Flux
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{{dablink|This article is about the concept of ''flux'' in science and mathematics. For other uses of the word, see [[Flux (disambiguation)]].}}
In the various subfields of [[physics]], there exist two common usages of the term '''flux''', both with rigorous mathematical frameworks.
*In the study of [[transport phenomena]] ([[heat transfer]], [[mass transfer]] and [[fluid dynamics]]), flux is defined as the amount that flows through a unit area ''per unit time''.<ref>{{cite book | first=R. Byron | last=Bird | coauthors=Stewart, Warren E., and Lightfoot, Edwin N. | year=1960 | title=Transport Phenomena | publisher=Wiley | id=ISBN 0-471-07392-X }}</ref> Flux in this definition is a [[vector (spatial)|vector]].
*In the field of [[electromagnetism]], flux is usually the [[integral]] of a [[Vector (spatial)|vector]] quantity over a finite surface. The result of this integration is a [[scalar (physics)|scalar]] quantity.<ref>{{cite book | first=Paul | last=Lorrain | coauthors=and Corson, Dale | year=1962 | title=Electromagnetic Fields and Waves }}</ref> The [[magnetic flux]] is thus the integral of the magnetic vector field B over a surface, and the electric flux is defined similarly. Using this definition, the flux of the [[Poynting vector]] over a specified surface is the rate at which electromagnetic energy flows through that surface. 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 ed. | publisher=Wiley | id=ISBN 0-471-81186-6 }} p.357</ref> It has units of [[watt]]s per [[square metre]] (Wm<sup>-2</sup>)
One could argue, based on the work of [[James Clerk Maxwell]]<ref name=Maxwell>{{cite book | last=Maxwell | first=James Clerk| authorlink=James Clerk Maxwell | year=1892 | title=Treatise on Electricity and Magnetism}}</ref>, that the transport definition precedes the more recent way the term is used in electromagnetism. The specific quote from Maxwell is "''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''".
In addition to these common mathematical definitions, there are many more loose usages found in fields such as biology.
==Transport phenomena==
===Flux definition and theorems===
Flux is surface bombardment rate. There are many fluxes used in the study of transport phenomena. Each type of flux has its own distinct unit of measurement along with distinct physical constants. Six of the most common forms of flux from the transport literature are defined as:
#''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,]])
#''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 convection]])<ref>{{cite book | last=Carslaw | first=H.S. | coauthors=and Jaeger, J.C. | title=Conduction of Heat in Solids | edition=Second Edition | year=1959 | publisher=Oxford University Press | id=ISBN 0-19-853303-9 }}</ref> (This definition of heat flux fits Maxwell's original definition.<ref name=Maxwell>{{cite book | last=Maxwell | first=James Clerk| authorlink=James Clerk Maxwell | year=1892 | title=Treatise on Electricity and Magnetism}}</ref>)
#''Chemical 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]])
#''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 moving 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 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.
#''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.
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===
Flux, or diffusion, for gaseous molecules can be related to the [[function (mathematics)|function]]:
:<math>\Phi = 2\pi\sigma_{ab}^2\sqrt{\frac{8kT}{\pi N}}</math>
where:
:*''N'' is the total number of gaseous particles,
:*''k'' is [[Boltzmann constant|Boltzmann's constant]],
:*''T'' is the relative temperature in kelvins,
:*<math>\sigma_{ab}</math> is the mean free path between the molecules ''a'' and ''b''.
Chemical molar flux of a component A in an [[isothermal]], [[isobaric]] [[system]] is also defined in [[Fick's law of diffusion|Ficks's first law]] as:
:<math>\overrightarrow{J_A} = -D_{AB} \nabla c_A</math>
where:
:*''<math>D_{AB}</math>'' is the molecular diffusion coefficient (m<sup>2</sup>/s) of component A diffusing through component B,
:*''<math>c_A</math>'' is the concentration ([[mole (unit)|mol]]/m<sup>3</sup>) of species A.<ref>{{cite book | last=Welty | authorlink= | coauthors=Wicks, Wilson and Rorrer | year=2001 | title=Fundamentals of Momentum, Heat, and Mass Transfer | edition=4th ed. | publisher=Wiley | id=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| authorlink=James Clerk Maxwell | year=1892 | title=Treatise on Electricity and Magnetism}}</ref>
Note: <math>\nabla</math> ("[[nabla symbol|nabla]]") denotes the [[del]] operator.
===Quantum mechanics===
{{main|Probability current}}
In [[quantum mechanics]], particles of mass m in the state <math>\psi(r,t)</math> have a probability density defined as
:<math>\rho = \psi^* \psi = |\psi|^2. \,</math>
So the probability of finding a particle in a unit of volume, say <math>d^3x</math>, is
:<math>|\psi|^2 d^3x. \,</math>
Then the number of particles passing through a perpendicular unit of area per unit time is
:<math>\mathbf{J} = -i \frac{h}{2m} \left(\psi^* \nabla \psi - \psi \nabla \psi^* \right). \,</math>
This is sometimes referred to as the "flux density".<ref>{{cite book | author=Sakurai, J. J. | title=Advanced Quantum Mechanics | publisher=Addison Wesley | year=1967 | id=ISBN 0-201-06710-2}}</ref>
==Electromagnetism==
===Flux definition and theorems===
An example of the second definition of flux is the magnitude of a river's current, that is, the amount of water that flows through a cross-section of the river each second. The amount of sunlight that lands on a patch of ground each second is also a kind of flux.
To better understand the concept of flux in Electromagnetism, imagine 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 opening is parallel to the wind, then no wind will be moving through the net. (These examples are not very good because they rely on a transport process and as stated in the introduction, transport flux is defined differently than E+M flux.) Perhaps the best way to think of flux abstractly is "How much stuff goes through your thing", where the stuff is a field and the thing is the imaginary surface.
[[Image:Flux diagram.png|right|frame|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]],
:<math>\Phi_f = \int_S \mathbf{E} \cdot \mathbf{dA}</math>
where:
:*''E'' is a [[vector field]] of Electric Force,
:*''dA'' is the [[vector area]] of the surface ''S'', directed as the [[surface normal]],
:*''<math>\Phi_f</math>'' is the resulting flux.
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 directed accordingly, usually 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 '''outflux''' is counted positive; the opposite is the '''influx'''.
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.
===Maxwell's equations===
The flux of [[electric field|electric]] and [[magnetic field]] lines is frequently discussed in [[electrostatics]]. This is because in [[Maxwell's equations]] in integral form involve integrals like above for electric and magnetic fields.
For instance, [[Gauss's law]] states that the flux of the electric field out of a closed surface is proportional to the [[electric charge]] enclosed in the surface (regardless of how that charge is distributed). The constant of proportionality is the reciprocal of the [[permittivity]] of free space.
Its integral form is:
: <math> \oint_A \epsilon_0 \mathbf{E} \cdot d\mathbf{A} = Q_A </math>
where:
:*''<math> \mathbf{E} </math>'' is the electric field,
:*''<math>d\mathbf{A}</math>'' is the area of a differential square on the surface ''A'' with an outward facing [[surface normal]] defining its direction,
:*''<math> Q_A \ </math>'' is the charge enclosed by the surface,
:*''<math> \epsilon_0 \ </math>'' is the [[permittivity]] of free space
:*''<math>\oint_A</math>'' is the integral over the surface ''A''.
Either <math> \oint_A \epsilon_0 \mathbf{E} \cdot d\mathbf{A} </math> or <math> \oint_A \mathbf{E} \cdot d\mathbf{A} </math> is called the '''electric flux'''.
[[Faraday's law of induction]] in integral form is:
:<math>\oint_C \mathbf{E} \cdot d\mathbf{l} = -\int_{\partial C} \ {d\mathbf{B}\over dt} \cdot d\mathbf{s} = - \frac{d \Phi_D}{ d t}</math>
where:
:*<math>\mathrm{d}\mathbf{l} </math> is an infinitesimal element ([[differential (mathematics)|differential]]) of the contour ''C'' (i.e. a [[vector]] with [[magnitude]] equal to the length of the [[infinitesimal]] line element, and [[direction]] equal to the direction of the [[contour]] ''C'').
The [[magnetic field]] is denoted by <math> \mathbf{B} </math>. Its flux is called the [[magnetic flux]]. 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 vector===
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.
==Biology==
In general, 'flux' in [[biology]] relates to movement of a substance between compartments. There are several cases where the concept of 'flux' is important.
* The movement of molecules across a membrane: in this case, flux is defined by the rate of [[diffusion]] or transport of a substance across a permeable [[biological membrane|membrane]]. Except in the case of active transport, net flux is directly proportional to the [[concentration]] difference across the membrane, the [[surface area]] of the membrane, and the membrane [[Semipermeable membrane|permeability]] constant.
* In [[ecology]], flux is often considered at the [[ecosystem]] level - for instance, accurate determination of [[Carbon flux|carbon fluxes]] using techniques like [[Eddy covariance|eddy covariance]] (at a regional and global level) is essential for modeling the causes and consequences of [[global warming]].
* [[Metabolic flux]] refers to the rate of flow of metabolites along a [[metabolic pathway]], or even through a single [[enzyme]]. A calculation may also be made of carbon (or other elements, e.g. nitrogen) flux. It is dependent on a number of factors, including: enzyme concentration; the concentration of precursor, product, and intermediate metabolites; [[post-translational modification]] of enzymes; and the presence of metabolic activators or repressors. [[Metabolic control analysis]] and [[flux balance analysis]] provide frameworks for understanding metabolic fluxes and their constraints.
==See also==
<div style="-moz-column-count:3; column-count:3;">
*[[Explosively pumped flux compression generator]]
*[[Fast Flux Test Facility]]
*[[Fluid dynamics]]
*[[Flux quantization]]
*[[Flux pinning]]
*[[Gauss's law]]
*[[Inverse-square law]]
*[[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 flow rate]]
*[[Fluence]] (flux for particle beams)
*[[Flux footprint]]
</div>
==References==
{{Reflist}}
==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 <!--Retrieved from CrossRef by DOI bot-->}}
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