Flux

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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.[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 (Wm-2)

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".

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

Contents

[edit] Transport phenomena

[edit] 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:

  1. Momentum flux, the rate of transfer of momentum across a unit area (N·s·m-2·s-1). (Newton's law of viscosity,)
  2. Heat flux, the rate of heat flow across a unit area (J·m-2·s-1). (Fourier's law of convection)[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 of groundwater flow)
  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 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.
  7. Energy flux, the rate of transfer of energy through a unit area (J·m-2·s-1). 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.

[edit] Chemical diffusion

Flux, or diffusion, for gaseous molecules can be related to the function:

\Phi = 2\pi\sigma_{ab}^2\sqrt{\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:

\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]

Note: \nabla ("nabla") denotes the del operator.

[edit] 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

\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]

[edit] Electromagnetism

[edit] 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.
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,

\Phi_f = \int_S \mathbf{E} \cdot \mathbf{dA}

where:

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.

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 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.

We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.

[edit] 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.

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:

\oint_A \epsilon_0 \mathbf{E} \cdot d\mathbf{A} = Q_A

where:

  • \mathbf{E} is the electric field,
  • d\mathbf{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 \oint_A \epsilon_0 \mathbf{E} \cdot d\mathbf{A} or \oint_A \mathbf{E} \cdot d\mathbf{A} is called the electric flux.

Faraday's law of induction in integral form is:

\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}

where:


The magnetic field is denoted by \mathbf{B} . 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.

[edit] 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.

[edit] 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 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 using techniques like 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.

[edit] See also

[edit] 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. 

[edit] Further reading