# Flux

**Flux** describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface.^{[1]}

The word *flux* comes from Latin: *fluxus* means "flow", and *fluere* is "to flow".^{[2]} As *fluxion*, this term was introduced into differential calculus by Isaac Newton.

The concept of heat flux was a key contribution of Joseph Fourier, in the analysis of heat transfer phenomena.^{[3]} His seminal treatise *Théorie analytique de la chaleur* (*The Analytical Theory of Heat*),^{[4]} defines *fluxion* 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 of James Clerk Maxwell,^{[5]} that the transport definition precedes the definition 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 the surface integral of the flux. It represents the quantity which passes through the surface.

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, flux *is* 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 corresponding **flux density**, if that term is used, refers to its derivative along the surface that was integrated. By the Fundamental theorem of calculus, the corresponding **flux density** is a flux according to the transport definition. Given a **current** 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 of *flux*, and the interchangeability of *flux*, *flow*, and *current* 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.

In transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the *rate of flow of a property per unit area,* which has the dimensions [quantity]·[time]^{−1}·[area]^{−1}.^{[6]} The area is of the surface the property is flowing "through" or "across". For example, the magnitude of a river's current, i.e. the amount of water that flows through a cross-section of the river each second, or the amount of sunlight energy that lands on a patch of ground each second, 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 symbol *j*, (or *J*) is used for flux, *q* for the physical quantity that flows, *t* for time, and *A* for area. These identifiers will be written in bold when and only when they are vectors.

In this case the surface in which flux is being measured is fixed, and has area *A*. 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 a scalar 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 of **p**, a point on the surface, and *A*, an area. Rather than measure the total flow through the surface, q measures the flow through the disk with area *A* centered at *p* along the surface.

These direct definitions, especially the last, are rather unwieldy. For example, the argmax construction is artificial from the perspective of empirical measurements, when with a Weathervane 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.

where **·** is the dot product of the unit vectors. This is, the component of flux passing through the surface (i.e. normal to it) is *j* cos *θ*, while the component of flux passing tangential to the area is *j* sin *θ*, but there is *no* flux actually passing *through* the area in the tangential direction. The *only* component of flux passing normal to the area is the cosine component.

For vector flux, the surface integral of **j** over a surface *S*, gives the proper flowing per unit of time through the surface.

Finally, we can integrate again over the time duration *t*_{1} to *t*_{2}, getting the total amount of the property flowing through the surface in that time (*t*_{2} − *t*_{1}):

Eight of the most common forms of flux from the transport phenomena literature are defined as follows:

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.

As mentioned above, chemical molar flux of a component A in an isothermal, isobaric system is defined in Fick's law of diffusion as:

where the nabla symbol ∇ denotes the gradient operator, *D _{AB}* is the diffusion coefficient (m

^{2}·s

^{−1}) of component A diffusing through component B,

*c*is the concentration (mol/m

_{A}^{3}) of component A.

^{[9]}

This flux has units of mol·m^{−2}·s^{−1}, and fits Maxwell's original definition of flux.^{[5]}

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.

In quantum mechanics, particles of mass *m* in the quantum state *ψ*(**r**, *t*) have a probability density defined as

So the probability of finding a particle in a differential volume element d^{3}**r** is

Then the number of particles passing perpendicularly through unit area of a cross-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 the surface integral of a vector field,^{[12]}

where **F** is a vector field, and d*A* is the vector area of the surface *A*, directed as the surface normal. For the second, **n** is the outward pointed unit normal vector to the surface.

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.

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.

One way to better understand the concept of flux in electromagnetism is by comparing it to 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 is 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 is parallel to the wind, then no wind will be moving through the net. The simplest way to think of flux is "how much air goes through the net", where the air is a velocity field and the net is the boundary of an imaginary surface.

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 radiating electric 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 the normal component of the electric field over a given area. Hence, units of electric flux are, in the MKS system, newtons per coulomb times meters squared, or N m^{2}/C. (Electric flux density is the electric flux per unit area, and is a measure of strength of the normal 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.)

This quantity arises in Gauss's law – which states that the flux of the electric field **E** out of a closed surface is proportional to the electric charge *Q _{A}* enclosed in the surface (independent of how that charge is distributed), the integral form is:

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 charge *q* is *q*/*ε*_{0}.^{[15]}

In free space the electric displacement is given by the constitutive relation **D** = *ε*_{0} **E**, so for any bounding surface the **D**-field flux equals the charge *Q _{A}* 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/m^{2} (Tesla) is denoted by **B**, and magnetic flux is defined analogously:^{[13]}^{[14]}

with the same notation above. The quantity arises in Faraday'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:

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.

Using this definition, the flux of the Poynting vector **S** over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before:^{[14]}

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.

Confusingly, the Poynting vector is sometimes called the *power flux*, which is an example of the first usage of flux, above.^{[16]} It has units of watts per square metre (W/m^{2}).