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Encyclopedia > Flux

In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks. Flux comes from Latin and means flowing. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ...

• 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, the volumetric flow rate.[1] Flux in this definition is a vector.
• In the field of electromagnetism, flux is usually the integral of a vector quantity over a finite surface. The result of this integration is a scalar quantity.[2] 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.[3] It has units of watts per square metre (W/m2)

One could argue, based on the work of James Clerk Maxwell[4], 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". The first edition of Transport Phenomena was published in 1960, two years after having been preliminarily published under the title Notes on Transport Phenomena based on mimeographed notes prepared for a chemical engineering course taught at the University of Wisconsin during the academic year 1957-1958. ... In thermal physics, heat transfer is the passage of thermal energy from a hot to a cold body. ... Mass transfer is the phrase commonly used in engineering for physical processes that involve molecular and convective transport of atoms and molecules within physical systems. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... In fluid dynamics, the volumetric flow rate, also volume flow rate and rate of fluid flow, is the volume of fluid which passes through a given volume per unit time (for example gallons per minute or squeaks per parsec). ... A vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but in this... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... In calculus, the integral of a function is an extension of the concept of a sum. ... A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ... Magnetic flux, represented by the Greek letter Î¦ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... The Poynting vector describes the energy flux (JÂ·mâˆ’2Â·sâˆ’1) of an electromagnetic field. ... Watts may refer to: Watt, the SI derived unit of power Watts and Co. ... A square metre (US spelling: square meter) is by definition the area enclosed by a square with sides each 1 metre long. ... James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist. ... In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ...

In addition to these common mathematically defined definitions, there are many more loose usages found in fields such as biology.

### Flux definition and theorems

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:

1. Momentum flux, the rate of transfer of momentum across a unit area (N·s·m-2·s-1). (Newtonian fluid, viscous flow)
2. Heat flux, the rate of heat flow across a unit area (J·m-2·s-1). (Fourier's Law)[5] (This definition of heat flux fits Maxwell's original definition.[4])
3. Chemical flux, the rate of movement of molecules across a unit area (mol·m-2·s-1). (Fick's law of diffusion)
4. Volumetric flux, the rate of volume flow across a unit area (m3·m-2·s-1). (Darcy's law)
5. Mass flux, the rate of mass flow across a unit area (kg·m-2·s-1). (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)
6. 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-2·s-1). Used in astronomy to determining 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.

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. In classical mechanics, momentum (pl. ... Viscosity is a measure of the resistance of a fluid to deform under shear stress. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another as a result of a difference in temperature. ... Heat flow along perfectly insulated wire Heat conduction is the transmission of heat across matter. ... Ficks laws of diffusion describe diffusion, and define the diffusion coefficient D. // Ficks laws of diffusion were derived by Adolf Fick in the year 1855. ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... In fluid dynamics, Darcys law is a phenomologically derived constitutive equation that describes the flow of a fluid through a porous medium. ... Unsolved problems in physics: What causes anything to have mass? The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. Mass is the property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... In physics, the photon (from Greek Ï†Ï‰Ï‚, phÅs, meaning light) is the quantum of the electromagnetic field; for instance, light. ... The steradian (ste from Greek stereos, solid) is the SI derived unit of solid angle, and the 3-dimensional equivalent of the radian. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In fluid mechanics, an incompressible fluid is a fluid whose density (often represented by the Greek letter &#961;) is constant: it is the same throughout the field and it does not change through time. ...

The fundamental laws that govern this process include:

• Newton's law of viscosity
• Fourier's law of convection
• Fick's law of diffusion.
• Darcy's law of groundwater flow

### Chemical diffusion

Flux, or diffusion, for gaseous molecules can be related to the function: Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

$Phi = 4pisigma_{ab}^2sqrt{frac{8kT}{pi N}}$

where

N is the total number of gaseous particles,
k is Boltzmann's constant,
T is the relative temperature in kelvins,
σab 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 Ficks's first law as: An isothermal process is a thermodynamic process in which the temperature of the system stays constant; &#916;T = 0. ... isobaric (meaning of the same weight or pressure) may refer to: in thermodynamics, an isobaric process, i. ... System (from Latin systÄ“ma, in turn from Greek systÄ“ma) is a set of entities, real or abstract, comprising a whole where each component interacts with or is related to at least one other component and they all serve a common objective. ... Ficks laws of diffusion describe diffusion, and define the diffusion coefficient D. // Ficks laws of diffusion were derived by Adolf Fick in the year 1855. ...

$overrightarrow{J_A} = -D_{AB} nabla c_A$

where

DAB is the molecular diffusion coefficient (m2/s) of component A diffusing through component B,
cA is the concentration (mol/m3) of species A.[6]

This flux has units of mol·m−2·s−1, and fits Maxwell's original definition of flux.[4] The mole (symbol: mol) is the SI base unit that measures an amount of substance. ...

Note: $nabla$ ("nabla") denotes the del operator. Nabla is a symbol, shown as . ... In vector calculus, del is a vector differential operator represented by the nabla symbol: âˆ‡. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ...

### Quantum mechanics

Main article: Probability current

In quantum mechanics, particles of mass m in the state ψ(r,t) have a probability density defined as In quantum mechanics, the probability current (sometimes called probability flux) is a useful concept which describes the flow of probability density. ... Fig. ...

$rho = psi^* psi = |psi|^2 ,$.

So the probability of finding a particle in a unit of volume, say d3x, is

$|psi|^2 d^3x ,$

Then the number of particles passing through a perpendicular unit of area per unit time is

$mathbf{J} = -i frac{h}{2m} left(psi^* nabla psi - psi nabla psi^* right) ,$

This is sometimes referred to as the "flux density".[7]

## 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.

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 of a vector field, Wikipedia does not have an article with this exact name. ... In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ...

$Phi_f = int_S mathbf{F} cdot mathbf{dA}$

where

F is a vector field,
dA is the vector area of the surface S, directed as the surface normal,
Φf  is the resulting flux.

The surface has to be 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. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In geometry, for a finite planar surface of scalar area , the vector area is defined as a vector whose magnitude is and whose direction is perpendicular to the plane, as determined by the right-hand screw rule on the rim. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ... The torus is an orientable surface. ...

The surface normal is directed accordingly, usually by the right-hand rule. The left-handed orientation is shown on the left, and the right-handed on the right. ...

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). In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

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. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

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). In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradskyâ€“Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

If the surface is not closed, it has an oriented curve as boundary. Stokes theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral is also called circulation, especially in fluid dynamics. Thus the curl is the circulation density. Stokes Theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In fluid dynamics, circulation is the path integral around a closed curve of the fluid velocity. ...

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 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. It has been suggested that optical field be merged into this article or section. ... This template is misplaced. ... Electrostatics (also known as Static Electricity) is the branch of physics that deals with the forces exerted by a static (i. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ...

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. In physics and mathematical analysis, Gausss law is the electrostatic application of the generalized Gausss theorem giving the equivalence relation between any flux, e. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

Its integral form is:

$oint_A epsilon_0 mathbf{E} cdot dmathbf{A} = Q_A$

where

$mathbf{E}$ is the electric field,
$dmathbf{A}$ is the area of a differential square on the surface A with an outward facing surface normal defining its direction,
$Q_A$ is the charge enclosed by the surface,
$epsilon_0$ is the permittivity of free space
$oint_A$ is the integral over the surface A.

Either or is called the electric flux. A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ... Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

Faraday's law of induction in integral form is: Faradays law of induction (more generally, the law of electromagnetic induction) states that the induced emf (electromotive force) in a closed loop equals the negative of the time rate of change of magnetic flux through the loop. ...

The magnetic field is denoted by . 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 inductors and many electric generators. This template is misplaced. ... Magnetic flux, represented by the Greek letter Î¦ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... Electromotive force (emf) is the amount of energy gained per unit charge that passes through a device in the opposite direction to the electric field existing across that device. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... Generator redirects here. ...

### Poynting vector

The flux of the Poynting vector through a surface is the electromagnetic 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. The Poynting vector describes the energy flux (JÂ·mâˆ’2Â·sâˆ’1) of an electromagnetic field. ... In physics, power (symbol: P) is the rate at which work is performed or energy is transferred. ... A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ...

## 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. This article or section does not cite any references or sources. ...

• 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 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 permeability constant.
• In ecology, flux is often considered at the ecosystem level - for instance, accurate determination of carbon fluxes (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.

This article or section does not cite any references or sources. ... A biological membrane or biomembrane is an enclosing or separating tissue which acts as a barrier within or around a cell. ... In chemistry, concentration is the measure of how much of a given substance there is mixed with another substance. ... Area is the measure of how much exposed area any two dimensional object has. ... Permeability has several meanings: In electromagnetism, permeability is the degree of magnetisation of a material in response to a magnetic field. ... This article or section does not cite any references or sources. ... A coral reef near the Hawaiian islands is an example of a complex marine ecosystem. ... Carbon flux is the net difference between sequestration and respiration of carbon dioxide. ... Global mean surface temperatures 1850 to 2006 Mean surface temperature anomalies during the period 1995 to 2004 with respect to the average temperatures from 1940 to 1980 Global warming is the observed increase in the average temperature of the Earths atmosphere and oceans in recent decades and the projected... Flux, or metabolic flux is the rate of turnover of molecules through a metabolic pathway or an enzyme. ... In biochemistry, a metabolic pathway is a series of chemical reactions occurring within a cell, catalyzed by enzymes, resulting in either the formation of a metabolic product to be used or stored by the cell, or the initiation of another metabolic pathway (then called a flux generating step). ... Ribbon diagram of the enzyme TIM, surrounded by the space-filling model of the protein. ... Posttranslational modification means the chemical modification of a protein after its translation. ... Metabolic control analysis is a computational method for analysing variation in fluxes and intermediate concentrations in a metabolic pathway relating to the effects of the different enzymes that constitute the pathway. ... Flux balance analysis (FBA) has been shown to be a very useful technique for analysis of metabolic capabilities of cellular systems. ...

A cutaway view of a flux compression generator. ... Aerial view of the Fast Flux Test Facility The Fast Flux Test Facility is a 400 MW nuclear test reactor owned by the U.S. Department of Energy. ... This article does not cite any references or sources. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... Flux quantization is a quantum phenomenon in which the magnetic field is quantized in the unit of , also known variously as flux quanta, fluxoids, vortices or fluxons. ... Flux pinning is the phenomenon where a magnets lines of force (called flux) become trapped or pinned inside a superconducting material. ... In physics and mathematical analysis, Gausss law is the electrostatic application of the generalized Gausss theorem giving the equivalence relation between any flux, e. ... This diagram shows how the law works. ... Latent heat flux is the flux of heat from the earths surface to the atmosphere that is associated with evaporation or condensation of water vapor at the surface; a component of the surface energy budget External links National Science Digital Library - Latent Heat Flux ... Luminous flux is a measure of the energy emitted by a light source in all directions. ... Magnetic flux, represented by the Greek letter Î¦ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... The magnetic flux quantum Î¦0 is the quantum of magnetic flux passing through a superconductor. ... neutron flux n : the rate of flow of neutrons; the number of neutrons passing through a unit area in unit time via dictionary. ... The Poynting flux (named after John Henry Poynting) gives the energy flux of an electromagnetic wave. ... The Poynting theorem is a statement due to John Henry Poynting about the conservation of energy for the electromagnetic field. ... Luminous flux or luminous power is the measure of the perceived power of light. ... In electronics, rapid single flux quantum (RSFQ) is a digital electronics technology that relies on quantum effects in superconducting materials to switch signals, instead of transistors. ... The sound energy flux is the average rate of flow of sound energy for one period through any specified area. ... In fluid dynamics, the volumetric flow rate, also volume flow rate and rate of fluid flow, is the volume of fluid which passes through a given volume per unit time (for example gallons per minute or squeaks per parsec). ... Fluence is defined as energy per unit area . ...

## References

1. ^ Bird, R. Byron; Stewart, Warren E., and Lightfoot, Edwin N. (1960). Transport Phenomena. Wiley. ISBN 0-471-07392-X.
2. ^ Lorrain, Paul; and Corson, Dale (1962). Electromagnetic Fields and Waves.
3. ^ Wangsness, Roald K. (1986). Electromagnetic Fields, 2nd ed., Wiley. ISBN 0-471-81186-6.  p.357
4. ^ a b c Maxwell, James Clerk (1892). Treatise on Electricity and Magnetism.
5. ^ Carslaw, H.S.; and Jaeger, J.C. (1959). Conduction of Heat in Solids, Second Edition, Oxford University Press. ISBN 0-19-853303-9.
6. ^ Welty; Wicks, Wilson and Rorrer (2001). Fundamentals of Momentum, Heat, and Mass Transfer, 4th ed., Wiley. ISBN 0-471-38149-7.
7. ^ Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley. ISBN 0-201-06710-2.

James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist. ...

• Stauffer, P.H. (2006). "Flux Flummoxed: A Proposal for Consistent Usage". Ground Water 44 (2): 125–128.

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