Flux describes any effect that appears to pass or travel (whether it actually moves or not) through asurface or substance. Flux is a concept inapplied mathematics andvector calculus which has many applications inphysics. Fortransport phenomena, flux is avector quantity, describing the magnitude and direction of the flow of a substance or property. Invector calculus flux is ascalar quantity, defined as thesurface integral of the perpendicular component of avector field over a surface.[1]
The wordflux comes fromLatin:fluxus means "flow", andfluere is "to flow".[2] Asfluxion, this term was introduced intodifferential calculus byIsaac Newton.
The concept ofheat flux was a key contribution ofJoseph Fourier, in the analysis of heat transfer phenomena.[3] His seminal treatiseThéorie analytique de la chaleur (The Analytical Theory of Heat),[4] definesfluxion 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 ofJames Clerk Maxwell,[5] that the transport definition precedes thedefinition of flux 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 thesurface 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, fluxis 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 correspondingflux density, if that term is used, refers to its derivative along the surface that was integrated. By theFundamental theorem of calculus, the correspondingflux density is a flux according to the transport definition. Given acurrent 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 offlux, and the interchangeability offlux,flow, andcurrent 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.
Intransport phenomena (heat transfer,mass transfer andfluid dynamics), flux is defined as therate of flow of a property per unit area, which has thedimensions [quantity]·[time]−1·[area]−1.[6] 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.
Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbolj, (orJ) is used for flux,q for thephysical quantity that flows,t for time, andA for area. These identifiers will be written in bold when and only when they are vectors.
First, flux as a (single) scalar:whereIn this case the surface in which flux is being measured is fixed and has areaA. 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 ascalar field defined along a surface, i.e. a function of points on the surface: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 ofp, a point on the surface, andA, an area. Rather than measure the total flow through the surface,q measures the flow through the disk with areaA centered atp along the surface.
Finally, flux as avector field: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), 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 anabuse of notation because the "arg max" cannot directly compare vectors; we take the vector with the biggest norm instead.)
These direct definitions, especially the last, are rather unwieldy[citation needed]. For example, the arg max construction is artificial from the perspective of empirical measurements, when with aweathervane 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 fluxj passes through the area at an angle θ to the area normal, then thedot productThat is, the component of flux passing through the surface (i.e. normal to it) isj cos θ, while the component of flux passing tangential to the area isj sin θ, but there isno flux actually passingthrough the area in the tangential direction. Theonly component of flux passing normal to the area is the cosine component.
For vector flux, thesurface integral ofj over asurfaceS, gives the proper flowing per unit of time through the surface:whereA (and its infinitesimal) is thevector area – combination of the magnitude of the areaA through which the property passes and aunit vector 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 durationt1 tot2, getting the total amount of the property flowing through the surface in that time (t2 − t1):
Eight of the most common forms of flux from the transport phenomena literature are defined as follows:[citation needed]
These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take thedivergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. Forincompressible flow, the divergence of the volume flux is zero.
As mentioned above, chemicalmolar flux of a component A in anisothermal,isobaric system is defined inFick's law of diffusion as:where thenabla symbol ∇ denotes thegradient operator,DAB is the diffusion coefficient (m2·s−1) of component A diffusing through component B,cA is theconcentration (mol/m3) of component A.[9]
This flux has units of mol·m−2·s−1, and fits Maxwell's original definition of flux.[5]
For dilute gases, kinetic molecular theory relates the diffusion coefficientD to the particle densityn =N/V, the molecular massm, the collisioncross section, and theabsolute temperatureT bywhere the second factor is themean free path and the square root (with theBoltzmann constantk) is themean velocity of the particles.
In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.
Inquantum mechanics, particles of massm in thequantum stateψ(r,t) have aprobability density defined asSo the probability of finding a particle in a differentialvolume element d3r isThen the number of particles passing perpendicularly through unit area of across-section per unit time is the probability flux;This is sometimes referred to as the probability current or current density,[10] or probability flux density.[11]
As a mathematical concept, flux is represented by thesurface integral of a vector field,[12]whereF is avector field, and dA is thevector area of the surfaceA, directed as thesurface normal. For the second,n is the outward pointedunit normal vector to the surface.
The surface has to beorientable, 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 theright-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 positivedivergence (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, thecurve 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 theinner product of the vector field with the surface normals.
If the surface encloses a 3D region, usually the surface is oriented such that theinflux is counted positive; the opposite is theoutflux.
Thedivergence 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 thedivergence).
If the surface is not closed, it has an oriented curve as boundary.Stokes' theorem states that the flux of thecurl of a vector field is theline integral of the vector field over this boundary. This path integral is also calledcirculation, 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.
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 radiatingelectric 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 thenormal component of the electric field over a given area. Hence, units of electric flux are, in theMKS system,newtons percoulomb times meters squared, or N m2/C. (Electric flux density is the electric flux per unit area, and is a measure of strength of thenormal 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 ofelectric flux are used, one for theE-field:[13][14]
and one for theD-field (called theelectric displacement):
This quantity arises inGauss's law – which states that the flux of theelectric fieldE out of aclosed surface is proportional to theelectric chargeQA enclosed in the surface (independent of how that charge is distributed), the integral form is:
whereε0 is thepermittivity 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 chargeq isq/ε0.[15]
In free space theelectric displacement is given by theconstitutive relationD =ε0E, so for any bounding surface theD-field flux equals the chargeQA 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.
The magnetic flux density (magnetic field) having the unit Wb/m2 (Tesla) is denoted byB, andmagnetic flux is defined analogously:[13][14]with the same notation above. The quantity arises inFaraday'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:wheredℓ is an infinitesimal vectorline element of theclosed curve, withmagnitude equal to the length of theinfinitesimal line element, anddirection given by the tangent to the curve, with the sign determined by the integration direction.
The time-rate of change of the magnetic flux through a loop of wire is minus theelectromotive 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 forinductors and manyelectric generators.
Using this definition, the flux of thePoynting vectorS over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before:[14]
The flux of thePoynting vector through a surface is the electromagneticpower, orenergy per unittime, passing through that surface. This is commonly used in analysis ofelectromagnetic radiation, but has application to other electromagnetic systems as well.
Confusingly, the Poynting vector is sometimes called thepower flux, which is an example of the first usage of flux, above.[16] It has units ofwatts persquare metre (W/m2).
| Quantity | Unit | Dimension | Notes | ||
|---|---|---|---|---|---|
| Name | Symbol[nb 1] | Name | Symbol | ||
| Radiant energy | Qe[nb 2] | joule | J | M⋅L2⋅T−2 | Energy of electromagnetic radiation. |
| Radiant energy density | we | joule per cubic metre | J/m3 | M⋅L−1⋅T−2 | Radiant energy per unit volume. |
| Radiant flux | Φe[nb 2] | watt | W = J/s | M⋅L2⋅T−3 | Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power", and calledluminosity in astronomy. |
| Spectral flux | Φe,ν[nb 3] | watt perhertz | W/Hz | M⋅L2⋅T −2 | Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm−1. |
| Φe,λ[nb 4] | watt per metre | W/m | M⋅L⋅T−3 | ||
| Radiant intensity | Ie,Ω[nb 5] | watt persteradian | W/sr | M⋅L2⋅T−3 | Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is adirectional quantity. |
| Spectral intensity | Ie,Ω,ν[nb 3] | watt per steradian per hertz | W⋅sr−1⋅Hz−1 | M⋅L2⋅T−2 | Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅nm−1. This is adirectional quantity. |
| Ie,Ω,λ[nb 4] | watt per steradian per metre | W⋅sr−1⋅m−1 | M⋅L⋅T−3 | ||
| Radiance | Le,Ω[nb 5] | watt per steradian per square metre | W⋅sr−1⋅m−2 | M⋅T−3 | Radiant flux emitted, reflected, transmitted or received by asurface, per unit solid angle per unit projected area. This is adirectional quantity. This is sometimes also confusingly called "intensity". |
| Spectral radiance Specific intensity | Le,Ω,ν[nb 3] | watt per steradian per square metre per hertz | W⋅sr−1⋅m−2⋅Hz−1 | M⋅T−2 | Radiance of asurface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. This is adirectional quantity. This is sometimes also confusingly called "spectral intensity". |
| Le,Ω,λ[nb 4] | watt per steradian per square metre, per metre | W⋅sr−1⋅m−3 | M⋅L−1⋅T−3 | ||
| Irradiance Flux density | Ee[nb 2] | watt per square metre | W/m2 | M⋅T−3 | Radiant fluxreceived by asurface per unit area. This is sometimes also confusingly called "intensity". |
| Spectral irradiance Spectral flux density | Ee,ν[nb 3] | watt per square metre per hertz | W⋅m−2⋅Hz−1 | M⋅T−2 | Irradiance of asurface per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density includejansky (1 Jy =10−26 W⋅m−2⋅Hz−1) andsolar flux unit (1 sfu =10−22 W⋅m−2⋅Hz−1 =104 Jy). |
| Ee,λ[nb 4] | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiosity | Je[nb 2] | watt per square metre | W/m2 | M⋅T−3 | Radiant fluxleaving (emitted, reflected and transmitted by) asurface per unit area. This is sometimes also confusingly called "intensity". |
| Spectral radiosity | Je,ν[nb 3] | watt per square metre per hertz | W⋅m−2⋅Hz−1 | M⋅T−2 | Radiosity of asurface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. This is sometimes also confusingly called "spectral intensity". |
| Je,λ[nb 4] | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiant exitance | Me[nb 2] | watt per square metre | W/m2 | M⋅T−3 | Radiant fluxemitted by asurface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity". |
| Spectral exitance | Me,ν[nb 3] | watt per square metre per hertz | W⋅m−2⋅Hz−1 | M⋅T−2 | Radiant exitance of asurface per unit frequency or wavelength. The latter is commonly measured in W⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity". |
| Me,λ[nb 4] | watt per square metre, per metre | W/m3 | M⋅L−1⋅T−3 | ||
| Radiant exposure | He | joule per square metre | J/m2 | M⋅T−2 | Radiant energy received by asurface per unit area, or equivalently irradiance of asurface integrated over time of irradiation. This is sometimes also called "radiant fluence". |
| Spectral exposure | He,ν[nb 3] | joule per square metre per hertz | J⋅m−2⋅Hz−1 | M⋅T−1 | Radiant exposure of asurface per unit frequency or wavelength. The latter is commonly measured in J⋅m−2⋅nm−1. This is sometimes also called "spectral fluence". |
| He,λ[nb 4] | joule per square metre, per metre | J/m3 | M⋅L−1⋅T−2 | ||
| See also: | |||||
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The dictionary definition offlux at Wiktionary