Gaussian units

Summary

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.[a] 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.

Carl Friedrich Gauss

SI units predominate in most fields, and continue to increase in popularity at the expense of Gaussian units.[1][b] Alternative unit systems also exist. Conversions between quantities in Gaussian and SI units are not 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 equations—will change depending on the system of units employed. As an example, quantities that are dimensionless in one system may have dimension in the other.

Alternative unit systems

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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.[1] In technical, scientific literature (such as theoretical physics and astronomy), Gaussian units were predominant until recent decades, but are now getting progressively less so.[1][b] The 8th SI Brochure acknowledges that the CGS-Gaussian unit system has advantages in classical and relativistic electrodynamics,[2] 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

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"Rationalized" unit systems

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One difference between Gaussian and SI units is in the factors of 4π in various formulas. With SI electromagnetic units, called rationalized,[3][4] Maxwell's equations have no explicit factors of 4π in the formulae, whereas the inverse-square force laws – Coulomb's law and the Biot–Savart lawdo have a factor of 4π attached to the r2. With Gaussian units, called unrationalized (and unlike Heaviside–Lorentz 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

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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) can be written entirely as a dimensional combination of the non-electrical base 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 centimetres, then the unit of F that is coherent with these units is the dyne.

The same law in the ISQ is:   where ε0 is the vacuum permittivity, a quantity that is not dimensionless: it has dimension (charge)2 (time)2 (mass)−1 (length)−3. Without ε0, the equation would be dimensionally inconsistent with the quantities as defined in the ISQ, 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 by the choice of definition of quantities. In the ISQ,   converts or scales flux density, D, to the corresponding electric field, E (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 c appears directly in electromagnetic formulas like Maxwell's equations (see below), whereas in the ISQ it appears via the product  .

Units for magnetism

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In the Gaussian system, unlike the ISQ, 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.[3] (The same factor applies to other magnetic quantities such as the magnetic field, H, and magnetization, M.) For example, in a planar light wave in vacuum, |EG(r, t)| = |BG(r, t)| in Gaussian units, while |EI(r, t)| = c |BI(r, t)| in the ISQ.

Polarization, magnetization

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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: EG, DG, PG, BG, HG, and MG. A further point is that the electric and magnetic susceptibility of a material is dimensionless in both 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

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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).[5] All formulas except otherwise noted are from Ref.[3]

Maxwell's equations

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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 ISQ
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

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Other electromagnetic laws in Gaussian system and ISQ
Name Gaussian system ISQ
Lorentz force    
Coulomb's law    
Electric field of
stationary point charge
   
Biot–Savart law[6]    
Poynting vector
(microscopic)
   

Dielectric and magnetic materials

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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 ISQ
   
   
   
   

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.

Expressions for fields in magnetic media
Gaussian system ISQ
   
   
   
   

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

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

Electromagnetic fields in Gaussian system and ISQ
Name Gaussian system ISQ
Electric field    
Magnetic B field    

Electrical circuit

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Electrical circuit values in Gaussian system and ISQ
Name Gaussian system ISQ
Charge conservation    
Lenz's law    
Ohm's law    
Capacitance    
Inductance    

where

Fundamental constants

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Fundamental constants in Gaussian system and ISQ
Name Gaussian system ISQ
Impedance of free space    
Electric constant    
Magnetic constant    
Fine-structure constant    
Magnetic flux quantum    
Conductance quantum    
Bohr radius    
Bohr magneton    

Electromagnetic unit names

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Table 1: Common electromagnetism units in SI vs Gaussian[7]
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 statA
(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)
 
Electric dipole moment p Cm Frcm
(cm5/2⋅g1/2⋅s−1)
 
Electric flux Φe C Fr
(cm3/2⋅g1/2⋅s−1)
 
Permittivity ε F/m cm/cm  
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 Mx
(cm3/2⋅g1/2⋅s−1)
 
Permeability μ H/m cm/cm  
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 coulombs (C) in SI base units, and expanding statcoulombs (or franklins, 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 t seconds, the half-life of the discharge is ~0.05 t 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

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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:[8]

Dimensionally equivalent units
Quantity Gaussian symbol In Gaussian
base units
Gaussian unit
of measure
Electric field EG cm−1/2⋅g1/2⋅s−1 statV/cm
Electric displacement field DG cm−1/2⋅g1/2⋅s−1 statC/cm2
Polarization density PG cm−1/2⋅g1/2⋅s−1 statC/cm2
Magnetic flux density BG cm−1/2⋅g1/2⋅s−1 G
Magnetizing field HG cm−1/2 g1/2⋅s−1 Oe
Magnetization MG cm−1/2⋅g1/2⋅s−1 dyn/Mx

General rules to translate a formula

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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 ISQ formula   where r is distance, and the "I" superscript indicates that the electric field and charge are defined as in the ISQ. 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-system formula:   which is the correct Gaussian-system 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 system to the ISQ 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.[9][10][11][c]

Table 2A: Replacement rules for translating formulas from Gaussian to ISQ
Name Gaussian system ISQ
electric field, electric potential, electromotive force    
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 scalar potential, magnetomotive force    
magnetic moment, magnetization, magnetic pole strength    
permittivity,
permeability
   
electric susceptibility,
magnetic susceptibility
   
conductivity, conductance, capacitance    
resistivity, resistance, inductance, memristance, impedance    
magnetic reluctance    
Table 2B: Replacement rules for translating formulas from ISQ to Gaussian
Name ISQ Gaussian system
electric field, electric potential, electromotive force    
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 scalar potential, magnetomotive force    
magnetic moment, magnetization, magnetic pole strength    
permittivity,
permeability
   
electric susceptibility,
magnetic susceptibility
   
conductivity, conductance, capacitance    
resistivity, resistance, inductance, memristance, impedance    
magnetic reluctance    

Once all occurrences of the product   have been replaced by  , there should be no remaining quantities in the equation that have an ISQ electromagnetic dimension (or, equivalently, that have an SI electromagnetic unit).

Notes

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  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 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.
  3. ^ 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.

References

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  1. ^ 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
  2. ^ 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 2021-06-04, retrieved 2021-12-16, p. 128
  3. ^ 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.
  4. ^ 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)
  5. ^ A. Garg, 2012, "Classical Electrodynamics in a Nutshell" (Princeton University Press).
  6. ^ Introduction to Electrodynamics by Capri and Panat, p180
  7. ^ 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.
  8. ^ Cohen, Douglas L. (2001). Demystifying Electromagnetic Equations. SPIE Press. p. 155. ISBN 9780819442345. Retrieved 2012-12-25.
  9. ^ Бредов, М. М.; Румянцев, В. В.; Топтыгин, И. Н. (1985). "Appendix 5: Units transform". Классическая электродинамика [Classical Electrodynamics] (in Russian). Nauka. p. 385.
  10. ^ Simpson, David. "SI / Gaussian Formula Conversion Table" (PDF). Prince George's Community College. Retrieved 23 February 2024.
  11. ^ Jackson, John. Classical Electrodynamics (3 ed.). John Wiley & Sons, Inc. p. 782. ISBN 0-471-30932-X.
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  • Comprehensive list of Gaussian unit names, and their expressions in base units
  • The evolution of the Gaussian Units Archived 2016-01-09 at the Wayback Machine by Dan Petru Danescu