# 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.

Mechanical CGS units
Dimension Unit Definition SI
length centimetre 1 cm = 10−2 m
mass gram 1 g = 10−3 kg
time second 1 s
force dyne 1 dyn = 1 g·cm/s² = 10−5 N
energy erg 1 erg = 1 g·cm²/s² = 10−7 J
power erg per second 1 erg/s = 1 g·cm²/s³ = 10−7 W
pressure barye 1 Ba = 1 dyn/cm² = 1 g/(cm·s²) = 10−1 Pa
viscosity poise 1 P = 1 g/(cm·s) = 10−1 Pa·s

The system goes back to a proposal made in 1832 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purposes, therefore the CGS system never gained wide general use outside the field of electrodynamics and was gradually superseded internationally starting in the 1880s but not to a significant extent until the mid-20th century by the more practical MKS (metre-kilogram-second) system,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 older 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.

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.

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.

## 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, electrostatic units (esu), 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 the Biot-Savart law), which describes the electrodynamic (or electromagnetic) force between currents. Each of these includes one proportionality constant, k1 or k2. The static definition of magnetic fields yields a third proportionality constant, α. The first two constants are related to each other through the speed of light, c (the ratio of k1 over k2 must equal c2).

We then have several choices:

k1 k2 α yields
1 c−1 1 electrostatic cgs system
c2 1 1 electromagnetic cgs system
1 c−2 c−1 Gaussian cgs system
(4·π·ε0)−1 µ0·(4·π)−1 1 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 electromagnetic units (emu, 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 $\displaystyle \epsilon_0$ and $\displaystyle \mu_0$ are $\displaystyle 1$ , and the only dimensional constant appearing in the equations is $\displaystyle c$ , 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 $\displaystyle 4 \pi$ 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. However, this also can be seen more as an application 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 1 centimetre.

Dimension Unit Definition SI
charge electrostatic unit of charge, franklin, statcoulomb 1 esu = 1 statC = 1 Fr = √(g·cm³/s²) = 3.3356 × 10−10 C
electric potential statvolt 1 statV = 1 erg/esu = 299.792458 V
electric field 1 statV/cm = 1 dyn/esu
magnetic field strength oersted 1 Oe = 1000/(4π) A/m = 79.577 A/m
magnetic flux density gauss 1 G = 10−4 T
magnetic flux maxwell 1 M = 1 G·cm² = 10−8 Wb
magnetic induction gauss 1 G = 1 M/cm²
resistance 1 s/cm
resistivity 1 s
capacitance 1 cm = 1.113 × 10−12 F
inductance 1 s²/cm = 8.988 × 1011 H

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 spheres of radii R and r is

$\displaystyle \frac{1}{\frac{1}{r}-\frac{1}{R}}$ .

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