).International System of Electric and Magnetic Units. Washington, D.C.: U.S. Government Printing Office.
  19. ^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.
  20. ^Gray, Truman S. (1954).Applied Electronics. New York: John Wiley & Sons, Inc. pp. 830–831, Appendix B.
  21. ^A.P. French; Edwind F. Taylor (1978).An Introduction to Quantum Physics. W.W. Norton & Company.

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