Editing Flux (section)

== Flux as a surface integral ==
=== General mathematical definition (surface integral) ===
[[Image:Flux diagram.png|thumb|upright=1.2|The flux visualized. The rings show the surface boundaries. The red arrows stand for the flow of charges, fluid particles, subatomic particles, photons, etc. The number of arrows that pass through each ring is the flux.]]
As a mathematical concept, flux is represented by the [[surface integral#Surface integrals of vector fields|surface integral of a vector field]],<ref>{{cite book |author1=Murray R. Spiegel |url=https://archive.org/details/vectoranalysis0000unse_t6w7 |title=Vector Analysis |title-link= |author2=S. Lipcshutz |author3=D. Spellman |publisher=McGraw Hill |year=2009 |isbn=978-0-07-161545-7 |edition=2nd |series=Schaum's Outlines |page=100}}</ref>
<math display=block>\Phi_F=\iint_A\mathbf{F}\cdot\mathrm{d}\mathbf{A}</math>
<math display=block>\Phi_F=\iint_A\mathbf{F}\cdot\mathbf{n}\,\mathrm{d}A</math>
where '''F''' is a [[vector field]], and d'''A''' is the [[vector area]] of the surface ''A'', directed as the [[Normal (geometry)|surface normal]]. For the second, '''n''' is the outward pointed [[unit normal vector]] to the surface.

The surface has to be [[orientability|orientable]], i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.

The surface normal is usually directed by the [[right-hand rule]].

Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density.

Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive [[divergence]] (sources) and end at areas of negative divergence (sinks).

See also the image at right: the number of red arrows passing through a unit area is the flux density, the [[curve]] encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the [[inner product]] of the vector field with the surface normals.

If the surface encloses a 3D region, usually the surface is oriented such that the '''influx''' is counted positive; the opposite is the '''outflux'''.

The [[divergence theorem]] states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the [[divergence]]).

If the surface is not closed, it has an oriented curve as boundary. [[Stokes' theorem]] states that the flux of the [[Curl (mathematics)|curl]] of a vector field is the [[line integral]] of the vector field over this boundary. This path integral is also called [[Circulation (fluid dynamics)|circulation]], especially in fluid dynamics. Thus the curl is the circulation density.

We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.

=== Electromagnetism ===

==== Electric flux ====

An electric "charge", such as a single proton in space, has a magnitude defined in coulombs. Such a charge has an electric field surrounding it. In pictorial form, the electric field from a positive point charge can be visualized as a dot radiating [[Field line|electric field lines]] (sometimes also called "lines of force"). Conceptually, electric flux can be thought of as "the number of field lines" passing through a given area.   Mathematically, electric flux is the integral of the [[Normal (geometry)|normal]] component of the electric field over a given area. Hence, units of electric flux are, in the [[MKS system of units|MKS system]], [[Newton (unit)|newtons]] per [[Coulomb (unit)|coulomb]] times meters squared, or N m<sup>2</sup>/C.  (Electric flux density is the electric flux per unit area, and is a measure of strength of the [[Normal (geometry)|normal]] component of the electric field averaged over the area of integration.  Its units are N/C, the same as the electric field in MKS units.)

Two forms of [[electric flux]] are used, one for the '''E'''-field:<ref name="Electromagnetism 2008">{{cite book|title=Electromagnetism |edition=2nd|author1=I.S. Grant |author2=W.R. Phillips |series=Manchester Physics|publisher=[[John Wiley & Sons]]|year=2008|isbn=978-0-471-92712-9}}</ref><ref name="Electrodynamics 2007">{{cite book|title=Introduction to Electrodynamics|edition=3rd|author=D.J. Griffiths|publisher=Pearson Education, [[Dorling Kindersley]]|year=2007|isbn=978-81-7758-293-2}}</ref>
: {{oiint
| preintegral = <math>\Phi_E=</math>
| intsubscpt = <math>{\scriptstyle A}</math>
| integrand = <math>\mathbf{E} \cdot {\rm d}\mathbf{A}</math>
}}
and one for the '''D'''-field (called the [[electric displacement]]):
: {{oiint
| preintegral = <math>\Phi_D=</math>
| intsubscpt = <math>{\scriptstyle A}</math>
| integrand = <math>\mathbf{D} \cdot {\rm d}\mathbf{A}</math>
}}

This quantity arises in [[Gauss's law]] – which states that the flux of the [[electric field]] '''E''' out of a [[closed surface]] is proportional to the [[electric charge]] ''Q<sub>A</sub>'' enclosed in the surface (independent of how that charge is distributed), the integral form is:
: {{oiint
| preintegral =
| intsubscpt = <math>{\scriptstyle A}</math>
| integrand = <math>\mathbf{E} \cdot {\rm d}\mathbf{A} = \frac{Q_A}{\varepsilon_0}</math>
}}
where ''ε''<sub>0</sub> is the [[permittivity of free space]].

If one considers the flux of the electric field vector, '''E''', for a tube near a point charge in the field of the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge ''q'' is ''q''/''ε''<sub>0</sub>.<ref>[https://feynmanlectures.caltech.edu/II_04.html#Ch4-S5-p7 The Feynman Lectures on Physics Vol. II Ch. 4: Electrostatics]</ref>

In free space the [[electric displacement]] is given by the [[constitutive relation]] '''D''' = ''ε''<sub>0</sub> '''E''', so for any bounding surface the '''D'''-field flux equals the charge ''Q<sub>A</sub>'' within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow", since nothing actually flows along electric field lines.

==== Magnetic flux ====
The magnetic flux density ([[magnetic field]]) having the unit Wb/m<sup>2</sup> ([[Tesla (unit)|Tesla]]) is denoted by '''B''', and [[magnetic flux]] is defined analogously:<ref name="Electromagnetism 2008"/><ref name="Electrodynamics 2007"/>
<math display=block>\Phi_B=\iint_A\mathbf{B}\cdot\mathrm{d}\mathbf{A}</math>
with the same notation above.  The quantity arises in [[Faraday's law of induction]], where the magnetic flux is time-dependent either because the boundary is time-dependent or magnetic field is time-dependent. In integral form:
<math display=block>- \frac{{\rm d} \Phi_B}{ {\rm d} t} = 
\oint_{\partial A} \mathbf{E} \cdot d \boldsymbol{\ell}</math>
where ''d'''''{{ell}}''' is an infinitesimal vector [[line element]] of the [[closed curve]] <math>\partial A</math>, with [[Magnitude (vector)|magnitude]] equal to the length of the [[infinitesimal]] line element, and [[Direction (geometry)|direction]] given by the tangent to the curve <math>\partial A</math>, with the sign determined by the integration direction.

The time-rate of change of the magnetic flux through a loop of wire is minus the [[electromotive force]] created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing a magnetic field opposite to the change.  This is the basis for [[inductor]]s and many [[electric generator]]s.

==== Poynting flux ====
Using this definition, the flux of the [[Poynting vector]] '''S''' over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before:<ref name="Electrodynamics 2007"/>

: {{oiint
| preintegral = <math>\Phi_S=</math>
| intsubscpt = <math>{\scriptstyle A}</math>
| integrand = <math>\mathbf{S} \cdot {\rm d}\mathbf{A}</math>
}}

The flux of the [[Poynting vector]] through a surface is the electromagnetic [[power (physics)|power]], or [[energy]] per unit [[time]], passing through that surface. This is commonly used in analysis of [[electromagnetic radiation]], but has application to other electromagnetic systems as well.

Confusingly, the Poynting vector is sometimes called the ''power flux'',  which is an example of the first usage of flux, above.<ref>{{cite book | first=Roald K. | last=Wangsness | year=1986 | title=Electromagnetic Fields | edition=2nd | publisher=Wiley | isbn=0-471-81186-6 }} p.357</ref> It has units of [[watt]]s per [[square metre]] (W/m<sup>2</sup>).