# Centimetre gram second system of units

CGS is an acronym for the centimetre-gram-second system of physical units. It goes back to a proposal made in 1832 by the German astronomer Carl Friedrich Gauss and was in 1874 extended by the British physicists James Clerk Maxwell and Thomson with a set of electromagnetic units. The sizes 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 quickly superseeded internationally in the 1880s by the more practical MKS (metre-kilogram-second) system, which led eventually to the modern SI standard units.

CGS units are is still occasionally encountered in older 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 were formulas concerning spheres dominate (for example, astronomy), it has been argued that the CGS system can be notationally slightly more convenient.

Starting from the international adoption of the MKSA standard in the 1940s and the SI standard in the 1960s, the use of CGS units has gradually disappeared wordwide, 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.

## Electromagnetic units

While for most units the difference between cgs and SI is a mere power of 10, the differences in electromagnetic units are considerable; so much so that formulas for physical laws need to be changed depending on what system of units one uses. In SI, electric current is defined via the magnetic force it exerts and charge is then defined as current multiplied with time. In one variant of the cgs system, esu, or electrostatic units, charge is defined via the force it exerts on other charges, and current is then defined as charge per time. One consequence of this approach is that Coulomb's law does not contain a constant of proportionality.

While the proportional constants in cgs simplify theoretical calcuations, they have the disadvantage that the units in cgs are hard to define through experiment. SI on the other hand starts with a unit of current, the ampere which is easy to determine through experiment, but which requires that the constants in the electromagnetic equations take on odd forms.

Ultimately, relating electromagnetic phenomena to time, length and mass relies on the forces observed on charges. There are two fundamental laws in action: Coulomb's law, which describes the electrostatic force between charges, and Ampère's law (also known as Biot-Savart's law), which describes the electrodynamic (or electromagnetic) force between currents. Each of these includes one proportionality constant, [itex]k_1\,\![itex] or [itex]k_2\,\![itex]. The static definition of magnetic fields yields a third proportionality constant, [itex]\alpha\,\![itex]. The first two constants are related to each other through the speed of light, [itex]c\,\![itex] (the ratio of [itex]k_1\,\![itex] over [itex]k_2\,\![itex] must equal [itex]c^2\,\![itex]).

We then have several choices:

[itex]k_1\,\![itex] [itex]k_2\,\![itex] [itex]\alpha\,\![itex] yields
[itex]1\,\![itex] [itex]1/c^2\,\![itex] [itex]1\,\![itex] electrostatic cgs system
[itex]c^2\,\![itex] [itex]1\,\![itex] [itex]1\,\![itex] electromagnetic cgs system
[itex]1\,\![itex] [itex]1/c^2\,\![itex] [itex]1/c\,\![itex] Gaussian cgs system
[itex]\frac{1}{4 \pi \epsilon_0}\,\![itex] [itex]\frac{\mu_0}{4 \pi}\,\![itex] [itex]1\,\![itex] SI

There were at various points in time about half a dozen systems of electromagnetic units in use, most based on the cgs system. These include emu, or electromagnetic units (chosen such that the Biot-Savart Law has no constant of proportionality), Gaussian, and Heaviside-Lorentz units. A key virtue of the Gaussian CGS system is that electric and magnetic fields have the same units, both [itex]\epsilon_0[itex] and [itex]\mu_0[itex] are [itex]1[itex], and the only dimensional constant appearing in the equations is [itex]c[itex], the speed of light. The Heaviside-Lorentz system has these desirable properties as well, 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 [itex]4 \pi[itex] appearing in the formulas, and it is in Heaviside-Lorentz units that the Maxwell equations take their simplest possible form.

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.

## Units

The units of cgs (specifically esu) are as follows:

The mantissas 2998, 3336, 1113, and 8988 are derived from the speed of light and 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 spheres of radii R and r is

[itex]\frac{1}{\frac{1}{r}-\frac{1}{R}}[itex]. By taking the limit as R goes to infinity we see C equals r.

## See also

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