From en.wikipedia.org:
[Mathematical concept applicable to physics] [the concept of flux in natural
science and mathematics]
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 in 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]
** Terminology
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,<ref name="Maxwell" /> 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.
** Flux as flow rate per unit area
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]<sup>−1</sup>·[area]<sup>−1</sup>.[5] 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.
*** General mathematical definition (transport)
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.
First, flux as a (single) scalar: <math display="block">j =
\frac{I}{A},</math> where <math display="block">I = \lim_{\Delta t \to
0}\frac{\Delta q}{\Delta t} = \frac{\mathrm{d}q}{\mathrm{d}t}.</math> 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: <math display="block">j(\mathbf{p}) = \frac{\partial
I}{\partial A}(\mathbf{p}),</math> <math display="block">I(A,\mathbf{p}) =
\frac{\mathrm{d}q}{\mathrm{d}t}(A, \mathbf{p}).</math> 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.
Finally, flux as a vector field: <math display="block">\mathbf{j}(\mathbf{p})
= \frac{\partial \mathbf{I}}{\partial A}(\mathbf{p}),</math> <math
display="block">\mathbf{I}(A,\mathbf{p}) =
\underset{\mathbf{\hat{n}}}{\operatorname{arg\,max}}\;
\mathbf{\hat{n}}_{\mathbf p} \frac{\mathrm{d}q}{\mathrm{d}t}(A,\mathbf{p},
\mathbf{\hat{n}}).</math> 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 <math>\mathbf{\hat{n}}</math>), 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 an abuse of notation
because the "arg[nnbsp]max" cannot directly compare vectors; we take the
vector with the biggest norm instead.)
**** Properties
These direct definitions, especially the last, are rather unwieldy [date=May
2025]. For example, the arg[nnbsp]max 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.
If the flux J passes through the area at an angle θ to the area normal
<math>\mathbf{\hat{n}}</math>, then the dot product <math
display="block">\mathbf{j} \cdot \mathbf{\hat{n}} = j\cos\theta.</math> That
is, the component of flux passing through the surface (i.e. normal to it) is
_j_[nnbsp]cos[nnbsp]_θ_, while the component of flux passing tangential to the
area is _j_[nnbsp]sin[nnbsp]_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: <math
display="block">\frac{\mathrm{d}q}{\mathrm{d}t} = \iint_S \mathbf{j} \cdot
\mathbf{\hat{n}}\, dA = \iint_S \mathbf{j} \cdot d\mathbf{A},</math> where A
(and its infinitesimal) is the vector area[snd] combination <math>\mathbf{A} =
A \mathbf{\hat{n}}</math> of the magnitude of the area _A_ through which the
property passes and a unit vector <math>\mathbf{\hat{n}}</math> 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 duration _t_<sub>1</sub> to
_t_<sub>2</sub>, getting the total amount of the property flowing through the
surface in that time (_t_<sub>2</sub> − _t_<sub>1</sub>): <math
display="block">q = \int_{t_1}^{t_2}\iint_S \mathbf{j}\cdot d\mathbf A\,
dt.</math>
*** Transport fluxes
Eight of the most common forms of flux from the transport phenomena literature
are defined as follows: [date=May 2025]
1. Momentum flux , the rate of transfer of momentum across a unit area (N·s·m <sup> −2 </sup> ·s <sup> −1 </sup> ). ( Newton's law of viscosity ) [6]
2. Heat flux , the rate of heat flow across a unit area (J·m <sup> −2 </sup> ·s <sup> −1 </sup> ). ( Fourier's law of conduction ) [7] (This definition of heat flux fits Maxwell's original definition.) <ref name="Maxwell" />
3. Diffusion flux , the rate of movement of molecules across a unit area (mol·m <sup> −2 </sup> ·s <sup> −1 </sup> ). ( Fick's law of diffusion ) <ref name="Physics P.M" />
4. Volumetric flux , the rate of volume flow across a unit area (m <sup> 3 </sup> ·m <sup> −2 </sup> ·s <sup> −1 </sup> ). ( Darcy's law of groundwater flow )
5. Mass flux , the rate of mass flow across a unit area (kg·m <sup> −2 </sup> ·s <sup> −1 </sup> ). (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 transferred in the form of photons at a certain distance from the source per unit area per second (J·m <sup> −2 </sup> ·s <sup> −1 </sup> ). 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 electromagnetic spectrum.
7. Energy flux , the rate of transfer of energy through a unit area (J·m <sup> −2 </sup> ·s <sup> −1 </sup> ). 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 <sup> −2 </sup> ·s <sup> −1 </sup> )
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
As mentioned above, chemical molar flux of a component A in an isothermal,
isobaric system is defined in Fick's law of diffusion as: <math
display="block">\mathbf{J}_A = -D_{AB} \nabla c_A</math> where the nabla
symbol ∇ denotes the gradient operator, _D<sub>AB</sub>_ is the diffusion
coefficient (m<sup>2</sup>·s<sup>−1</sup>) of component A diffusing through
component B, _c<sub>A</sub>_ is the concentration (mol/m<sup>3</sup>) of
component A.[8]
This flux has units of mol·m<sup>−2</sup>·s<sup>−1</sup>, and fits Maxwell's
original definition of flux.[9]
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 <math>\sigma</math>, and the absolute temperature _T_
by <math display="block">D = \frac{2}{3 n\sigma}\sqrt{\frac{kT}{\pi m}}</math>
where the second factor is the mean free path and the square root (with the
Boltzmann 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
[Probability current]
In quantum mechanics, particles of mass _m_ in the quantum state _ψ_( R, _t_)
have a probability density defined as <math display="block">\rho = \psi^* \psi
= |\psi|^2. </math> So the probability of finding a particle in a
differential volume element d<sup>3</sup> R is <math display="block"> dP =
|\psi|^2 \, d^3\mathbf{r}. </math> Then the number of particles passing
perpendicularly through unit area of a cross-section per unit time is the
probability flux; <math display="block">\mathbf{J} = \frac{i \hbar}{2m}
\left(\psi \nabla \psi^* - \psi^* \nabla \psi \right). </math> This is
sometimes referred to as the probability current or current density,[10] or
probability flux density.[11]
** Flux as a surface integral
*** General mathematical definition (surface integral)
As a mathematical concept, flux is represented by the surface integral of a
vector field,[12] <math
display=block>\Phi_F=\iint_A\mathbf{F}\cdot\mathrm{d}\mathbf{A}</math> <math
display=block>\Phi_F=\iint_A\mathbf{F}\cdot\mathbf{n}\,\mathrm{d}A</math>
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.
The surface normal is usually directed 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.
*** Electromagnetism
**** Electric flux
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<sup>2</sup>/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.)
Two forms of electric flux are used, one for the E-field:[13][14]
{{ oiint
| preintegral = <math>\Phi_E=</math> | intsubscpt = <math>{\scriptstyle
A}</math> | integrand = <math>\mathbf{E} \cdot {\rm d}\mathbf{A}</math> }} and
one for the D-field (called the electric displacement):
{{ oiint
| preintegral = <math>\Phi_D=</math> | intsubscpt = <math>{\scriptstyle
A}</math> | integrand = <math>\mathbf{D} \cdot {\rm d}\mathbf{A}</math> }}
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<sub>A</sub>_ enclosed in the surface (independent of how that charge
is distributed), the integral form is:
{{ oiint
| preintegral = | intsubscpt = <math>{\scriptstyle A}</math> | integrand =
<math>\mathbf{E} \cdot {\rm d}\mathbf{A} = \frac{Q_A}{\varepsilon_0}</math> }}
where _ε_<sub>0</sub> is the permittivity 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 charge _q_ is _q_/_ε_<sub>0</sub>.[15]
In free space the electric displacement is given by the constitutive relation
D = _ε_<sub>0</sub> E, so for any bounding surface the D-field flux equals the
charge _Q<sub>A</sub>_ 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.
**** Magnetic flux
The magnetic flux density (magnetic field) having the unit Wb/m<sup>2</sup>
(Tesla) is denoted by B, and magnetic flux is defined analogously:<ref
name="Electromagnetism 2008"/><ref name="Electrodynamics 2007"/> <math
display=block>\Phi_B=\iint_A\mathbf{B}\cdot\mathrm{d}\mathbf{A}</math> 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: <math
display=block>- \frac{{\rm d} \Phi_B}{ {\rm d} t} = \oint_{\partial A}
\mathbf{E} \cdot d \boldsymbol{\ell}</math> where _d_ [ELL] is an
infinitesimal vector line element of the closed curve <math>\partial A</math>,
with magnitude equal to the length of the infinitesimal line element, and
direction given by the tangent to the curve <math>\partial A</math>, with the
sign determined by the integration direction.
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 flux
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:<ref name="Electrodynamics 2007"/>
{{ oiint
| preintegral = <math>\Phi_S=</math> | intsubscpt = <math>{\scriptstyle
A}</math> | integrand = <math>\mathbf{S} \cdot {\rm d}\mathbf{A}</math> }}
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<sup>2</sup>).
** SI radiometry units
[SI radiometry units]
** See also
[Mathematics] [colwidth=28em]
- AB magnitude
- Explosively pumped flux compression generator
- Eddy covariance flux (aka, eddy correlation, eddy flux)
- Fast Flux Test Facility
- Fluence (flux of the first sort for particle beams)
- Fluid dynamics
- Flux footprint
- Flux pinning
- Flux quantization
- Gauss's law
- Inverse-square law
- Jansky (non SI unit of spectral flux density)
- Latent heat flux
- Luminous flux
- Magnetic flux
- Magnetic flux quantum
- Neutron flux
- Poynting flux
- Poynting theorem
- Radiant flux
- Rapid single flux quantum
- Sound energy flux
- Volumetric flux (flux of the first sort for fluids)
- Volumetric flow rate (flux of the second sort for fluids)
[div col end]
** Notes
[reflist]
- [last= Browne ]
- [ last= Purcell ]
** Further reading
- [ author=Stauffer, P.H. ]
** External links
- [Wiktionary-inline]
Category:Physical quantities Category:Vector calculus Category:Rates