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8.1 CGS Units

The SI—an extension of the MKS (meter–kilogram–second) system—has largely supplanted the older CGS (centimeter–gram–second) system, but CGS units are still used in a few specialized fields, especially in physics where they lead to a more elegant formulation of Maxwell’s equations. Conversions between SI and CGS involving mechanical units are straightforward, involving powers of 10 (e.g., 1 m = 100 cm). Conversions involving electromagnetic units are more complicated, and units supports four different systems of CGS units: electrostatic units (ESU), electromagnetic units (EMU), the Gaussian system and the Heaviside–Lorentz system. The differences between these systems arise from different choices made for proportionality constants in electromagnetic equations. Coulomb’s law gives electrostatic force between two charges separated by a distance \(r\):

\[ F = k_{\rm C} { {q_1 q_2} \over {r^2} }. \]

Ampere’s law gives the electromagnetic force per unit length between two current-carrying conductors separated by a distance \(r\):

\[ { F \over \ell } = 2 k_{\rm A} { {I_1 I_2} \over {r} } . \]

The two constants, \(k_{\rm C}\) and \(k_{\rm A}\), are related by the square of the speed of light:

\[k_{\rm A} = k_{\rm C} / c^{2}. \]

In the SI, the constants have dimensions, and an additional base unit, the ampere, measures electric current. The CGS systems do not define new base units, but express charge and current as derived units in terms of mass, length, and time. In the ESU system, the constant for Coulomb’s law is chosen to be unity and dimensionless, which defines the unit of charge. In the EMU system, the constant for Ampere’s law is chosen to be unity and dimensionless, which defines a unit of current. The Gaussian system usually uses the ESU units for charge and current; it chooses another constant so that the units for the electric and magnetic fields are the same. The Heaviside–Lorentz system is “rationalized” so that factors of \(4\pi\) do not appear in Maxwell’s equations. The SI system is similarly rationalized, but the other CGS systems are not. In the Heaviside–Lorentz (HLU) system the factor of \(4\pi\) appears in Coulomb’s law instead; this system differs from the Gaussian system by factors of \(\sqrt{4\pi}\).

The dimensions of electrical quantities in the various CGS systems are different from the SI dimensions for the same units; strictly, conversions between these systems and SI are not possible. But units in different systems relate to the same physical quantities, so there is a correspondence between these units. The units program defines the units so that you can convert between corresponding units in the various systems.

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