Gaussian units

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Gaussian units constitute a metric system 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.[1] 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 predominate in most fields, and continue to increase in popularity at the expense of Gaussian units.[2][3] Alternative unit systems also exist. Conversions between quantities in the Gaussian unit system and the SI unit system are not as straightforward as direct unit conversions because the quantities themselves are defined differently in the different systems, which has the effect that the equations expressing physical laws of electromagnetism (such as Maxwell's equations) change depending on what system of units is being used. As an example, quantities that are dimensionless in one system may have dimension in another.

History[edit]

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[edit]

The Gaussian unit system is just one of several electromagnetic unit systems within CGS. Others include "electrostatic units", "electromagnetic units", and Lorentz–Heaviside units.

Some other unit systems are called "natural units", a category that includes Hartree atomic units, Planck units, and others.

SI units are by far the most common system of units today. In engineering and practical areas, SI is nearly universal and has been for decades.[2] In technical, scientific literature (such as theoretical physics and astronomy), Gaussian units were predominant until recent decades, but are now getting progressively less so.[2][3] The 8th SI Brochure acknowledges that the CGS-Gaussian unit system has advantages in classical and relativistic electrodynamics,[4] 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 units[edit]

"Rationalized" unit systems[edit]

One difference between Gaussian and SI units is in the factors of 4π in various formulas. SI electromagnetic units are called "rationalized",[5][6] because Maxwell's equations have no explicit factors of 4π in the formulae. On the other hand, the inverse-square force laws – Coulomb's law and the Biot–Savart lawdo have a factor of 4π attached to the r2. In unrationalized Gaussian units (not Lorentz–Heaviside units) the situation is reversed: two of Maxwell's equations have factors of 4π in the formulas, while both of the inverse-square force laws, Coulomb's law and the Biot–Savart law, have no factor of 4π attached to r2 in the denominator.

(The quantity 4π appears because 4πr2 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.)

Unit of charge[edit]

A major difference between Gaussian and SI units is in the definition of the unit of charge. In SI, a separate base unit (the ampere) is associated with electromagnetic phenomena, with the consequence that something like electrical charge (1 coulomb = 1 ampere × 1 second) is a unique dimension of physical quantity and is not 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, statC) can be written entirely as a dimensional combination of the mechanical units (gram, centimetre, second), as:

1 statC = 1 g1/2⋅cm3/2⋅s−1

For example, Coulomb's law in Gaussian units has no constant:

where F is the repulsive force between two electrical charges, QG
1
and QG
2
are the two charges in question, and r is the distance separating them. If QG
1
and QG
2
are expressed in statC and r in cm, then F will come out expressed in dyne.

The same law in SI units is:

where ε0 is the vacuum permittivity, a quantity with dimension, namely (charge)2 (time)2 (mass)−1 (length)−3. Without ε0, the two sides would not have consistent dimensions in SI, whereas the quantity ε0 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 simply by the judicious choice of units. In SI, 1/ε0, converts or scales flux density, D, to electric field, E (the latter has dimension of force per charge), while in rationalized Gaussian units, electric flux density is the same quantity as electric field strength in free space.

In Gaussian units, the speed of light c appears explicitly in electromagnetic formulas like Maxwell's equations (see below), whereas in SI it appears only via the product .

Units for magnetism[edit]

In Gaussian units, unlike SI units, the electric field EG and the magnetic field BG 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.[5] (The same factor applies to other magnetic quantities such as H and M.) For example, in a planar light wave in vacuum, |EG(r, t)| = |BG(r, t)| in Gaussian units, while |ESI(r, t)| = c |BSI(r, t)| in SI units.

Polarization, magnetization[edit]

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: EG, DG, PG, BG, HG, and MG. Another important point is that the electric and magnetic susceptibility 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[edit]

This section has a list of the basic formulae of electromagnetism, given in both Gaussian and SI units. 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.[7] All formulas except otherwise noted are from Ref.[5]

Maxwell's equations[edit]

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.

Name Gaussian units SI units
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[edit]

Name Gaussian units SI units
Lorentz force
Coulomb's law
Electric field of
stationary point charge
Biot–Savart law [8]
Poynting vector
(microscopic)

Dielectric and magnetic materials[edit]

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.

Gaussian quantities SI quantities

where

The quantities and are both dimensionless, and they have the same numeric value. By contrast, the electric susceptibility and are both unitless, but have different numeric values 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.

Gaussian quantities SI quantities

where

The quantities and are both dimensionless, and they have the same numeric value. By contrast, the magnetic susceptibility and are both unitless, but has different numeric values in the two systems for the same material:

Vector and scalar potentials[edit]

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

Name Gaussian units SI units
Electric field
Magnetic B field

Electromagnetic unit names[edit]

(For non-electromagnetic units, see Centimetre–gram–second system of units.)

Table 1: Common electromagnetism units in SI vs Gaussian
2.998 is shorthand for exactly 2.99792458 (see speed of light)[9]
Quantity Symbol SI unit Gaussian unit
(in base units)
Conversion factor
electric charge q C Fr
(cm3/2⋅g1/2⋅s−1)
electric current I A Fr/s
(cm3/2⋅g1/2⋅s−2)
electric potential
(voltage)
φ
V
V statV
(cm1/2⋅g1/2⋅s−1)
electric field E V/m statV/cm
(cm−1/2⋅g1/2⋅s−1)
electric
displacement field
D C/m2 Fr/cm2
(cm−1/2g1/2s−1)
magnetic B field B T G
(cm−1/2⋅g1/2⋅s−1)
magnetic H field H A/m Oe
(cm−1/2⋅g1/2⋅s−1)
magnetic dipole
moment
m Am2 erg/G
(cm5/2⋅g1/2⋅s−1)
magnetic flux Φm Wb Gcm2
(cm3/2⋅g1/2⋅s−1)
resistance R Ω s/cm
resistivity ρ Ωm s
capacitance C F cm
inductance L H s2/cm
Note: The SI 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 dimensional analysis. For example, the top row says , a relation which can be verified with dimensional analysis, by expanding and C in SI base units, 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 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[edit]

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 and Hz, or between newton-metre and joule.) 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:[10]

Quantity In Gaussian
base units
Gaussian unit
of measure
EG cm−1/2⋅g1/2⋅s−1 statV/cm
DG cm−1/2⋅g1/2⋅s−1 statC/cm2
PG cm−1/2⋅g1/2⋅s−1 statC/cm2
BG cm−1/2⋅g1/2⋅s−1 G
HG cm−1/2 g1/2⋅s−1 Oe
MG cm−1/2⋅g1/2⋅s−1 dyn/Mx

General rules to translate a formula[edit]

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

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:

Therefore, after substituting and simplifying, we get the Gaussian-units formula:

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.[11] For some examples of how to use this table, see:[12]

Table 2A: Replacement rules for translating formulas from Gaussian to SI
Name Gaussian units SI units
electric field, electric potential
electric displacement field
charge, charge density, current,
current density, polarization density,
electric dipole moment
magnetic B field, magnetic flux,
magnetic vector potential
magnetic H field
magnetic moment, magnetization
permittivity,
permeability
electric susceptibility,
magnetic susceptibility
conductivity, conductance, capacitance
resistivity, resistance, inductance
Table 2B: Replacement rules for translating formulas from SI to Gaussian
Name SI units Gaussian units
electric field, electric potential
electric displacement field
charge, charge density, current,
current density, polarization density,
electric dipole moment
magnetic B field, magnetic flux,
magnetic vector potential
magnetic H field
magnetic moment, magnetization
permittivity,
permeability
electric susceptibility,
magnetic susceptibility
conductivity, conductance, capacitance
resistivity, resistance, inductance

Once all occurrences of the product have been replaced by , there should be no remaining quantities in the equation with an SI electromagnetic dimension remaining.

Notes and references[edit]

  1. ^ One of many examples of using the term "cgs units" to refer to Gaussian units is: Lecture notes from Stanford University
  2. ^ a b c "CGS", in How Many? A Dictionary of Units of Measurement, by Russ Rowlett and the University of North Carolina at Chapel Hill
  3. ^ a b 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.
  4. ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2017-08-14, p. 128
  5. ^ a b c Littlejohn, Robert (Fall 2017). "Gaussian, SI and Other Systems of Units in Electromagnetic Theory" (PDF). Physics 221A, University of California, Berkeley lecture notes. Retrieved 2018-04-18.
  6. ^ Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity, Archived 2009-04-29 at the Wayback Machine" The Physics Teacher 24(2): 97–99. Alternate web link (subscription required)
  7. ^ A. Garg, "Classical Electrodynamics in a Nutshell" (Princeton University Press, 2012).
  8. ^ Introduction to Electrodynamics by Capri and Panat, p180
  9. ^ Cardarelli, F. (2004). Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins (2nd ed.). Springer. pp. 20–25. ISBN 978-1-85233-682-0.
  10. ^ Cohen, Douglas L. (2001). Demystifying Electromagnetic Equations. p. 155. ISBN 9780819442345. Retrieved 2012-12-25.
  11. ^ Бредов М.М.; Румянцев В.В.; Топтыгин И.Н. (1985). "Appendix 5: Units transform (p.385)". Классическая электродинамика. Nauka.
  12. ^ Units in Electricity and Magnetism. See the section "Conversion of Gaussian formulae into SI" and the subsequent text.

External links[edit]