Centimetre gram second system of units
Encyclopedia
The centimetre–gram–second system (abbreviated CGS or cgs) is a metric system
of physical units
based on centimetre
as the unit of length
, gram
as a unit of mass
, and second
as a unit of time
. All CGS mechanical unit
s are unambiguously derived from these three base units, but there are several different ways of extending the CGS system to cover electromagnetism
.
The CGS system has been largely supplanted by the MKS system
, based on metre
, kilogram
, and second
. MKS was in turn extended and replaced by the International System of Units
(SI). The latter adopts the three base units of MKS, plus the ampere
, mole
, candela
and kelvin
. In many fields of science and engineering, SI is the only system of units in use. However, there remain certain subfields where CGS is prevalent.
In measurements of purely mechanical systems (involving units of length
, mass
, force
, energy
, pressure
, and so on.), the differences between CGS and SI are straightforward and rather trivial; the unitconversion factors are all powers of 10 arising from the relations 100 cm = 1 m and 1000 g = 1 kg. For example, the CGS derived unit of force is the dyne
, equal to 1 g·cm/s^{2}, while the SI derived unit of force is the newton, 1 kg·m/s^{2}. Thus it is straightforward to show that 1 dyne=10^{−5} newtons.
On the other hand, in measurements of electromagnetic phenomena (involving units of charge
, electric and magnetic fields, voltage
, and so on), converting between CGS and SI is much more subtle and involved. In fact, formulas for physical laws of electromagnetism (such as Maxwell's equations
) need to be adjusted depending on what system of units one uses. This is because there is no onetoone correspondence between electromagnetic units in SI and those in CGS, as is the case for mechanical units. Furthermore, within CGS, there are several plausible choices of electromagnetic units, leading to different unit "subsystems", including Gaussian
, "ESU", "EMU", and Heaviside–Lorentz. Among these choices, Gaussian units are the most common today, and in fact the phrase "CGS units" is often used to refer specifically to CGSGaussian units
.
. In 1874, it was extended by the British physicists James Clerk Maxwell
and William Thomson
with a set of electromagnetic units.
The values (by order of magnitude
) of many CGS units turned out to be inconvenient for practical purposes. For example, many everyday length measurements yield hundreds or thousands of centimetres, such as those of human height
and sizes of rooms and buildings. Thus the CGS system never gained wide general use outside the field of electrodynamics and laboratory science. Starting in the 1880s, and more significantly by the mid20th century, CGS was gradually superseded internationally by the MKS (metre–kilogram–second) system, which in turn became the modern SI
standard.
From the international adoption of the MKS standard in the 1940s and the SI standard in the 1960s, the technical use of CGS units has gradually declined worldwide, in the United States
more slowly than elsewhere. CGS units are today no longer accepted by the house styles of most scientific journals, textbook publishers, or standards bodies, although they are commonly used in astronomical journals such as the Astrophysical Journal
. CGS units are still occasionally encountered in technical literature, especially in the United States
in the fields of material science, electrodynamics and astronomy
. The continued usage of CGS units is most prevalent in magnetism and related fields, as the primary MKS unit, the tesla, is inconvenienently large, leading to the continued common use of the gauss
, the CGS equivalent.
The units gram
and centimetre
remain useful as prefix
ed units within the SI system, especially for instructional physics and chemistry experiments, where they match the small scale of tabletop setups. However, where derived unit
s are needed, the SI ones are generally used and taught instead of the CGS ones today. For example, a physics lab course might ask students to record lengths in centimeters, and masses in grams, but force (a derived unit) in newtons, a usage consistent with the SI system.
as the unit of time) is the same in both systems.
There is a onetoone correspondence between the base units of mechanics in CGS and SI, and the laws of mechanics are not affected by the choice of units. The definitions of all derived units in terms of the three base units are therefore the same in both systems, and there is an unambiguous onetoone correspondence of derived units:
(definition of velocity
) (Newton's second law of motion
) (energy
defined in terms of work
) (pressure
defined as force per unit area) (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:
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:
units in the CGS and SI systems are much more involved – so much so that formulas for physical laws of electromagnetism are adjusted depending on what system of units one uses. This illustrates the fundamental difference in the ways the two systems are built:
(electric current) to a mechanical quantity such as force. They can be written in systemindependent form as follows:
Maxwell's theory of electromagnetism
relates these two laws to each other. It states that the ratio of proportionality constants and must obey , where c is the speed of light
. Therefore, if one derives the unit of charge from the Coulomb's law by setting , it is obvious that the Ampère's force law will contain a prefactor . Alternatively, deriving the unit of current, and therefore the unit of charge, from the Ampère's force law by setting or , will lead to a constant prefactor in the Coulomb's law.
Indeed, both of these mutuallyexclusive approaches have been practiced by the users of CGS system, leading to the two independent and mutuallyexclusive 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 derive Ampère's force law
, resulting in the relationship: . Therefore, if the unit of charge is based on the Ampère's force law
such that , it is natural to derive the unit of magnetic field by setting . 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 a vacuum, we need to also define the constants ε_{0} and μ_{0}, which are the vacuum permittivity and permeability, respectively. Then we have (generally) and , where P and M are polarization density
and magnetization
vectors. The factors λ and λ′ are rationalization constants, which are usually chosen to be 4πk_{C}ε_{0}, a dimensionless quantity. If λ = λ′ = 1, the system is said to be "rationalized": the laws for systems of spherical geometry
contain factors of 4π (for example, point charges), those of cylindrical geometry – factors of 2π (for example, wire
s), and those of planar geometry contain no factors of π (for example, parallelplate capacitor
s). However, the original CGS system used λ = λ′ = 4π, or, equivalently, k_{C}ε_{0} = 1. Therefore, Gaussian, ESU, and EMU subsystems of CGS (described below) are not rationalized.
The constant b in SI system is a unitbased scaling factor defined as: .
Also, note the following correspondence of the above constants to those in Jackson and Leung:
In systemindependent form, Maxwell's equations
in vacuum
can be written as:
Note that of all these variants, only in Gaussian and Heaviside–Lorentz systems equals rather than 1. As a result, vectors and of an electromagnetic wave propagating in vacuum have the same units and are equal in magnitude in these two variants of CGS.
does not contain an explicit prefactor
.
The ESU unit of charge, franklin (Fr), also known as statcoulomb
or esu charge, is therefore defined as follows: Therefore, in electrostatic CGS units, a franklin is equal to a centimetre times square root of dyne:
The unit of current is defined as:
Dimensionally in the ESU CGS system, charge q is therefore equivalent to m^{1/2}L^{3/2}t^{−1}. Neither charge nor current are therefore an independent dimension of physical quantity in ESU CGS. This reduction of units is an application of the Buckingham π theorem.
as well). In the EMU CGS subsystem, is done by setting the Ampere force constant , so that Ampère's force law
simply contains 2 as an explicit prefactor
(this prefactor 2 is itself a result of integrating a more general formulation of Ampère's law over the length of the infinite wire).
The EMU unit of current, biot (Bi), also known as abampere
or emu current, is therefore defined as follows:
Therefore, in electromagnetic CGS units, a biot is equal to a square root of dyne:
The unit of charge in CGS EMU is:
Dimensionally in the EMU CGS system, charge q is therefore equivalent to m^{1/2}L^{1/2}. Neither charge nor current are therefore an independent dimension of physical quantity in EMU CGS.
Metric system
The metric system is an international decimalised system of measurement. France was first to adopt a metric system, in 1799, and a metric system is now the official system of measurement, used in almost every country in the world...
of physical units
Units of measurement
A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of...
based on centimetre
Centimetre
A centimetre is a unit of length in the metric system, equal to one hundredth of a metre, which is the SI base unit of length. Centi is the SI prefix for a factor of . Hence a centimetre can be written as or — meaning or respectively...
as the unit of length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
, gram
Gram
The gram is a metric system unit of mass....
as a unit of mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
, and second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
as a unit of time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
. All CGS mechanical unit
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
s are unambiguously derived from these three base units, but there are several different ways of extending the CGS system to cover electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
.
The CGS system has been largely supplanted by the MKS system
Mks system of units
The MKS system of units is a physical system of units that expresses any given measurement using fundamental units of the metre, kilogram, and/or second ....
, based on metre
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one tenmillionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...
, kilogram
Kilogram
The kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...
, and second
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
. MKS was in turn extended and replaced by the International System of Units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...
(SI). The latter adopts the three base units of MKS, plus the ampere
Ampere
The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after AndréMarie Ampère , French mathematician and physicist, considered the father of electrodynamics...
, mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon12 , the isotope of carbon with atomic weight 12. This corresponds to a value...
, candela
Candela
The candela is the SI base unit of luminous intensity; that is, power emitted by a light source in a particular direction, weighted by the luminosity function . A common candle emits light with a luminous intensity of roughly one candela...
and kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
. In many fields of science and engineering, SI is the only system of units in use. However, there remain certain subfields where CGS is prevalent.
In measurements of purely mechanical systems (involving units of length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
, mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
, force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
, energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
, pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure. Definition :...
, and so on.), the differences between CGS and SI are straightforward and rather trivial; the unitconversion factors are all powers of 10 arising from the relations 100 cm = 1 m and 1000 g = 1 kg. For example, the CGS derived unit of force is the dyne
Dyne
In physics, the dyne is a unit of force specified in the centimetregramsecond system of units, a predecessor of the modern SI. One dyne is equal to exactly 10 µN...
, equal to 1 g·cm/s^{2}, while the SI derived unit of force is the newton, 1 kg·m/s^{2}. Thus it is straightforward to show that 1 dyne=10^{−5} newtons.
On the other hand, in measurements of electromagnetic phenomena (involving units of charge
Charge (physics)
In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.Formal definition:...
, electric and magnetic fields, voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
, and so on), converting between CGS and SI is much more subtle and involved. In fact, formulas for physical laws of electromagnetism (such as Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
) need to be adjusted depending on what system of units one uses. This is because there is no onetoone correspondence between electromagnetic units in SI and those in CGS, as is the case for mechanical units. Furthermore, within CGS, there are several plausible choices of electromagnetic units, leading to different unit "subsystems", including Gaussian
Gaussian units
Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussiancgs units, or often just cgs units...
, "ESU", "EMU", and Heaviside–Lorentz. Among these choices, Gaussian units are the most common today, and in fact the phrase "CGS units" is often used to refer specifically to CGSGaussian units
Gaussian units
Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussiancgs units, or often just cgs units...
.
History
The CGS system goes back to a proposal made in 1832 by the German mathematician Carl Friedrich GaussCarl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
. In 1874, it was extended by the British physicists James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
and William Thomson
William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...
with a set of electromagnetic units.
The values (by order of magnitude
Order of magnitude
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...
) of many CGS units turned out to be inconvenient for practical purposes. For example, many everyday length measurements yield hundreds or thousands of centimetres, such as those of human height
Human height
Human height is the distance from the bottom of the feet to the top of the head in a human body standing erect.When populations share genetic background and environmental factors, average height is frequently characteristic within the group...
and sizes of rooms and buildings. Thus the CGS system never gained wide general use outside the field of electrodynamics and laboratory science. Starting in the 1880s, and more significantly by the mid20th century, CGS was gradually superseded internationally by the MKS (metre–kilogram–second) system, which in turn became the modern SI
Si
Si, si, or SI may refer to : Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
standard.
From the international adoption of the MKS standard in the 1940s and the SI standard in the 1960s, the technical use of CGS units has gradually declined worldwide, in the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
more slowly than elsewhere. CGS units are today no longer accepted by the house styles of most scientific journals, textbook publishers, or standards bodies, although they are commonly used in astronomical journals such as the Astrophysical Journal
Astrophysical Journal
The Astrophysical Journal is a peerreviewed scientific journal covering astronomy and astrophysics. It was founded in 1895 by the American astronomers George Ellery Hale and James Edward Keeler. It publishes three 500page issues per month....
. CGS units are still occasionally encountered in technical literature, especially in the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
in the fields of material science, electrodynamics and astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
. The continued usage of CGS units is most prevalent in magnetism and related fields, as the primary MKS unit, the tesla, is inconvenienently large, leading to the continued common use of the gauss
Gauss (unit)
The gauss, abbreviated as G, is the cgs unit of measurement of a magnetic field B , named after the German mathematician and physicist Carl Friedrich Gauss. One gauss is defined as one maxwell per square centimeter; it equals 1 tesla...
, the CGS equivalent.
The units gram
Gram
The gram is a metric system unit of mass....
and centimetre
Centimetre
A centimetre is a unit of length in the metric system, equal to one hundredth of a metre, which is the SI base unit of length. Centi is the SI prefix for a factor of . Hence a centimetre can be written as or — meaning or respectively...
remain useful as prefix
SI prefix
The International System of Units specifies a set of unit prefixes known as SI prefixes or metric prefixes. An SI prefix is a name that precedes a basic unit of measure to indicate a decadic multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol...
ed units within the SI system, especially for instructional physics and chemistry experiments, where they match the small scale of tabletop setups. However, where derived unit
SI derived unit
The International System of Units specifies a set of seven base units from which all other units of measurement are formed, by products of the powers of base units. These other units are called SI derived units, for example, the SI derived unit of area is square metre , and of density is...
s are needed, the SI ones are generally used and taught instead of the CGS ones today. For example, a physics lab course might ask students to record lengths in centimeters, and masses in grams, but force (a derived unit) in newtons, a usage consistent with the SI system.
Definition of CGS units in mechanics
In mechanics, the CGS and SI systems of units are built in an identical way. The two systems differ only in the scale of two out of the three base units (centimetre versus metre and gram versus kilogram, respectively), while the third unit (secondSecond
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
as the unit of time) is the same in both systems.
There is a onetoone correspondence between the base units of mechanics in CGS and SI, and the laws of mechanics are not affected by the choice of units. The definitions of all derived units in terms of the three base units are therefore the same in both systems, and there is an unambiguous onetoone correspondence of derived units:
(definition of velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
) (Newton's second law of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
) (energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
defined in terms of work
Mechanical work
In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...
) (pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure. Definition :...
defined as force per unit area) (dynamic viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
defined as shear stress
Shear stress
A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...
per unit velocity gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
).
Thus, for example, the CGS unit of pressure, barye
Barye
The barye , or sometimes barad, barrie, bary, baryd, baryed, or barie, is the centimetregramsecond unit of pressure. It is equal to 1 dyne per square centimetre....
, is related to the CGS base units of length, mass, and time in the same way as the SI unit of pressure, pascal
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...
, 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·s^{2})
 1 Pa = 1 kg/(m·s^{2}).
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·s^{2}) = 10^{3} kg/(10^{2 }m·s^{2}) = 10^{1} kg/(m·s^{2}) = 10^{1} Pa.
Definitions and conversion factors of CGS units in mechanics
Quantity  Symbol  CGS unit  CGS unit abbreviation  Definition  Equivalent in SI units 

length Length In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire... , position 
L, x  centimetre Centimetre A centimetre is a unit of length in the metric system, equal to one hundredth of a metre, which is the SI base unit of length. Centi is the SI prefix for a factor of . Hence a centimetre can be written as or — meaning or respectively... 
cm  1/100 of metre Metre The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one tenmillionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology... 
= 10^{−2} m 
mass Mass Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:... 
m  gram Gram The gram is a metric system unit of mass.... 
g  1/1000 of kilogram Kilogram The kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water... 
= 10^{−3} kg 
time Time Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects.... 
t  second Second The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock.... 
s  1 second Second The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock.... 
= 1 s 
velocity Velocity In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ... 
v  centimetre per second  cm/s  cm/s  = 10^{−2} m/s 
acceleration Acceleration In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ... 
a  gal Gal (unit) The gal, sometimes called galileo, is a unit of acceleration used extensively in the science of gravimetry. The gal is defined as 1 centimeter per second squared .... 
Gal  cm/s^{2}  = 10^{−2} m/s^{2} 
force  F  dyne Dyne In physics, the dyne is a unit of force specified in the centimetregramsecond system of units, a predecessor of the modern SI. One dyne is equal to exactly 10 µN... 
dyn  g·cm/s^{2}  = 10^{−5} N 
energy Energy In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems... 
E  erg Erg An erg is the unit of energy and mechanical work in the centimetregramsecond system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work".... 
erg  g·cm^{2}/s^{2}  = 10^{−7} J Joule The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second... 
power Power (physics) In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit... 
P  erg Erg An erg is the unit of energy and mechanical work in the centimetregramsecond system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work".... per second Second The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock.... 
erg/s  g·cm^{2}/s^{3}  = 10^{−7} W Watt The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.Definition:... 
pressure Pressure Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure. Definition :... 
p  barye Barye The barye , or sometimes barad, barrie, bary, baryd, baryed, or barie, is the centimetregramsecond unit of pressure. It is equal to 1 dyne per square centimetre.... 
Ba  g/(cm·s^{2})  = 10^{1} Pa Pascal (unit) The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre... 
dynamic viscosity Viscosity Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity... 
μ  poise Poise The poise is the unit of dynamic viscosity in the centimetre gram second system of units. It is named after Jean Louis Marie Poiseuille .... 
P  g/(cm·s)  = 10^{1} Pa·s 
wavenumber Wavenumber In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector... 
k  kayser Wavenumber In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector... 
cm^{−1}  cm^{−1}  = 100 m^{−1} 
CGS approach to electromagnetic units
The conversion factors relating electromagneticElectromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
units in the CGS and SI systems are much more involved – so much so that formulas for physical laws of electromagnetism are adjusted depending on what system of units one uses. This illustrates the fundamental difference in the ways the two systems are built:
 In SI, the unit of electric currentElectric currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
is chosen to be 1 ampereAmpereThe ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after AndréMarie Ampère , French mathematician and physicist, considered the father of electrodynamics...
(A). It is a base unit of the SI system, along with meter, kilogram, and second. The ampere is not dimensionally equivalent to any combination of other base units, so electromagnetic laws written in SI require an additional constant of proportionality (see Vacuum permittivity) to bridge electromagnetic units to kinematic units. All other electric and magnetic units are derived from these four base units using the most basic common definitions: for example, electric chargeCharge (physics)In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.Formal definition:...
q is defined as current I multiplied by time t,

 ,
 therefore unit of electric charge, coulomb (C), is defined as 1 C = 1 A·s.
 CGS system avoids introducing new base units and instead derives all electric and magnetic units from centimeter, gram, and second based on the physics laws that relate electromagnetic phenomena to mechanics.
Alternate derivations of CGS units in electromagnetism
Electromagnetic relationships to length, time and mass may be derived by equally appealing methods. Two of them rely on the forces observed on charges. Two fundamental laws relate (independently of each other) the electric charge or its rate of changeDerivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
(electric current) to a mechanical quantity such as force. They can be written in systemindependent form as follows:
 The first is Coulomb's lawCoulomb's lawCoulomb's law or Coulomb's inversesquare law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
, , which describes the electrostatic force F between electric charges and , separated by distance d. Here is a constant which depends on how exactly the unit of charge is derived from the CGS base units.
 The second is Ampère's force lawAmpère's force lawIn magnetostatics, the force of attraction or repulsion between two currentcarrying wires is often called Ampère's force law...
, , 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 wires' diameters. Since and , the constant also depends on how the unit of charge is derived from the CGS base units.
Maxwell's theory of electromagnetism
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
relates these two laws to each other. It states that the ratio of proportionality constants and must obey , where c is the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
. Therefore, if one derives the unit of charge from the Coulomb's law by setting , it is obvious that the Ampère's force law will contain a prefactor . Alternatively, deriving the unit of current, and therefore the unit of charge, from the Ampère's force law by setting or , will lead to a constant prefactor in the Coulomb's law.
Indeed, both of these mutuallyexclusive approaches have been practiced by the users of CGS system, leading to the two independent and mutuallyexclusive branches of CGS, described in the subsections below. However, the freedom of choice in deriving electromagnetic units from the units of length, mass, and time is not limited to the definition of charge. While the electric field can be related to the work performed by it on a moving electric charge, the magnetic force is always perpendicular to the velocity of the moving charge, and thus the work performed by the magnetic field on any charge is always zero. This leads to a choice between two laws of magnetism, each relating magnetic field to mechanical quantities and electric charge:
 The first law describes the Lorentz forceLorentz forceIn physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...
produced by a magnetic field B on a charge q moving with velocity v:


 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 BiotSavart lawBiotSavart lawThe Biot–Savart law is an equation in electromagnetism that describes the magnetic field B generated by an electric current. The vector field B depends on the magnitude, direction, length, and proximity of the electric current, and also on a fundamental constant called the magnetic constant...
:
 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 BiotSavart law
 where r and are the length and the unit vector in the direction of vector r.

These two laws can be used to derive Ampère's force law
Ampère's force law
In magnetostatics, the force of attraction or repulsion between two currentcarrying wires is often called Ampère's force law...
, resulting in the relationship: . Therefore, if the unit of charge is based on the Ampère's force law
Ampère's force law
In magnetostatics, the force of attraction or repulsion between two currentcarrying wires is often called Ampère's force law...
such that , it is natural to derive the unit of magnetic field by setting . 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
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
H in a medium other than a vacuum, we need to also define the constants ε_{0} and μ_{0}, which are the vacuum permittivity and permeability, respectively. Then we have (generally) and , where P and M are polarization density
Polarization density
In classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is...
and magnetization
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...
vectors. The factors λ and λ′ are rationalization constants, which are usually chosen to be 4πk_{C}ε_{0}, a dimensionless quantity. If λ = λ′ = 1, the system is said to be "rationalized": the laws for systems of spherical geometry
Spherical geometry
Spherical geometry is the geometry of the twodimensional surface of a sphere. It is an example of a geometry which is not Euclidean. Two practical applications of the principles of spherical geometry are to navigation and astronomy....
contain factors of 4π (for example, point charges), those of cylindrical geometry – factors of 2π (for example, wire
Wire
A wire is a single, usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads and to carry electricity and telecommunications signals. Wire is commonly formed by drawing the metal through a hole in a die or draw plate. Standard sizes are determined by various...
s), and those of planar geometry contain no factors of π (for example, parallelplate capacitor
Capacitor
A capacitor is a passive twoterminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s). However, the original CGS system used λ = λ′ = 4π, or, equivalently, k_{C}ε_{0} = 1. Therefore, Gaussian, ESU, and EMU subsystems of CGS (described below) are not rationalized.
Various extensions of the CGS system to electromagnetism
The table below shows the values of the above constants used in some common CGS subsystems:system        Electrostatic CGS (ESU, esu, or stat)  1  c^{−2}  1  c^{−2}  c^{−2}  1  4π  4π  

Electromagnetic CGS (EMU, emu, or ab) 
c^{2}  1  c^{−2}  1  1  1  4π  4π  
Gaussian Gaussian units Gaussian units comprise a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs units. It is also called the Gaussian unit system, Gaussiancgs units, or often just cgs units... CGS 
1  c^{−1}  1  1  c^{−2}  c^{−1}  4π  4π  
Lorentz–Heaviside CGS  1  1  c^{−1}  1  1  
SI Si Si, si, or SI may refer to : Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system... 
1  1  1 
The constant b in SI system is a unitbased scaling factor defined as: .
Also, note the following correspondence of the above constants to those in Jackson and Leung:
In systemindependent form, Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
in vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
can be written as:
Note that of all these variants, only in Gaussian and Heaviside–Lorentz systems equals rather than 1. As a result, vectors and of an electromagnetic wave propagating in vacuum have the same units and are equal in magnitude in these two variants of CGS.
Electrostatic units (ESU)
In one variant of the CGS system, Electrostatic units (ESU), charge is defined via the force it exerts on other charges, and current is then defined as charge per time. It is done by setting the Coulomb force constant , so that Coulomb's lawCoulomb's law
Coulomb's law or Coulomb's inversesquare law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
does not contain an explicit prefactor
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
.
The ESU unit of charge, franklin (Fr), also known as statcoulomb
Statcoulomb
The statcoulomb or franklin or electrostatic unit of charge is the physical unit for electrical charge used in the centimetregramsecond system of units and Gaussian units. It is a derived unit given by...
or esu charge, is therefore defined as follows: Therefore, in electrostatic CGS units, a franklin is equal to a centimetre times square root of dyne:
 .
The unit of current is defined as:
 .
Dimensionally in the ESU CGS system, charge q is therefore equivalent to m^{1/2}L^{3/2}t^{−1}. Neither charge nor current are therefore an independent dimension of physical quantity in ESU CGS. This reduction of units is an application of the Buckingham π theorem.
ESU notation
All electromagnetic units in ESU CGS system that do not have proper names are denoted by a corresponding SI name with an attached prefix "stat" or with a separate abbreviation "esu".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 ampereAmpere
The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after AndréMarie Ampère , French mathematician and physicist, considered the father of electrodynamics...
as well). In the EMU CGS subsystem, is done by setting the Ampere force constant , so that Ampère's force law
Ampère's force law
In magnetostatics, the force of attraction or repulsion between two currentcarrying wires is often called Ampère's force law...
simply contains 2 as an explicit prefactor
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
(this prefactor 2 is itself a result of integrating a more general formulation of Ampère's law over the length of the infinite wire).
The EMU unit of current, biot (Bi), also known as abampere
Abampere
The abampere , also called the biot after JeanBaptiste Biot, is the basic electromagnetic unit of electric current in the emucgs system of units . One abampere is equal to ten amperes in the SI system of units...
or emu current, is therefore defined as follows:
Therefore, in electromagnetic CGS units, a biot is equal to a square root of dyne:
 .
The unit of charge in CGS EMU is:
 .
Dimensionally in the EMU CGS system, charge q is therefore equivalent to m^{1/2}L^{1/2}. Neither charge nor current are therefore an independent dimension of physical quantity in EMU CGS.