# Centimetre gram second system of units

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Centimetre gram second system of units

The centimetre-gram-second system (CGS) is a system of physical units. It is always the same for mechanical units, but there are several variants of electric additions. It was replaced by the MKS, or metre-kilogram-second system, which in turn was replaced by the International System of Units (SI), which has the three base units of MKS plus the ampere, mole, candela and kelvin.

History

The system goes back to a proposal made in 1833 by the German mathematician Carl Friedrich Gauss and was in 1874 extended by the British physicists James Clerk Maxwell and William Thomson with a set of electromagnetic units. The sizes (order of magnitude) of many CGS units turned out to be inconvenient for practical purposes, therefore the CGS system never gained wide general use outside the field of electrodynamics and was gradually superseded internationally starting in the 1880s but not to a significant extent until the mid-20th century by the more practical MKS (metre-kilogram-second) system, which led eventually to the modern SI standard units.

CGS units are still occasionally encountered in technical literature, especially in the United States in the fields of electrodynamics and astronomy. SI units were chosen such that electromagnetic equations concerning spheres contain 4π, those concerning coils contain 2π and those dealing with straight wires lack π entirely, which was the most convenient choice for electrical-engineering applications. In those fields where formulas concerning spheres dominate (for example, astronomy), it has been argued that the CGS system can be notationally slightly more convenient. it can also sometimes be a willy.

Starting 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 disappeared worldwide, in the United States more slowly than in the rest of the world. CGS units are today no longer accepted by the house styles of most scientific journals, textbook publishers and standards bodies, although they are commonly used in astronomical journals such as the Astrophysical Journal.

The units gram and centimetre remain useful within the SI, especially for instructional physics and chemistry experiments, where they match well the small scales of table-top setups. In these uses, they are occasionally referred to as the system of “LAB” units. However, where derived units are needed, the SI ones are generally used and taught today instead of the CGS ones.

CGS units in mechanics

In mechanics, both CGS and SI systems are built in an identical way. The only difference between the two systems is the scale of two out of the three base units needed in mechanics (centimetre versus metre and gram versus kilogram), while the third unit (measure of time: second) is the same in both systems . The laws and definitions of mechanics that are used to obtain all derived units from the three base units are the same in both systems, for example:

:$v=frac\left\{x\right\}\left\{t\right\}$ (deifnition of velocity)

:$F=mfrac\left\{d^2x\right\}\left\{dt^2\right\}$ (Newton's first law of motion)

:$E = Fcdot dx$ (energy defined in terms of work)

:$p = frac\left\{F\right\}\left\{L^2\right\}$ (pressure defined as force per unit area)

:$eta = au/frac\left\{dv\right\}\left\{dx\right\}$ (dynamic viscosity defined as shear stress per unit velocity gradient).

This explains why, 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 Ba = 1 g/(cm·s2)

:1 Pa = 1 kg/(m·s2).

However, expressing a CGS derived unit in terms of the SI base units involves a combination of 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.

The mantissas derived from the speed of light are more precisely 299792458, 333564095198152, 1112650056, and 89875517873681764.

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" is: $frac\left\{1\right\}\left\{frac\left\{1\right\}\left\{r\right\}-frac\left\{1\right\}\left\{R$.By taking the limit as "R" goes to infinity we see "C" equals "r".

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 $k_C = 1$, so that Coulomb’s law does not contain an explicit prefactor.

The ESU unit of charge, statcoulomb or esu charge, is therefore defined as follows: Bquote|two equal point charges spaced 1 centimetre apart are said to be of 1 statcoulomb each if the electrostatic force between them is precisely 1 dyne. In CGS electrostatic units, a statcoulomb is equal to a centimetre times square root of dyne:: $\left(mathrm\left\{1,statcoulomb = 1,esu, charge = 1,cmsqrt\left\{dyne\right\}=1,g^\left\{1/2\right\} cdot cm^\left\{3/2\right\} cdot s^\left\{-1\right)$.Dimensionally in the CGS ESU system, charge "q" is therefore equivalent to m1/2L3/2t−1 and is not an independent dimension of physical quantity. This reduction of units is an application of the Buckingham π theorem.

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. [cite journal
author = Bennett, L. H.; Page, C. H.; and Swartzendruber, L. J.
title = Comments on units in magnetism
year = 1978
journal = Journal of Research of the National Bureau of Standards
volume = 83
issue = 1
pages = 9–12
doi =
] These include electromagnetic units (emu, chosen such that the Biot-Savart law has no explicit prefactor), Gaussian units, and Heaviside-Lorentz units.

Further complicating matters is the fact that some physicists and engineers in the United States use hybrid units, such as volts per centimetre 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. More difficult is to translate electromagnetic quantities from SI to cgs, which is also not hard, e.g. by using the three relations $q\text{'}=q/sqrt\left\{4pi epsilon_0\right\}$, $mathbf E\text{'}=mathbf Ecdot sqrt\left\{4pi epsilon_0\right\}$, and $mathbf B\text{'}=mathbf Bcdotsqrt\left\{4pi/ mu_0\right\}$, where $epsilon_0\left(,,equiv 1/\left(c^2mu_0\right)\right)$ and $mu_0$ are the well-known vacuum permittivities and "c" the corresponding light velocity, whereas $q, ,,mathbf E$ and $mathbf B$ are the electrical charge, electric field, and magnetic induction, respectively, "without primes" in a SI system and "with primes" in a CGS system.

However, the above-mentioned example of "hybrid units" can also simply be seen as a practical example of the previously described "LAB" units usage since electric fields near small circuit devices would be measured across distances on the order of magnitude of one centimetre.

= Physical constants in CGS unitsCite book | year=1978 | author= A.P. French, Edwind F. Taylor| title= An Introduction to Quantum Physics | publisher=W.W. Norton & Company] =

Pro and contra

A key virtue of the Gaussian CGS system is that electric and magnetic fields have the same units, $4piepsilon_0$ is replaced by $1$, and the only dimensional constant appearing in the equations is $c$, the speed of light. The Heaviside-Lorentz system has these desirable properties as well (with $epsilon_0$ equalling 1), but 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 $4 pi$ appearing in the formulas, and it is in Heaviside-Lorentz units that the Maxwell equations take their simplest possible form.

At the same time, the elimination of $epsilon_0$ and $mu_0$ can also be viewed as a major disadvantage of all the variants of the CGS system. Within classical electrodynamics, this elimination makes sense because it greatly simplifies the Maxwell equations. In quantum electrodynamics, however, the vacuum is no longer just empty space, but it is filled with virtual particles that interact in complicated ways. The fine structure constant in Gaussian CGS is given as $alpha=e^2/hbar c$ and it has been cause to much mystification how its numerical value $alpha approx 1/137.036$ should be explained. In SI units with $alpha = e^2/4 pi epsilon_0hbar c$ it may be clearer that it is in fact the complicated quantum structure of the vacuum that gives rise to a non-trivial vacuum permittivity. However, the advantage would be purely pedagogical, and in practice, SI units are essentially never used in quantum electrodynamics calculations. In fact the high energy community uses a system where every quantity is expressed by only one unit, namely by "eV", i.e. lengths "L" by the corresponding reciprocal quantity $frac\left\{hbar \right\}\left\{m_Lcdot c\right\} equiv L=frac\left\{hbar c\right\}\left\{E_L\right\}$, where the Einstein expression corresponding to $m_L$, $E_L=m_L,,c^2$, is an "energy", which thus can naturally be expressed in "eV" ($hbar$ is Planck's constant divided by $2pi$).

* Scientific units named after people
* Units of measurement
* SI electromagnetism units
* SI units

References and notes

General literature

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