Gaussian units
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Gaussian units constitute a metric system of units of measurement. This system is the most common of the several electromagnetic unit systems based on the centimetre–gram–second system of units (CGS). It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units.Template:Efn The term "cgs units" is ambiguous and therefore to be avoided if possible: there are several variants of CGS, which have 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.<ref name=Rowlett/>Template:Efn Alternative unit systems also exist. Conversions between quantities in the Gaussian and SI systems are Template:Em direct unit conversions, because the quantities themselves are defined differently in each system. This means that the equations that express physical laws of electromagnetism—such as Maxwell's equations—will change depending on the system of quantities that is employed. As an example, quantities that are dimensionless in one system may have dimension in the other.
Alternative unit systemsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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.<ref name=Rowlett>"CGS", in How Many? A Dictionary of Units of Measurement, by Russ Rowlett and the University of North Carolina at Chapel Hill</ref> In technical, scientific literature (such as theoretical physics and astronomy), Gaussian units were predominant until recent decades, but are now getting progressively less so.<ref name=Rowlett/>Template:Efn The 8th SI Brochure mentions the CGS-Gaussian unit system,<ref>Template:SIbrochure8th, p. 128</ref> 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 systemsEdit
"Rationalized" unit systemsEdit
One difference between the Gaussian and SI systems is in the factor Template:Math in various formulas that relate the quantities that they define. With SI electromagnetic units, called rationalized,<ref name=Littlejohn>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name=Kowalski>Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity", Template:Webarchive The Physics Teacher 24(2): 97–99. Alternate web link (subscription required)</ref> Maxwell's equations have no explicit factors of Template:Math in the formulae, whereas the inverse-square force laws – Coulomb's law and the Biot–Savart law – Template:Em have a factor of Template:Math attached to the Template:Math. With Gaussian units, called unrationalized (and unlike Heaviside–Lorentz units), the situation is reversed: two of Maxwell's equations have factors of Template:Math in the formulas, while both of the inverse-square force laws, Coulomb's law and the Biot–Savart law, have no factor of Template:Math attached to Template:Math in the denominator.
(The quantity Template:Math appears because Template:Math is the surface area of the sphere of radius Template:Mvar, 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 chargeEdit
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) Template:Em be written entirely as a dimensional combination of the non-electrical base units (gram, centimetre, second), as:
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 Template:Mvar is the repulsive force between two electrical charges, Template:Math and Template:Math are the two charges in question, and Template:Mvar is the distance separating them. If Template:Math and Template:Math are expressed in statC and Template:Mvar in centimetres, then the unit of Template:Mvar 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 Template:Math is the vacuum permittivity, a quantity that is not dimensionless: it has dimension (charge)2 (time)2 (mass)−1 (length)−3. Without Template:Math, the equation would be dimensionally inconsistent with the quantities as defined in the ISQ, whereas the quantity Template:Math 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, Template:Math converts or scales electric flux density, Template:Math, to the corresponding electric field, Template:Math (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 Template:Mvar appears directly in electromagnetic formulas like Maxwell's equations (see below), whereas in the ISQ it appears via the product Template:Math.
Units for magnetismEdit
In the Gaussian system, unlike the ISQ, the electric field Template:Math and the magnetic field Template:Math have the same dimension. This amounts to a factor of [[speed of light|Template:Mvar]] between how Template:Math 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, Template:Math, and magnetization, Template:Math.) For example, in a planar light wave in vacuum, Template:Math in Gaussian units, while Template:Math in the ISQ.
Polarization, magnetizationEdit
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: Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, and Template:Math. A further point is that the 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.)
List of equationsEdit
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).<ref name=Garg>A. Garg, 2012, "Classical Electrodynamics in a Nutshell" (Princeton University Press).</ref> All formulas except otherwise noted are from Ref.<ref name=Littlejohn/>
Maxwell's equationsEdit
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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.
| Name | Gaussian system | Template:Abbr |
|---|---|---|
| Gauss's lawTemplate:Br(macroscopic) | <math>\nabla \cdot \mathbf{D}^{_\mathrm{G}} = 4\pi\rho_\mathrm{f}^{_\mathrm{G}}</math> | <math>\nabla \cdot \mathbf{D}^{_\mathrm{I}} = \rho_\mathrm{f}^{_\mathrm{I}}</math> |
| Gauss's law (microscopic) |
<math>\nabla \cdot \mathbf{E}^{_\mathrm{G}} = 4\pi\rho^{_\mathrm{G}}</math> | <math>\nabla \cdot \mathbf{E}^{_\mathrm{I}} = \frac{1}{\varepsilon_0} \rho^{_\mathrm{I}}</math> |
| Gauss's law for magnetism | <math>\nabla \cdot \mathbf{B}^{_\mathrm{G}} = 0</math> | <math>\nabla \cdot \mathbf{B}^{_\mathrm{I}} = 0</math> |
| Maxwell–Faraday equation (Faraday's law of induction) |
<math>\nabla \times \mathbf{E}^{_\mathrm{G}} + \frac{1}{c}\frac{\partial \mathbf{B}^{_\mathrm{G}}} {\partial t} = 0</math> | <math>\nabla \times \mathbf{E}^{_\mathrm{I}} + \frac{\partial \mathbf{B}^{_\mathrm{I}}} {\partial t} = 0</math> |
| Ampère–Maxwell equation (macroscopic) |
<math>\nabla \times \mathbf{H}^{_\mathrm{G}} - \frac{1}{c} \frac{\partial \mathbf{D}^{_\mathrm{G}}} {\partial t} = \frac{4\pi}{c}\mathbf{J}_\mathrm{f}^{_\mathrm{G}}</math> | <math>\nabla \times \mathbf{H}^{_\mathrm{I}} - \frac{\partial \mathbf{D}^{_\mathrm{I}}} {\partial t}= \mathbf{J}_\mathrm{f}^{_\mathrm{I}}</math> |
| Ampère–Maxwell equation (microscopic) |
<math>\nabla \times \mathbf{B}^{_\mathrm{G}} - \frac{1}{c}\frac{\partial \mathbf{E}^{_\mathrm{G}}} {\partial t} = \frac{4\pi}{c}\mathbf{J}^{_\mathrm{G}}</math> | <math>\nabla \times \mathbf{B}^{_\mathrm{I}} - \frac{1}{c^2}\frac{\partial \mathbf{E}^{_\mathrm{I}}} {\partial t} = \mu_0\mathbf{J}^{_\mathrm{I}}</math> |
Other basic lawsEdit
| Name | Gaussian system | Template:Abbr |
|---|---|---|
| Lorentz force | <math>\mathbf{F} = q^{_\mathrm{G}}\,\left(\mathbf{E}^{_\mathrm{G}}+\tfrac{1}{c}\,\mathbf{v}\times\mathbf{B}^{_\mathrm{G}}\right)</math> | <math>\mathbf{F} = q^{_\mathrm{I}}\,\left(\mathbf{E}^{_\mathrm{I}}+\mathbf{v}\times\mathbf{B}^{_\mathrm{I}}\right)</math> |
| Coulomb's law | <math>\mathbf{F} = \frac{q^{_\mathrm{G}}_1 q^{_\mathrm{G}}_2}{r^2}\,\mathbf{\hat r}</math> | <math>\mathbf{F} = \frac{1}{4\pi\varepsilon_0}\,\frac{q^{_\mathrm{I}}_1 q^{_\mathrm{I}}_2}{r^2}\, \mathbf{\hat r}</math> |
| Electric field of stationary point charge |
<math>\mathbf{E}^{_\mathrm{G}} = \frac{q^{_\mathrm{G}}}{r^2}\,\mathbf{\hat r}</math> | <math>\mathbf{E}^{_\mathrm{I}} = \frac{1}{4\pi\varepsilon_0}\,\frac{q^{_\mathrm{I}}}{r^2}\,\mathbf{\hat r}</math> |
| Biot–Savart law<ref>Introduction to Electrodynamics by Capri and Panat, p180</ref> | <math> \mathbf{B}^{_\mathrm{G}} = \frac{1}{c}\!\oint\frac{I^{_\mathrm{G}} \times \mathbf{\hat r}}{r^2}\,\operatorname{d}\!\mathbf{\boldsymbol{\ell}}</math> | <math> \mathbf{B}^{_\mathrm{I}} = \frac{\mu_0}{4\pi}\!\oint\frac{I^{_\mathrm{I}} \times \mathbf{\hat r}}{r^2}\,\operatorname{d}\!\mathbf{\boldsymbol{\ell}}</math> |
| Poynting vector (microscopic) |
<math>\mathbf{S} = \frac{c}{4\pi}\,\mathbf{E}^{_\mathrm{G}} \times \mathbf{B}^{_\mathrm{G}}</math> | <math>\mathbf{S} = \frac{1}{\mu_0}\,\mathbf{E}^{_\mathrm{I}} \times \mathbf{B}^{_\mathrm{I}}</math> |
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 system | Template:Abbr |
|---|---|
| <math>\mathbf{D}^{_\mathrm{G}} = \mathbf{E}^{_\mathrm{G}}+4\pi\mathbf{P}^{_\mathrm{G}}</math> | <math>\mathbf{D}^{_\mathrm{I}} = \varepsilon_0 \mathbf{E}^{_\mathrm{I}}+\mathbf{P}^{_\mathrm{I}}</math> |
| <math>\mathbf{P}^{_\mathrm{G}} = \chi^{_\mathrm{G}}_\mathrm{e}\mathbf{E}^{_\mathrm{G}}</math> | <math>\mathbf{P}^{_\mathrm{I}} = \chi^{_\mathrm{I}}_\mathrm{e}\varepsilon_0\mathbf{E}^{_\mathrm{I}}</math> |
| <math>\mathbf{D}^{_\mathrm{G}} = \varepsilon^{_\mathrm{G}}\mathbf{E}^{_\mathrm{G}}</math> | <math>\mathbf{D}^{_\mathrm{I}} = \varepsilon^{_\mathrm{I}}\mathbf{E}^{_\mathrm{I}}</math> |
| <math>\varepsilon^{_\mathrm{G}} = 1+4\pi\chi^{_\mathrm{G}}_\mathrm{e}</math> | <math>\varepsilon^{_\mathrm{I}}/\varepsilon_0 = 1+\chi^{_\mathrm{I}}_\mathrm{e}</math> |
where
- Template:Math and Template:Math are the electric field and displacement field, respectively;
- Template:Math is the polarization density;
- <math>\varepsilon</math> is the permittivity;
- <math>\varepsilon_0</math> is the permittivity of vacuum (used in the SI system, but meaningless in Gaussian units); and
- <math>\chi_\mathrm{e}</math> is the electric susceptibility.
The quantities <math>\varepsilon^{_\mathrm{G}}</math> and <math>\varepsilon^{_\mathrm{I}}/\varepsilon_0</math> are both dimensionless, and they have the same numeric value. By contrast, the electric susceptibility <math>\chi_\mathrm{e}^{_\mathrm{G}}</math> and <math>\chi_\mathrm{e}^{_\mathrm{I}}</math> are both unitless, but have Template:Em for the same material: <math display="block">4\pi \chi_\mathrm{e}^{_\mathrm{G}} = \chi_\mathrm{e}^{_\mathrm{I}}\,.</math>
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 system | Template:Abbr |
|---|---|
| <math>\mathbf{B}^{_\mathrm{G}} = \mathbf{H}^{_\mathrm{G}}+4\pi\mathbf{M}^{_\mathrm{G}}</math> | <math>\mathbf{B}^{_\mathrm{I}} = \mu_0 (\mathbf{H}^{_\mathrm{I}}+\mathbf{M}^{_\mathrm{I}})</math> |
| <math>\mathbf{M}^{_\mathrm{G}} = \chi^{_\mathrm{G}}_\mathrm{m}\mathbf{H}^{_\mathrm{G}}</math> | <math>\mathbf{M}^{_\mathrm{I}} = \chi^{_\mathrm{I}}_\mathrm{m}\mathbf{H}^{_\mathrm{I}}</math> |
| <math>\mathbf{B}^{_\mathrm{G}} = \mu^{_\mathrm{G}}\mathbf{H}^{_\mathrm{G}}</math> | <math>\mathbf{B}^{_\mathrm{I}} = \mu^{_\mathrm{I}}\mathbf{H}^{_\mathrm{I}}</math> |
| <math>\mu^{_\mathrm{G}} = 1+4\pi\chi^{_\mathrm{G}}_\mathrm{m}</math> | <math>\mu^{_\mathrm{I}}/\mu_0 = 1+\chi^{_\mathrm{I}}_\mathrm{m}</math> |
where
- Template:Math and Template:Math are the magnetic fields;
- Template:Math is magnetization;
- <math>\mu</math> is magnetic permeability;
- <math>\mu_0</math> is the permeability of vacuum (used in the SI system, but meaningless in Gaussian units); and
- <math>\chi_\mathrm{m}</math> is the magnetic susceptibility.
The quantities <math>\mu^{_\mathrm{G}}</math> and <math>\mu^{_\mathrm{I}}/\mu_0</math> are both dimensionless, and they have the same numeric value. By contrast, the magnetic susceptibility <math>\chi_\mathrm{m}^{_\mathrm{G}}</math> and <math>\chi_\mathrm{m}^{_\mathrm{I}}</math> are both unitless, but has Template:Em in the two systems for the same material: <math display="block">4\pi \chi_\mathrm{m}^{_\mathrm{G}} = \chi_\mathrm{m}^{_\mathrm{I}}</math>
Vector and scalar potentialsEdit
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The electric and magnetic fields can be written in terms of a vector potential Template:Math and a scalar potential Template:Mvar:
| Name | Gaussian system | Template:Abbr |
|---|---|---|
| Electric field | <math>\mathbf{E}^{_\mathrm{G}} = -\nabla\phi^{_\mathrm{G}}-\frac{1}{c}\frac{\partial \mathbf{A}^{_\mathrm{G}}}{\partial t}</math> | <math>\mathbf{E}^{_\mathrm{I}} = -\nabla\phi^{_\mathrm{I}}-\frac{\partial \mathbf{A}^{_\mathrm{I}}}{\partial t}</math> |
| Magnetic B field | <math>\mathbf{B}^{_\mathrm{G}} = \nabla \times \mathbf{A}^{_\mathrm{G}}</math> | <math>\mathbf{B}^{_\mathrm{I}} = \nabla \times \mathbf{A}^{_\mathrm{I}}</math> |
Electrical circuitEdit
| Name | Gaussian system | Template:Abbr |
|---|---|---|
| Charge conservation | <math>I^{_\mathrm{G}} = \frac{\mathrm{d}Q^{_\mathrm{G}}}{\mathrm{d}t}</math> | <math>I^{_\mathrm{I}} = \frac{\mathrm{d}Q^{_\mathrm{I}}}{\mathrm{d}t}</math> |
| Lenz's law | <math>V^{_\mathrm{G}} = \frac{1}{c}\frac{\mathrm{d}\mathrm{\Phi}^{_\mathrm{G}}}{\mathrm{d}t}</math> | <math>V^{_\mathrm{I}} = -\frac{\mathrm{d}\mathrm{\Phi}^{_\mathrm{I}}}{\mathrm{d}t}</math> |
| Ohm's law | <math>V^{_\mathrm{G}} = R^{_\mathrm{G}} I^{_\mathrm{G}}</math> | <math>V^{_\mathrm{I}} = R^{_\mathrm{I}} I^{_\mathrm{I}}</math> |
| Capacitance | <math>Q^{_\mathrm{G}} = C^{_\mathrm{G}} V^{_\mathrm{G}}</math> | <math>Q^{_\mathrm{I}} = C^{_\mathrm{I}} V^{_\mathrm{I}}</math> |
| Inductance | <math>\mathrm{\Phi}^{_\mathrm{G}} = cL^{_\mathrm{G}} I^{_\mathrm{G}}</math> | <math>\mathrm{\Phi}^{_\mathrm{I}} = L^{_\mathrm{I}} I^{_\mathrm{I}}</math> |
where
- Template:Mvar is the electric charge
- Template:Mvar is the electric current
- Template:Mvar is the electric potential
- Template:Math is the magnetic flux
- Template:Mvar is the electrical resistance
- Template:Mvar is the capacitance
- Template:Mvar is the inductance
Fundamental constantsEdit
| Name | Gaussian system | Template:Abbr |
|---|---|---|
| Impedance of free space | <math>Z_0^{_\mathrm{G}} = \frac{4\pi}{c}</math> | <math>Z_0^{_\mathrm{I}} = \sqrt{\frac{\mu_0}{\varepsilon_0}}</math> |
| Electric constant | <math>1 = \frac{4\pi}{Z_0^{_\mathrm{G}}c}</math> | <math>\varepsilon_0 = \frac{1}{Z_0^{_\mathrm{I}}c}</math> |
| Magnetic constant | <math>1 = \frac{Z_0^{_\mathrm{G}}c}{4\pi}</math> | <math>\mu_0 = \frac{Z_0^{_\mathrm{I}}}{c}</math> |
| Fine-structure constant | <math>\alpha = \frac{(e^{_\mathrm{G}})^2}{\hbar c}</math> | <math>\alpha = \frac{1}{4\pi\varepsilon_0} \frac{(e^{_\mathrm{I}})^2}{\hbar c}</math> |
| Magnetic flux quantum | <math>\phi_0^{_\mathrm{G}} = \frac{hc}{2e^{_\mathrm{G}}}</math> | <math>\phi_0^{_\mathrm{I}} = \frac{h}{2e^{_\mathrm{I}}}</math> |
| Conductance quantum | <math>G_0^{_\mathrm{G}} = \frac{2(e^{_\mathrm{G}})^2}{h}</math> | <math>G_0^{_\mathrm{I}} = \frac{2(e^{_\mathrm{I}})^2}{h}</math> |
| Bohr radius | <math>a_\mathrm{B} =\frac{\hbar^2}{m_\mathrm{e}(e^{_\mathrm{G}})^2}</math> | <math>a_\mathrm{B} =\frac{4\pi\varepsilon_0\hbar^2}{m_\mathrm{e}(e^{_\mathrm{I}})^2}</math> |
| Bohr magneton | <math>\mu_\mathrm{B}^{_\mathrm{G}} =\frac{e^{_\mathrm{G}}\hbar}{2m_\mathrm{e}c}</math> | <math>\mu_\mathrm{B}^{_\mathrm{I}} =\frac{e^{_\mathrm{I}}\hbar}{2m_\mathrm{e}}</math> |
Electromagnetic unit namesEdit
| Quantity | Symbol | SI unit | Gaussian unitTemplate:Br(in base units) | Conversion factor |
|---|---|---|---|---|
| Electric charge | Template:Mvar | C | FrTemplate:Br(cm3/2⋅g1/2⋅s−1) | <math>\frac{q^{_\mathrm{G}}}{q^{_\mathrm{I}}} = \frac{1}{\sqrt{4\pi\varepsilon_0}} \approx \frac{2.998 \times 10^9 \, \mathrm{Fr}}{1\, \mathrm{C}}</math> |
| Electric current | Template:Mvar | A | statATemplate:Br(cm3/2⋅g1/2⋅s−2) | <math>\frac{I^{_\mathrm{G}}}{I^{_\mathrm{I}}} = \frac{1}{\sqrt{4\pi\varepsilon_0}} \approx \frac{2.998 \times 10^9 \, \mathrm{statA}}{1\, \mathrm{A}}</math> |
| Electric potential,Template:BrVoltage | Template:MvarTemplate:BrTemplate:Mvar | V | statVTemplate:Br(cm1/2⋅g1/2⋅s−1) | <math>\frac{V^{_\mathrm{G}}}{V^{_\mathrm{I}}} = \sqrt{4\pi\varepsilon_0} \approx \frac{1\, \mathrm{statV}}{2.998 \times 10^2 \, \mathrm{V}}</math> |
| Electric field | Template:Math | V/m | statV/cmTemplate:Br(cm−1/2⋅g1/2⋅s−1) | <math>\frac{\mathbf{E}^{_\mathrm{G}}}{\mathbf{E}^{_\mathrm{I}}} = \sqrt{4\pi\varepsilon_0} \approx \frac{1 \, \mathrm{statV/cm}}{2.998 \times 10^4 \, \mathrm{V/m}}</math> |
| Electric displacement field | Template:Math | C/m2 | Fr/cm2Template:Br(cm−1/2g1/2s−1) | <math>\frac{\mathbf{D}^{_\mathrm{G}}}{\mathbf{D}^{_\mathrm{I}}} = \sqrt{\frac{4\pi}{\varepsilon_0}} \approx \frac{4\pi\times 2.998 \times 10^5 \, \mathrm{Fr/cm}^2}{ 1 \, \mathrm{C/m}^2}</math> |
| Electric dipole moment | Template:Math | C⋅m | Fr⋅cmTemplate:Br(cm5/2⋅g1/2⋅s−1) | <math>\frac{\mathbf{p}^{_\mathrm{G}}}{\mathbf{p}^{_\mathrm{I}}} = \frac{1}{\sqrt{4\pi\varepsilon_0}} \approx \frac{2.998 \times 10^{11} \, \mathrm{Fr}{\cdot}\mathrm{cm}}{1 \, \mathrm{C}{\cdot}\mathrm{m}}</math> |
| Electric fluxTemplate:Efn | Template:Math | C | FrTemplate:Br(cm3/2⋅g1/2⋅s−1) | <math>\frac{\Phi^{_\mathrm{G}}_{\mathrm{e}}}{\Phi^{_\mathrm{I}}_{\mathrm{e}}} = \sqrt{\frac{4\pi}{\varepsilon_0}} \approx \frac{4\pi\times 2.998 \times 10^9 \, \mathrm{Fr}}{1 \, \mathrm{C}}</math> |
| Permittivity | Template:Mvar | F/m | cm/cm | <math>\frac{\varepsilon^{_\mathrm{G}}}{\varepsilon^{_\mathrm{I}}} = \frac{1}{\varepsilon_0} \approx \frac{4\pi \times 2.998^2 \times 10^{9} \, \mathrm{cm/cm}}{1 \, \mathrm{F/m}}</math> |
| Magnetic B field | Template:Math | T | GTemplate:Br(cm−1/2⋅g1/2⋅s−1) | <math>\frac{\mathbf{B}^{_\mathrm{G}}}{\mathbf{B}^{_\mathrm{I}}} = \sqrt{\frac{4\pi}{\mu_0}} \approx \frac{10^4 \, \mathrm{G}}{1 \, \mathrm{T}}</math> |
| Magnetic H field | Template:Math | A/m | OeTemplate:Br(cm−1/2⋅g1/2⋅s−1) | <math>\frac{\mathbf{H}^{_\mathrm{G}}}{\mathbf{H}^{_\mathrm{I}}} = \sqrt{4\pi\mu_0} \approx \frac{4\pi \times 10^{-3} \, \mathrm{Oe}}{1 \, \mathrm{A/m}}</math> |
| Magnetic dipole moment | Template:Math | A⋅m2 | erg/GTemplate:Br(cm5/2⋅g1/2⋅s−1) | <math>\frac{\mathbf{m}^{_\mathrm{G}}}{\mathbf{m}^{_\mathrm{I}}} = \sqrt{\frac{\mu_0}{4\pi}} \approx \frac{10^3 \, \mathrm{erg/G}}{1 \, \mathrm{A}{\cdot}\mathrm{m}^2}</math> |
| Magnetic flux | Template:Math | Wb | MxTemplate:Br(cm3/2⋅g1/2⋅s−1) | <math>\frac{\Phi^{_\mathrm{G}}_{\mathrm{m}}}{\Phi^{_\mathrm{I}}_{\mathrm{m}}} = \sqrt{\frac{4\pi}{\mu_0}} \approx \frac{10^8 \, \mathrm{Mx}}{1 \, \mathrm{Wb}}</math> |
| Permeability | Template:Mvar | H/m | cm/cm | <math>\frac{\mu^{_\mathrm{G}}}{\mu^{_\mathrm{I}}} = \frac{1}{\mu_0} \approx \frac{1 \, \mathrm{cm/cm}}{4\pi \times 10^{-7} \, \mathrm{H/m}}</math> |
| Magnetomotive force | <math>\mathcal F</math> | A | GiTemplate:Br(cm1/2⋅g1/2⋅s−1) | <math>\frac{\mathcal F^{_\mathrm{G}}}{\mathcal F^{_\mathrm{I}}} = \sqrt{4\pi\mu_0} \approx \frac{4\pi \times 10^{-1} \, \mathrm{Gi}}{1 \, \mathrm{A}}</math> |
| Magnetic reluctance | <math>\mathcal R</math> | H−1 | Gi/MxTemplate:Br(cm−1) | <math>\frac{\mathcal R^{_\mathrm{G}}}{\mathcal R^{_\mathrm{I}}} = \mu_0 \approx \frac{4\pi \times 10^{-9} \, \mathrm{Gi/Mx}}{1 \, \mathrm{H}^{-1}}</math> |
| Resistance | Template:Mvar | Ω | s/cm | <math>\frac{R^{_\mathrm{G}}}{R^{_\mathrm{I}}} = 4\pi\varepsilon_0 \approx \frac{1 \, \mathrm{s/cm}}{2.998^2 \times 10^{11} \, \Omega}</math> |
| Resistivity | Template:Mvar | Ω⋅m | s | <math>\frac{\rho^{_\mathrm{G}}}{\rho^{_\mathrm{I}}} = 4\pi\varepsilon_0 \approx \frac{1 \, \mathrm{s}}{2.998^2 \times 10^{9} \, \Omega{\cdot}\mathrm{m}}</math> |
| Capacitance | Template:Mvar | F | cm | <math>\frac{C^{_\mathrm{G}}}{C^{_\mathrm{I}}} = \frac{1}{4\pi\varepsilon_0} \approx \frac{2.998^2 \times 10^{11} \, \mathrm{cm}}{1 \, \mathrm{F}}</math> |
| Inductance | Template:Mvar | H | s2/cm | <math>\frac{L^{_\mathrm{G}}}{L^{_\mathrm{I}}} = 4\pi\varepsilon_0 \approx \frac{1 \, \mathrm{s}^2/\mathrm{cm}}{2.998^2 \times 10^{11} \, \mathrm{H}}</math> |
Note: The SI quantities <math>\varepsilon_0</math> and <math>\mu_0</math> satisfy Template:Tmath.
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 Template:Nowrap \approx {2.998 \times 10^9 \,\mathrm{Fr}} \,/\, {1\,\mathrm{C}}</math>,}} a relation which can be verified with dimensional analysis, by expanding <math>\varepsilon_0</math> 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 Template:Mvar seconds, the half-life of the discharge is Template:Math 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 newton-metre and joule.) The different names help avoid ambiguities and misunderstandings as to what physical quantity is being measured. In particular, Template:Em of the following quantities are dimensionally equivalent in Gaussian units, but they are nevertheless given different unit names as follows:<ref>Template:Cite book</ref>
| Quantity | Gaussian symbol | In Gaussian base units |
Gaussian unit of measure |
|---|---|---|---|
| Electric field | Template:Math | cm−1/2⋅g1/2⋅s−1 | statV/cm |
| Electric displacement field | Template:Math | cm−1/2⋅g1/2⋅s−1 | statC/cm2 |
| Polarization density | Template:Math | cm−1/2⋅g1/2⋅s−1 | statC/cm2 |
| Magnetic flux density | Template:Math | cm−1/2⋅g1/2⋅s−1 | G |
| Magnetizing field | Template:Math | cm−1/2⋅g1/2⋅s−1 | Oe |
| Magnetization | Template:Math | 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 ISQ formula <math display="block">\mathbf{E}^{_\mathrm{I}} = \frac{q^{_\mathrm{I}}}{4\pi \varepsilon_0 r^2} \hat{\mathbf{r}} ,</math> where Template:Mvar is distance, and the "Template:Smaller" 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: <math display="block">\begin{align} \frac{\mathbf{E}^{_\mathrm{G}}}{\mathbf{E}^{_\mathrm{I}}} &= \sqrt{4\pi\varepsilon_0}\,, \\ \frac{q^{_\mathrm{G}}}{q^{_\mathrm{I}}} &= \frac{1}{\sqrt{4\pi\varepsilon_0}}\,. \end{align}</math>
Therefore, after substituting and simplifying, we get the Gaussian-system formula: <math display="block">\mathbf{E}^{_\mathrm{G}} = \frac{q^{_\mathrm{G}}}{r^2}\hat{\mathbf{r}}\,,</math> 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 the 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). Replace <math>1/c^2</math> by <math>\varepsilon_0 \mu_0</math> (or vice versa). 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.<ref>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="Jackson">Template:Cite book</ref>Template:Efn
| Name | Gaussian system | Template:Abbr |
|---|---|---|
| electric field, electric potential, electromotive force | <math> \left(\mathbf{E}^{_\mathrm{G}}, \varphi^{_\mathrm{G}}, \mathcal E^{_\mathrm{G}}\right) </math> | <math> \sqrt{4\pi\varepsilon_0}\left(\mathbf{E}^{_\mathrm{I}}, \varphi^{_\mathrm{I}}, \mathcal E^{_\mathrm{I}}\right) </math> |
| electric displacement field | <math> \mathbf{D}^{_\mathrm{G}} </math> | <math> \sqrt{\frac{4\pi}{\varepsilon_0}}\mathbf{D}^{_\mathrm{I}} </math> |
| charge, charge density, current, current density, polarization density, electric dipole moment |
<math> \left(q^{_\mathrm{G}}, \rho^{_\mathrm{G}}, I^{_\mathrm{G}}, \mathbf{J}^{_\mathrm{G}},\mathbf{P}^{_\mathrm{G}}, \mathbf{p}^{_\mathrm{G}}\right) </math> | <math> \frac{1}{\sqrt{4\pi\varepsilon_0}}\left(q^{_\mathrm{I}}, \rho^{_\mathrm{I}}, I^{_\mathrm{I}}, \mathbf{J}^{_\mathrm{I}},\mathbf{P}^{_\mathrm{I}},\mathbf{p}^{_\mathrm{I}}\right) </math> |
| [[Magnetic field|magnetic Template:Math field]], magnetic flux, magnetic vector potential |
<math> \left(\mathbf{B}^{_\mathrm{G}}, \Phi_\mathrm{m}^{_\mathrm{G}},\mathbf{A}^{_\mathrm{G}}\right) </math> | <math> \sqrt{\frac{4\pi}{\mu_0}}\left(\mathbf{B}^{_\mathrm{I}}, \Phi_\mathrm{m}^{_\mathrm{I}},\mathbf{A}^{_\mathrm{I}}\right) </math> |
| [[Magnetic field|magnetic Template:Math field]], magnetic scalar potential, magnetomotive force | <math> \left(\mathbf{H}^{_\mathrm{G}}, \psi^{_\mathrm{G}}, \mathcal F^{_\mathrm{G}}\right) </math> | <math> \sqrt{4\pi\mu_0}\left(\mathbf{H}^{_\mathrm{I}}, \psi^{_\mathrm{I}}, \mathcal F^{_\mathrm{I}}\right) </math> |
| magnetic moment, magnetization, magnetic pole strength | <math> \left(\mathbf{m}^{_\mathrm{G}}, \mathbf{M}^{_\mathrm{G}}, p^{_\mathrm{G}}\right) </math> | <math> \sqrt{\frac{\mu_0}{4\pi}}\left(\mathbf{m}^{_\mathrm{I}}, \mathbf{M}^{_\mathrm{I}}, p^{_\mathrm{I}}\right) </math> |
| permittivity, permeability |
<math> \left(\varepsilon^{_\mathrm{G}}, \mu^{_\mathrm{G}}\right) </math> | <math> \left(\frac{\varepsilon^{_\mathrm{I}}}{\varepsilon_0}, \frac{\mu^{_\mathrm{I}}}{\mu_0}\right) </math> |
| electric susceptibility, magnetic susceptibility |
<math> \left(\chi_\mathrm{e}^{_\mathrm{G}}, \chi_\mathrm{m}^{_\mathrm{G}}\right) </math> | <math> \frac{1}{4\pi}\left(\chi_\mathrm{e}^{_\mathrm{I}}, \chi_\mathrm{m}^{_\mathrm{I}}\right) </math> |
| conductivity, conductance, capacitance | <math> \left(\sigma^{_\mathrm{G}}, S^{_\mathrm{G}}, C^{_\mathrm{G}}\right) </math> | <math> \frac{1}{4\pi\varepsilon_0}\left(\sigma^{_\mathrm{I}},S^{_\mathrm{I}},C^{_\mathrm{I}}\right) </math> |
| resistivity, resistance, inductance, memristance, impedance | <math> \left(\rho^{_\mathrm{G}},R^{_\mathrm{G}},L^{_\mathrm{G}},M^{_\mathrm{G}},Z^{_\mathrm{G}}\right) </math> | <math> 4\pi\varepsilon_0\left(\rho^{_\mathrm{I}},R^{_\mathrm{I}},L^{_\mathrm{I}},M^{_\mathrm{I}},Z^{_\mathrm{I}}\right) </math> |
| magnetic reluctance | <math> \mathcal{R}^{_\mathrm{G}}</math> | <math> \mu_0\mathcal{R}^{_\mathrm{I}} </math> |
| Name | Template:Abbr | Gaussian system |
|---|---|---|
| electric field, electric potential, electromotive force | <math> \left(\mathbf{E}^{_\mathrm{I}}, \varphi^{_\mathrm{I}}, \mathcal E^{_\mathrm{I}}\right) </math> | <math> \frac{1}{\sqrt{4\pi\varepsilon_0}}\left(\mathbf{E}^{_\mathrm{G}}, \varphi^{_\mathrm{G}}, \mathcal E^{_\mathrm{G}}\right) </math> |
| electric displacement field | <math> \mathbf{D}^{_\mathrm{I}} </math> | <math> \sqrt{\frac{\varepsilon_0}{4\pi}}\mathbf{D}^{_\mathrm{G}} </math> |
| charge, charge density, current, current density, polarization density, electric dipole moment |
<math> \left(q^{_\mathrm{I}}, \rho^{_\mathrm{I}}, I^{_\mathrm{I}}, \mathbf{J}^{_\mathrm{I}},\mathbf{P}^{_\mathrm{I}}, \mathbf{p}^{_\mathrm{I}}\right) </math> | <math> \sqrt{4\pi\varepsilon_0}\left(q^{_\mathrm{G}}, \rho^{_\mathrm{G}}, I^{_\mathrm{G}}, \mathbf{J}^{_\mathrm{G}},\mathbf{P}^{_\mathrm{G}},\mathbf{p}^{_\mathrm{G}}\right) </math> |
| [[Magnetic field|magnetic Template:Math field]], magnetic flux, magnetic vector potential |
<math> \left(\mathbf{B}^{_\mathrm{I}}, \Phi_\mathrm{m}^{_\mathrm{I}},\mathbf{A}^{_\mathrm{I}}\right) </math> | <math> \sqrt{\frac{\mu_0}{4\pi}}\left(\mathbf{B}^{_\mathrm{G}}, \Phi_\mathrm{m}^{_\mathrm{G}},\mathbf{A}^{_\mathrm{G}}\right) </math> |
| [[Magnetic field|magnetic Template:Math field]], magnetic scalar potential, magnetomotive force | <math> \left(\mathbf{H}^{_\mathrm{I}}, \psi^{_\mathrm{I}}, \mathcal F^{_\mathrm{I}}\right) </math> | <math> \frac{1}{\sqrt{4\pi\mu_0}}\left(\mathbf{H}^{_\mathrm{G}}, \psi^{_\mathrm{G}}, \mathcal F^{_\mathrm{G}}\right) </math> |
| magnetic moment, magnetization, magnetic pole strength | <math> \left(\mathbf{m}^{_\mathrm{I}}, \mathbf{M}^{_\mathrm{I}}, p^{_\mathrm{I}}\right) </math> | <math> \sqrt{\frac{4\pi}{\mu_0}}\left(\mathbf{m}^{_\mathrm{G}}, \mathbf{M}^{_\mathrm{G}}, p^{_\mathrm{G}}\right) </math> |
| permittivity, permeability |
<math> \left(\varepsilon^{_\mathrm{I}}, \mu^{_\mathrm{I}}\right) </math> | <math> \left(\varepsilon_0\varepsilon^{_\mathrm{G}}, \mu_0\mu^{_\mathrm{G}}\right) </math> |
| electric susceptibility, magnetic susceptibility |
<math> \left(\chi_\mathrm{e}^{_\mathrm{I}}, \chi_\mathrm{m}^{_\mathrm{I}}\right) </math> | <math> 4\pi \left(\chi_\mathrm{e}^{_\mathrm{G}}, \chi_\mathrm{m}^{_\mathrm{G}}\right) </math> |
| conductivity, conductance, capacitance | <math> \left(\sigma^{_\mathrm{I}}, S^{_\mathrm{I}}, C^{_\mathrm{I}}\right) </math> | <math> 4\pi\varepsilon_0\left(\sigma^{_\mathrm{G}},S^{_\mathrm{G}},C^{_\mathrm{G}}\right) </math> |
| resistivity, resistance, inductance, memristance, impedance | <math> \left(\rho^{_\mathrm{I}},R^{_\mathrm{I}},L^{_\mathrm{I}},M^{_\mathrm{I}},Z^{_\mathrm{I}}\right) </math> | <math> \frac{1}{4\pi\varepsilon_0}\left(\rho^{_\mathrm{G}},R^{_\mathrm{G}},L^{_\mathrm{G}},M^{_\mathrm{G}},Z^{_\mathrm{G}}\right) </math> |
| magnetic reluctance | <math> \mathcal{R}^{_\mathrm{I}}</math> | <math> \frac{1}{\mu_0}\mathcal{R}^{_\mathrm{G}} </math> |
After the rules of the table have been applied and the resulting formula has been simplified, replace all combinations <math>\varepsilon_0 \mu_0</math> by <math>1/c^2</math>.