Centimetre gram second system of units

The centimetre-gram-second system (abbreviated CGS or cgs) is a metric system of physical units based on centimetre as the unit of length, gram as a unit of mass, and second as a unit of time. All CGS mechanical units are unambiguously derived from these three base units, but there are several different ways of extending the CGS system to cover electromagnetism.

The CGS system has been largely supplanted by the MKS system, based on metre, kilogram, and second. MKS was in turn extended and replaced by the International System of Units (SI). The latter adopts the three base units of MKS, plus the ampere, mole, candela and kelvin. In many fields of science and engineering, SI is the only system of units in use. However, there remain certain subfields where CGS is prevalent.

In measurements of purely mechanical systems (involving units of length, mass, force, energy, pressure, etc.), the differences between CGS and SI are straightforward and rather trivial; the unit-conversion factors are all powers of 10 arising from the relations 100 cm = 1 m and 1000 g = 1 kg. For example, the CGS derived unit of force is the dyne, equal to 1 g·cm/s², while the SI derived unit of force is the newton, 1 kg·m/s². Thus it is straightforward to show that 1 dyne=10⁻⁵ newton.

On the other hand, in measurements of electromagnetic phenomena (involving units of charge, electric and magnetic fields, voltage, etc.), converting between CGS and SI is much more subtle and involved. In fact, formulas for physical laws of electromagnetism (such as Maxwell's equations) need to be adjusted depending on what system of units one uses. This is because there is no one-to-one correspondence between electromagnetic units in SI and those in CGS, as is the case for mechanical units. Furthermore, within CGS, there are several plausible choices of electromagnetic units, leading to different unit "sub-systems", including Gaussian, "ESU", "EMU", and Heaviside-Lorentz. Among these choices, Gaussian units are the most common today, and in fact the phrase "CGS units" is often used to refer specifically to CGS-Gaussian units.

History

The CGS system goes back to a proposal made in 1832 by the German mathematician Carl Friedrich Gauss. In 1874, it was extended by the British physicists James Clerk Maxwell and William Thomson with a set of electromagnetic units.

The values (by order of magnitude) of many CGS units turned out to be inconvenient for practical purposes. For example, many everyday length measurements yield hundreds or thousands of centimetres, such as those of human height and sizes of rooms and buildings. Thus the CGS system never gained wide general use outside the field of electrodynamics and laboratory science. Starting in the 1880s, and more significantly by the mid-20th century, CGS was gradually superseded internationally by the MKS (metre-kilogram-second) system, which in turn became the modern SI standard.

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 declined worldwide, in the United States more slowly than elsewhere. CGS units are today no longer accepted by the house styles of most scientific journals, textbook publishers, or standards bodies, although they are commonly used in astronomical journals such as the Astrophysical Journal. CGS units are still occasionally encountered in technical literature, especially in the United States in the fields of material science, electrodynamics and astronomy.

The units gram and centimetre remain useful as prefixed units within the SI system, especially for instructional physics and chemistry experiments, where they match the small scale of table-top setups. However, where derived units are needed, the SI ones are generally used and taught instead of the CGS ones today. For example, a physics lab course might ask students to record lengths in centimeters, and masses in grams, but force (a derived unit) in newtons, a usage consistent with the SI system.

Definition of CGS units in mechanics

In mechanics, the CGS and SI systems of units are built in an identical way. The two systems differ only in the scale of two out of the three base units (centimetre versus metre and gram versus kilogram, respectively), while the third unit (second as the unit of time) is the same in both systems.

There is a one-to-one correspondence between the base units of mechanics in CGS and SI, and the laws of mechanics are not affected by the choice of units. The definitions of all derived units in terms of the three base units are therefore the same in both systems, and there is an unambiguous one-to-one correspondence of derived units:

:$v\; =\; \backslash frac\{dx\}\{dt\}$  (definition of velocity) :$F=m\backslash frac\{d^2x\}\{dt^2\}$  (Newton's second law of motion) :$E\; =\; \backslash int\; \backslash vec\{F\}\backslash cdot\; \backslash vec\{dx\}$  (energy defined in terms of 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.

Definitions and conversion factors of CGS units in mechanics

{|class="wikitable" style="text-align: left;" ! Quantity ! Symbol !! CGS unit !! CGS unitabbreviation!! Definition !! Equivalentin SI units |- ! length, position |align="center" | L, x|| centimetre ||align="center" | cm || 1/100 of metre || = 10⁻² m |- ! mass |align="center" | m|| gram || align="center" |g || 1/1000 of kilogram || = 10⁻³ kg |- ! time |align="center" | t|| second||align="center" |s|| 1 second || = 1 s |- ! velocity |align="center" | v|| centimetre per second ||align="center" |cm/s || cm/s || = 10⁻² m/s |- ! force |align="center" | F|| dyne ||align="center" |dyn || g cm / s² || = 10⁻⁵ N |- ! energy |align="center" | E|| erg ||align="center" |erg || g cm² / s² || = 10⁻⁷ J |- ! power |align="center" | P|| erg per second||align="center" |erg/s || g cm² / s³ || = 10⁻⁷ W |- ! pressure |align="center" | p|| barye ||align="center" | Ba|| g / (cm s²) || = 10⁻¹ Pa |- ! dynamic viscosity |align="center" | μ|| poise ||align="center" |P|| g / (cm s) || = 10⁻¹ Pa·s |- ! wavenumber | align="center" | k || kayser ||align="center" |cm⁻¹|| cm⁻¹ || = 100 m⁻¹ |}

Derivation of CGS units in electromagnetism

CGS approach to electromagnetic units

The conversion factors relating electromagnetic units in the CGS and SI systems are much more involved — so much so that formulas for physical laws of electromagnetism are adjusted depending on what system of units one uses. This illustrates the fundamental difference in the ways the two systems are built: In SI, the unit of electric current is chosen to be 1 ampere (A). It is a base unit of the SI system, along with meter, kilogram, and second. The ampere is not dimensionally equivalent to any combination of other base units, so electromagnetic laws written in SI require an additional constant of proportionality (see Vacuum permittivity) to bridge electromagnetic units to kinematic units. 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, ::$q=I\backslash cdot\; t$, :therefore unit of electric charge, coulomb (C), is defined as 1 C = 1 A·s.

CGS system avoids introducing new base units and instead derives all electric and magnetic units from centimeter, gram, and second based on the physics laws that relate electromagnetic phenomena to mechanics.

Alternate derivations of CGS units in electromagnetism

Electromagnetic relationships to length, time and mass may be derived by equally appealing methods. Two of them rely on the forces observed on charges. Two fundamental laws relate (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 in system-independent form as follows:

The first is Coulomb's law, $F\; =\; k\_C\; \backslash frac\{q\; \backslash cdot\; q^\backslash prime\}\{d^2\}$, which describes the electrostatic force F between electric charges $q$ and $q^\backslash prime$, separated by distance d. Here $k\_C$ is a constant which depends on how exactly the unit of charge is derived from the CGS base units.

The second is Ampère's force law, $\backslash frac\{dF\}\{dL\}\; =\; 2\; k\_A\backslash frac\{I\; \backslash ,\; I^\backslash prime\}\{d\}$, which describes the magnetic force F per unit length L between currents I and I' flowing in two straight parallel wires of infinite length, separated by a distance d that is much greater than the wires' diameters. Since $I=q/t\backslash ,$ and $I^\backslash prime=q^\backslash prime/t$, the constant $k\_A$ also depends on how the unit of charge is derived from the CGS base units.

Maxwell's theory of electromagnetism relates these two laws to each other. It states that the ratio of proportionality constants $k\_C$ and $k\_A$ must obey $k\_C\; /\; k\_A\; =\; c^2$, where c is the speed of light. Therefore, if one derives the unit of charge from the Coulomb's law by setting $k\_C=1$, it is obvious that the Ampère's force law will contain a prefactor $2/c^2$. Alternatively, deriving the unit of current, and therefore the unit of charge, from the Ampère's force law by setting $k\_A\; =\; 1$ or $k\_A\; =\; 1/2$, will lead to a constant prefactor in the Coulomb's law.

Indeed, both of these mutually-exclusive approaches have been practiced by the users of CGS system, leading to the two independent and mutually-exclusive branches of CGS, described in the subsections below. However, the freedom of choice in deriving electromagnetic units from the units of length, mass, and time is not limited to the definition of charge. While the electric field can be related to the work performed by it on a moving electric charge, the magnetic force is always perpendicular to the velocity of the moving charge, and thus the work performed by the magnetic field on any charge is always zero. This leads to a choice between two laws of magnetism, each relating magnetic field to mechanical quantities and electric charge:

The first law describes the Lorentz force produced by a magnetic field B on a charge q moving with velocity v:

:: $\backslash mathbf\{F\}\; =\; \backslash alpha\_L\; q\backslash ;\backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\}\backslash ;.$

The second describes the creation of a static magnetic field B by an electric current I of finite length dl at a point displaced by a vector r, known as Biot-Savart law:

:: $d\backslash mathbf\{B\}\; =\; \backslash alpha\_B\backslash frac\{I\; d\backslash mathbf\{l\}\; \backslash times\; \backslash mathbf\{\backslash hat\; r\}\}\{r^2\}\backslash ;,$ where r and $\backslash mathbf\{\backslash hat\; r\}$ are the length and the unit vector in the direction of vector r. These two laws can be used to derive Ampère's force law, resulting in the relationship: $k\_A\; =\; \backslash alpha\_L\; \backslash cdot\; \backslash alpha\_B\backslash ;$. Therefore, if the unit of charge is based on the Ampère's force law such that $k\_A\; =\; 1$, it is natural to derive the unit of magnetic field by setting $\backslash alpha\_L\; =\; \backslash alpha\_B=1\backslash ;$. However, if it is not the case, a choice has to be made as to which of the two laws above is a more convenient basis for deriving the unit of magnetic field.

Furthermore, if we wish to describe the electric displacement field D and the magnetic field 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. Then we have the laws for systems of spherical geometry contain factors of 4π (e.g. point charges), those of cylindrical geometry — factors of 2π (e.g. wires), and those of planar geometry contain no factors of π (e.g. parallel-plate capacitors). However, the original CGS system used λ = λ′ = 4π, or, equivalently, k_Cε₀ = 1. Therefore, Gaussian, ESU, and EMU subsystems of CGS (described below) are not rationalized.

Various extensions of the CGS system to electromagnetism

The table below shows the values of the above constants used in some common CGS subsystems:

Pro and contra

While the absence of explicit prefactors in some CGS subsystems simplifies some theoretical calculations, it has the disadvantage that sometimes the units in CGS are hard to define through experiment. Also, lack of unique unit names leads to a great confusion: thus “15 emu” may mean either 15 abvolt, or 15 emu units of electric dipole moment, or 15 emu units of magnetic susceptibility, sometimes (but not always) per gram or per mole. On the other hand, SI starts with a unit of current, the ampere, which is easier to determine through experiment, but which requires extra prefactors in the electromagnetic equations. With its system of unique named units, SI also removes any confusion in usage: 1 ampere is a fixed quantity of a specific variable, and so are 1 henry and 1 ohm.

A key virtue of the Gaussian CGS system is that electric and magnetic fields have the same units, $4\backslash pi\backslash epsilon\_0$ 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 $\backslash epsilon\_0$ equaling 1), but it 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 $4\; \backslash pi$ appearing in the formulas, and it is in Heaviside-Lorentz units that the Maxwell equations take their simplest form.

In SI, and other rationalized systems (e.g. Heaviside-Lorentz), the unit of current was chosen such that electromagnetic equations concerning charged spheres contain 4π, those concerning coils of current and straight wires contain 2π and those dealing with charged surfaces 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 non-rationalized CGS system can be somewhat more convenient notationally.

In fact, in certain fields, specialized unit systems are used to simplify formulas even further than either SI or CGS, by using some system of natural units. For example, the particle physics community uses a system where every quantity is expressed by only one unit, the eV, with lengths, times, etc. all converted into eV's by inserting factors of c and $\backslash hbar$. This unit system is very convenient for particle-physics calculations, but would be impractical in other contexts.

See also

Scientific units named after people

SI electromagnetism units

SI units

Units of measurement

References and notes

General literature

Category:Systems of units