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]
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
Contents 1 History 2 Definition of CGS units in mechanics 2.1 Definitions and conversion factors of CGS units in mechanics 3 Derivation of CGS units in electromagnetism 3.1 CGS approach to electromagnetic units
3.2 Alternate derivations of CGS units in electromagnetism
3.3 Various extensions of the CGS system to electromagnetism
3.4
3.4.1 ESU notation 3.5 Electromagnetic units (EMU) 3.5.1 EMU notation 3.6 Relations between ESU and EMU units 3.7 Practical cgs units 3.8 Other variants 4 Electromagnetic units in various CGS systems 5 Physical constants in CGS units 6 Advantages and disadvantages 7 See also 8 References and notes 9 General literature History[edit]
The CGS system goes back to a proposal in 1832 by the German
mathematician
v = d x d t displaystyle v= frac dx dt (definition of velocity) F = m d 2 x d t 2 displaystyle F=m frac d^ 2 x dt^ 2 (Newton's second law of motion) E = ∫ F → ⋅ d x → displaystyle E=int vec F cdot vec dx (energy defined in terms of work) p = F L 2 displaystyle p= frac F L^ 2 (pressure defined as force per unit area) η = τ / d v d x 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[edit] 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[6] cm−1 = 100 m−1 Derivation of CGS units in electromagnetism[edit] CGS approach to electromagnetic units[edit] 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 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, q = I ⋅ t displaystyle q=Icdot 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[edit] 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[7] in system-independent form as follows: The first is Coulomb's law, F = k C q ⋅ q ′ d 2 displaystyle F=k_ rm C frac qcdot q^ prime d^ 2 , which describes the electrostatic force F between electric charges q displaystyle q and q ′ displaystyle q^ prime , separated by distance d. Here k C 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, d F d L = 2 k A I I ′ d 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 I = q / t displaystyle I=q/t, and I ′ = q ′ / t displaystyle I^ prime =q^ prime /t , the constant k A 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 k C displaystyle k_ rm C and k A displaystyle k_ rm A must obey k C / k A = c 2 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
k C = 1 displaystyle k_ rm C =1 then
2 / c 2 displaystyle 2/c^ 2 . Alternatively, deriving the unit of current, and therefore the unit
of charge, from the
k A = 1 displaystyle k_ rm A =1 or k A = 1 / 2 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
F = α L q v × B . 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: d B = α B I d l × r ^ r 2 , displaystyle dmathbf B =alpha _ rm B frac Idmathbf l times mathbf hat r r^ 2 ;, where r and r ^ 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
k A = α L ⋅ α B 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 k A = 1 displaystyle k_ rm A =1 , it is natural to derive the unit of magnetic field by setting α L = α B = 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 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[7] (generally) D = ϵ 0 E + λ P displaystyle mathbf D =epsilon _ 0 mathbf E +lambda mathbf P and H = B / μ 0 − λ ′ M 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" 4 π k C ϵ 0 displaystyle 4pi k_ rm C epsilon _ 0 and 4 π α B / ( μ 0 α L ) displaystyle 4pi alpha _ rm B /(mu _ 0 alpha _ rm L ) , respectively. If the rationalization constants are equal, then c 2 = 1 / ( ϵ 0 μ 0 α L 2 ) 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":[8] 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, k C ϵ 0 = α B / ( μ 0 α L ) = 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 rationalized. Various extensions of the CGS system to electromagnetism[edit] The table below shows the values of the above constants used in some common CGS subsystems: system k C displaystyle k_ rm C α B displaystyle alpha _ rm B ϵ 0 displaystyle epsilon _ 0 μ 0 displaystyle mu _ 0 k A = k C c 2 displaystyle k_ rm A = frac k_ rm C c^ 2 α L = k C α B c 2 displaystyle alpha _ rm L = frac k_ rm C alpha _ rm B c^ 2 λ = 4 π k C ϵ 0 displaystyle lambda =4pi k_ rm C epsilon _ 0 λ ′ = 4 π α B μ 0 α L displaystyle lambda '= frac 4pi alpha _ rm B mu _ 0 alpha _ rm L Electrostatic[7] CGS (ESU, esu, or stat-) 1 c−2 1 c−2 c−2 1 4π 4π Electromagnetic[7] CGS (EMU, emu, or ab-) c2 1 c−2 1 1 1 4π 4π Gaussian[7] CGS 1 c−1 1 1 c−2 c−1 4π 4π Lorentz–Heaviside[7] CGS 1 4 π displaystyle frac 1 4pi 1 4 π c displaystyle frac 1 4pi c 1 1 1 4 π c 2 displaystyle frac 1 4pi c^ 2 c−1 1 1 SI c 2 b displaystyle frac c^ 2 b 1 b displaystyle frac 1 b b 4 π c 2 displaystyle frac b 4pi c^ 2 4 π b displaystyle frac 4pi b 1 b displaystyle frac 1 b 1 1 1 The constant b in SI system is a unit-based scaling factor defined as: b = 10 7 A 2 / N = 10 7 m / H = 4 π / μ 0 = 4 π ϵ 0 c 2 = c 2 / k C displaystyle b=10^ 7 ,mathrm A ^ 2 /mathrm N =10^ 7 ,mathrm m/H =4pi /mu _ 0 =4pi epsilon _ 0 c^ 2 =c^ 2 /k_ rm C ; . Also, note the following correspondence of the above constants to those in Jackson[7] and Leung:[9] k C = k 1 = k E displaystyle k_ rm C =k_ 1 =k_ rm E α B = α ⋅ k 2 = k B displaystyle alpha _ rm B =alpha cdot k_ 2 =k_ rm B k A = k 2 = k E / c 2 displaystyle k_ rm A =k_ 2 =k_ rm E /c^ 2 α L = k 3 = k F displaystyle alpha _ rm L =k_ 3 =k_ rm F In system-independent form,
∇ → ⋅ E → = 4 π k C ρ ∇ → ⋅ B → = 0 ∇ → × E → = − α L ∂ B → ∂ t ∇ → × B → = 4 π α B J → + α B k C ∂ E → ∂ t displaystyle begin array ccl vec nabla cdot vec E &=&4pi 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 4pi 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 α L displaystyle alpha _ rm L equals c − 1 displaystyle c^ -1 rather than 1. As a result, vectors E → displaystyle vec E and B → 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.
k C = 1 displaystyle k_ rm C =1 , so that
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: 1 F r = 1 s t a t c o u l o m b = 1 e s u c h a r g e = 1 c m d y n e = 1 g 1 / 2 ⋅ c m 3 / 2 ⋅ s − 1 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: 1 F r / s = 1 s t a t a m p e r e = 1 e s u c u r r e n t = 1 ( c m / s ) d y n e = 1 g 1 / 2 ⋅ c m 3 / 2 ⋅ s − 2 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[edit]
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".[10]
Electromagnetic units (EMU)[edit]
In another variant of the CGS system, electromagnetic units (EMU),
current is defined via the force existing between two thin, parallel,
infinitely long wires carrying it, and charge is then defined as
current multiplied by time. (This approach was eventually used to
define the SI unit of ampere as well). In the EMU CGS subsystem, this
is done by setting the
k A = 1 displaystyle k_ rm A =1 , so that
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: 1 B i = 1 a b a m p e r e = 1 e m u c u r r e n t = 1 d y n e = 1 g 1 / 2 ⋅ c m 1 / 2 ⋅ s − 1 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: 1 B i ⋅ s = 1 a b c o u l o m b = 1 e m u c h a r g e = 1 s ⋅ d y n e = 1 g 1 / 2 ⋅ c m 1 / 2 displaystyle mathrm 1,Bicdot s=1,abcoulomb=1,emu,charge=1,scdot 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[edit] 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".[10] Relations between ESU and EMU units[edit] The ESU and EMU subsystems of CGS are connected by the fundamental relationship k C / k A = c 2 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
1 s t a t c o u l o m b 1 a b c o u l o m b = 1 s t a t a m p e r e 1 a b a m p e r e = c − 1 displaystyle mathrm frac 1,statcoulomb 1,abcoulomb =mathrm frac 1,statampere 1,abampere =c^ -1 and 1 s t a t v o l t 1 a b v o l t = 1 s t a t t e s l a 1 g a u s s = c 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: 1 s t a t o h m 1 a b o h m = 1 s t a t v o l t 1 a b v o l t × 1 a b a m p e r e 1 s t a t a m p e r e = c 2 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[edit]
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 amp
had been adopted as international standard units by the International
Electrical Congress of 1881.[11] 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 amps per centimetre, and resistivity is in
ohm-cm.[12]
Some physicists and electrical engineers in North America still use
these hybrid units.[13]
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.[14] These
also include the
Conversion of SI units in electromagnetism to ESU, EMU, and Gaussian subsystems of CGS[10] c = 29,979,245,800 Quantity Symbol SI unit ESU unit EMU unit Gaussian unit electric charge / flux q / ΦE 1 C ↔ (10−1 c) or (4π×10−1 c) statC ↔ (10−1) abC ↔ (10−1 c) or (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 C⋅m ↔ (10 c) statC⋅cm ↔ (10) abC⋅cm ↔ (1019 c) D magnetic dipole moment μ 1 A⋅m2 ↔ (103 c) statA⋅cm2 ↔ (103) abA⋅cm2 ↔ (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 Ω⋅m ↔ (1011 c−2) s ↔ (1011) abΩ⋅cm ↔ (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−1⋅s2 ↔ (109) abH ↔ (109 c−2) cm−1⋅s2 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
k C = 1 4 π ϵ 0 = μ 0 ( c / 100 ) 2 4 π = 10 − 7 N / A 2 ⋅ 10 − 4 ⋅ c 2 = 10 − 11 N ⋅ c 2 / A 2 . displaystyle k_ rm C = frac 1 4pi epsilon _ 0 = frac mu _ 0 (c/100)^ 2 4pi =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: 1 1 r − 1 R 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[edit] Commonly used physical constants in CGS units[15] 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
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 cm3/(g⋅s2) Planck constant h 6.626 068 85 × 10−27 erg⋅s ħ 1.054 5716 × 10−27 erg⋅s
Advantages and disadvantages[edit]
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
List of scientific units named after people Metre–tonne–second system of units United States customary units References and notes[edit] ^ "Centimetre-gram-second system physics". Encyclopedia Britannica.
Retrieved 2018-03-27. [not in citation given]
^ "The Centimeter-Gram-
General literature[edit] Griffiths, David J. (1999). "Appendix C: Units". Introduction to
v t e Systems of measurement Current General
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