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| {{about|the concept of flux in [[natural science]] and [[mathematics]]}}
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| [[File:General flux diagram.svg|thumb|400px|Flux '''F''' through a [[surface]], d'''S''' is the [[Differential (infinitesimal)|differential]] [[vector area]] element, '''n''' is the [[unit normal]] to the surface. '''Left:''' No flux passes in the surface, the maximum amount flows normal to the surface. '''Right:''' The reduction in flux passing through a surface can be visualized by reduction in '''F''' or d'''S''' equivalently (resolved into [[Euclidean vector#Decomposition|components]], θ is angle to normal '''n'''). '''F'''·d'''S''' is the component of flux passing though the surface, multiplied by the area of the surface (see [[dot product]]). For this reason flux represents physically a flow ''per unit area''.]]
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| In the various subfields of [[physics]], there exist two common usages of the term '''flux''', both with rigorous mathematical frameworks. A simple and ubiquitous concept throughout [[physics]] and [[applied mathematics]] is the flow of a physical property in space, frequently also with time variation. It is the basis of the field concept in [[field (physics)|physics]] and mathematics, with two principal applications: in [[transport phenomena]] and [[surface integral]]s. The terms '''"flux"''', '''"current"''', '''"flux density"''', '''"current density"''', can sometimes be used interchangeably and ambiguously, though the terms used below match those of the contexts in the literature.
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| == Origin of the term ==
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| {{Wiktionary}}
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| 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 | postscript=<!--None--> }}</ref> As ''[[Method of Fluxions|fluxion]]'', this term was introduced into [[differential calculus]] by [[Isaac Newton]].
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| == Flux as flow rate per unit area ==
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| 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 | authorlink=Robert Byron Bird|coauthors=Stewart, Warren E., and [[Edwin N. Lightfoot|Lightfoot, Edwin N.]]| year=1960 | title=Transport Phenomena | publisher=Wiley | isbn=0-471-07392-X }}</ref> For example, the magnitude of a river's current, i.e. the amount of water that flows through a cross-section of the river each second, or the amount of sunlight that lands on a patch of ground each second is also a kind of flux.
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| === General mathematical definition (transport) ===
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| In this definition, flux is generally a [[vector (geometry)|vector]] due to the widespread and useful definition of [[vector area]], although there are some cases where only the magnitude is important (like in number fluxes, see below). The frequent symbol is ''j'' (or ''J''), and a definition for scalar flux of [[physical quantity]] ''q'' is the [[Limit of a function|limit]]:
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| :<math>j = \lim \limits_{A \rightarrow 0}\frac{I}{A}=\frac{dI}{dA}</math>
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| where:
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| :<math>I = \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta q}{ \Delta t} = \frac{dq}{dt}</math>
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| is the flow of quantity ''q'' per unit time ''t'', and ''A'' is the area through which the quantity flows.
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| For vector flux, the [[surface integral]] of '''j''' over a [[surface]] ''S'', followed by an integral over the time duration ''t''<sub>1</sub> to ''t''<sub>2</sub>, gives the total amount of the property flowing through the surface in that time (''t''<sub>2</sub> − ''t''<sub>1</sub>):
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| :<math>q=\int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot\mathbf{\hat{n}}{\rm d}A{\rm d}t </math>
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| The [[area]] required to calculate the flux is real or imaginary, flat or curved, either as a cross-sectional area or a surface. The [[vector area]] is a combination of the magnitude of the area through which the mass passes through, ''A'', and a [[unit vector]] normal to the area, <math>\mathbf{\hat{n}}</math>. The relation is <math>\mathbf{A} = A \mathbf{\hat{n}}</math>.
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| If the flux '''j''' passes through the area at an angle θ to the area normal <math>\mathbf{\hat{n}}</math>, then
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| :<math>\mathbf{j}\cdot\mathbf{\hat{n}}= j\cos\theta </math>
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| where '''·''' is the [[dot product]] of the unit vectors. This is, the component of flux passing through the surface (i.e. normal to it) is ''j'' cos θ, while the component of flux passing tangential to the area is ''j'' sin 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.
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| One could argue, based on the work of [[James Clerk Maxwell]],<ref name = Maxwell/> that the transport definition precedes the more recent way the term is used in electromagnetism. The specific quote from Maxwell is:
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| {{quote|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}}
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| === Transport fluxes ===
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| Eight of the most common forms of flux from the transport phenomena literature are defined as follows:
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| # [[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>). ([[viscosity|Newton's law of viscosity]],)<ref name="Physics P.M">{{cite book|title=Essential Principles of Physics|author=P.M. Whelan, M.J. Hodgeson|edition=2nd|year=1978|publisher=John Murray|isbn=0 7195 3382 1}}</ref>
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| # [[Heat flux]], the rate of [[heat]] flow across a unit area (J·m<sup>−2</sup>·s<sup>−1</sup>). ([[Heat conduction|Fourier's law of conduction]])<ref>{{cite book | last=Carslaw | first=H.S. | coauthors=and 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/>
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| # [[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"/>
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| # [[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]])
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| # [[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.)
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| # [[Radiative flux]], the amount of energy transferred in the form of [[photons]] at a certain distance from the source per [[steradian]] 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 infrared spectrum.
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| # [[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.
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| # Particle flux, the rate of transfer of particles through a unit area ([number of particles] m<sup>−2</sup>·s<sup>−1</sup>)
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| 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.
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| ==== Chemical diffusion ====
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| 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:
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| :<math>\mathbf{J}_A = -D_{AB} \nabla c_A</math>
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| 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 | authorlink= | coauthors=Wicks, Wilson and Rorrer | year=2001 | title=Fundamentals of Momentum, Heat, and Mass Transfer | edition=4th | publisher=Wiley | isbn=0-471-38149-7 }}</ref>
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| 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| authorlink=James Clerk Maxwell | year=1892 | title=Treatise on Electricity and Magnetism | isbn=0-486-60636-8}}</ref>
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| 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
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| :<math>D = \frac{2}{3 n\sigma}\sqrt{\frac{kT}{\pi M}}</math>
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| where the second factor is the [[mean free path]] and the square root (with [[Boltzmann constant|Boltzmann's constant]] ''k'') is the [[Maxwell–Boltzmann distribution#Typical speeds|mean velocity]] of the particles.
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| In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.
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| === Quantum mechanics ===
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| {{Main|Probability current}}
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| In [[quantum mechanics]], particles of mass ''m'' in the [[quantum state]] ψ('''r''', t) have a [[probability amplitude|probability density]] defined as
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| :<math>\rho = \psi^* \psi = |\psi|^2. \,</math>
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| So the probability of finding a particle in a differential [[volume element]] d<sup>3</sup>'''r''' is
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| :<math>{\rm d}P = |\psi|^2 {\rm d}^3\mathbf{r}. \,</math>
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| Then the number of particles passing perpendicularly through unit area of a [[Cross section (geometry)|cross-section]] per unit time is the probability flux;
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| :<math>\mathbf{J} = \frac{i \hbar}{2m} \left(\psi \nabla \psi^* - \psi^* \nabla \psi \right). \,</math>
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| This is sometimes referred to as the probability current or current density,<ref>{{cite book|title=Quantum Mechanics Demystified|author=D. McMahon|series=Demystified|publisher=Mc Graw Hill|year=2006|isbn=0-07-145546 9}}</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>
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| == Flux as a surface integral ==
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| [[Image:Flux diagram.png|thumb|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.]]
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| === General mathematical definition (surface integral) ===
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| 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|title=Vector Analysis|edition=2nd|author=M.R. Spiegel, S. Lipcshutz, D. Spellman|series=Schaum’s Outlines|page=100|publisher=McGraw Hill|year=2009|isbn=978-0-07-161545-7}}</ref>
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| :{{oiint
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| | preintegral = <math>\Phi_F=</math>
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| | intsubscpt = <math>{\scriptstyle A}</math>
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| | integrand = <math>\mathbf{F} \cdot {\rm d}\mathbf{A}</math>
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| }}
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| where '''F''' is a [[vector field]], and d''A'' is the [[vector area]] of the surface ''A'', directed as the [[Normal (geometry)|surface normal]].
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| 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.
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| The surface normal is directed usually by the [[right-hand rule]].
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| Conversely, one can consider the flux the more fundamental quantity and call the vector field the '''flux density'''.
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| 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).
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| 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.
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| If the surface encloses a 3D region, usually the surface is oriented such that the '''influx''' is counted positive; the opposite is the '''outflux'''.
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| 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]]).
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| 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.
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| We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.
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| === Electromagnetism ===
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| One way to better understand the concept of flux in electromagnetism, is by comparing it to a butterfly net. The amount of air moving through the net at any given instant in time is the flux. If the wind speed is high, then the flux through the net is large. If the net is made bigger, then the flux would be larger even though the wind speed is the same. For the most air to move through the net, the opening of the net must be facing the direction the wind is blowing. If the net is parallel to the wind, then no wind will be moving through the net. The simplest way to think of flux is "how much air goes through the net", where the air is a (velocity) field and the net is the boundary of an imaginary surface.
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| ==== Electric flux ====
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| Two forms of [[electric flux]] are used, one for the '''E'''-field:<ref name="Electromagnetism 2008">{{cite book|title=Electromagnetism |edition=2nd|author=I.S. Grant, W.R. Phillips|series=Manchester Physics|publisher=John Wiley & Sons|year=2008|isbn=9-780471-927129}}</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=81-7758-293-3}}</ref>
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| :{{oiint
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| | preintegral = <math>\Phi_E=</math>
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| | intsubscpt = <math>{\scriptstyle A}</math>
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| | integrand = <math>\mathbf{E} \cdot {\rm d}\mathbf{A}</math>
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| }}
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| and one for the '''D'''-field (called the [[electric displacement]]):
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| :{{oiint
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| | preintegral = <math>\Phi_D=</math>
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| | intsubscpt = <math>{\scriptstyle A}</math>
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| | integrand = <math>\mathbf{D} \cdot {\rm d}\mathbf{A}</math>
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| }}
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| 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:
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| :{{oiint
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| | preintegral =
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| | intsubscpt = <math>{\scriptstyle A}</math>
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| | integrand = <math>\mathbf{E} \cdot {\rm d}\mathbf{A} = \frac{Q_A}{\varepsilon_0}</math>
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| }}
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| where ε<sub>0</sub> is the [[permittivity of free space]].
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| If one considers the flux of the electric field vector, '''E''', for a tube near a point charge in the field 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>{{cite book|last=Feynman|first=Richard P|title=The Feynman Lectures on Physics|publisher=Addison-Wesley|year=1964|volume=II|pages=4–8, 9|isbn=0-7382-0008-5}}</ref>
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| 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.
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| ==== Magnetic flux ====
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| 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"/> | |
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| :{{oiint
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| | preintegral = <math>\Phi_B=</math>
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| | intsubscpt = <math>{\scriptstyle A}</math>
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| | integrand = <math>\mathbf{B} \cdot {\rm d}\mathbf{A}</math>
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| }}
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| with the same notation above. The quantity arises in [[Faraday's law of induction]], in integral form:
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| :<math>\oint_C \mathbf{E} \cdot d \boldsymbol{\ell} = -\int_{\partial C} {\partial \mathbf{B}\over \partial t} \cdot {\rm d}\mathbf{s} = - \frac{{\rm d} \Phi_D}{ {\rm d} t}</math>
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| where ''d'''''{{ell}}''' is an infinitesimal vector [[line element]] of the [[closed curve]] ''C'', with [[Magnitude (vector)|magnitude]] equal to the length of the [[infinitesimal]] line element, and [[Direction (geometry)|direction]] given by the tangent to the curve ''C'', with the sign determined by the integration direction.
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| 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.
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| ==== Poynting flux ====
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| 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"/>
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| :{{oiint
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| | preintegral = <math>\Phi_S=</math>
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| | intsubscpt = <math>{\scriptstyle A}</math>
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| | integrand = <math>\mathbf{S} \cdot {\rm d}\mathbf{A}</math>
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| }}
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| 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.
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| 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>).
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| == See also ==
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| {{multicol}}
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| * [[AB magnitude]]
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| * [[Explosively pumped flux compression generator]]
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| * [[Eddy covariance]] flux (aka, eddy correlation, eddy flux)
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| * [[Fast Flux Test Facility]]
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| * [[Fluence]] (flux for particle beams)
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| * [[Fluid dynamics]]
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| * [[Flux footprint]]
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| * [[Flux pinning]]
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| * [[Flux quantization]]
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| * [[Gauss's law]]
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| * [[Inverse-square law]]
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| {{multicol-break}}
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| * [[Jansky]] (non SI unit of spectral flux density)
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| * [[Latent heat flux]]
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| * [[Luminous flux]]
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| * [[Magnetic flux]]
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| * [[Magnetic flux quantum]]
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| * [[Neutron flux]]
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| * [[Poynting flux]]
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| * [[Poynting theorem]]
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| * [[Radiant flux]]
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| * [[Rapid single flux quantum]]
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| * [[Sound energy flux]]
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| * [[Volumetric flow rate]]
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| {{multicol-end}}
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| {{Portal|Mathematics}}
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| == Notes ==
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| {{Reflist}}
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| == Further reading ==
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| * {{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}}
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| [[Category:Physical quantities]]
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| [[Category:Vector calculus]]
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| [[ar:تدفق]]
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| [[ru:Поток векторного поля]]
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