Editing Maxwell's equations (section)

=== Formulation in the Gaussian system ===
{{main|Gaussian units}}

The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing [[dimensional analysis|dimensioned]] factors of {{math|''ε''<sub>0</sub>}} and {{math|''μ''<sub>0</sub>}} into the units (and thus redefining these). With a corresponding change in the values of the quantities for the [[Lorentz force]] law this yields the same physics, i.e. trajectories of charged particles, or [[work (physics)|work]] done by an electric motor. These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the [[electromagnetic tensor]]: the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension.<ref name=Jackson>{{cite book|author=J. D. Jackson|title=Classical Electrodynamics|edition=3rd|isbn=978-0-471-43132-9|date=1975-10-17|publisher=Wiley |url=https://archive.org/details/classicalelectro00jack_0}}</ref>{{rp|vii}}  Such modified definitions are conventionally used with the Gaussian ([[Centimetre gram second system of units#Alternate derivations of CGS units in electromagnetism|CGS]]) units. Using these definitions, colloquially "in Gaussian units",<ref name=Littlejohn>
{{cite web
 | url=http://bohr.physics.berkeley.edu/classes/221/0708/notes/emunits.pdf
 | title=Gaussian, SI and Other Systems of Units in Electromagnetic Theory
 | work=Physics 221A, University of California, Berkeley lecture notes
 | author=Littlejohn, Robert|author-link1=Robert Grayson Littlejohn
 | date=Fall 2007
 | access-date=2008-05-06
}}</ref>
the Maxwell equations become:<ref name=Griffiths>
{{cite book
 | author=David J Griffiths
 | title=Introduction to electrodynamics
 | year=1999
 | edition=Third
 | pages=[https://archive.org/details/introductiontoel00grif_0/page/559 559–562]
 | publisher=Prentice Hall
 | isbn=978-0-13-805326-0
 | url=https://archive.org/details/introductiontoel00grif_0/page/559
}}</ref>

{| class="wikitable"
|-
! scope="col" style="width: 15em;" | Name
! scope="col" | Integral equations
! scope="col" | Differential equations
|-
| [[Gauss's law]]
| {{oiint}}<math>\vphantom{\oint}_{\scriptstyle\partial \Omega }\mathbf{E}\cdot\mathrm{d}\mathbf{S} = 4\pi \iiint_\Omega \rho \,\mathrm{d}V</math>
| <math>\nabla \cdot \mathbf{E} = 4\pi\rho </math>
|-
| [[Gauss's law for magnetism]]
| {{oiint}}<math>\vphantom{\oint}_{\scriptstyle \partial \Omega }\mathbf{B}\cdot\mathrm{d}\mathbf{S} = 0</math>
| <math>\nabla \cdot \mathbf{B} = 0</math>
|-
| Maxwell–Faraday equation ([[Faraday's law of induction]])
| <math>\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} = -\frac{1}{c}\frac{\mathrm{d}}{\mathrm{d}t}\iint_\Sigma \mathbf{B}\cdot\mathrm{d}\mathbf{S}</math>
| <math>\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}</math>
|-
| [[Ampère–Maxwell law]]
| <math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B}\cdot\mathrm{d}\boldsymbol{\ell} = \frac{1}{c} \left( 4\pi \iint_\Sigma  \mathbf{J}\cdot\mathrm{d}\mathbf{S} + \frac{\mathrel{\mathrm{d}}}{\mathrm{d}t} \iint_\Sigma \mathbf{E}\cdot \mathrm{d}\mathbf{S}\right)
\end{align}
</math>
| <math>\nabla \times \mathbf{B} = \frac{1}{c}\left( 4\pi\mathbf{J} + \frac{\partial \mathbf{E}}{\partial t}\right)</math>
|}

The equations simplify slightly when a system of quantities is chosen in the speed of light, ''c'', is used for [[nondimensionalization]], so that, for example, seconds and lightseconds are interchangeable, and ''c'' = 1.

Further changes are possible by absorbing factors of {{math|4''π''}}. This process, called rationalization, affects whether [[Coulomb's law]] or [[Gauss's law]] includes such a factor (see ''[[Heaviside–Lorentz units]]'', used mainly in [[particle physics]]).