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{{Short description|Mathematical concept applicable to physics}}
{{about|the concept of flux in natural science and mathematics}}

'''Flux''' describes any effect that appears to pass or travel (whether it actually moves or not) through a [[surface]] or substance. Flux is a concept in [[applied mathematics]] and [[vector calculus]] which has many applications in [[physics]].  For [[transport phenomena]], flux is a [[Euclidean vector|vector]] quantity, describing the magnitude and direction of the flow of a substance or property. In [[vector calculus]] flux is a [[Scalar (physics)|scalar]] quantity, defined as the [[surface integral]] of the perpendicular component of a [[vector field]] over a surface.<ref>Purcell, p. 22–26</ref>

== Terminology ==
The word ''flux'' comes from [[Latin]]: ''fluxus'' means "flow", and ''fluere'' is "to flow".<ref>{{cite book | title=An Etymological Dictionary of Modern English | first=Ernest | last=Weekley | publisher=Courier Dover Publications | year=1967 | isbn=0-486-21873-2 | page=581 }}</ref> As ''[[Method of Fluxions|fluxion]]'', this term was introduced into [[differential calculus]] by [[Isaac Newton]].

The concept of [[heat flux]] was a key contribution of [[Joseph Fourier]], in the analysis of heat transfer phenomena.<ref>{{cite book |last1=Herivel |first1=John |title=Joseph Fourier: the man and the physicist |date=1975 |publisher=Clarendon Press |location=Oxford |isbn=0-19-858149-1 |pages=181–191}}</ref> His seminal treatise ''Théorie analytique de la chaleur'' (''The Analytical Theory of Heat''),<ref>{{cite book | last = Fourier | first = Joseph | title = Théorie analytique de la chaleur | publisher = Firmin Didot Père et Fils | year = 1822 | location = Paris | language = fr | url=https://archive.org/details/bub_gb_TDQJAAAAIAAJ | oclc=2688081 }}</ref> defines ''fluxion'' as a central quantity and proceeds to derive the now well-known expressions of flux in terms of temperature differences across a slab, and then more generally in terms of temperature gradients or differentials of temperature, across other geometries. One could argue, based on the work of [[James Clerk Maxwell]],<ref name="Maxwell" /> that the transport definition precedes the [[Magnetic flux|definition of flux used in electromagnetism]]. The specific quote from Maxwell is:
{{blockquote|In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the [[surface integral]] of the flux. It represents the quantity which passes through the surface. |James Clerk Maxwell}}

According to the transport definition, flux may be a single vector, or it may be a vector field / function of position. In the latter case flux can readily be integrated over a surface. By contrast, according to the electromagnetism definition, flux ''is'' the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux" is being used according to the transport definition (and furthermore is a vector field rather than single vector). This is ironic because Maxwell was one of the major developers of what we now call "electric flux" and "magnetic flux" according to the electromagnetism definition. Their names in accordance with the quote (and transport definition) would be "surface integral of electric flux" and "surface integral of magnetic flux", in which case "electric flux" would instead be defined as "electric field" and "magnetic flux" defined as "magnetic field". This implies that Maxwell conceived of these fields as  flows/fluxes of some sort.

Given a flux according to the electromagnetism definition, the corresponding '''flux density''', if that term is used, refers to its derivative along the surface that was integrated. By the [[Fundamental theorem of calculus]], the corresponding '''flux density''' is a flux according to the transport definition. Given a '''current''' such as electric current—charge per time, '''current density''' would also be a flux according to the transport definition—charge per time per area. Due to the conflicting definitions of ''flux'', and the interchangeability of ''flux'', ''flow'', and ''current'' in nontechnical English, all of the terms used in this paragraph are sometimes used interchangeably and ambiguously. Concrete fluxes in the rest of this article will be used in accordance to their broad acceptance in the literature, regardless of which definition of flux the term corresponds to.

== Flux as flow rate per unit area ==
In [[transport phenomena]] ([[heat transfer]], [[mass transfer]] and [[fluid dynamics]]), flux is defined as the ''rate of flow of a property per unit area'', which has the [[dimensional analysis|dimensions]] [quantity]·[time]<sup>−1</sup>·[area]<sup>−1</sup>.<ref>{{cite book | first=R. Byron | last=Bird | author-link=Robert Byron Bird | author2=Stewart, Warren E. | author3=Lightfoot, Edwin N. | author3-link=Edwin N. Lightfoot | year=1960 | title=Transport Phenomena | publisher=Wiley | isbn=0-471-07392-X | url-access=registration | url=https://archive.org/details/transportphenome00bird }}</ref> The area is of the surface the property is flowing "through" or "across". For example, the amount of water that flows through a cross section of a river each second divided by the area of that cross section, or the amount of sunlight energy that lands on a patch of ground each second divided by the area of the patch, are kinds of flux.

=== General mathematical definition (transport) ===
[[File:General flux diagram.svg|thumb|upright=1.5|The [[field line]]s of a [[vector field]] {{math|'''F'''}} through surfaces with [[unit vector|unit]] normal {{math|'''n'''}}, the angle from {{math|'''n'''}} to {{math|'''F'''}} is {{mvar|θ}}. Flux is a measure of how much of the field passes through a given surface. {{math|'''F'''}} is decomposed into components perpendicular (⊥) and parallel {{nowrap|( ‖ )}} to {{math|'''n'''}}. Only the parallel component contributes to flux because it is the maximum extent of the field passing through the surface at a point, the perpendicular component does not contribute. <br />'''Top:''' Three field lines through a plane surface, one normal to the surface, one parallel, and one intermediate. <br />'''Bottom:''' Field line through a [[curved surface]], showing the setup of the unit normal and surface element to calculate flux.]]
[[Image:Surface integral - definition.svg|thumb|upright=1.5|To calculate the flux of a vector field {{math|'''F'''}} ''(red arrows)'' through a surface {{mvar|S}} the surface is divided into small patches {{mvar|dS}}.  The flux through each patch is equal to the normal (perpendicular) component of the field, the [[dot product]] of {{math|'''F'''('''x''')}} with the unit normal vector {{math|'''n'''('''x''')}} ''(blue arrows)'' at the point {{math|'''x'''}} multiplied by the area {{mvar|dS}}.  The sum of {{math|'''F''' · '''n''', ''dS''}} for each patch on the surface is the flux through the surface.]]
Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbol ''j'', (or ''J'') is used for flux, ''q'' for the [[physical quantity]] that flows, ''t'' for time, and ''A'' for area. These identifiers will be written in bold when and only when they are vectors.

First, flux as a (single) scalar:
<math display="block">j = \frac{I}{A},</math>
where
<math display="block">I = \lim_{\Delta t \to 0}\frac{\Delta q}{\Delta t} = \frac{\mathrm{d}q}{\mathrm{d}t}.</math>
In this case the surface in which flux is being measured is fixed and has area ''A''. The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position and perpendicular to the surface.

Second, flux as a [[scalar field]] defined along a surface, i.e. a function of points on the surface:
<math display="block">j(\mathbf{p}) = \frac{\partial I}{\partial A}(\mathbf{p}),</math>
<math display="block">I(A,\mathbf{p}) = \frac{\mathrm{d}q}{\mathrm{d}t}(A, \mathbf{p}).</math>
As before, the surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. ''q'' is now a function of '''p''', a point on the surface, and ''A'', an area. Rather than measure the total flow through the surface, ''q'' measures the flow through the disk with area ''A'' centered at ''p'' along the surface.

Finally, flux as a [[vector field]]:
<math display="block">\mathbf{j}(\mathbf{p}) = \frac{\partial \mathbf{I}}{\partial A}(\mathbf{p}),</math>
<math display="block">\mathbf{I}(A,\mathbf{p}) = \underset{\mathbf{\hat{n}}}{\operatorname{arg\,max}} \mathbf{\hat{n}}_{\mathbf p} \frac{\mathrm{d}q}{\mathrm{d}t}(A,\mathbf{p}, \mathbf{\hat{n}}).</math>
In this case, there is no fixed surface we are measuring over. ''q'' is a function of a point, an area, and a direction (given by a unit vector <math>\mathbf{\hat{n}}</math>), and measures the flow through the disk of area A perpendicular to that unit vector. ''I'' is defined picking the unit vector that maximizes the flow around the point, because the true flow is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. (Strictly speaking, this is an [[abuse of notation]] because the "arg{{nnbsp}}max" cannot directly compare vectors; we take the vector with the biggest norm instead.)

==== Properties ====

These direct definitions, especially the last, are rather unwieldy. For example, the arg{{nnbsp}}max construction is artificial from the perspective of empirical measurements, when with a [[weathervane]] or similar one can easily deduce the direction of flux at a point. Rather than defining the vector flux directly, it is often more intuitive to state some properties about it. Furthermore, from these properties the flux can uniquely be determined anyway.

If the flux '''j''' passes through the area at an angle θ to the area normal <math>\mathbf{\hat{n}}</math>, then the [[dot product]]
<math display="block">\mathbf{j} \cdot \mathbf{\hat{n}} = j\cos\theta.</math>
That is, the component of flux passing through the surface (i.e. normal to it) is ''j''{{nnbsp}}cos{{nnbsp}}''θ'', while the component of flux passing tangential to the area is ''j''{{nnbsp}}sin{{nnbsp}}''θ'', but there is ''no'' flux actually passing ''through'' the area in the tangential direction. The ''only'' component of flux passing normal to the area is the cosine component.

For vector flux, the [[surface integral]] of '''j''' over a [[Surface (mathematics)|surface]] ''S'', gives the proper flowing per unit of time through the surface:
<math display="block">\frac{\mathrm{d}q}{\mathrm{d}t} = \iint_S \mathbf{j} \cdot \mathbf{\hat{n}}\, dA = \iint_S \mathbf{j} \cdot d\mathbf{A},</math>
where '''A''' (and its infinitesimal) is the [[vector area]]{{snd}} combination <math>\mathbf{A} = A \mathbf{\hat{n}}</math> of the magnitude of the area ''A'' through which the property passes and a [[unit vector]] <math>\mathbf{\hat{n}}</math> normal to the area.
Unlike in the second set of equations, the surface here need not be flat.

Finally, we can integrate again over the time duration ''t''<sub>1</sub> to ''t''<sub>2</sub>, getting the total amount of the property flowing through the surface in that time (''t''<sub>2</sub>&nbsp;−&nbsp;''t''<sub>1</sub>):
<math display="block">q = \int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot d\mathbf A\, dt.</math>

=== Transport fluxes ===
Eight of the most common forms of flux from the transport phenomena literature are defined as follows:
# [[Transport phenomena#Momentum transfer|Momentum flux]], the rate of transfer of [[momentum]] across a unit area (N·s·m<sup>−2</sup>·s<sup>−1</sup>). ([[Newton's law of viscosity]])<ref name="Physics P.M">{{cite book|title=Essential Principles of Physics |author1=P.M. Whelan |author2=M.J. Hodgeson |edition=2nd|year=1978|publisher=John Murray|isbn=0-7195-3382-1}}</ref>
# [[Heat flux]], the rate of [[heat]] flow across a unit area (J·m<sup>−2</sup>·s<sup>−1</sup>). ([[Fourier's law|Fourier's law of conduction]])<ref>{{cite book | last=Carslaw | first=H.S. |author2=Jaeger, J.C. | title=Conduction of Heat in Solids  | edition=Second | year=1959 | publisher=Oxford University Press | isbn=0-19-853303-9 }}</ref> (This definition of heat flux fits Maxwell's original definition.)<ref name="Maxwell" />
# [[Diffusion flux]], the rate of movement of molecules across a unit area (mol·m<sup>−2</sup>·s<sup>−1</sup>). ([[Fick's law of diffusion]])<ref name="Physics P.M" />
# [[Volumetric flux]], the rate of [[volume]] flow across a unit area (m<sup>3</sup>·m<sup>−2</sup>·s<sup>−1</sup>). ([[Darcy's law|Darcy's law of groundwater flow]])
# [[Mass flux]], the rate of [[mass]] flow across a unit area (kg·m<sup>−2</sup>·s<sup>−1</sup>). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density.)
# [[Radiative flux]], the amount of energy transferred in the form of [[photons]] at a certain distance from the source per unit area per second (J·m<sup>−2</sup>·s<sup>−1</sup>). Used in astronomy to determine the [[Magnitude (astronomy)|magnitude]] and [[spectral class]] of a star.  Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the electromagnetic spectrum.
# [[Energy flux]], the rate of transfer of [[energy]] through a unit area (J·m<sup>−2</sup>·s<sup>−1</sup>). The radiative flux and heat flux are specific cases of energy flux.
# [[Particle flux]], the rate of transfer of particles through a unit area ([number of particles] m<sup>−2</sup>·s<sup>−1</sup>)

These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the [[divergence]] of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space.  For [[incompressible flow]], the divergence of the volume flux is zero.

==== Chemical diffusion ====
As mentioned above, chemical [[mass flux#Molar fluxes|molar flux]] of a component A in an [[isothermal]], [[Isobaric process|isobaric system]] is defined in [[Fick's law of diffusion]] as:
<math display="block">\mathbf{J}_A = -D_{AB} \nabla c_A</math>
where the [[nabla symbol]] ∇ denotes the [[gradient]] operator, ''D<sub>AB</sub>'' is the diffusion coefficient (m<sup>2</sup>·s<sup>−1</sup>) of component A diffusing through component B, ''c<sub>A</sub>'' is the [[concentration]] ([[mole (unit)|mol]]/m<sup>3</sup>) of component A.<ref>{{cite book | last=Welty |author2=Wicks, Wilson and Rorrer | year=2001 | title=Fundamentals of Momentum, Heat, and Mass Transfer | edition=4th | publisher=Wiley | isbn=0-471-38149-7 }}</ref>

This flux has units of mol·m<sup>−2</sup>·s<sup>−1</sup>, and fits Maxwell's original definition of flux.<ref name="Maxwell">{{cite book | last=Maxwell | first=James Clerk| author-link=James Clerk Maxwell | year=1892 | title=Treatise on Electricity and Magnetism | isbn=0-486-60636-8}}</ref>

For dilute gases, kinetic molecular theory relates the diffusion coefficient ''D'' to the particle density ''n'' = ''N''/''V'', the molecular mass ''m'', the collision [[Cross section (physics)|cross section]] <math>\sigma</math>, and the [[Thermodynamic temperature|absolute temperature]] ''T'' by
<math display="block">D = \frac{2}{3 n\sigma}\sqrt{\frac{kT}{\pi m}}</math>
where the second factor is the [[mean free path]] and the square root (with the [[Boltzmann constant]] ''k'') is the [[Maxwell–Boltzmann distribution#Typical speeds|mean velocity]] of the particles.

In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.

=== Quantum mechanics ===
{{Main|Probability current}}

In [[quantum mechanics]], particles of mass ''m'' in the [[quantum state]] ''ψ''('''r''', ''t'') have a [[probability amplitude|probability density]] defined as
<math display="block">\rho = \psi^* \psi = |\psi|^2. </math>
So the probability of finding a particle in a differential [[volume element]] d<sup>3</sup>'''r''' is
<math display="block"> dP = |\psi|^2 \, d^3\mathbf{r}. </math>
Then the number of particles passing perpendicularly through unit area of a [[Cross section (geometry)|cross-section]] per unit time is the probability flux;
<math display="block">\mathbf{J} = \frac{i \hbar}{2m} \left(\psi \nabla \psi^* - \psi^* \nabla \psi \right). </math>
This is sometimes referred to as the probability current or current density,<ref>{{cite book |author=D. McMahon |url=https://archive.org/details/quantumfieldtheo0000mcma |title=Quantum Mechanics Demystified |publisher=Mc Graw Hill |year=2008 |isbn=978-0-07-145546-6 |edition=2nd |url-access=registration}}</ref> or probability flux density.<ref>{{cite book | author=Sakurai, J. J. | title=Advanced Quantum Mechanics | publisher=Addison Wesley | year=1967 | isbn=0-201-06710-2}}</ref>

== 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>).

== SI radiometry units ==
{{SI radiometry units}}

== See also ==
{{Portal|Mathematics}}
{{div col|colwidth=28em}}
* [[AB magnitude]]
* [[Explosively pumped flux compression generator]]
* [[Eddy covariance]] flux (aka, eddy correlation, eddy flux)
* [[Fast Flux Test Facility]]
* [[Fluence]] (flux of the first sort for particle beams)
* [[Fluid dynamics]]
* [[Flux footprint]]
* [[Flux pinning]]
* [[Flux quantization]]
* [[Gauss's law]]
* [[Inverse-square law]]
* [[Jansky]] (non SI unit of spectral flux density)
* [[Latent heat flux]]
* [[Luminous flux]]
* [[Magnetic flux]]
* [[Magnetic flux quantum]]
* [[Neutron flux]]
* [[Poynting flux]]
* [[Poynting theorem]]
* [[Radiant flux]]
* [[Rapid single flux quantum]]
* [[Sound energy flux]]
* [[Volumetric flux]] (flux of the first sort for fluids)
* [[Volumetric flow rate]] (flux of the second sort for fluids)
{{div col end}}

== Notes ==
{{reflist}}
* {{cite book |last= Browne | first= Michael | title=Physics for Engineering and Science, 2nd Edition. | publisher=[[McGraw-Hill Education|McGraw-Hill Publishing]] | series=Schaum Outlines | location=New York, Toronto | year= 2010 | isbn= 978-0-0716-1399-6 }}
* {{cite book | last= Purcell | first= Edward | title=Electricity and Magnetism, 3rd Edition | publisher=[[Cambridge University Press]] | location=Cambridge, UK | year=2013 | isbn=978110-7014022 }}

== Further reading ==
* {{cite journal | author=Stauffer, P.H. | title=Flux Flummoxed: A Proposal for Consistent Usage | journal=Ground Water | year=2006 | volume=44 | issue=2 | pages= 125–128 | doi = 10.1111/j.1745-6584.2006.00197.x | pmid=16556188| bibcode=2006GrWat..44..125S | s2cid=21812226 | doi-access=free }}

== External links ==
* {{Wiktionary-inline}}

[[Category:Physical quantities]]
[[Category:Vector calculus]]
[[Category:Rates]]