Centimetre–gram–second system of units

 ( Metrology ) 

The centimetre–gram–second system of units ( CGS or cgs) is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time. All CGS mechanics units are unambiguously derived from these three base units, but there are several different ways in which the CGS system was extended to cover electromagnetism.

The CGS system has been largely supplanted by the MKS units based on the metre, kilogram, and second, which was in turn extended and replaced by the International System of Units (SI). In many fields of science and engineering, SI is the only system of units in use, but CGS is still prevalent in certain subfields.

In measurements of purely mechanical systems (involving units of length, mass, force, energy, pressure, and so on), the differences between CGS and SI are straightforward: the Unit conversion are all powers of 10 as and . For example, the CGS unit of force is the dyne, which is defined as , so the SI unit of force, the newton (), is equal to .

On the other hand, in measurements of electromagnetic phenomena (involving units of charge, electric and magnetic fields, voltage, and so on), converting between CGS and SI is less straightforward. Formulas for physical laws of electromagnetism (such as Maxwell's equations) take a form that depends on which system of units is being used, because the electromagnetic quantities are defined differently in SI and in CGS. Furthermore, within CGS, there are several plausible ways to define electromagnetic quantities, leading to different "sub-systems", including Gaussian units, "ESU", "EMU", and Heaviside–Lorentz units. Among these choices, Gaussian units are the most common today, and "CGS units" is often intended to refer to CGS-Gaussian units.

The sizes of many CGS units turned out to be inconvenient for practical purposes. For example, many everyday objects are hundreds or thousands of centimetres long, such as humans, rooms and buildings. Thus the CGS system never gained wide use outside the field of science. Starting in the 1880s, and more significantly by the mid-20th century, CGS was gradually superseded internationally for scientific purposes by the MKS (metre–kilogram–second) system, which in turn developed into the modern SI standard.

Since 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 declined worldwide. CGS units have been deprecated in favor of SI units by NIST, as well as organizations such as the American Physical Society and the International Astronomical Union. SI units are predominantly used in engineering applications and physics education, while Gaussian CGS units are still commonly used in theoretical physics, describing microscopic systems, relativistic electrodynamics, and astrophysics.

The units gram and centimetre remain useful as noncoherent units within the SI system, as with any other Metric prefix SI units.

There is a direct correspondence between the base units of mechanics in CGS and SI. Since the formulae expressing the laws of mechanics are the same in both systems and since both systems are coherent, the definitions of all coherent in terms of the base units are the same in both systems, and there is an unambiguous relationship between derived units:

• $v\; =\; \backslash frac\{dx\}\{dt\}$  (definition of velocity)
• $F\; =\; m\backslash frac\{d^2x\}\{dt^2\}$  (Newton's second law of motion)
• $E\; =\; \backslash int\; \backslash vec\{F\}\backslash cdot\; d\backslash vec\{x\}$  (energy defined in terms of Mechanical work)
• $p\; =\; \backslash frac\{F\}\{L^2\}$  (pressure defined as force per unit area)
• $\backslash eta\; =\; \backslash tau/\backslash frac\{dv\}\{dx\}$  (dynamic viscosity defined as shear stress per unit velocity gradient).

Thus, 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 unit of pressure = 1 unit of force / (1 unit of length)² = 1 unit of mass / (1 unit of length × (1 unit of time)²)

1 Ba = 1 g/(cm⋅s²)

1 Pa = 1 kg/(m⋅s²).

Expressing a CGS derived unit in terms of the SI base units, or vice versa, requires combining 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.

• In SI, the unit of electric current, the ampere (A), was historically defined such that the magnetism force exerted by two infinitely long, thin, parallel wires 1 metre apart and carrying a current of 1 ampere is exactly . This definition results in all SI electromagnetic units being numerically consistent (subject to factors of some integer powers of 10) with those of the CGS-EMU system described in further sections. The ampere is a base unit of the SI system, with the same status as the metre, kilogram, and second. Thus the relationship in the definition of the ampere with the metre and newton is disregarded, and the ampere is not treated as dimensionally equivalent to any combination of other base units. As a result, electromagnetic laws in SI require an additional constant of proportionality (see Vacuum permeability) to relate electromagnetic units to kinematic units. (This constant of proportionality is derivable directly from the above definition of the ampere.) All other electric and magnetic units are derived from these four base units using the most basic common definitions: for example, electric charge q is defined as current I multiplied by time t, $$q\; =\; I\; \backslash ,\; t,$$ resulting in the unit of electric charge, the coulomb (C), being defined as 1 C = 1 A⋅s.
• The CGS system variant avoids introducing new base quantities and units, and instead defines all electromagnetic quantities by expressing the physical laws that relate electromagnetic phenomena to mechanics with only dimensionless constants, and hence all units for these quantities are directly derived from the centimetre, gram, and second.

In each of these systems the quantities called "charge" etc. may be a different quantity; they are distinguished here by a superscript. The corresponding quantities of each system are related through a proportionality constant.

Maxwell's equations can be written in each of these systems as:

! System
! width=175

$F=\{q^\backslash text\{ESU\}\_1\; q^\backslash text\{ESU\}\_2\; \backslash over\; r^2\}\; .$

The ESU unit of charge, franklin ( Fr), also known as statcoulomb or esu charge, is therefore defined as follows:

$\backslash mathrm\{1\backslash ,Fr\; =\; 1\backslash ,statcoulomb\; =\; 1\backslash ,esu\backslash ;\; charge\; =\; 1\backslash ,dyne^\{1/2\}\{\backslash cdot\}cm=1\backslash ,g^\{1/2\}\{\backslash cdot\}cm^\{3/2\}\{\backslash cdot\}s^\{-1\}\}\; .$

$\backslash mathrm\{1\backslash ,Fr/s\; =\; 1\backslash ,statampere\; =\; 1\backslash ,esu\backslash ;\; current\; =\; 1\backslash ,dyne^\{1/2\}\{\backslash cdot\}cm\{\backslash cdot\}s^\{-1\}=1\backslash ,g^\{1/2\}\{\backslash cdot\}cm^\{3/2\}\{\backslash cdot\}s^\{-2\}\}\; .$

In the CGS-ESU system, charge q therefore has the dimension of M^1/2L^3/2T⁻¹.

Other units in the CGS-ESU system include the statampere (1 statC/s) and statvolt (1 erg/statC).

In CGS-ESU, all electric and magnetic quantities are dimensionally expressible in terms of length, mass, and time, and none has an independent dimension. Such a system of units of electromagnetism, in which the dimensions of all electric and magnetic quantities are expressible in terms of the mechanical dimensions of mass, length, and time, is traditionally called an 'absolute system'.

Therefore, in '''electromagnetic CGS units''', a biot is equal to a square root of dyne: