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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 • 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 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. Five of the most common forms of flux from the transport literature are defined as: 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. The fundamental laws that govern this process include: Chemical diffusion Flux, or diffusion, for gaseous molecules can be related to the function: where N is the total number of gaseous particles, k is Boltzmann's constant, T is the relative temperature in kelvins, 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: abla c_A where is the molecular diffusion coefficient (m2/s) of component A diffusing through component B, is the concentration (mol/m3) of species A •. This flux has units of mol·m−2·s−1, and fits Maxwell's original definition of flux•. Note: ("nabla") denotes the del operator. Quantum mechanics In quantum mechanics, particles of mass m in the state have a probability density defined as ho = psi^ Then the number of particles passing through a perpendicular unit of area per unit time is abla psi - psi abla psi^ This is sometimes referred to as the "flux density". • 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. As a mathematical concept, flux is represented by the surface integral of a vector field, where F is a vector field, dA is the vector area of the surface S, directed as the surface normal, 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. 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. Maxwells 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: where is the electric field, is the area of a differential square on the surface A with an outward facing surface normal defining its direction, is the charge enclosed by the surface, is the permittivity of free space is the integral over the surface A. Either or is called the electric flux. Faraday's law of induction in integral form is: The magnetic field density, also called magnetic flux density, 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. 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 Further reading | ||||||||
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