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Centimetre gram second system of units
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The centimetre-gram-second system (abbreviated CGS or cgs) is a metric system of physical units based on centimetre, gram, and second. All of CGS mechanical units are unambiguously derived from these three base units, but there are several alternative variants of extending the CGS system in electromagnetism. The CGS system has been largely supplanted by MKS, or metre-kilogram-second system, which in turn was replaced by the International System of Units (SI). The latter has the three base units of MKS plus the ampere, mole, candela and kelvin.
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The centimetre-gram-second system (abbreviated CGS or cgs) is a metric system of physical units based on centimetre, gram, and second. All of CGS mechanical units are unambiguously derived from these three base units, but there are several alternative variants of extending the CGS system in electromagnetism. The CGS system has been largely supplanted by MKS, or metre-kilogram-second system, which in turn was replaced by the International System of Units (SI). The latter has the three base units of MKS plus the ampere, mole, candela and kelvin. SI is the only system used in many fields of science and engineering, although there remain certain subfields where CGS is prevalent.
In mechanics (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 that ultimately arise from the relations 100 cm = 1 m and 1000 g = 1 kg. On the other hand, in electromagnetism (involving units for charges, fields, fluxes, voltages, etc.), converting between different systems 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. Even within cgs, there is a number of different electromagnetic unit "sub-systems", including Gaussian, "ESU", "EMU", and Heaviside-Lorentz. Gaussian units are the most common within cgs, and in fact the phrase "cgs units" is often used to refer specifically to cgs-Gaussian units.
History
The system goes back to a proposal made in 1833 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 (e.g. 1000 cm is not quite compatible with the human scale and would lead to an awkward "kilo-centimetre"), and thus the CGS system never gained wide general use outside the field of electrodynamics and laboratory science. Starting in the 1880s, but not to a significant extent until the mid-20th century, CGS was gradually superseded internationally by a more practical MKS (metre-kilogram-second) system, which eventually led to the modern SI standard.
CGS units are still occasionally encountered in technical literature, especially in the United States in the fields of material science, electrodynamics and astronomy. SI unit of ampere was chosen such that electromagnetic equations concerning charged spheres contain 4p, those concerning coils of current contain 2p and those dealing with straight wires lack p 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 somewhat more convenient notationally.
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 declined 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, or 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 scale 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.
Definition of CGS units in mechanics
In mechanics, both CGS and SI systems are built in an identical way. The only difference between the two systems is the scale of two out of the three base units needed in mechanics (centimetre versus metre and gram versus kilogram), while the third unit (measure of time: second) is the same in both systems . The laws and definitions of mechanics that are used to obtain all derived units from the three base units are the same in both systems, for example:
(definition of velocity)
(Newton's second law of motion)
(energy defined in terms of work)
(pressure defined as force per unit area)
(dynamic viscosity defined as shear stress per unit velocity gradient).
This explains why, 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 Ba = 1 g/(cm·s2)
- 1 Pa = 1 kg/(m·s2).
However, expressing a CGS derived unit in terms of the SI base units involves a combination of 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.
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 arbitrarily chosen to be 1 ampere (A). It is set as a base unit of the SI system along with metre, kilogram, and second. 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,
- ,
- therefore unit of electric charge, coulomb (C), is defined as 1 C = 1 A·s.
- CGS system takes a rather different, and a much more direct approach to the definition of electromagnetic units: it avoids introducing new base units and instead derives all electric and magnetic units from centimetre, gram, and second based on the physics laws that relate electromagnetic phenomena to mechanics.
Possible alternatives for deriving electromagnetic units
Relating electromagnetic quantities to length, time and mass, however, can be done in a variety of equally appealing ways. Two of them rely on the forces observed on charges. There are two fundamental laws that 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, , which describes the electrostatic force F between electric charges and , separated by distance d. Here 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, , which describes the magnetic force F per unit length L between currents I and I flowing in two long parallel wires, separated by distance d. Since and , the constant 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 and must obey , where c is the speed of light. Therefore, if one derives the unit of charge from the Coulomb's law by setting , it is obvious that the Ampère's force law will contain a prefactor . Alternatively, deriving the unit of current, and therefore the unit of charge, from the Ampère's force law by setting or , 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:
- 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:
where r and 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: . Therefore, if the unit of charge is based on the Ampère's force law such that , it is natural to derive the unit of magnetic field by setting . 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 and the magnetic field in a medium other than a vacuum, we need to also define the constants e0 and µ0, which are the vacuum permittivity and permeability, respectively. Then we have (generally) and . The factors ? and ?' are rationalization constants, which are usually chosen to be 4pkCe0, which is dimensionless. If this quantity equals 1, the system is said to be rationalized. The original CGS system, however, used ? = ?' = 4p, or, equivalently, kCe0 = 1.
Various extensions of the CGS system to electromagnetism
While the absence of some explicit prefactors in CGS simplifies theoretical calculations, it has the disadvantage that the units in CGS are hard to define through experiment. On the other hand, SI starts with a unit of current, the ampere, which is easy to determine through experiment, but which requires extra prefactors in the electromagnetic equations.
The table below shows the values of the above constants used in some common systems:
| system | | | | | | | | |
|---|
| electrostatic (esu or ESU) | 1 | c-2 | 1 | c-2 | c-2 | 1 | 4p | 4p | | electromagnetic (emu or EMU) | c2 | 1 | c-2 | 1 | 1 | 1 | 4p | 4p | | Gaussian | 1 | c-1 | 1 | 1 | c-2 | c-1 | 4p | 4p | | Heaviside-Lorentz | | | 1 | 1 | | c-1 | 1 | 1 | | SI | | | | | | 1 | 1 | 1 |
The constant b in SI system is a unit-based scaling factor defined as: .
Also, note the following correspondence of the above constants to those in Jackson and Leung:
-
In system-independent form, Maxwell's equations in vacuum can be written as:
Note that of all these variants, only in Gaussian and Heaviside-Lorentz systems equals rather than 1. As a result, vectors and of an electromagnetic wave propagating in vacuum have the same units and are equal in magnitude in these two variants of CGS.
Electrostatic units (ESU)
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. It is done by setting the Coulomb force constant , so that Coulomb’s law does not contain an explicit prefactor.
The ESU unit of charge, statcoulomb or esu charge, is therefore defined as follows: In CGS electrostatic units, a statcoulomb is equal to a centimetre times square root of dyne:
- .
Dimensionally in the CGS ESU system, charge q is therefore equivalent to m1/2L3/2t-1 and is not an independent dimension of physical quantity. This reduction of units is an application of the Buckingham p theorem.
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 explicit prefactor), 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 , , and , where and are the well-known vacuum permittivities and c the corresponding light velocity, whereas and 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 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.
Electromagnetic units in various CGS systems
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
- .
By taking the limit as R goes to infinity we see C equals r.
Physical constants in CGS units
Pro and contra
A key virtue of the Gaussian CGS system is that electric and magnetic fields have the same units, is replaced by , and the only dimensional constant appearing in the equations is , the speed of light. The Heaviside-Lorentz system has these desirable properties as well (with 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 appearing in the formulas, and it is in Heaviside-Lorentz units that the Maxwell equations take their simplest form.
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 . This unit system is very convenient for particle-physics calculations, but would be impractical in other contexts.
See also
General literature
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