Gaussian units constitute a metric system of units of measurement. This system is the most common of the several electromagnetic unit systems based on the centimetre–gram–second system of units (CGS). It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units. The term "cgs units" is ambiguous and therefore to be avoided if possible: there are several variants of CGS, which have conflicting definitions of electromagnetic quantities and units.
SI units predominate in most fields, and continue to increase in popularity at the expense of Gaussian units. Alternative unit systems also exist. Conversions between quantities in the Gaussian and SI systems are direct unit conversions, because the quantities themselves are defined differently in each system. This means that the equations that express physical laws of electromagnetism—such as Maxwell's equations—will change depending on the system of quantities that is employed. As an example, quantities that are dimensionless in one system may have dimension in the other.
Alternative unit systems
The Gaussian unit system is just one of several electromagnetic unit systems within CGS. Others include "electrostatic units", "electromagnetic units", and Heaviside–Lorentz units.
Some other unit systems are called "natural units", a category that includes atomic units, Planck units, and others.
The International System of Units (SI), with the associated International System of Quantities (ISQ), is by far the most common system of units today. In engineering and practical areas, SI is nearly universal and has been for decades."CGS", in How Many? A Dictionary of Units of Measurement, by Russ Rowlett and the University of North Carolina at Chapel Hill In technical, scientific literature (such as theoretical physics and astronomy), Gaussian units were predominant until recent decades, but are now getting progressively less so. The 8th SI Brochure mentions the CGS-Gaussian unit system,, p. 128 but the 9th SI Brochure makes no mention of CGS systems.
Natural units may be used in more theoretical and abstract fields of physics, particularly particle physics and string theory.
Major differences between Gaussian and SI systems
Rationalized unit systems
One difference between the Gaussian and SI systems is in the factor in various formulas that relate the quantities that they define. With SI electromagnetic units, called rationalized,Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity", The Physics Teacher 24(2): 97–99. Alternate web link (subscription required) Maxwell's equations have no explicit factors of in the formulae, whereas the inverse-square force laws – Coulomb's law and the Biot–Savart law – have a factor of attached to the . With Gaussian units, called unrationalized (and unlike Heaviside–Lorentz units), the situation is reversed: two of Maxwell's equations have factors of in the formulas, while both of the inverse-square force laws, Coulomb's law and the Biot–Savart law, have no factor of attached to in the denominator.
(The quantity appears because is the surface area of the sphere of radius , which reflects the geometry of the configuration. For details, see the articles Relation between Gauss's law and Coulomb's law and Inverse-square law.)
Unit of charge
A major difference between the Gaussian system and the ISQ is in the respective definitions of the quantity charge. In the ISQ, a separate base dimension, electric current, with the associated SI unit, the ampere, is associated with electromagnetic phenomena, with the consequence that a unit of electrical charge (1 coulomb = 1 ampere × 1 second) is a physical quantity that cannot be expressed purely in terms of the mechanical units (kilogram, metre, second). On the other hand, in the Gaussian system, the unit of electric charge (the statcoulomb, statC) be written entirely as a dimensional combination of the non-electrical base units (gram, centimetre, second), as:
For example, Coulomb's law in Gaussian units has no constant:
where is the repulsive force between two electrical charges, and are the two charges in question, and is the distance separating them. If and are expressed in statC and in centimetres, then the unit of that is coherent with these units is the dyne.
The same law in the ISQ is:
where is the vacuum permittivity, a quantity that is not dimensionless: it has dimension (charge)2 (time)2 (mass)−1 (length)−3. Without , the equation would be dimensionally inconsistent with the quantities as defined in the ISQ, whereas the quantity does not appear in Gaussian equations. This is an example of how some dimensional physical constants can be eliminated from the expressions of physical law by the choice of definition of quantities. In the ISQ, converts or scales electric flux density, , to the corresponding electric field, (the latter has dimension of force per charge), while in the Gaussian system, electric flux density is the same quantity as electric field strength in free space aside from a dimensionless constant factor.
In the Gaussian system, the speed of light appears directly in electromagnetic formulas like Maxwell's equations (see below), whereas in the ISQ it appears via the product .
Units for magnetism
In the Gaussian system, unlike the ISQ, the electric field and the magnetic field have the same dimension. This amounts to a factor of between how is defined in the two unit systems, on top of the other differences. (The same factor applies to other magnetic quantities such as the magnetic field, , and magnetization, .) For example, in a planar light wave in vacuum, in Gaussian units, while in the ISQ.
Polarization, magnetization
There are further differences between Gaussian system and the ISQ in how quantities related to polarization and magnetization are defined. For one thing, in the Gaussian system, all of the following quantities have the same dimension: , , , , , and . A further point is that the electric and magnetic susceptibility of a material is dimensionless in both the Gaussian system and the ISQ, but a given material will have a different numerical susceptibility in the two systems. (Equation is given below.)
List of equations
This section has a list of the basic formulae of electromagnetism, given in both the Gaussian system and the International System of Quantities (ISQ). Most symbol names are not given; for complete explanations and definitions, please click to the appropriate dedicated article for each equation. A simple conversion scheme for use when tables are not available may be found in Garg (2012).A. Garg, 2012, "Classical Electrodynamics in a Nutshell" (Princeton University Press). All formulas except otherwise noted are from Ref.
Maxwell's equations
Here are Maxwell's equations, both in macroscopic and microscopic forms. Only the "differential form" of the equations is given, not the "integral form"; to get the integral forms, apply the divergence theorem or Kelvin–Stokes theorem.
+ Maxwell's equations in Gaussian system and ISQ Name Gaussian system Gauss's law(macroscopic) Gauss's law(microscopic) Gauss's law for magnetism Maxwell–Faraday equation(Faraday's law of induction) Ampère–Maxwell equation (macroscopic) Ampère–Maxwell equation (microscopic)
Other basic laws
+ Other electromagnetic laws in Gaussian system and ISQ Name Gaussian system Lorentz force Coulomb's law Electric field ofstationary point charge Biot–Savart lawIntroduction to Electrodynamics by Capri and Panat, p180 Poynting vector(microscopic)
Dielectric and magnetic materials
Below are the expressions for the various fields in a dielectric medium. It is assumed here for simplicity that the medium is homogeneous, linear, isotropic, and nondispersive, so that the permittivity is a simple constant.
+ Expressions for fields in dielectric media Gaussian system where and are the electric field and the displacement field, respectively; is the polarization density; is the permittivity; is the permittivity of vacuum (used in the SI system, but meaningless in Gaussian units); and is the electric susceptibility.
The quantities and are both dimensionless, and they have the same numeric value. By contrast, the electric susceptibility and are both unitless, but have for the same material:
Next, here are the expressions for the various fields in a magnetic medium. Again, it is assumed that the medium is homogeneous, linear, isotropic, and nondispersive, so that the permeability is a simple constant.
+ Expressions for fields in magnetic media Gaussian system where and are the magnetic fields; is the magnetization; is the magnetic permeability; is the permeability of vacuum (used in the SI system, but meaningless in Gaussian units); and is the magnetic susceptibility.
The quantities and are both dimensionless, and they have the same numeric value. By contrast, the magnetic susceptibility and are both unitless, but has in the two systems for the same material:
Vector and scalar potentials
The electric and magnetic fields can be written in terms of a vector potential and a scalar potential :
+ Electromagnetic fields in Gaussian system and ISQ Name Gaussian system Electric field Magnetic B field
Electrical circuit
+ Electrical circuit values in Gaussian system and ISQ Name Gaussian system Charge conservation Lenz's law Ohm's law Capacitance Inductance where is the electric charge is the electric current is the electric potential is the magnetic flux is the electrical resistance is the capacitance is the inductance
Fundamental constants
+ Fundamental constants in Gaussian system and ISQ Name Gaussian system Impedance of free space Electric constant Magnetic constant Fine-structure constant Magnetic flux quantum Conductance quantum Bohr radius Bohr magneton
Electromagnetic unit names
+ Table 1: Common electromagnetism units in SI vs Gaussian Quantity Symbol SI unit Gaussian unit(in base units) Conversion factor Electric charge C Fr(cm3/2⋅g1/2⋅s−1) Electric current A statA(cm3/2⋅g1/2⋅s−2) Electric potential,Voltage V statV(cm1/2⋅g1/2⋅s−1) Electric field V/m statV/cm(cm−1/2⋅g1/2⋅s−1) Electric displacement field C/m2 Fr/cm2(cm−1/2g1/2s−1) Electric dipole moment C⋅m Fr⋅cm(cm5/2⋅g1/2⋅s−1) Electric flux C Fr(cm3/2⋅g1/2⋅s−1) Permittivity F/m cm/cm Magnetic B field T G(cm−1/2⋅g1/2⋅s−1) Magnetic H field A/m Oe(cm−1/2⋅g1/2⋅s−1) Magnetic dipole moment A⋅m2 erg/G(cm5/2⋅g1/2⋅s−1) Magnetic flux Wb Mx(cm3/2⋅g1/2⋅s−1) Permeability H/m cm/cm Magnetomotive force A Gi(cm1/2⋅g1/2⋅s−1) Magnetic reluctance H−1 Gi/Mx(cm−1) Resistance Ω s/cm Resistivity Ω⋅m s Capacitance F cm Inductance H s2/cm Note: All the equations are in SI, and the quantities and satisfy .
The conversion factors are written both symbolically and numerically. The numerical conversion factors can be derived from the symbolic conversion factors by substitution of equals for equals. For example, the top row says a relation which can be verified by substituting for and Fr their values as given in the permittivity row and Gaussian unit column, respectively, and expressing them and C in SI base units.
The equations of the table, including those in the Gaussian unit column and the note but excluding those containing a numerical conversion factor, constitute an interpretation (or model) in the SI of the Gaussian system, both its quantity and unit equations. An interpreted unit equation, however, is not a conversion, which makes use of the excluded equations. For example, the equation qG = 1 Fr is interpreted to mean qI/(4ε0)1/2 = 10−9/2 m3/2⋅kg1/2⋅s−1 but is converted to qI = 3.3356 × 10−10 C.