Centimetre gram second system of units
Encyclopedia
The centimetre–gram–second system (abbreviated CGS or cgs) is a metric system
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 ten-millionth 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 André-Marie 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 carbon-12 , 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 unit-conversion 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 centimetre-gram-second system of units, a predecessor of the modern SI. One dyne is equal to exactly 10 µN...

, equal to 1 g·cm/s2, while the SI derived unit of force is the newton, 1 kg·m/s2. 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 one-to-one 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 "sub-systems", 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, Gaussian-cgs 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 CGS-Gaussian 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, Gaussian-cgs units, or often just cgs units...

.

## History

The CGS system goes back to a proposal made in 1832 by the German mathematician Carl Friedrich Gauss
Carl 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 mid-20th 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 peer-reviewed 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 500-page 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 table-top 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 (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 the unit of time) is the same in both systems.

There is a one-to-one 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 one-to-one 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...

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 centimetre-gram-second 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·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

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 ten-millionth 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/s2 = 10−2 m/s2
force F dyne
Dyne
In physics, the dyne is a unit of force specified in the centimetre-gram-second system of units, a predecessor of the modern SI. One dyne is equal to exactly 10 µN...

dyn g·cm/s2 = 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 centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

erg g·cm2/s2 = 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 centimetre-gram-second 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·cm2/s3 = 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 centimetre-gram-second unit of pressure. It is equal to 1 dyne per square centimetre....

Ba g/(cm·s2) = 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 electromagnetic
Electromagnetism
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 current
Electric current
Electric 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 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 André-Marie 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 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:...

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 change
Derivative
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 system-independent form as follows:
• The first is Coulomb's law
Coulomb's law
Coulomb's law or Coulomb's inverse-square 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 law
Ampère's force law
In magnetostatics, the force of attraction or repulsion between two current-carrying 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 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 Lorentz force
Lorentz force
In 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 Biot-Savart law
Biot-Savart law
The 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...

:
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 current-carrying 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 current-carrying 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 ε0 and μ0, 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 4πkCε0, 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 two-dimensional 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, parallel-plate capacitor
Capacitor
A capacitor is a passive two-terminal 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, kCε0 = 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-) c2 1 c−2 1 1 1 4π 4π GaussianGaussian unitsGaussian 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, Gaussian-cgs 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 SISiSi, 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 unit-based scaling factor defined as: .

Also, note the following correspondence of the above constants to those in Jackson and Leung:

In system-independent 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 law
Coulomb's law
Coulomb's law or Coulomb's inverse-square 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 centimetre-gram-second 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 m1/2L3/2t−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 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 André-Marie 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 current-carrying 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 Jean-Baptiste Biot, is the basic electromagnetic unit of electric current in the emu-cgs 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 m1/2L1/2. Neither charge nor current are therefore an independent dimension of physical quantity in EMU CGS.

#### EMU notation

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".

### Relations between ESU and EMU units

The ESU and EMU subsystems of CGS are connected by the fundamental relationship (see above), where c = 29,979,245,800 ≈ 3·1010 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...

in vacuum in cm/s. 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 Coulomb's law
Coulomb's law
Coulomb's law or Coulomb's inverse-square 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...

or Ampère's force law
Ampère's force law
In magnetostatics, the force of attraction or repulsion between two current-carrying wires is often called Ampère's force law...

) is equal either to c-1 or c:
and.
Units derived from these may have ratios equal to higher powers of c, for example:.

### 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 also include Gaussian 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, Gaussian-cgs units, or often just cgs units...

, and Heaviside–Lorentz units.

Further complicating matters is the fact that some physicists and engineers
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

in the United States use hybrid units, such as volt
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...

s per 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...

for electric field. In fact, this is essentially the same as the SI unit system, by the variant to translate all lengths used into cm, e.g. 1 m = 100 cm.

## Electromagnetic units in various CGS systems

Conversion of SI units in electromagnetism to ESU, EMU, and Gaussian subsystems of CGS
c = 29,979,245,800 ≈ 3·1010
Quantity Symbol SI unit ESU unit EMU unit Gaussian unit
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, Gaussian-cgs units, or often just cgs units...

electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

q 1 C  = (10−1 c) statC
Statcoulomb
The statcoulomb or franklin or electrostatic unit of charge is the physical unit for electrical charge used in the centimetre-gram-second system of units and Gaussian units. It is a derived unit given by...

= (10−1) abC
Abcoulomb
The abcoulomb or electromagnetic unit of charge is the basic physical unit of electric charge in the cgs-emu system of units...

= (10−1 c) Fr
Statcoulomb
The statcoulomb or franklin or electrostatic unit of charge is the physical unit for electrical charge used in the centimetre-gram-second system of units and Gaussian units. It is a derived unit given by...

electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

I 1 A
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 André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

= (10−1 c) statA = (10−1) abA
Abampere
The abampere , also called the biot after Jean-Baptiste Biot, is the basic electromagnetic unit of electric current in the emu-cgs system of units . One abampere is equal to ten amperes in the SI system of units...

= (10−1 c) Fr
Statcoulomb
The statcoulomb or franklin or electrostatic unit of charge is the physical unit for electrical charge used in the centimetre-gram-second system of units and Gaussian units. It is a derived unit given by...

·s−1
electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

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...

φ
V
1 V
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...

= (108 c−1) statV
Statvolt
The statvolt is a unit of voltage and electrical potential used in the cgs system of units. The conversion to the SI system isIt is a useful unit for electromagnetism because one statvolt per centimetre is equal in magnitude to one gauss. Thus, for example, an electric field of one statvolt/cm has...

= (108) abV
Abvolt
The abvolt is one option for the unit of potential difference in the CGS system of units, and is equal to 10−8 volts in the SI system.A potential difference of 1 abV will drive a current of one abampere through a resistance of one abohm....

= (108 c−1) statV
Statvolt
The statvolt is a unit of voltage and electrical potential used in the cgs system of units. The conversion to the SI system isIt is a useful unit for electromagnetism because one statvolt per centimetre is equal in magnitude to one gauss. Thus, for example, an electric field of one statvolt/cm has...

electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

E 1 V
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...

/m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

= (106 c−1) statV
Statvolt
The statvolt is a unit of voltage and electrical potential used in the cgs system of units. The conversion to the SI system isIt is a useful unit for electromagnetism because one statvolt per centimetre is equal in magnitude to one gauss. Thus, for example, an electric field of one statvolt/cm has...

/cm
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...

= (106) abV
Abvolt
The abvolt is one option for the unit of potential difference in the CGS system of units, and is equal to 10−8 volts in the SI system.A potential difference of 1 abV will drive a current of one abampere through a resistance of one abohm....

/cm
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...

= (106 c−1) statV
Statvolt
The statvolt is a unit of voltage and electrical potential used in the cgs system of units. The conversion to the SI system isIt is a useful unit for electromagnetism because one statvolt per centimetre is equal in magnitude to one gauss. Thus, for example, an electric field of one statvolt/cm has...

/cm
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...

magnetic B 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;...

B 1 T
Tesla (unit)
The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...

= (104 c−1) statT = (104) G
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...

= (104) G
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...

magnetic H 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 1 A
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 André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

/m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

= (4π 10−3 c) statA/cm
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...

= (4π 10−3) Oe
Oersted
Oersted is the unit of magnetizing field in the CGS system of units.-Difference between cgs and SI systems:...

= (4π 10−3) Oe
Oersted
Oersted is the unit of magnetizing field in the CGS system of units.-Difference between cgs and SI systems:...

magnetic dipole moment μ 1 A
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 André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

·
Square metre
The square metre or square meter is the SI derived unit of area, with symbol m2 . It is defined as the area of a square whose sides measure exactly one metre...

= (103 c) statA·cm² = (103) abA
Abampere
The abampere , also called the biot after Jean-Baptiste Biot, is the basic electromagnetic unit of electric current in the emu-cgs system of units . One abampere is equal to ten amperes in the SI system of units...

·cm²
= (103) erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

/G
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...

magnetic flux
Magnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...

Φm 1 Wb
Weber (unit)
In physics, the weber is the SI unit of magnetic flux. A flux density of one Wb/m2 is one tesla.The weber is named for the German physicist Wilhelm Eduard Weber .- Definition :...

= (108 c−1) statT·cm² = (108) Mw
Maxwell (unit)
The maxwell, abbreviated as Mx, is the compound derived CGS unit of magnetic flux. The unit was previously called a line. The unit name honours James Clerk Maxwell, who presented the unified theory of electromagnetism, and was established by the IEC in 1930.In a magnetic field of strength one...

= (108) G
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...

·cm²
resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

R 1 Ω
Ohm
The ohm is the SI unit of electrical resistance, named after German physicist Georg Simon Ohm.- Definition :The ohm is defined as a resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere,...

= (109 c−2) s
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....

/cm
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...

= (109) abΩ
Abohm
The abohm is the basic unit of electrical resistance in the emu-cgs system of units . One abohm is equal to 10-9 ohms in the SI system of units; one abohm is a nanoohm....

= (109 c−2) s
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....

/cm
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...

resistivity ρ 1 Ω
Ohm
The ohm is the SI unit of electrical resistance, named after German physicist Georg Simon Ohm.- Definition :The ohm is defined as a resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere,...

·m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

= (1011 c−2) s
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....

= (1011) abΩ
Abohm
The abohm is the basic unit of electrical resistance in the emu-cgs system of units . One abohm is equal to 10-9 ohms in the SI system of units; one abohm is a nanoohm....

·cm
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...

= (1011 c−2) s
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....

capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

C 1 F
The farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second...

= (10−9 c2) cm
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...

= (10-9) abF  = (10-9 c2) cm
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...

inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

L 1 H  = (109 c−2) cm
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...

−1·s
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....

−2
= (109) abH
Abhenry
Abhenry is the centimeter-gram-second electromagnetic unit of inductance, equal to one billionth of a henry....

= (109 c−2) cm
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...

−1·s
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....

2

In this table, c = 29,979,245,800 ≈ 3·1010 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...

in vacuum in the CGS units of cm/s.

One can think of the SI value of the Coulomb constant kC as:
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 between a sphere of radius 1 cm in vacuum and infinity. The capacitance C between two concentric spheres of radii R and r in ESU CGS system is:
.

By taking the limit as R goes to infinity we see C equals r.

## Physical constants in CGS units

Commonly used physical constants in CGS units
Constant Symbol Value
Atomic mass unit
Atomic mass unit
The unified atomic mass unit or dalton is a unit that is used for indicating mass on an atomic or molecular scale. It is defined as one twelfth of the rest mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state, and has a value of...

u 1.660 538 782 × 10−24 g
Gram
The gram is a metric system unit of mass....

Bohr magneton μB 9.274 009 15 × 10−21 erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

/G
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...

(EMU, Gaussian)
2.780 278 00 × 10−10 statA·cm2 (ESU)
The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom...

a0 5.291 772 0859 × 10−9 cm
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...

Boltzmann constant k 1.380 6504 × 10−16 erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

/K
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...

Electron mass
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

me 9.109 382 15 × 10−28 g
Gram
The gram is a metric system unit of mass....

Elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

e 4.803 204 27 × 10−10 Fr
Statcoulomb
The statcoulomb or franklin or electrostatic unit of charge is the physical unit for electrical charge used in the centimetre-gram-second system of units and Gaussian units. It is a derived unit given by...

(ESU, Gaussian)
1.602 176 487 × 10−20 abC
Abcoulomb
The abcoulomb or electromagnetic unit of charge is the basic physical unit of electric charge in the cgs-emu system of units...

(EMU)
Fine-structure constant
Fine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...

α ≈ 1/137 7.297 352 570 × 10−3
Gravitational constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

G 6.674 28 × 10−8 cm
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...

3/(g
Gram
The gram is a metric system unit of mass....

·s
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....

2)
Planck constant
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...

h 6.626 068 85 × 10−27 erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

·s
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....

1.054 5716 × 10−27 erg
Erg
An erg is the unit of energy and mechanical work in the centimetre-gram-second system of units, symbol "erg". Its name is derived from the Greek ergon, meaning "work"....

·s
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....

Speed of light in vacuum
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...

c ≡ 2.997 924 58 × 1010 cm
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...

/s
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....

## Pro and contra

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 abvolt
Abvolt
The abvolt is one option for the unit of potential difference in the CGS system of units, and is equal to 10−8 volts in the SI system.A potential difference of 1 abV will drive a current of one abampere through a resistance of one abohm....

, or 15 emu units of electric dipole moment
Electric dipole moment
In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity with SI units of Coulomb-meter...

, or 15 emu units of magnetic susceptibility
Magnetic susceptibility
In electromagnetism, the magnetic susceptibility \chi_m is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field...

, sometimes (but not always) per gram
Gram
The gram is a metric system unit of mass....

or per 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 carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

. On the other hand, SI starts with a unit of current, 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 André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

, which is easier to determine through experiment, but which requires extra prefactors in the electromagnetic equations. With its system of unique named units, SI also removes any confusion in usage: 1 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 André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

is a fixed quantity of a specific variable, and so are 1 henry and 1 ohm
Ohm
The ohm is the SI unit of electrical resistance, named after German physicist Georg Simon Ohm.- Definition :The ohm is defined as a resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere,...

.

A key virtue of the Gaussian CGS system
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, Gaussian-cgs units, or often just cgs units...

is that electric and magnetic fields have the same units, is replaced by , and the only dimensional constant appearing in the equations is , the speed of light. The Heaviside–Lorentz system has these desirable properties as well (with equaling 1), but it is a "rationalized" system (as is SI) in which the charges and fields are defined in such a way that there are many fewer factors of appearing in the formulas, and it is in Heaviside–Lorentz units that the Maxwell equations take their simplest form.

In SI, and other rationalized systems (for example, Heaviside–Lorentz), the unit of current was chosen such that electromagnetic equations concerning charged spheres contain 4π, those concerning coils of current and straight wires contain 2π and those dealing with charged surfaces lack π entirely, which was the most convenient choice for electrical-engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

applications. In those fields where formulas concerning spheres dominate (for example, astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

), it has been argued that the non-rationalized CGS system can be somewhat more convenient notationally.

In fact, in certain fields, specialized unit systems are used to simplify formulas even further than either SI or CGS, by using some system of natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...

. For example, the particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

community uses a system where every quantity is expressed by only one unit, the eV, with lengths, times, and so on. all converted into eV's by inserting factors of c
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...

and
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...

. This unit system is very convenient for particle-physics calculations, but would be impractical in other contexts.