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The centimetre–gram–second system of units (CGS or cgs) is a variant of the
metric system The metric system is a system of measurement that standardization, standardizes a set of base units and a nomenclature for describing relatively large and small quantities via decimal-based multiplicative unit prefixes. Though the rules gover ...
based on the centimetre as the unit of
length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
, the gram as the unit of
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
, and the second as the unit of
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
. All CGS mechanical units are unambiguously derived from these three base units, but there are several different ways in which the CGS system was extended to cover
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
. The CGS system has been largely supplanted by the MKS system based on the
metre The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
, kilogram, and second, which was in turn extended and replaced by the
International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
(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 In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
,
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
,
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
, and so on), the differences between CGS and SI are straightforward: the unit-conversion factors are all powers of 10 as and . For example, the CGS unit of force is the
dyne The dyne (symbol: dyn; ) is a derived units of measurement, unit of force (physics), force specified in the centimetre–gram–second system of units, centimetre–gram–second (CGS) system of units, a predecessor of the modern International S ...
, which is defined as , so the SI unit of force, the newton (), is equal to . On the other hand, in measurements of electromagnetic phenomena (involving units of charge, electric and magnetic fields,
voltage Voltage, also known as (electrical) potential difference, electric pressure, or electric tension, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), ...
, and so on), converting between CGS and SI is less straightforward. Formulas for physical laws of electromagnetism (such as
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
) take a form that depends on which system of units is being used, because the electromagnetic quantities are defined differently in SI and in CGS. Furthermore, within CGS, there are several plausible ways to define electromagnetic quantities, leading to different "sub-systems", including
Gaussian units Gaussian units constitute a metric system of units of measurement. This system is the most common of the several electromagnetic unit systems based on the centimetre–gram–second system of units (CGS). It is also called the Gaussian unit syst ...
, "ESU", "EMU", and Heaviside–Lorentz units. Among these choices, Gaussian units are the most common today, and "CGS units" is often intended to refer to CGS-Gaussian units.


History

The 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. Gauss chose the units of millimetre, milligram and second. In 1873, a committee of the British Association for the Advancement of Science, including physicists
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
and William 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 ...". The sizes of many CGS units turned out to be inconvenient for practical purposes. For example, many everyday objects are hundreds or thousands of centimetres long, such as humans, rooms and buildings. Thus the CGS system never gained wide use outside the field of science. Starting in the 1880s, and more significantly by the mid-20th century, CGS was gradually superseded internationally for scientific purposes by the MKS (metre–kilogram–second) system, which in turn developed into the modern SI standard. Since the international adoption of the MKS standard in the 1940s and the SI standard in the 1960s, the technical use of CGS units has gradually declined worldwide. CGS units have been deprecated in favor of SI units by NIST, as well as organizations such as the American Physical Society and the
International Astronomical Union The International Astronomical Union (IAU; , UAI) is an international non-governmental organization (INGO) with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreach, education, and developmen ...
. SI units are predominantly used in
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
applications and physics education, while Gaussian CGS units are still commonly used in theoretical physics, describing microscopic systems, relativistic electrodynamics, and astrophysics. The units gram and centimetre remain useful as noncoherent units within the SI system, as with any other
prefix A prefix is an affix which is placed before the stem of a word. Particularly in the study of languages, a prefix is also called a preformative, because it alters the form of the word to which it is affixed. Prefixes, like other affixes, can b ...
ed SI units.


Definition of CGS units in mechanics

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 are coherent, the definitions of all coherent derived units in terms of the base units are the same in both systems, and there is an unambiguous relationship between derived units: * v = \frac  (definition of
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
) * F = m\frac  (
Newton's second law of motion Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...
) * E = \int \vec\cdot d\vec  (
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
defined in terms of work) * p = \frac   (
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
defined as force per unit area) * \eta = \tau/\frac  (dynamic
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
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


Derivation of CGS units in electromagnetism


CGS approach to electromagnetic units

The conversion factors relating
electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
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 of
electric current An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge c ...
, the ampere (A), was historically defined such that the magnetic force exerted by two infinitely long, thin, parallel wires 1 
metre The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
apart and carrying a current of 1  ampere is exactly . This definition results in all SI electromagnetic units being numerically consistent (subject to factors of some
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
powers of 10) with those of the CGS-EMU system described in further sections. The ampere is a base unit of the SI system, with the same status as the metre, kilogram, and second. Thus the relationship in the definition of the ampere with the metre and newton is disregarded, and the ampere is not treated as dimensionally equivalent to any combination of other base units. As a result, electromagnetic laws in SI require an additional constant of proportionality (see ''
Vacuum permeability The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum'', ''magnetic constant'') is the magnetic permeability in a classical vacuum. It is a physical constant, conventionally ...
'') 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 Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
''q'' is defined as current ''I'' multiplied by time ''t'', q = I \, t, resulting in the unit of electric charge, the
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). It is defined to be equal to the electric charge delivered by a 1 ampere current in 1 second, with the elementary charge ''e'' as a defining c ...
(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 Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
can be written in each of these systems as:


Electrostatic units (ESU)

In the electrostatic units variant of the CGS system, (CGS-ESU), charge is defined as the quantity that obeys a form of
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
without a multiplying constant (and current is then defined as charge per unit time): : F= . The ESU unit of charge, franklin (Fr), also known as statcoulomb or esu charge, is therefore defined as follows: Therefore, in CGS-ESU, a franklin is equal to a centimetre times square root of dyne: : \mathrm . The unit of current is defined as: : \mathrm . In the CGS-ESU system, charge ''q'' therefore has the dimension of M1/2L3/2T−1. Other units in the CGS-ESU system include the statampere (1 statC/s) and statvolt (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'.3
/sup>


Unit symbols

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.


Electromagnetic units (EMU)

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). The EMU unit of current, biot (Bi), also known as abampere or emu current, is therefore defined as follows: Therefore, in electromagnetic CGS units, a biot is equal to a square root of dyne: : \mathrm. The unit of charge in CGS EMU is: : \mathrm. Dimensionally in the CGS-EMU system, charge ''q'' is therefore equivalent to M1/2L1/2. Hence, neither charge nor current is an independent physical quantity in the CGS-EMU system.


EMU notation

All electromagnetic units in the CGS-EMU system that do not have proper names are denoted by a corresponding SI name with an attached prefix "ab" or with a separate abbreviation "emu".


Practical CGS units

The practical CGS system is a hybrid system that uses the
volt The volt (symbol: V) is the unit of electric potential, Voltage#Galvani potential vs. electrochemical potential, electric potential difference (voltage), and electromotive force in the International System of Units, International System of Uni ...
and the ampere 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. As well as the volt and ampere, the
farad The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units, International System of Units (SI), equivalent to 1 coulomb per volt (C/V). It is named afte ...
(capacitance), ohm (resistance),
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). It is defined to be equal to the electric charge delivered by a 1 ampere current in 1 second, with the elementary charge ''e'' as a defining c ...
(electric charge), and henry (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. 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., and , but the above formula for is invalid. A closely related system is the International System of Electric and Magnetic Units, which has a different unit of mass so that the formula for ′ 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., and ). Inevitably, the powers of ten reappeared in other contexts, but the effect was to make the familiar joule and watt the units of work and power respectively. The ampere-turn system is constructed in a similar way by considering magnetomotive force and magnetic field strength to be electrical quantities and rationalizing the system by dividing the units of magnetic pole strength and magnetization by 4. The units of the first two quantities are the ampere and the ampere per centimetre respectively. The unit of magnetic permeability is that of the emu system, and the magnetic constitutive equations are and . Magnetic reluctance is given a hybrid unit to ensure the validity of Ohm's law for magnetic circuits. In all the practical systems ''ε''0 = 8.8542 × 10−14 A⋅s/(V⋅cm), ''μ''0 = 1 V⋅s/(A⋅cm), and ''c''2 = 1/(4''π'' × 10−9 ''ε''0''μ''0).


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. These include the
Gaussian units Gaussian units constitute a metric system of units of measurement. This system is the most common of the several electromagnetic unit systems based on the centimetre–gram–second system of units (CGS). It is also called the Gaussian unit syst ...
and the Heaviside–Lorentz units.


Electromagnetic units in various CGS systems

In this table, ''c'' = is the numeric value of the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
in vacuum when expressed in units of centimetres per second. The symbol "≘" is used instead of "=" as a reminder that the units are ''corresponding'' but not ''equal''. 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 ''c''2) cm in ESU; ''but'' it is incorrect to replace "1 F" with "(10−9 ''c''2) cm" within an equation or formula. (This warning is a special aspect of electromagnetism units. By contrast it is ''always'' correct to replace, e.g., "1 m" with "100 cm" within an equation or formula.)


Physical constants in CGS units


Advantages and disadvantages

Lack of unique unit names leads to potential confusion: "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. 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 1 henry, 1  ohm, and 1 volt. In the CGS-Gaussian system, electric and magnetic fields have the same units, 40 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 properties as well (with ''ε''0 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 4, those concerning coils of current and straight wires contain 2 and those dealing with charged surfaces lack entirely, which was the most convenient choice for applications in
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
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 of natural units. For example, in
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
a system is in use where every quantity is expressed by only one unit of energy, the
electronvolt In physics, an electronvolt (symbol eV), also written electron-volt and electron volt, is the measure of an amount of kinetic energy gained by a single electron accelerating through an Voltage, electric potential difference of one volt in vacuum ...
, with lengths, times, and so on all converted into units of energy by inserting factors of
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
''c'' and the reduced Planck constant ''ħ''. This unit system is convenient for calculations in
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, but is impractical in other contexts.


See also

* Outline of metrology and measurement *
International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
* International System of Electrical and Magnetic Units * List of metric units * List of scientific units named after people * Metre–tonne–second system of units *
United States customary units United States customary units form a system of measurement units commonly used in the United States and most U.S. territories since being standardized and adopted in 1832. The United States customary system developed from English units that ...
* Foot–pound–second system of units


References and notes


General literature

* * * * {{DEFAULTSORT:Centimetre-gram-second system of units Metrology Systems of units Metric system British Science Association