Editing Gaussian units (section)

== Major differences between Gaussian and SI systems ==

=== "Rationalized" unit systems ===

One difference between the Gaussian and SI systems is in the factor {{math|4''π''}} in various formulas that relate the quantities that they define. With SI electromagnetic units, called ''rationalized'',<ref name=Littlejohn>{{cite web | url=http://bohr.physics.berkeley.edu/classes/221/1112/notes/emunits.pdf | title=Gaussian, SI and Other Systems of Units in Electromagnetic Theory | work=Physics 221A, University of California, Berkeley lecture notes|author-link1=Robert Grayson Littlejohn | author=Littlejohn, Robert | date=Fall 2017 | access-date=2018-04-18 }}</ref><ref name=Kowalski>Kowalski, Ludwik, 1986, [http://alpha.montclair.edu/~kowalskiL/SI/SI_PAGE.HTML "A Short History of the SI Units in Electricity"], {{webarchive |url=https://web.archive.org/web/20090429035624/http://alpha.montclair.edu/~kowalskiL/SI/SI_PAGE.HTML |date=2009-04-29 }}  ''The Physics Teacher'' 24(2): 97–99. [https://dx.doi.org/10.1119/1.2341955 Alternate web link (subscription required)]</ref> [[Maxwell's equations]] have no explicit factors of {{math|4''π''}} in the formulae, whereas the [[inverse-square law|inverse-square]] force laws &ndash; [[Coulomb's law]] and the [[Biot–Savart law]] &ndash; {{em|do}} have a factor of {{math|4''π''}} attached to the {{math|''r''{{i sup|2}}}}. With Gaussian units, called ''unrationalized'' (and unlike [[Heaviside–Lorentz units]]), the situation is reversed: two of Maxwell's equations have factors of {{math|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 {{math|4''π''}} attached to {{math|''r''{{i sup|2}}}} in the denominator.

(The quantity {{math|4''π''}} appears because {{math|4''πr''{{i sup|2}}}} is the [[sphere#Surface area|surface area of the sphere]] of radius {{mvar|r}}, which reflects the geometry of the configuration. For details, see the articles ''[[Gauss's law#Relation to Coulomb's law|Relation between Gauss's law and Coulomb's law]]'' and ''[[Inverse-square law]]''.)

=== Unit of charge ===

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&nbsp;[[coulomb]]&nbsp;= 1&nbsp;ampere&nbsp;× 1&nbsp;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) {{em|can}} be written entirely as a dimensional combination of the non-electrical base units (gram, centimetre, second), as:

{{block indent|1={{val|1|u=statC}} = {{val|1|u=g<sup>1/2</sup>⋅cm<sup>3/2</sup>⋅s<sup>−1</sup>}}.}}

For example, [[Coulomb's law]] in Gaussian units has no constant:
<math display="block">F = \frac{Q^{_\mathrm{G}}_1 Q^{_\mathrm{G}}_2}{r^2} ,</math>
where {{mvar|F}} is the repulsive force between two electrical charges, {{math|''Q''{{su|p={{small|G}}|b=1|lh=0.8em}}}} and {{math|''Q''{{su|p={{small|G}}|b=2|lh=0.8em}}}} are the two charges in question, and {{mvar|r}} is the distance separating them. If {{math|''Q''{{su|p={{small|G}}|b=1|lh=0.8em}}}} and {{math|''Q''{{su|p={{small|G}}|b=2|lh=0.8em}}}} are expressed in [[statC]] and {{mvar|r}} in [[centimetre]]s, then the unit of {{mvar|F}} that is coherent with these units is the [[dyne]].

The same law in the ISQ is:
<math display="block">F = \frac{1}{4\pi\varepsilon_0} \frac{Q^{_\mathrm{I}}_1 Q^{_\mathrm{I}}_2}{r^2}</math>
where {{math|''ε''{{sub|0}}}} is the [[vacuum permittivity]], a quantity that is not dimensionless: it has dimension ([[electric charge|charge]])<sup>2</sup> ([[time]])<sup>2</sup> ([[mass]])<sup>−1</sup> ([[length]])<sup>−3</sup>. Without {{math|''ε''{{sub|0}}}}, the equation would be dimensionally inconsistent with the quantities as defined in the ISQ, whereas the quantity {{math|''ε''{{sub|0}}}} does not appear in Gaussian equations.  This is an example of how some dimensional [[physical constant]]s can be eliminated from the expressions of [[physical law]] by the choice of definition of quantities.  In the ISQ, {{math|1/''ε''<sub>0</sub>}} converts or scales [[Electric displacement field|electric flux density]], {{math|'''D'''}}, to the corresponding [[electric field]], {{math|'''E'''}} (the latter has dimension of [[force]] per [[electric charge|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]] {{mvar|c}} appears directly in electromagnetic formulas like [[Maxwell's equations]] (see below), whereas in the ISQ it appears via the product {{math|1=''μ''<sub>0</sub>''ε''<sub>0</sub> = 1/''c''<sup>2</sup>}}.

=== Units for magnetism ===

In the Gaussian system, unlike the ISQ, the electric field {{math|'''E'''{{ssup|G}}}} and the [[magnetic field]] {{math|'''B'''{{ssup|G}}}} have the same dimension. This amounts to a factor of [[speed of light|{{mvar|c}}]] between how {{math|'''B'''}} is defined in the two unit systems, on top of the other differences.<ref name=Littlejohn/> (The same factor applies to other magnetic quantities such as the [[magnetic field]], {{math|'''H'''}}, and [[magnetization]], {{math|'''M'''}}.) For example, in a [[Sinusoidal plane-wave solutions of the electromagnetic wave equation|planar light wave in vacuum]], {{math|1={{abs|'''E'''{{ssup|G}}('''r''', ''t'')}} = {{abs|'''B'''{{ssup|G}}('''r''', ''t'')}}}} in Gaussian units, while {{math|1={{abs|'''E'''{{ssup|I}}('''r''', ''t'')}} = ''c''&nbsp;{{abs|'''B'''{{ssup|I}}('''r''', ''t'')}}}} in the ISQ.

=== Polarization, magnetization ===

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: {{math|'''[[electric field|E]]'''{{ssup|G}}}}, {{math|'''[[electric displacement field|D]]'''{{ssup|G}}}}, {{math|'''[[polarization density|P]]'''{{ssup|G}}}}, {{math|[[magnetic field|'''B''']]{{ssup|G}}}}, {{math|[[magnetic field|'''H''']]{{ssup|G}}}}, and {{math|'''[[magnetization|M]]'''{{ssup|G}}}}. A further point is that the [[electric susceptibility|electric]] and [[magnetic susceptibility]] of a material is dimensionless in both the Gaussian system and the ISQ, but a given material will have a different numerical susceptibility in the two systems. (Equation is given below.)