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Carl Gauss

Gaussian unitsFrom Wikipedia, the free encyclopedia

Gaussian units comprise a metric system of physical units. Thissystem is the most common of the several electromagnetic unitsystems based on cgs (centimetre–gram–second) units. It is alsocalled the Gaussian unit system, Gaussian-cgs units, or often

just cgs units.[1] The term "cgs units" is ambiguous and therefore tobe avoided if possible: cgs contains within it several conflicting setsof electromagnetism units, not just Gaussian units, as describedbelow.

The most common alternative to Gaussian units are SI units. SI unitsare predominant in most fields, and continue to increase in

popularity at the expense of Gaussian units.[2][3] (Other alternativeunit systems also exist, as discussed below.) Conversions betweenGaussian units and SI units are not as simple as normal unitconversions. For example, the formulas for physical laws ofelectromagnetism (such as Maxwell's equations) need to beadjusted depending on what system of units one uses. As anotherexample, quantities that are dimensionless (loosely "unitless") in onesystem may have dimension in another.

Contents

1 History

2 Alternative unit systems

3 Major differences between Gaussian and SI units3.1 "Rationalized" unit systems

3.2 Unit of charge

3.3 Units for magnetism

3.4 Polarization, magnetization

4 List of equations

4.1 Maxwell's equations

4.2 Other basic laws

4.3 Dielectric and magnetic materials4.4 Vector and scalar potentials

5 Electromagnetic unit names

5.1 Dimensionally equivalent units

6 General rules to translate a formula

7 Notes and references

8 External links

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History

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

Alternative unit systems

Main article: Alternative CGS units in electromagnetism

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 near-universal and has been for

decades,[2] while 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 andstring theory.

Major differences between Gaussian and SI units

"Rationalized" unit systems

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 theother hand, the inverse-square force laws, Coulomb's law and the Biot–Savart law, do have a factor of 4π attached

to the r2. In unrationalized Gaussian units (not Lorentz–Heaviside units) the situation is reversed: Two ofMaxwell'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. For details, see the articlesRelation between Gauss's law and Coulomb's law and Inverse-square law.)

Unit of charge

A major difference between Gaussian and SI units is in the definition of the unit of charge. In SI, a separate baseunit (the ampere) is associated with electrical phenomena, with the consequence that something like electricalcharge (1 coulomb = 1 ampere × 1 second) is a unique dimension of physical quantity and is not expressed purely

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in terms of the mechanical units (kilogram, metre, second). On the other hand, in Gaussian units, the unit of electricalcharge (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 appears 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 permitivity, a quantity with dimension, namely (charge)2 (time)2 (mass)−1 (length)−3.

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 theexpressions 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 isthe 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 unitsand electromagnetic phenomena is clearer in Gaussian units than in SI. In particular, in Gaussian units, the speed oflight c shows up directly in electromagnetic formulas like Maxwell's equations (see below), whereas in SI it onlyshows up implicitly via the relation .

Units for magnetism

In Gaussian units, unlike SI units, the electric field E and the magnetic field B have the same dimension. Thisamounts 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 planarlight wave in vacuum, |E(r,t)|=|B(r,t)| in Gaussian units, while |E(r,t)|=c|B(r,t)| in SI units.

Polarization, magnetization

There are further differences between Gaussian and SI units in how quantities related to polarization andmagnetization 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 isdimensionless in both Gaussian and SI units, but a given material will have a different numerical susceptibility in thetwo systems. (Equation is given below.)

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List of equations

This section has a list of the basic formulae of electromagnetism, given in both Gaussian and SI units. Most symbolnames are not given; for complete explanations and definitions, please click to the appropriate dedicated article for

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

Maxwell's equations

Main article: Maxwell's equations

Here are Maxwell's equations, both in macroscopic and microscopic forms. Only the "differential form" of theequations is given, not the "integral form"; to get the integral forms apply the divergence theorem or Kelvin–Stokestheorem.

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

Name Gaussian units SI units

Lorentz force

Coulomb's law

Electric field ofstationary point charge

Biot–Savart law

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 themedium is homogeneous, linear, isotropic, and nondispersive, so that the permittivity is a simple constant.

Gaussian units SI units

where

E and D are the electric field and displacement field, respectively;

P is the polarization density; is the permittivity;

is the permittivity of vacuum (used in the SI system, but meaningless in Gaussian units);

is the electric susceptibility

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 twosystems for the same material:

Next, here are the expressions for the various fields in a magnetic medium. Again, it is assumed that the medium ishomogeneous, linear, isotropic, and nondispersive, so that the permeability is a simple constant.

Gaussian units SI units

where

B and H are the magnetic fields

M is magnetization

is magnetic permeability is the permeability of vacuum (used in the SI system, but meaningless in Gaussian units);

is the magnetic susceptibility

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 twosystems for the same material:

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Vector and scalar potentials

Main articles: Magnetic vector potential and Electric potential

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 names

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

Conversion of SI units in electromagnetism to Gaussian

subsystem of CGS[6]

c = 29,979,245,800 ≈ 3·1010

Quantity Symbol SI unit Gaussian unit

electric charge q 1 C ↔ (10−1 c) Fr

electric current I 1 A ↔ (10−1 c) Fr/s

electric potential

voltage

φ

V1 V ↔ (108 c−1) statV

electric field E 1 V/m ↔ (106 c−1) statV/cm

magnetic induction B 1 T ↔ (104) Gs

magnetic field strength H 1 A/m ↔ (4π 10−3) Oe

magnetic dipole moment μ 1 A·m² ↔ (103) erg/Gs

magnetic flux Φm 1 Wb ↔ (108) Gs·cm²

resistance R 1 Ω ↔ (109 c−2) s/cm

resistivity ρ 1 Ω·m ↔ (1011 c−2) s

capacitance C 1 F ↔ (10−9 c2) cm

inductance L 1 H ↔ (109 c−2) s2/cm

In this table, the letter c represents the number 29,979,245,800 ≈ 3·1010, the numerical value of the speed of light

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In this table, the letter c represents the number 29,979,245,800 ≈ 3·1010, the numerical value of the speed of lightexpressed in cm/s. The symbol "↔" was used instead of "=" as a reminder that the SI and Gaussian units arecorresponding but not equal because they have incompatible dimensions. For example, according to the top row

of the table, something with a charge of 1 C also has a charge of (10−1 c) Fr, but it is usually incorrect to replace "1

C" with "(10−1 c) Fr" within an equation or formula, unless all other units in the formula are also replaced by theirGaussian equivalents.

It is surprising to think of measuring capacitance in centimetres. One useful example is that a centimetre ofcapacitance 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-platecapacitor, which has a "leaky" dielectric with permittivity 1 but a finite resistivity. After charging it up, the capacitorwill 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 chargeof the capacitor, and therefore this example illuminates the fundamental connection between resistivity and timeunits.

Dimensionally equivalent units

A number of the units defined by the table have different names but are in fact dimensionally equivalent—i.e., theyhave the same expression in terms of the base units cm, g, s. (This is analogous to the distinction in SI betweenbecquerel and Hz, or between newton metre and joule.) The different names help avoid ambiguities andmisunderstandings 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:[7]

QuantityIn 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−1Mx/cm2

or emu/cm3

[8]

General rules to translate a formula

To convert any formula from Gaussian units to SI units, replace the quantity in the Gaussian column by the quantityin 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] It may also be necessary to

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

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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

Relative permittivity,

Relative permeability

Electric susceptibility,

Magnetic susceptibility

Conductivity, Conductance, Capacitance

Resistivity, Resistance, Inductance

Notes and references

1. ^ One of many examples of using the term "cgs units" to refer to Gaussian units is: Lecture notes from StanfordUniversity (http://nlpc.stanford.edu/nleht/Science/reference/conversion.pdf)

2. ̂a b c d "CGS" (http://www.unc.edu/~rowlett/units/cgsmks.html), in How Many? A Dictionary of Units ofMeasurement, 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 in1998, uses mostly SI units.

4. ̂a b c Littlejohn, Robert (Fall 2011). "Gaussian, SI and Other Systems of Units in Electromagnetic Theory"(http://bohr.physics.berkeley.edu/classes/221/1112/notes/emunits.pdf) (PDF). Physics 221A, University ofCalifornia, Berkeley lecture notes. Retrieved 2008-05-06.

5. ^ Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity,(http://alpha.montclair.edu/~kowalskiL/SI/SI_PAGE.HTML)" The Physics Teacher 24(2): 97–99. Alternate web link(subscription required) (http://dx.doi.org/10.1119/1.2341955)

6. ^ Cardarelli, F. (2004). Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences andOrigins (http://books.google.com/books?id=6KCx8Ww75VkC) (2nd ed.). Springer. pp. 20–25. ISBN 1-85233-682-X.

7. ^ ''Demystifying Electromagnetic Equations'' (http://books.google.com/books?id=CQMsK5xW1DcC&pg=PA155).Books.google.com. p. 155. Retrieved 2012-12-25.

8. ^ Despite this usage, "emu" on its own is not a unit; see CRC handbook of chemistry and physics

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8. ^ Despite this usage, "emu" on its own is not a unit; see CRC handbook of chemistry and physics(http://books.google.com/books?id=kTnxSi2B2FcC&pg=PT46)

9. ^ lecture notes on units in electrodynamics (http://fatcat.ftj.agh.edu.pl/~bartekw/downloads/units_eld.pdf)

10. ^ Бредов М.М., Румянцев В.В., Топтыгин И.Н. (1985). "Appendix 5: Units transform (p.385)". Классическаяэлектродинамика. Nauka.

11. ^ Units in Electricity and Magnetism (http://www.qsl.net/g4cnn/units/units.htm). See the section "Conversion ofGaussian formulae into SI" and the subsequent text.

External links

Comprehensive list of Gaussian unit names, and their expressions in base units

(http://www.pgccphy.net/1030/gaussian.html)The evolution of the Gaussian Units (http://www.gsjournal.net/old/science/danescu.pdf) by Dan Petru

Danescu

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