# CGS system of units

The CGS system of units (also CGS system , cgs system , CGS or cgs , from the English " c entimetre g ram s econd") is a metric , coherent system of units based on the units centimeter , gram and second . The CGS units of the mechanics can be clearly derived from these basic units , but there are several competing extensions of the CGS system for electromagnetic units . The four most common variants are:

Only the Gaussian system of units has significant significance today, with “CGS unit” in modern literature mostly a Gaussian CGS unit is meant.

## overview

The CGS system was introduced in 1874 by the British Association for the Advancement of Science and replaced in 1889 by the MKS system of units , based on the base units of the meter , kilogram and second . The MKS for its part was expanded to include the electromagnetic base unit amperes (then often referred to as the MKSA system ) and finally merged into the Système International d'Unités (SI) in 1960 , which today also includes the base units mol , candela and kelvin . In most fields, the SI is the only system of units in use, but there are areas in which the CGS - especially its expanded forms - is still used.

Since CGS and MKS (or the SI) in the field of mechanics are based on the same system of sizes with the basic sizes of length , mass and time , the dimensional products of the derived units are the same in both systems. A conversion between units is limited to multiplication with a pure number factor. To simplify matters, there is also the fact that conversion factors only occur in powers of 10, as results from the relationships 100 cm = 1 m and 1000 g = 1 kg. An example: For the force , the derived CGS unit is dyn (corresponds to 1 g · cm · s −2 ) and the derived MBS unit is the newton (corresponds to 1 kg · m · s −2 ). So the conversion is 1 dyn = 10 −5  N.

On the other hand, conversions between electromagnetic units of the CGS and those of the MKSA are quite cumbersome. While the MKSA introduces the ampere as a unit for the electrical current , none of the extensions to the CGS requires an additional base unit. Instead, the proportionality constants are specified by definition in Coulomb's law ( electrical permittivity ), Ampère's law and Faraday's law of induction . The various sensible options in the definition have led to the various forms of the CGS system. In any case, all electromagnetic units can be traced back to the three purely mechanical base units. However, this not only changes the dimensional products of those derived units, but also the form of physical equations of magnitude in electrodynamics (see e.g. Maxwell's equations ). There is therefore no one-to-one correspondence between the electromagnetic units of the MKSA (or the SI) and the CGS, not even between the different CGS variants. In addition to a pure numerical factor, conversions also include the size values ​​of the above constants saved in the CGS.

The principle of fixing natural constants (instead of introducing basic units) can also be transferred to other areas of physics and has led to the development of other systems of units such as the atomic system of units . The SI has also been using this method since the changes in 2019 ; In contrast to the CGS and other systems of units, the previous base units are still continued as such.

## CGS units of mechanics

As in other systems of units, the CGS units comprise two groups of units, the base units and the derived units. The latter can be written as the product of powers (power product) of the base units. Since the system is coherent (“connected”), there are no further numerical factors in the power products. For the CGS unit of any size G this means mathematically:

${\ displaystyle [G] = \ mathrm {cm} ^ {\ alpha} \, \ mathrm {g} ^ {\ beta} \, \ mathrm {s} ^ {\ gamma}}$

Here cm, g and s are the unit symbols of the basic units centimeter, gram and second. The exponents α , β and γ are each positive or negative integers or zero. The unit equation above can also be represented as a corresponding dimension equation:

${\ displaystyle \ dim G = L ^ {\ alpha} \, M ^ {\ beta} \, T ^ {\ gamma}}$

Here, L, M and T dimension of the sign of the base quantities length, mass and time (English time ).

Since the MKS system of units uses the same basic quantities, the dimension of a quantity is the same in both systems (the same bases and the same exponents in the dimension product). Because of the two different base units, only the exponents in the unit equation match the base s . Formally the conversion is:

${\ displaystyle [G] _ {\ text {MKS}} = \ mathrm {m} ^ {\ alpha} \, \ mathrm {kg} ^ {\ beta} \, \ mathrm {s} ^ {\ gamma} = 10 ^ {2 \ alpha +3 \ beta} \, \ mathrm {cm} ^ {\ alpha} \, \ mathrm {g} ^ {\ beta} \, \ mathrm {s} ^ {\ gamma} = 10 ^ {2 \ alpha +3 \ beta} \, [G] _ {\ text {CGS}}}$

Each CGS unit therefore clearly corresponds to an MKS unit, they only differ by a numerical factor.

### CGS derived units with special names

Some derived CGS units have their own names and symbols (symbols) that can be combined with all base and derived units. For example, the CGS unit of force, the dyn (= g · cm / s 2 ), is suitable for expressing the unit of energy, the erg , as dyn times centimeters (dyn · cm). The following table lists the named units.

size unit unit-
sign
expressed in
SI- CGS CGS base
units
Gravity acceleration Gal Gal 10 −2  m s −2 cm / s 2 cm · s −2
force Dyn dyn 10 −5  N g cm / s 2 cm g s −2
pressure Barye Ba 10 −1  Pa dyn / cm 2 cm −1 g s −2
Energy , work erg erg 10 −7  yrs dyn · cm cm 2 g s −2
Kinematic viscosity Stokes St. 10 −4  m 2 s −1 cm 2 / s cm 2 s −1
Dynamic viscosity Poise P 10 −1  Pa s g / (cm s) cm −1 g s −1
Wavenumber Kayser kayser 10 2  m −1 1 cm cm −1

## CGS units of electrodynamics

### General formulation of electrodynamics

Electrodynamic quantities are linked to mechanical quantities via several laws of nature. The electrodynamics itself is fully described by Maxwell's equations , which can be formulated independently of the system of units with the help of two proportionality constants and : ${\ displaystyle \ alpha _ {1}}$${\ displaystyle \ alpha _ {2}}$

{\ displaystyle {\ begin {aligned} \ operatorname {div} \, {\ vec {E}} & = \ alpha _ {1} \, \ rho \;, & \ operatorname {div} \, {\ vec { B}} & = 0 \;, \\\ operatorname {red} \, {\ vec {E}} & = - {\ frac {\ alpha _ {1}} {\ alpha _ {2}}} \, {\ frac {\ partial {\ vec {B}}} {\ partial t}} \;, & \ operatorname {red} \, {\ vec {B}} & = {\ frac {1} {c ^ { 2}}} \ alpha _ {2} \, {\ vec {j}} + {\ frac {1} {c ^ {2}}} \, {\ frac {\ alpha _ {2}} {\ alpha _ {1}}} \, {\ frac {\ partial {\ vec {E}}} {\ partial t}} \;, \ end {aligned}}}

where means the charge density and the electrical current density . As can be seen from the above equations, the constant relates the electric charge to the electric field strength ( Coulomb's law ) and the constant relates the electric current to the magnetic flux density ( Ampère's law ). The constant ratio and its reciprocal value describes the dependence of the electric and magnetic fields when these change over time ( displacement current and induction law ). ${\ displaystyle \ rho}$${\ displaystyle {\ vec {j}}}$${\ displaystyle \ alpha _ {1}}$ ${\ displaystyle Q}$ ${\ displaystyle {\ vec {E}}}$${\ displaystyle \ alpha _ {2}}$ ${\ displaystyle I}$ ${\ displaystyle {\ vec {B}}}$${\ displaystyle \ alpha _ {2} / \ alpha _ {1}}$

Every system of units of mechanics can be extended to describe electrodynamics by defining the values ​​of 2 of the 3 constants , and . In principle, there are three ways to do this: ${\ displaystyle \ alpha _ {1}}$${\ displaystyle \ alpha _ {2}}$${\ displaystyle \ alpha _ {2} / \ alpha _ {1}}$

• Introduction of two new basic units for electric charge and electric current . As a result, the above constants become measured variables that are afflicted with a measurement uncertainty .${\ displaystyle Q}$${\ displaystyle I}$
• Choice of a new base unit either for or for and the explicit definition of a constant. The remaining constants are then faulty measured variables.${\ displaystyle Q}$${\ displaystyle I}$
• Dispensing with new base units by explicitly defining two constants. The third constant is also fixed and not subject to errors.

All extensions of the CGS system rely on the third way. In the SI, on the other hand, the second route was taken with the introduction of the ampere as a unit of and definition . As a result of the changes to the base units decided on at the 26th General Conference on Weights and Measures , a paradigm shift to the first path takes place in the SI system: Due to the derivation of the base unit ampere from the elementary charge, electrical charge and electrical current have been explicit in the SI since May 20, 2019 Are defined. The constants and thus also become measured variables with measurement uncertainty . ${\ displaystyle I}$${\ displaystyle \ alpha _ {2} / \ alpha _ {1} = 1}$ ${\ displaystyle e}$${\ displaystyle Q}$ ${\ displaystyle I}$${\ displaystyle \ alpha _ {1} = \ alpha _ {2}}$${\ displaystyle \ mu _ {0}}$

The following table summarizes the different systems of units:

System of units ${\ displaystyle \ alpha _ {1}}$ ${\ displaystyle \ alpha _ {2}}$ ${\ displaystyle \ alpha _ {2} / \ alpha _ {1}}$
Electrostatic CGS system ${\ displaystyle 4 \ pi}$ ${\ displaystyle 4 \ pi}$ 1
Electromagnetic CGS system ${\ displaystyle 4 \ pi c ^ {2}}$ ${\ displaystyle 4 \ pi c ^ {2}}$ 1
Gaussian CGS system ${\ displaystyle 4 \ pi}$ ${\ displaystyle 4 \ pi c}$ ${\ displaystyle c}$
Heaviside-Lorentz unit system 1 ${\ displaystyle c}$ ${\ displaystyle c}$
International System of Units (SI) (a) ${\ textstyle 1 {,} 256 \, 637 \, 062 \, 12 (19) \ cdot 10 ^ {- 6} \ mathrm {\ frac {N} {A ^ {2}}} \ cdot c ^ {2 } \,}$(b) 1
(a)The SI introduces the ampere (A) as an independent base unit. The following applies: .${\ displaystyle \ alpha _ {1} = \ alpha _ {2} = \ mu _ {0} c ^ {2}}$
(b) CODATA Recommended Values. NIST , accessed June 4, 2019 . Numerical value of the magnetic field constant μ0(approximately 4π · 10−7).

### Electromagnetic units in various CGS systems

Electromagnetic SI units and the corresponding units in three variants of the CGS system
Electromagnetic
quantity
unit Gaussian unit in cgs
SI ESU EMU Gauss
charge Q 1 C 10 −1 c statC 10 −1 ABC 10 −1 c Fr. Fr = statC = g 1/2 cm 3/2 s −1
Amperage I. 1 A 10 −1 c statA 10 −1 abA 10 −1 c statA statA = g 1/2 cm 3/2 s −2
tension U 1 V 10 8 c −1 statV 10 8 abV 10 8 c −1 statV statV = g 1/2 cm 1/2 s −1
electric field strength E. 1 V / m 10 6 c −1 statV / cm 10 6 abV / cm 10 6 c −1 statV / cm statV / cm = g 1/2 cm −1/2 s −1
electric dipole moment p 1 C · m 10 1 c statC · cm 10 1 abC cm 10 19 c D. D = g 1/2 cm 5/2 s −1
magnetic flux density B. 1 T 10 4 c −1 instead of 10 4 G 10 4 G G = g 1/2 cm −1/2 s −1
magnetic field strength H 1 A / m 4π · 10 −3  c statA / cm 4π · 10 −3 Oe 4π · 10 −3 Oe Oe = g 1/2 cm −1/2 s −1
magnetic dipole moment m, μ A · m 2 10 3 c statA cm 2 10 3 abA cm 2 10 3 erg / G G = g 1/2 cm 5/2 s −1
magnetic flooding Θ 1 A 4π · 10 −1 c statA 4π · 10 −1 abA 4π · 10 −1 Gb Gb = g 1/2 cm 1/2 s −1
magnetic river Φ 1 Wb 10 8 c −1 statT cm 2 10 8 G cm 2 10 8 Mx Mx = g 1/2 cm 3/2 s −1
resistance R. 1 Ω 10 9 c −2 s / cm 10 9 abΩ 10 9 c −2 s / cm cm −1 s
specific resistance ρ 1 Ω · m 10 11 c −2 s 10 11 abΩ cm 10 11  c −2 s s
capacity C. 1 F. 10 −9 c 2 cm 10 −9 abF 10 −9 c 2 cm cm
Inductance L. 1 H. 10 9 c −2 cm −1 s 2 10 9 fromH 10 9 c −2 cm −1 s 2 cm −1 s 2
electrical power P 1 V * A = 1 W = 10 7 erg / s 10 7 erg / s 10 7 erg / s erg / s = g cm 2 s −3

The "≙" symbol indicates that this is not a simple conversion of units of measure. The CGS sizes generally have a different dimension than the corresponding size in the SI . That is why it is usually not allowed to simply replace the units in formulas. c is the speed of light .

## literature

• Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins . 3. Edition. Springer, 2004, ISBN 1-85233-682-X .