TheInfoList

OR: Gaussian units constitute a
metric system The metric system is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the Interna ...
of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. 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 with conflicting definitions of electromagnetic quantities and units.
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
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 Gaussian and SI units are direct unit conversions, because the quantities themselves are defined differently in each system. This means that the equations expressing physical laws of electromagnetism—such as
Maxwell's Maxwell's, last known as Maxwell's Tavern, was a bar/restaurant and music club in Hoboken, New Jersey. Over several decades the venue attracted a wide variety of acts looking for a change from the New York City concert spaces across the river. Ma ...
—will change depending on the system of units employed. As an example, quantities that are
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
in one system may have dimension in the other.

# History

Gaussian units existed before the CGS system. The British Association report of 1873 that proposed the CGS contains gaussian units derived from the foot–grain–second and metre–gram–second as well. There are also references to foot–pound–second gaussian units.

# Alternative unit systems

The Gaussian unit system is just one of several electromagnetic unit systems within CGS. Others include "
electrostatic units The electrostatic system of units (CGS-ESU) is a system of units used to measure quantities of electric charge, electric current, and voltage within the centimetre–gram–second (or "CGS") system of metric units. In electrostatic units, electrica ...
", " electromagnetic units", and
Heaviside–Lorentz units Heaviside–Lorentz units (or Lorentz–Heaviside units) constitute a system of units (particularly electromagnetic units) within CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that th ...
. Some other unit systems are called "
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ...
", a category that includes Hartree atomic units, Planck units, and others.
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
are by far the most common system of units today. In
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
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 Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experime ...
and
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxies ...
), Gaussian units were predominant until recent decades, but are now getting progressively less so.For example, one widely used graduate electromagnetism textbook is ''Classical Electrodynamics'' by J.D. Jackson. The second edition, published in 1975, used Gaussian units exclusively, but the third edition, published in 1998, uses mostly SI units. Similarly, ''Electricity and Magnetism'' by Edward Purcell is a popular undergraduate textbook. The second edition, published in 1984, used Gaussian units, while the third edition, published in 2013, switched to SI units. The 8th SI Brochure acknowledges that the CGS-Gaussian unit system has advantages in classical and relativistic electrodynamics, 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 Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) a ...
and
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and inter ...
.

# Major differences between Gaussian and SI units

## "Rationalized" unit systems

One difference between Gaussian and SI units is in the factors of 4''π'' in various formulas. SI electromagnetic units are called "rationalized",Kowalski, Ludwik, 1986,
A Short History of the SI Units in Electricity
" ''The Physics Teacher'' 24(2): 97–99
/ref> because
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
have no explicit factors of 4''π'' in the formulae. On the other hand, the
inverse-square In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understo ...
force laws –
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...
and the
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
– ''do'' have a factor of 4''π'' attached to the ''r''. In ''unrationalized'' Gaussian units (not
Heaviside–Lorentz units Heaviside–Lorentz units (or Lorentz–Heaviside units) constitute a system of units (particularly electromagnetic units) within CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that th ...
) the situation is reversed: two of Maxwell's equations have factors of 4''π'' in the formulas, while both of the inverse-square force laws, Coulomb's law and the Biot–Savart law, have no factor of 4''π'' attached to ''r'' in the denominator. (The quantity 4''π'' appears because 4''πr'' is the surface area of the sphere of radius ''r'', 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 In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be unders ...
.)

## Unit of charge

A major difference between Gaussian and SI systems is in the respective definitions of the of charge quantity. In SI, a separate base unit (the
ampere The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elec ...
) is associated with electromagnetic phenomena, with the consequence that a unit of electrical charge (1
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary ch ...
= 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 electrical charge (the
statcoulomb The franklin (Fr) or statcoulomb (statC) electrostatic unit of charge (esu) is the physical unit for electrical charge used in the cgs-esu and Gaussian units. It is a derived unit given by : 1 statC = 1 dyn1/2⋅cm = 1 cm3/2⋅g1/2⋅s−1. Th ...
, statC) ''can'' be written entirely as a dimensional combination of the mechanical units (gram, centimetre, second), as: : = . For example,
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...
in Gaussian units has no constant: :$F = \frac ,$ where ''F'' is the repulsive force between two electrical charges, ''Q'' and ''Q'' are the two charges in question, and ''r'' is the distance separating them. If ''Q'' and ''Q'' are expressed in
statC The franklin (Fr) or statcoulomb (statC) electrostatic unit of charge (esu) is the physical unit for electrical charge used in the cgs-esu and Gaussian units. It is a derived unit given by : 1 statC = 1 dyn1/2⋅cm = 1 cm3/2⋅g1/2⋅s−1. That ...
and ''r'' in cm, then the unit of ''F'' that is coherent with these units is the
dyne The dyne (symbol: dyn; ) is a derived unit of force specified in the centimetre–gram–second (CGS) system of units, a predecessor of the modern SI. History The name dyne was first proposed as a CGS unit of force in 1873 by a Committee o ...
. The same law in SI units is: :$F = \frac \frac$ where ''ε''0 is the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
, a quantity that is not dimensionless, namely (
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aq ...
)2 ( time)2 (
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementa ...
)−1 (
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inter ...
)−3. Without ''ε''0, the equation would be dimensionally inconsistent with the SI quantities, whereas the quantity ''ε''0 does not appear in Gaussian equations. This is an example of how some dimensional
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant ...
s can be eliminated from the expressions of
physical law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow ...
by the choice of quantities. In SI, 1/''ε''0, converts or scales
flux density Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
, D, to the corresponding
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
, E (the latter has dimension of
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...
per
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * '' Charge!!'', an album by The Aq ...
), while in Gaussian units, electric flux density is the same quantity as electric field strength in
free space A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
aside from a dimensionless constant factor. In Gaussian units, the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit for ...
''c'' appears explicitly in electromagnetic formulas like
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
(see below), whereas in SI it appears via the product $\mu_0 \varepsilon_0=1/c^2$.

## Units for magnetism

In Gaussian units, unlike SI units, the electric field E and the
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
B have the same dimension. This amounts to a factor of '' c'' between how B is defined in the two unit systems, on top of the other differences. (The same factor applies to other magnetic quantities such as H and M.) For example, in a planar light wave in vacuum, in Gaussian units, while in SI units.

## Polarization, magnetization

There are further differences between Gaussian and SI units in how quantities related to polarization and magnetization are defined. For one thing, in Gaussian units, ''all'' of the following quantities have the same dimension: E, D, P, B, H, and M. Another important point is that the
electric Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described b ...
and
magnetic susceptibility In electromagnetism, the magnetic susceptibility (Latin: , "receptive"; denoted ) is a measure of how much a material will become magnetized in an applied magnetic field. It is the ratio of magnetization (magnetic moment per unit volume) to the ...
of a material is dimensionless in both Gaussian and SI units, 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 Gaussian and 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 Ref.A. Garg, "Classical Electrodynamics in a Nutshell" (Princeton University Press, 2012). 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 In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the '' divergence'' of the field in the ...
or Kelvin–Stokes theorem.

## 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 In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more ...
is a simple constant. where *E and D are the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
and displacement field, respectively; *P is the
polarization density In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
; *$\varepsilon$ is the
permittivity In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' ( epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more ...
; *$\varepsilon_0$ is the permittivity of vacuum (used in the SI system, but meaningless in Gaussian units); *$\chi_\text$ is the electric susceptibility The quantities $\varepsilon^\text$ and $\varepsilon^\text/\varepsilon_0$ are both dimensionless, and they have the same numeric value. By contrast, the electric susceptibility $\chi_\text^\text$ and $\chi_\text^\text$ are both unitless, but have ''different numeric values'' for the same material: ::$4\pi \chi_\text^\text = \chi_\text^\text$ 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. where *B and H are the
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
s *M is
magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or D ...
*$\mu$ is
magnetic permeability In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by Willia ...
*$\mu_0$ is the permeability of vacuum (used in the SI system, but meaningless in Gaussian units); *$\chi_\text$ is the
magnetic susceptibility In electromagnetism, the magnetic susceptibility (Latin: , "receptive"; denoted ) is a measure of how much a material will become magnetized in an applied magnetic field. It is the ratio of magnetization (magnetic moment per unit volume) to the ...
The quantities $\mu^\text$ and $\mu^\text/\mu_0$ are both dimensionless, and they have the same numeric value. By contrast, the
magnetic susceptibility In electromagnetism, the magnetic susceptibility (Latin: , "receptive"; denoted ) is a measure of how much a material will become magnetized in an applied magnetic field. It is the ratio of magnetization (magnetic moment per unit volume) to the ...
$\chi_\text^\text$ and $\chi_\text^\text$ are both unitless, but has ''different numeric values'' in the two systems for the same material: ::$4\pi \chi_\text^\text = \chi_\text^\text$

## Vector and scalar potentials

The electric and magnetic fields can be written in terms of a vector potential A and a scalar potential ''φ'':

## Electrical circuit

where * ''Q'' is the quantity of electricity * ''I'' is the
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving p ...
* ''V'' is the
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
* ''Φ'' is the
magnetic flux In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the webe ...
* ''R'' is the
electrical resistance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallel ...
* ''C'' is the
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
* ''L'' is the
inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...

# Electromagnetic unit names

(For non-electromagnetic units, see
Centimetre–gram–second system of units The centimetre–gram–second system of units (abbreviated CGS or cgs) is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time. All CGS mechanical unit ...
.) ::Note: The SI quantities $\epsilon_0$ and $\mu_0$ satisfy $\epsilon_0\mu_0 = 1/c^2$. The conversion factors are written both symbolically and numerically. The numerical conversion factors can be derived from the symbolic conversion factors by
dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such a ...
. For example, the top row says $\frac = \frac$, a relation which can be verified with dimensional analysis, by expanding $\epsilon_0$ and C in
SI base units The SI base units are the standard units of measurement defined by the International System of Units (SI) for the seven base quantities of what is now known as the International System of Quantities: they are notably a basic set from which all ...
, and expanding Fr in Gaussian base units. It is surprising to think of measuring capacitance in centimetres. One useful example is that a centimetre of capacitance is the capacitance between a sphere of radius 1 cm in vacuum and infinity. Another surprising unit is measuring
resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
in units of seconds. A physical example is: Take a parallel-plate capacitor, which has a "leaky" dielectric with permittivity 1 but a finite resistivity. After charging it up, the capacitor will discharge itself over time, due to current leaking through the dielectric. If the resistivity of the dielectric is "X" seconds, the half-life of the discharge is ~0.05X seconds. This result is independent of the size, shape, and charge of the capacitor, and therefore this example illuminates the fundamental connection between resistivity and time units.

## Dimensionally equivalent units

A number of the units defined by the table have different names but are in fact dimensionally equivalent – i.e., they have the same expression in terms of the base units cm, g, s. (This is analogous to the distinction in SI between
becquerel The becquerel (; symbol: Bq) is the unit of radioactivity in the International System of Units (SI). One becquerel is defined as the activity of a quantity of radioactive material in which one nucleus decays per second. For applications relati ...
and Hz, or between
newton-metre The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is the unit of torque (also called ) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton applie ...
and
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
.) The different names help avoid ambiguities and misunderstandings as to what physical quantity is being measured. In particular, ''all'' of the following quantities are dimensionally equivalent in Gaussian units, but they are nevertheless given different unit names as follows:

# General rules to translate a formula

Any formula can be converted between Gaussian and SI units by using the symbolic conversion factors from Table 1 above. For example, the electric field of a stationary point charge has the SI formula :$\mathbf^ = \frac \hat ,$ where ''r'' is distance, and the "SI" subscripts indicate that the electric field and charge are defined using SI definitions. If we want the formula to instead use the Gaussian definitions of electric field and charge, we look up how these are related using Table 1, which says: :$\frac = \sqrt \quad , \quad \frac = \frac \, .$ Therefore, after substituting and simplifying, we get the Gaussian-units formula: :$\mathbf^ = \frac\hat ,$ which is the correct Gaussian-units formula, as mentioned in a previous section. For convenience, the table below has a compilation of the symbolic conversion factors from Table 1. To convert any formula from Gaussian units to SI units using this table, replace each symbol in the Gaussian column by the corresponding expression in the SI column (vice versa to convert the other way). This will reproduce any of the specific formulas given in the list above, such as Maxwell's equations, as well as any other formula not listed. For some examples of how to use this table, see:Units in Electricity and Magnetism
See the section "Conversion of Gaussian formulae into SI" and the subsequent text.
Once all occurrences of the product $\epsilon_0 \mu_0$ have been replaced by $1/c^2$, there should be no remaining quantities in the equation with an SI electromagnetic dimension remaining.