The centimetre–gram–second system of units (CGS or cgs) is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time. All CGS mechanical units are unambiguously derived from these three base units, but there are several different ways in which the CGS system was extended to cover electromagnetism.
The CGS system has mainly been supplanted by the MKS system based on the metre, kilogram, and second, which was in turn extended and replaced by the International System of Units (SI). In many fields of science and engineering, SI is the only system of units in use, but CGS is still prevalent in certain subfields.
In measurements of purely mechanical systems (involving units of length, mass, force, energy, pressure, and so on), the differences between CGS and SI are straightforward: the unit-conversion factors are all powers of 10 as and . For example, the CGS unit of force is the dyne, which is defined as , so the SI unit of force, the newton (), is equal to .
In contrast, converting measurements of electromagnetic quantities—such as electric charge, electric and magnetic fields, and voltage—between CGS and SI systems is considerably more complex. This is because the form of the equations governing electromagnetic phenomena, including Maxwell's equations, depends on the system of units employed; electromagnetic quantities are defined differently in SI and in CGS. Moreover, several distinct versions of the CGS system exist, each defining electromagnetic units differently. These include the electrostatic (ESU), electromagnetic (EMU), Gaussian units, and Heaviside–Lorentz units. Gaussian units are the most widely used in modern scientific literature,Jackson, J. D. Classical Electrodynamics, 3rd ed. Wiley, 1999. and the term "CGS units" is often understood to refer specifically to the CGS–Gaussian system.Mohr, Peter J.; Newell, David B.; Taylor, Barry N. "CODATA Recommended Values of the Fundamental Physical Constants: 2014." Rev. Mod. Phys. 88 (3), 035009 (2016).
History The CGS system goes back to a proposal in 1832 by the German mathematician Carl Friedrich Gauss to base a system of absolute units on the three fundamental units of length, mass and time.. English translation Gauss chose the units of millimetre, milligram and second. In 1873, a committee of the British Association for the Advancement of Science, including physicists James Clerk Maxwell and William Thomson, 1st Baron Kelvin recommended the general adoption of centimetre, gram and second as fundamental units, and to express all derived electromagnetic units in these fundamental units, using the prefix "C.G.S. unit of ...".
The sizes of many CGS units turned out to be inconvenient for practical purposes. For example, many everyday objects are hundreds or thousands of centimetres long, such as humans, rooms and buildings. Thus the CGS system never gained wide use outside the field of science. Starting in the 1880s, and more significantly by the mid-20th century, CGS was gradually superseded internationally for scientific purposes by the MKS (metre–kilogram–second) system, which in turn developed into the modern SI standard.
Since the international adoption of the MKS standard in the 1940s and the SI standard in the 1960s, the technical use of CGS units has gradually declined worldwide. CGS units have been deprecated in favour of SI units by NIST, as well as organisations such as the American Physical Society and the International Astronomical Union. SI units are predominantly used in engineering applications and physics education, while Gaussian CGS units are still commonly used in theoretical physics, describing microscopic systems, relativistic electrodynamics, and astrophysics.
The units gram and centimetre remain useful as noncoherent units within the SI system, as with any other prefixed SI units.
Definition of CGS units in mechanics In mechanics, the quantities in the CGS and SI systems are defined identically. The two systems differ only in the scale of the three base units (centimetre versus metre and gram versus kilogram, respectively), with the third unit (second) being the same in both systems.
There is a direct correspondence between the base units of mechanics in CGS and SI. Since the formulae expressing the laws of mechanics are the same in both systems and since both systems are coherent, the definitions of all coherent derived units in terms of the base units are the same in both systems, and there is an unambiguous relationship between derived units: (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: 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 Quantity symbol CGS unit name Unit symbol Unit definition In SI units length, position L, x centimetre cm 1/100 of metre 10−2 m mass m gram g 1/1000 of kilogram 10−3 kg time t seconds 1 second 1 s velocity v centimetre per second cm/s cm/s 10−2 m/s acceleration a gal Gal cm/s2 10−2 m/s2 force F dyne dyn g⋅cm/s2 10−5 N energy E erg erg g⋅cm2/s2 10−7 J power P erg per seconderg/s g⋅cm2/s3 10−7 W pressure p barye Ba g/(cm⋅s2) 10−1 Pa dynamic viscosity μ poise P g/(cm⋅s) 10−1 Pa⋅s kinematic viscosity ν stokes St cm2/s 10−4 m2/s wavenumber k kayser cm−1 or K cm−1 100 m−1
Derivation of CGS units in electromagnetism
CGS approach to electromagnetic units The laws of electromagnetism (specifically, the four Maxwell's equations) are formulated with fundamentally different assumptions in SI and CGS units. The SI system introduces new units to represent concepts such as electric charge, current, and magnetic flux, while CGS avoids adding new units. Rather, CGS represents all electromagnetic quantities by expressing the laws of electromagnetism in purely mechanical units, without introducing further units beyond the centimetre, gram, and second.
For example, in SI the unit of electric current, the ampere (A), was historically defined such that the magnetic force exerted by two infinitely long, thin, parallel wires 1 metre apart and carrying a current of 1 ampere is exactly . This definition results in most SI electromagnetic units being consistent (subject to factors of some integer powers of 10) with those of the CGS-EMU system described in further sections. The ampere is a base unit of the SI system, with the same status as the metre, kilogram, and second. Thus the relationship in the definition of the ampere with the metre and newton is disregarded, and the ampere is not treated as dimensionally equivalent to any combination of other base units. As a result, electromagnetic laws in SI require an additional constant of proportionality (see Vacuum permeability) to relate electromagnetic units to mechanical units. (This constant of proportionality is derivable directly from the above definition of the ampere.) All other electric and magnetic units are derived from these four base units using the most basic common definitions: for example, electric charge q is defined as current I multiplied by time t, resulting in the unit of electric charge, the coulomb (C), being defined as 1 C = 1 A⋅s. Alternative derivations of CGS units in electromagnetism Electromagnetic relationships to length, time and mass may be derived by several equally appealing methods. Two of them rely on the forces observed on charges. Two fundamental laws relate (seemingly independently of each other) the electric charge or its rate of change (electric current) to a mechanical quantity such as force. They can be written in system-independent form as follows:
The first is Coulomb's law, , 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 base units. The second is 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 wire diameters. Since and , the constant also depends on how the unit of charge is derived from the base units.
Maxwell's theory of electromagnetism relates these two laws to each other. It states that the ratio of the proportionality constants and must obey , where c is the speed of light in vacuum. Therefore, if one derives the unit of charge from Coulomb's law by setting then Ampère's force law will contain a factor . Alternatively, deriving the unit of current, and therefore the unit of charge, from Ampère's force law by setting or , will lead to a constant factor in Coulomb's law.
Indeed, both of these mutually exclusive approaches have been practiced by users of the 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 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 the Biot–Savart law: where r and are the length and the unit vector in the direction of vector r respectively. These two laws can be used to derive Ampère's force law above, resulting in the relationship: . Therefore, if the unit of charge is based on 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 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 polarisation density and magnetisation vectors. The units of P and M are usually so chosen that the factors and ′ are equal to the "rationalisation constants" and , respectively. If the rationalisation constants are equal, then . If they are equal to one, then the system is said to be "rationalised": the laws for systems of spherical geometry contain factors of 4 (for example, point charges), those of cylindrical geometry factors of 2 (for example, wires), and those of planar geometry contain no factors of (for example, parallel-plate capacitors). However, the modern CGS systems, except Heaviside–Lorentz, use = ′ = 4, or, equivalently, . Therefore, Gaussian, ESU, and EMU subsystems of CGS (described below) are not rationalised.