Open main menu

Wikipedia β

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: cgs contains within it several conflicting sets of electromagnetism units, not just Gaussian units, as described below.

The most common alternative to Gaussian units are SI units. SI units are predominant in most fields, and continue to increase in popularity at the expense of Gaussian units.[2][3] (Other alternative unit systems also exist, as discussed below.) Conversions between Gaussian units and SI units are not as simple as normal unit conversions. For example, the formulas for physical laws of electromagnetism (such as Maxwell's equations) need to be adjusted depending on what system of units one uses. As another example, quantities that are dimensionless (loosely "unitless") in one system may have dimension in another.

Contents

HistoryEdit

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 systemsEdit

The main alternative to the Gaussian unit system is SI units, historically also called the MKSA system of units for metre–kilogram–second–ampere.[2]

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 atomic units, Planck units, and others.

SI units are by far the most common 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]

Natural units are most common in more theoretical and abstract fields of physics, particularly particle physics and string theory.

Major differences between Gaussian and SI unitsEdit

"Rationalized" unit systemsEdit

One difference between Gaussian and SI units is in the factors of 4π in various formulas. SI electromagnetic units are called "rationalized",[4][5] 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 r 2. 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 r 2 in the denominator.

(The quantity 4π appears because 4π r 2 is the surface area of the sphere of radius r. For details, see the articles Relation between Gauss's law and Coulomb's law and Inverse-square law.)

Unit of chargeEdit

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 Gaussian units, 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 is simple:

 

where F is the repulsive force between two electrical charges, Q1 and Q2 are the two charges in question, and r is the distance separating them. If Q1 and Q2 are expressed in statC and r in cm, then F will come out expressed in dyne.

By contrast, 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, and ke is Coulomb's constant. Without ε0, the two sides could not have consistent dimensions in SI, and in fact the quantity ε0 does not even exist in Gaussian units. 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, flux density is the very same as electric field in free space, not just a scaled copy.

Since the unit of charge is built out of mechanical units (mass, length, time), the relation between mechanical units and electromagnetic phenomena is clearer in Gaussian units than in SI. In particular, in Gaussian units, the speed of light c shows up directly in electromagnetic formulas like Maxwell's equations (see below), whereas in SI it only shows up implicitly via the relation  .

Units for magnetismEdit

In Gaussian units, unlike SI units, the electric field E and the magnetic field B have the same dimension. This amounts to a factor of c difference between how B is defined in the two unit systems, on top of the other differences.[4] (The same factor applies to other magnetic quantities such as H and M.) For example, in a planar light wave in vacuum, |E(r, t)| = |B(r, t)| in Gaussian units, while |E(r, t)| = c|B(r, t)| in SI units.

Polarization, magnetizationEdit

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 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 equationsEdit

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.[6] All formulas except otherwise noted are from Ref.[4]

Maxwell's equationsEdit

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 lawsEdit

Name Gaussian units SI units
Lorentz force    
Coulomb's law    
Electric field of
stationary point charge
   
Biot–Savart law  [7]  
Poynting vector
(microscopic)
   

Dielectric and magnetic materialsEdit

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 units SI units
   
   
   
   

where

The quantities   in Gaussian units and   in SI are both dimensionless, and they have the same numeric value. By contrast, the electric susceptibility   is unitless in both systems, but has different numeric values in the two systems 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 units SI units
   
   
   
   

where

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

 

Vector and scalar potentialsEdit

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
(static)
   
Electric field
(general)
   
Magnetic B field    

Electromagnetic unit namesEdit

(For non-electromagnetic units, see main cgs article.)

Table 1: Common electromagnetism units in SI vs Gaussian
2.998 is shorthand for exactly 2.99792458 (see speed of light)[8]
Quantity Symbol SI unit Gaussian unit
(in base units)
Conversion factor
electric charge q C Fr
(cm3/2g1/2s−1)
 
electric current I A Fr/s
(cm3/2g1/2s−2)
 
electric potential
(voltage)
φ
V
V statV
(cm1/2g1/2s−1)
 
electric field E V/m statV/cm
(cm−1/2g1/2s−1)
 
electric
displacement field
D C/m2 Fr/cm2
(cm−1/2g1/2s−1)
 
magnetic B field B T G
(cm−1/2g1/2s−1)
 
magnetic H field H A/m Oe
(cm−1/2g1/2s−1)
 
magnetic dipole
moment
m Am2 erg/G
(cm5/2g1/2s−1)
 
magnetic flux Φm Wb Gcm2
(cm3/2g1/2s−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 unitsEdit

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:[9]

Quantity In Gaussian
base units
Gaussian unit
of measure
E cm−1/2 g1/2 s−1 statV/cm
D cm−1/2 g1/2 s−1 statC/cm2
P cm−1/2 g1/2 s−1 statC/cm2
B cm−1/2 g1/2 s−1 Gs
H cm−1/2 g1/2 s−1 Oe
M cm−1/2 g1/2 s−1 dyn/Mx

General rules to translate a formulaEdit

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

Table 2A: Replacement rules for translating formulas from Gaussian to SI
Name Gaussian units SI units
Speed of light    
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    
Relative permittivity,
Relative 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
Final substitution A    
Final substitution B    
Speed of light    
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    
Relative permittivity,
Relative permeability
   
Vacuum permittivity,
Vacuum permeability
   
Absolute permittivity,
Absolute permeability
   
Electric susceptibility,
Magnetic susceptibility
   
Conductivity, Conductance, Capacitance    
Resistivity, Resistance, Inductance    

It may be necessary to apply either Final substitution A or Final substitution B (but not both) after all the other rules have been applied and the resulting formula has already been simplified as much as possible.

Notes and referencesEdit

  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 d "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. ^ a b c Littlejohn, Robert (Fall 2011). "Gaussian, SI and Other Systems of Units in Electromagnetic Theory" (PDF). Physics 221A, University of California, Berkeley lecture notes. Retrieved 2008-05-06. 
  5. ^ 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)
  6. ^ A. Garg, "Classical Electrodynamics in a Nutshell" (Princeton University Press, 2012).
  7. ^ Introduction to Electrodynamics by Capri and Panat, p180
  8. ^ Cardarelli, F. (2004). Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins (2nd ed.). Springer. pp. 20–25. ISBN 1-85233-682-X. 
  9. ^ ''Demystifying Electromagnetic Equations''. Books.google.com. p. 155. Retrieved 2012-12-25. 
  10. ^ Бредов М.М.; Румянцев В.В.; Топтыгин И.Н. (1985). "Appendix 5: Units transform (p.385)". Классическая электродинамика. Nauka. 
  11. ^ Units in Electricity and Magnetism. See the section "Conversion of Gaussian formulae into SI" and the subsequent text.

External linksEdit