# 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.

Mechanical CGS units
Dimension Unit Definition SI
length centimetre 1/100 of metre = 10−2 m
mass gram 1/1000 of kilogram = 10−3 kg
time second 1 second = 1 s
force dyne g cm / s2 = 10−5 N
energy erg g cm2 / s2 = 10−7 J
power erg per second g cm2 / s3 = 10−7 W
pressure barye g / (cm s2) = 10−1 Pa
dynamic viscosity poise g / (cm s) = 10−1 Pa·s

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therefore the CGS system never gained wide general use outside the field of electrodynamics and was gradually superseded internationally starting in the 1880s but not to a significant extent until the mid-20th century by the more practical MKS (metre-kilogram-second) system,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

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 electromagnetism

While for most units the difference between cgs and SI are just powers of 10, the differences in electromagnetic units are more involved — so much so that formulas for physical laws of electromagnetism are adjusted 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. What this means specifically is that in cgs electrostatic units, the unit of charge or statcoulomb, is defined as such a quantity of charge that the Coulomb force constant is set to 1. That is, for two point charges, each with 1 statcoulomb spaced apart by 1 centimetre, the electrostatic force between them will be, by definition, precisely one dyne. This also has the effect of eliminating a separate dimension or fundamental unit for electric charge. In cgs electrostatic units, a statcoulomb is the same as a centimetre times square root of dyne. Dimensionally in the cgs esu system, charge Q is equivalent to M1/2L3/2T−1 and not an independent dimension of physical quantity. This reduction of units is an application of the Buckingham π theorem.

While the proportional constants in cgs simplify theoretical calculations, 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. The first is Coulomb's law, which describes the electrostatic force between charges $\left( F = k_C q q^\prime / r^2 \right)$. The second is Ampère's force law, which describes the electrodynamic (or electromagnetic) force between currents $( dF / dl = 2 k_A I I^\prime / d$ for two long parallel wires). The proportionality constants in these two equations are related by $k_C / k_A = c^2$, where c is the speed of light. The static definition of magnetic fields (Biot-Savart law) yields a third proportionality constant, α, which establishes convenient dimensions.

If we wish to describe the electric displacement field $\vec D$ and the magnetic field $\vec H$ 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. These two values are related by $\sqrt{\mu_0\epsilon_0}=\alpha / c$. Then we have (generally) $\vec D = \epsilon_0 \vec E + \lambda \vec P$ and $\vec H = \vec B / \mu_0 - \lambda^\prime \vec M$. The factors λ and λ′ are rationalization constants, which are usually chosen to both be equal to 4πkCε0, which is dimensionless. If this quantity equals 1, the system is said to be rationalized.

The table below shows the constant values used in some common systems:

system kC α ε0
electrostatic (esu) 1 c 1/(4π)
electromagnetic (emu) c2 1 c−2
Gaussian 1 c 1
Heaviside-Lorentz 1/4π c 1
SI c2/b 1 b/(4πc2)

(The b in SI is a scaling factor equal to 107 A2/N = 107 m/H.)

In system-independent form, Maxwell's equations can be written

$\begin{array}{ccl} \vec \nabla \cdot \vec E & = & 4 \pi k_C \rho \\ \vec \nabla \cdot \vec B & = & 0 \\ \vec \nabla \times \vec E & = & \displaystyle{- \frac{1}{\alpha} \frac{\partial \vec B}{\partial t}} \\ \vec \nabla \times \vec B & = & \displaystyle{\frac{\alpha}{c^2} \left(4 \pi k_C \vec J + \frac{\partial \vec E}{\partial t}\right)} \end{array}$

Dimension Unit Definition SI
charge electrostatic unit of charge, franklin, statcoulomb 1 esu = 1 statC = 1 Fr = √(g·cm³/s²) = 3.33564 × 10−10 C
electric current biot 1 esu/s = 3.33564 × 10−10 A
electric potential statvolt 1 statV = 1 erg/esu = 299.792458 V
electric field 1 statV/cm = 1 dyn/esu = 2.99792458 × 104 V/m
magnetic field strength H oersted 1 Oe = 1000/(4π) A/m = 79.577 A/m
magnetic flux maxwell 1 Mw = 1 G·cm² = 10−8 Wb
magnetic induction B gauss 1 G = 1 Mw/cm² = 10−4 T
resistance 1 s/cm = 8.988 × 1011 Ω
resistivity 1 s = 8.988 × 109 Ω·m
capacitance 1 cm = 1.113 × 10−12 F
inductance statH = 8.988 × 1011 H
wavenumber kayser 1 /cm = 100 /m

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

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

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

## Physical constants in CGS units[1]

Constant Symbol Value
Atomic mass unit u 1.660 × 10−24 g
Avogadro constant NA 6.022 × 1023 (mol)−1
Bohr magneton μB 9.274 × 10−21 erg/G
Bohr radius a0 5.291 × 10−9 cm
Boltzmann constant k 1.380 × 10−16 erg/K
Electron mass me 9.109 × 10−28 g
Elementary charge e 4.803 × 10−10 esu of charge
1.602 × 10−20 emu of charge
Fine-structure constant α 7.297 × 10−3
Gravitational constant G 6.674 × 10−8 cm3/(g·s2)
Planck constant h 6.626 × 10−27 erg·s
Speed of light in vacuum c 2.998 × 1010 cm/s

## 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 include electromagnetic units (emu, chosen such that the Biot-Savart law has no constant of proportionality), 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 $q'=q/\sqrt{4\pi \epsilon_0}$,   $\mathbf E'=\mathbf E\cdot \sqrt{4\pi \epsilon_0}$, and $\mathbf B'=\mathbf B\cdot\sqrt{4\pi/ \mu_0}$, where $\epsilon_0(\,\,\equiv 1/(c^2\mu_0))$ 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 be 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.

## Pro and contra

A key virtue of the Gaussian CGS system is that electric and magnetic fields have the same units, $4\pi\epsilon_0$ 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 $\alpha = e^2/4 \pi \epsilon_0\hbar c$ 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 $\frac{\hbar }{m_L\cdot c} \equiv L=\frac{\hbar c}{E_L}$, 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 Plancks constant divided by $2\pi$).

## References

1. A.P. French, Edwind F. Taylor (1978). An Introduction to Quantum Physics (in English). W.W. Norton & Company.