Flux

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.

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 few common mathematical definitions, there are many more looser, but equally valid, usages to describe observations from other fields such as biology, the arts, history, and humanities.

Contents

Transport phenomena

Origin of the term

The word flux comes from Latin: fluxus means "flow", and fluere is "to flow".[5] As fluxion, this term was introduced into differential calculus by Isaac Newton.

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. Seven 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 conduction)[6] (This definition of heat flux fits Maxwell's original definition.)[4]
  3. Diffusion 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.
  8. Particle flux, the rate of transfer of particles through a unit area ([number of particles] m-2·s-1)

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

Chemical molar flux of a component A in an isothermal, isobaric system is defined in above-mentioned Fick's first law as:

\mathbf{J}_A = -D_{AB} \nabla c_A

where:

  • D_{AB} is the diffusion coefficient (m2/s) of component A diffusing through component B,
  • c_A is the concentration (mol/m3) of species A.[7]

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.

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 \sigma, and the absolute temperature T by

D = \frac{2}{3 n\sigma}\sqrt{\frac{kT}{\pi M}}

where the second factor is the mean free path and the square root (with Boltzmann's constant k) is the mean velocity of the particles.

In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.

Quantum mechanics

Main article: Probability current

In quantum mechanics, particles of mass m in the state \psi(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 d^3x, 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{\hbar}{2m} \left(\psi^* \nabla \psi - \psi \nabla \psi^* \right). \,

This is sometimes referred to as the flux density.[8]

Electromagnetism

Flux definition and theorems

An example of the first 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. 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 virtual surface.

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 influx is counted positive; the opposite is the outflux.

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.

Maxwell's equations

The flux of electric and magnetic field lines is frequently discussed in electrostatics. This is because 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.

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/ε0.[9]

In free space the electric displacement vector D = ε0 E so for any bounding surface the flux of D = q, the charge within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow".

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:

  • \mathrm{d}\mathbf{l} is an infinitesimal element (differential) of the closed curve C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C, with the sign determined by the integration direction).

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.

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.

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.

See also

Notes

  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. ISBN 0716703319. 
  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. ISBN 0486606368. 
  5. ^ Weekley, Ernest (1967). An Etymological Dictionary of Modern English. Courier Dover Publications. p. 581. ISBN 0486218732 
  6. ^ Carslaw, H.S.; and Jaeger, J.C. (1959). Conduction of Heat in Solids (Second ed.). Oxford University Press. ISBN 0-19-853303-9. 
  7. ^ Welty; Wicks, Wilson and Rorrer (2001). Fundamentals of Momentum, Heat, and Mass Transfer (4th ed.). Wiley. ISBN 0-471-38149-7. 
  8. ^ Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley. ISBN 0-201-06710-2. 
  9. ^ Feynman, Richard P (1964). The Feynman Lectures on Physics. II. Addison-Wesley. pp. 4–8,9. ISBN 0738200085. 

Further reading