Editing Gaussian units (section)

== List of equations ==

This section has a list of the basic formulae of electromagnetism, given in both the Gaussian system and the [[International System of Quantities|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 equations ===
{{main|Maxwell's equations}}

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 [[Stokes' theorem|Kelvin–Stokes theorem]].

{| class="wikitable plainrowheaders"
|+ Maxwell's equations in Gaussian system and ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[Gauss's law]]{{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>
|-
! scope="row" | [[Gauss's law]]<br />(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>
|-
! scope="row" | [[Gauss's law for magnetism]]
|<math>\nabla \cdot \mathbf{B}^{_\mathrm{G}} = 0</math>
|<math>\nabla \cdot \mathbf{B}^{_\mathrm{I}} = 0</math>
|-
! scope="row" | Maxwell–Faraday equation<br />([[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>
|-
! scope="row" | [[Ampère–Maxwell equation]]<br /> (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>
|-
! scope="row" | [[Ampère–Maxwell equation]]<br /> (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 laws ===

{| class="wikitable plainrowheaders"
|+ Other electromagnetic laws in Gaussian system and ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[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>
|-
! scope="row" | [[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> <br />
|-
! scope="row" | Electric field of<br />[[Coulomb's law|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>
|-
! scope="row" | [[Biot–Savart law]]<ref>[https://books.google.com/books?id=RvZFuqw6mXEC&pg=PA180 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>
|-
! scope="row" | [[Poynting vector]]<br />(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 materials ===

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.

{| class="wikitable"
|+ Expressions for fields in dielectric media
|-
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
| <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
* {{math|'''E'''}} and {{math|'''D'''}} are the [[electric field]] and [[Electric displacement field|displacement field]], respectively;
* {{math|'''P'''}} 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 {{em|different numeric values}} 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 (electromagnetism)|permeability]] is a simple constant.

{| class="wikitable"
|+ Expressions for fields in magnetic media
|-
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
| <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
* {{math|'''B'''}} and {{math|'''H'''}} are the [[magnetic field]]s;
* {{math|'''M'''}} 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 {{em|different numeric values}} 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 potentials ===
{{main|Magnetic vector potential|Electric potential}}

The electric and magnetic fields can be written in terms of a vector potential {{math|'''A'''}} and a scalar potential {{mvar|ϕ}}:

{| class="wikitable plainrowheaders"
|+ Electromagnetic fields in Gaussian system and ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[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>
|-
! scope="row" | [[Magnetic field|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 circuit ===

{| class="wikitable plainrowheaders"
|+ Electrical circuit values in Gaussian system and ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[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>
|-
! scope="row" | [[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>
|-
! scope="row" | [[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>
|-
! scope="row" | [[Capacitance]]
| <math>Q^{_\mathrm{G}} = C^{_\mathrm{G}} V^{_\mathrm{G}}</math>
| <math>Q^{_\mathrm{I}} = C^{_\mathrm{I}} V^{_\mathrm{I}}</math>
|-
! scope="row" | [[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
* {{mvar|Q}} is the [[electric charge]]
* {{mvar|I}} is the [[electric current]]
* {{mvar|V}} is the [[electric potential]]
* {{math|&Phi;}} is the [[magnetic flux]]
* {{mvar|R}} is the [[electrical resistance]]
* {{mvar|C}} is the [[capacitance]]
* {{mvar|L}} is the [[inductance]]

=== Fundamental constants ===

{| class="wikitable plainrowheaders"
|+ Fundamental constants in Gaussian system and ISQ
|-
! scope="col" | Name
! scope="col" | Gaussian system
! scope="col" | {{abbr|ISQ|International System of Quantities}}
|-
! scope="row" | [[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>
|-
! scope="row" | [[Electric constant]]
| <math>1 = \frac{4\pi}{Z_0^{_\mathrm{G}}c}</math>
| <math>\varepsilon_0 = \frac{1}{Z_0^{_\mathrm{I}}c}</math>
|-
! scope="row" | [[Magnetic constant]]
| <math>1 = \frac{Z_0^{_\mathrm{G}}c}{4\pi}</math>
| <math>\mu_0 = \frac{Z_0^{_\mathrm{I}}}{c}</math>
|-
! scope="row" | [[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>
|-
! scope="row" | [[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>
|-
! scope="row" | [[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>
|-
! scope="row" | [[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>
|-
! scope="row" | [[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>
|}