View source for Physics:Electric flux

{{short description|Measure of electric field through surface}}
{{Electromagnetism|cTopic=Electrostatics}}
In [[Physics:Electromagnetism|electromagnetism]], '''electric flux''' is the measure of the [[Physics:Electric field|electric field]] through a given surface,<ref>Purcell, pp. 22–26</ref> although an electric field in itself cannot flow.

The electric field '''E''' can exert a force on an electric charge at any point in space. The electric field is the [[Gradient|gradient]] of the potential.

==Overview==
An electric charge, such as a single [[Physics:Electron|electron]] in space, has an electric field surrounding it. In pictorial form, this electric field is shown as "lines of flux" being radiated from a dot (the charge). These are called Gauss lines.<ref>Purcell, pp. 5–6.</ref> Note that field lines are a graphic illustration of field strength and direction and have no physical meaning. The density of these lines corresponds to the electric field strength, which could also be called the electric flux density: the number of "lines" per unit area. Electric flux is proportional to the total number of electric [[Field line|field line]]s going through a surface.  For simplicity in calculations it is often convenient to consider a surface perpendicular to the flux lines. If the electric field is uniform, the electric flux passing through a surface of [[Vector area|vector area]] {{math|'''S'''}} is
<math display="block">\Phi_E = \mathbf{E} \cdot \mathbf{S} = ES \cos \theta,</math>
where {{math|'''E'''}} is the electric field (having units of {{math|V/m}}), {{math|''E''}} is its magnitude, {{math|''S''}} is the area of the surface, and {{math|''θ''}} is the angle between the electric field lines and the normal (perpendicular) to {{math|'''S'''}}.

For a non-uniform electric field, the electric flux {{math|dΦ<sub>'''''E'''''</sub>}} through a small surface area {{math|d'''S'''}} is given by
<math display="block">\textrm d\Phi_E = \mathbf{E} \cdot \textrm d\mathbf{S}</math>
(the electric field, {{math|'''E'''}}, multiplied by the component of area perpendicular to the field).  The electric flux over a surface {{math|'''S'''}} is therefore given by the [[Surface integral|surface integral]]:
<math display="block">\Phi_E = \iint_S \mathbf{E} \cdot \textrm{d}\mathbf{S}</math>
where {{math|'''E'''}} is the electric field and {{math|d'''S'''}} is a differential area on the closed surface {{math|''S''}} with an outward facing surface normal defining its direction.

For a closed [[Gaussian surface]], electric flux is given by:
{{block indent|em=1.6|text={{oiint
| preintegral = <math>\Phi_E =\,\!</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>\mathbf{E}\cdot \textrm{d}\mathbf{S} = \frac{Q}{\varepsilon_0}\,\!</math>}}}}
where
* {{math|'''E'''}} is the [[Physics:Electric field|electric field]],
* {{math|'''S'''}} is any closed surface,
* {{math|''Q''}} is the total [[Physics:Electric charge|electric charge]] inside the surface {{math|''S''}},
* {{math|''ε''<sub>0</sub>}} is the electric constant (a universal constant, also called the "[[Physics:Permittivity|permittivity]] of free space") ({{math|1=''ε''<sub>0</sub> ≈ {{val|8.854187817|e=-12|u=[[Farad|F]]/[[Metre|m]]}}}})
This relation is known as Gauss' law for electric fields in its [[Integral|integral]] form and it is one of [[Physics:Maxwell's equations|Maxwell's equations]].

While the electric flux is not affected by charges that are not within the closed surface, the net electric field, {{math|'''E'''}} can be affected by charges that lie outside the closed surface. While Gauss's law holds for all situations, it is most useful for "by hand" calculations when high degrees of symmetry exist in the electric field.  Examples include spherical and cylindrical symmetry.

The [SI] unit of electric flux is the volt-meter ({{math|V·m}}), or Dimenstion is m⁰t‐¹a‐¹, or, equivalently, [[Newton (unit)|newton]]-meter squared per [[Coulomb|coulomb]] ({{math|N·m<sup>2</sup>·C<sup>−1</sup>}}). Thus, the unit of electric flux expressed in terms of SI base units is {{math|kg·m<sup>3</sup>·s<sup>−3</sup>·A<sup>−1</sup>}}. Its dimensional formula is <math>\mathsf{L}^3\mathsf{MT}^{-3}\mathsf{I}^{-1}</math>.

== See also ==
* [[Physics:Magnetic flux|Magnetic flux]]
* [[Physics:Maxwell's equations|Maxwell's equations]]
* [[Physics:Electric field|Electric field]]
* [[Physics:Magnetic field|Magnetic field]]
* [[Physics:Electromagnetic field|Electromagnetic field]]

== Citations ==
<references />

== References ==
* Purcell, Edward, Morin, David; Electricity and Magnetism, 3rd Edition; Cambridge University Press, New York. 2013 {{ISBN|9781107014022}}.
* Browne, Michael, PhD; Physics for Engineering and Science, 2nd Edition; McGraw Hill/Schaum, New York; 2010. {{ISBN|0071613994}}

== External links ==
* [http://hyperphysics.phy-astr.gsu.edu/Hbase/electric/gaulaw.html#c3 Electric flux] – HyperPhysics

[[Category:Electrostatics]]
[[Category:Electromagnetic quantities]]

{{Sourceattribution|Electric flux}}