Centimetre–gram–second system of units

The CGS system goes back to a proposal in 1832 by the German mathematician Carl Friedrich Gauss to base a system of absolute units on the three fundamental units of length, mass and time. Gauss chose the units of millimetre, milligram and second. In 1873, a committee of the British Association for the Advancement of Science, including physicists James Clerk Maxwell and William Thomson, 1st Baron Kelvin recommended the general adoption of centimetre, gram and second as fundamental units, and to express all derived electromagnetic units in these fundamental units, using the prefix "C.G.S. unit of ...". 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 Starting in the 1880s, and more significantly by the mid-20th century, CGS was gradually superseded internationally for scientific purposes by the MKS (metre–kilogram–second) system, which in turn developed into the modern SI standard.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 favour of SI units by NIST, as well as organisations 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 prefixed SI units. == Definition of CGS units in mechanics ==

In mechanics, the quantities in the CGS and SI systems are defined identically. The two systems differ only in the scale of the three base units (centimetre versus metre and gram versus kilogram, respectively), with the third unit (second) being the same in both systems. 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 derived units in terms of the base units are the same in both systems, and there is an unambiguous relationship between derived units: • v = \frac{dx}{dt}  (definition of velocity) • F = m\frac{d^2x}{dt^2}  (Newton's second law of motion) • E = \int \vec{F}\cdot d\vec{x}  (energy defined in terms of work) • p = \frac{F}{L^2}   (pressure defined as force per unit area) • \eta = \tau/\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. Definitions and conversion factors of CGS units in mechanics == Derivation of CGS units in electromagnetism ==

CGS approach to electromagnetic units The laws of electromagnetism (specifically, the four Maxwell's equations) are formulated with fundamentally different assumptions in SI and CGS units. The SI system introduces new units to represent concepts such as electric charge, current, and magnetic flux, while CGS avoids adding new units. Rather, CGS represents all electromagnetic quantities by expressing the laws of electromagnetism in purely mechanical units, without introducing further units beyond the centimetre, gram, and second. For example, in SI the unit of electric current, the ampere (A), was historically defined such that the magnetic 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 most SI electromagnetic units being 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 mechanical 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 \, t, resulting in the unit of electric charge, the coulomb (C), being defined as 1 C = 1 A⋅s. Alternative derivations of CGS units in electromagnetism Electromagnetic relationships to length, time and mass may be derived by several equally appealing methods. Two of them rely on the forces observed on charges. Two fundamental laws relate (seemingly independently of each other) the electric charge or its rate of change (electric current) to a mechanical quantity such as force. They can be written the laws for systems of spherical geometry contain factors of 4 (for example, point charges), those of cylindrical geometry factors of 2 (for example, wires), and those of planar geometry contain no factors of (for example, parallel-plate capacitors). However, the modern CGS systems, except Heaviside–Lorentz, use = ′ = 4, or, equivalently, k_{\rm C} \epsilon_0 = \alpha_{\rm B} / (\mu_0 \alpha_{\rm L}) = 1. Therefore, Gaussian, ESU, and EMU subsystems of CGS (described below) are not rationalised. Various extensions of the CGS system to electromagnetism The table below shows the values of the above constants used in some common CGS subsystems: In the CGS systems c = 2.9979 × 1010 cm/s, and in the SI system c = 2.9979 × 108 m/s and b ≈ 107 A2/N = 107 m/H. 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: Electrostatic units (ESU) In the electrostatic units variant of the CGS system, (CGS-ESU), charge is defined as the quantity that obeys a form of Coulomb's law without a multiplying constant (and current is then defined as charge per unit time): : F={q^\text{ESU}_1 q^\text{ESU}_2 \over r^2} . The ESU unit of charge, franklin (Fr), also known as statcoulomb or esu charge, is therefore defined as follows: Therefore, in CGS-ESU, a franklin is equal to a centimetre times square root of dyne: : \mathrm{1\,Fr = 1\,statcoulomb = 1\,esu\; charge = 1\,dyne^{1/2}{\cdot}cm=1\,g^{1/2}{\cdot}cm^{3/2}{\cdot}s^{-1}} . The unit of current is defined as: : \mathrm{1\,Fr/s = 1\,statampere = 1\,esu\; current = 1\,dyne^{1/2}{\cdot}cm{\cdot}s^{-1}=1\,g^{1/2}{\cdot}cm^{3/2}{\cdot}s^{-2}} . In the CGS-ESU system, charge q therefore has the dimension of M1/2L3/2T−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'.:3 ESU notation All electromagnetic units in the CGS-ESU system that do not have proper names are denoted by a corresponding SI name with an attached prefix "stat" or with a separate abbreviation "esu". As well as the volt and ampere, the farad (capacitance), ohm (resistance), coulomb (electric charge), and henry (inductance) are consequently also used in the practical system and are the same as the SI units. The magnetic units are those of the emu system. The electrical units, other than the volt and ampere, are determined by the requirement that any equation involving only electrical and kinematical quantities that is valid in SI should also be valid in the system. For example, since electric field strength is voltage per unit length, its unit is the volt per centimetre, which is one hundred times the SI unit. The system is electrically rationalised and magnetically unrationalised; i.e., and , but the above formula for is invalid. A closely related system is the International System of Electric and Magnetic Units, which has a different unit of mass so that the formula for ′ is invalid. The unit of mass was chosen to remove powers of ten from contexts in which they were considered to be objectionable (e.g., and ). Inevitably, the powers of ten reappeared in other contexts, but the effect was to make the familiar joule and watt the units of work and power respectively. The ampere-turn system is constructed in a similar way by considering magnetomotive force and magnetic field strength to be electrical quantities and rationalising the system by dividing the units of magnetic pole strength and magnetisation by 4. The units of the first two quantities are the ampere and the ampere per centimetre respectively. The unit of magnetic permeability is that of the emu system, and the magnetic constitutive equations are and . Magnetic reluctance is given a hybrid unit to ensure the validity of Ohm's law for magnetic circuits. In all the practical systems ε0 = 8.8542 × 10−14 A⋅s/(V⋅cm), μ0 = 1 V⋅s/(A⋅cm), and c2 = 1/(4 × 10−9 ε0μ0). Maxwell's equations in free space are also the same in all the systems. In the practical systems inductance is considered to be an electrical quantity and is defined by L = 10−8 NΦB/I. Its unit is the henry, symbolized by H. The mathematical unit of permeability is 1 H/cm, although the physical unit is 4 × 10−9 henry per centimetre. Here we use words for the physical unit and symbols for the mathematical unit, which is the symbol for the physical unit in the system. 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 the Gaussian units and the Heaviside–Lorentz units. == Electromagnetic units in various CGS systems ==

In this table, c = is the numeric value of the speed of light in vacuum when expressed in units of centimetres per second. The correspondence symbol "≘" is used instead of the equality sign "=" as a reminder that the conversion statements in the table are not equations. For example, according to the capacitance row of the table, if a capacitor has a capacitance of 1 F in SI, then it has a capacitance of (10−9 c2) cm in ESU; this statement is not an equation, and hence it is incorrect to replace "1 F" with "(10−9 c2) cm" within an equation or formula. (This warning is a special aspect of electromagnetism units. By contrast it is always correct to replace, e.g., "1 m" with "100 cm" within an equation or formula.) == Physical constants in CGS units ==

== Advantages and disadvantages ==

Lack of unique unit names leads to potential confusion: "15 emu" may mean either 15 abvolts, or 15 emu units of electric dipole moment, or 15 emu units of magnetic susceptibility, sometimes (but not always) per gram, or per mole. With its system of uniquely named units, the SI removes any confusion in usage: 1 ampere is a fixed value of a specified quantity, and so are 1 henry, 1 ohm, and 1 volt. In the CGS-Gaussian system, electric and magnetic fields have the same units, 4ε0 is replaced by 1, and the only dimensional constant appearing in the Maxwell equations is c, the speed of light. The Heaviside–Lorentz system has these properties as well (with ε0 equaling 1). In SI, and other rationalised systems (for example, Heaviside–Lorentz), the unit of current was chosen such that electromagnetic equations concerning charged spheres contain 4, those concerning coils of current and straight wires contain 2 and those dealing with charged surfaces lack entirely, which was the most convenient choice for applications in electrical engineering and relates directly to the geometric symmetry of the system being described by the equation. Specialised unit systems are used to simplify formulas further than either SI or CGS do, by eliminating constants through a convention of normalising quantities with respect to some system of natural units. For example, in particle physics a system is in use where every quantity is expressed by only one unit of energy, the electronvolt, with lengths, times, and so on all converted into units of energy by inserting factors of speed of light c and the reduced Planck constant ħ. This unit system is convenient for calculations in particle physics, but is impractical in other contexts. == See also ==