Editing Maxwell's equations (section)

== Formulation in terms of electric and magnetic fields (microscopic or in vacuum version) ==
<!-- please do not change to Electromagnetic field: we want to (modestly) stress that in this formulation Electric and Magnetic fields play an intertwined but separate role -->
In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate [[Physical law|law of nature]], the [[Lorentz force]] law, describes how the electric and magnetic fields act on charged particles and currents. By convention, a version of this law in the original equations by Maxwell is no longer included. The [[vector calculus]] formalism below, the work of [[Oliver Heaviside]],<ref>Bruce J. Hunt (1991) ''[[The Maxwellians]]'', chapter 5 and appendix, [[Cornell University Press]]</ref><ref>{{cite web|url=http://ethw.org/Maxwell's_Equations|title=Maxwell's Equations |date=29 October 2019 |publisher=Engineering and Technology History Wiki |access-date=2021-12-04}}</ref> has become standard. It is rotationally invariant, and therefore mathematically more transparent than Maxwell's original 20 equations in ''x'', ''y'' and ''z'' components. The [[#Relativistic formulations|relativistic formulations]] are more symmetric and Lorentz invariant. For the same equations expressed using tensor calculus or differential forms (see ''{{section link||Alternative formulations}}'').

The differential and integral formulations are mathematically equivalent; both are useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential equations are purely ''local'' and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using [[finite element analysis]].<ref>{{cite book |title=Partial differential equations and the finite element method |last=Šolín |first=Pavel |year=2006 |publisher=John Wiley and Sons |isbn=978-0-471-72070-6 |page=273 |url=https://books.google.com/books?id=-hIG3NZrnd8C&pg=PA273}}</ref>

=== Key to the notation ===
<!---please do not make this list much longer – we used to have a gigantic table of all the constants, variables, terminology, and units, which was converted into prose as it probably should for an encyclopedia (see the history in pre-2013). Editors (by all means in good faith) may add the units, alternative names and symbols, etc. to the list and make it longer and denser, then eventually there would be a good reason to resurrect the big table format again...--->

Symbols in '''bold''' represent [[Vector (geometric)|vector]] quantities, and symbols in ''italics'' represent [[scalar (physics)|scalar]] quantities, unless otherwise indicated.
The equations introduce the [[electric field]], {{math|'''E'''}}, a [[vector field]], and the [[magnetic field]], {{math|'''B'''}}, a [[pseudovector]] field, each generally having a time and location dependence.
The sources are
* the total electric [[charge density]] (total charge per unit volume), {{math|''ρ''}}, and
* the total electric [[current density]] (total current per unit area), {{math|'''J'''}}.

The [[universal constant]]s appearing in the equations (the first two ones explicitly only in the SI formulation) are:
* the [[permittivity of free space]], {{math|''ε''<sub>0</sub>}}, and
* the [[permeability of free space]], {{math|''μ''<sub>0</sub>}}, and
* the [[speed of light]], <math>c = ({\varepsilon_0\mu_0})^{-1/2}</math>

==== Differential equations ====
In the differential equations, 
*the [[nabla symbol]], {{math|∇}}, denotes the three-dimensional [[gradient]] operator, [[del]],
*the {{math|∇⋅}} symbol (pronounced "del dot") denotes the [[divergence]] operator,
*the {{math|∇×}} symbol (pronounced "del cross") denotes the [[curl (mathematics)|curl]] operator.

==== Integral equations ====
In the integral equations, 
* {{math|Ω}} is any volume with closed [[boundary (topology)|boundary]] surface {{math|∂Ω}}, and
* {{math|Σ}} is any surface with closed boundary curve {{math|∂Σ}},
The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over a given time interval. For example, since the surface is time-independent, we can bring the [[differentiation under the integral sign]] in Faraday's law:
<math display="block"> \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} = \iint_{\Sigma}  \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\mathbf{S}\,,</math>
Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using the differential version and using Gauss' and Stokes' theorems as appropriate. 
* {{oiint
   | intsubscpt=
   | integrand=}}<math>{\vphantom{\int}}_{\scriptstyle\partial \Omega}</math> is a [[surface integral]] over the boundary surface {{math|∂Ω}}, with the loop indicating the surface is closed
* <math>\iiint_\Omega</math> is a [[volume integral]] over the volume {{math|Ω}},
* <math>\oint_{\partial \Sigma}</math> is a [[line integral]] around the boundary curve {{math|∂Σ}}, with the loop indicating the curve is closed.
* <math>\iint_\Sigma</math> is a [[surface integral]] over the surface {{math|Σ}},
* The ''total'' [[electric charge]] {{math|''Q''}} enclosed in {{math|Ω}} is the [[volume integral]] over {{math|Ω}} of the [[charge density]] {{math|''ρ''}} (see the "macroscopic formulation" section below): <math display="block">Q = \iiint_\Omega \rho \ \mathrm{d}V,</math> where {{math|d''V''}} is the [[volume element]].
* The ''net'' [[magnetic flux]] {{math|Φ<sub>''B''</sub>}} is the [[surface integral]] of the magnetic field {{math|'''B'''}} passing through a fixed surface, {{math|Σ}}: <math display="block">\Phi_B = \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d} \mathbf{S},</math>
* The ''net'' [[electric flux]] {{math|Φ<sub>''E''</sub>}} is the surface integral of the electric field {{math|'''E'''}} passing through {{math|Σ}}: <math display="block">\Phi_E = \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d} \mathbf{S},</math>
* The ''net'' [[electric current]] {{math|''I''}} is the surface integral of the [[electric current density]] {{math|'''J'''}} passing through {{math|Σ}}: <math display="block">I = \iint_{\Sigma} \mathbf{J} \cdot \mathrm{d} \mathbf{S},</math> where {{math|d'''S'''}} denotes the differential [[vector area|vector element]] of surface area {{math|''S''}}, [[Normal (geometry)|normal]] to surface {{math|Σ}}. (Vector area is sometimes denoted by {{math|'''A'''}} rather than {{math|'''S'''}}, but this conflicts with the notation for [[magnetic vector potential]]).

=== Formulation in the SI ===

{| class="wikitable"
|-
! scope="col" style="width: 15em;" | Name
! scope="col" | [[Integral]] equations 
! scope="col" | [[Partial differential equation|Differential]] equations
|-
|  [[Gauss's law]]
| {{oiint}}<math>\vphantom{\oint}_{\scriptstyle\partial \Omega}\mathbf{E}\cdot\mathrm{d}\mathbf{S} = \frac{1}{\varepsilon_0} \iiint_\Omega \rho \,\mathrm{d}V</math>
| <math>\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}</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{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S} </math>
| <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
|-
| [[Ampère–Maxwell law]]
| <math>
\begin{align}
\oint_{\partial \Sigma} & \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 \left(\iint_{\Sigma} \mathbf{J} \cdot \mathrm{d}\mathbf{S} + \varepsilon_0 \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S} \right) \\
\end{align}
</math>
| <math>\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) </math>
|}

=== 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]]).