![General flux diagram](https://upload.wikimedia.org/wikipedia/commons/a/ad/General_flux_diagram.svg)
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
or substance. Flux is a concept in
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
and
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
which has many applications to
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. For
transport phenomena
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechan ...
, flux is a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
quantity, describing the magnitude and direction of the flow of a substance or property. In
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
flux is a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
quantity, defined as the
surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
of the perpendicular component of a
vector field over a surface.
Terminology
The word ''flux'' comes from
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
: ''fluxus'' means "flow", and ''fluere'' is "to flow". As ''
fluxion
A fluxion is the instantaneous rate of change, or gradient, of a fluent (a time-varying quantity, or function) at a given point. Fluxions were introduced by Isaac Newton to describe his form of a time derivative (a derivative with respect to ti ...
'', this term was introduced into
differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
by
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
.
The concept of heat flux was a key contribution of
Joseph Fourier
Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French people, French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier an ...
, in the analysis of heat transfer phenomena. His seminal treatise ''Théorie analytique de la chaleur'' (''The Analytical Theory of Heat''), 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
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
,
[ that the transport definition precedes the definition of flux used in electromagnetism. The specific quote from Maxwell is:
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 fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
, 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
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechan ...
(heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
, mass transfer
Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtration, ...
and fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
), flux is defined as the ''rate of flow of a property per unit area,'' which has the dimensions
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
uantity imesup>−1·rea
REA or Rea may refer to:
Places
* Rea, Lombardy, in Italy
* Rea, Missouri, United States
* River Rea, a river in Birmingham, England
* River Rea, Shropshire, a river in Shropshire, England
* Rea, Hungarian name of Reea village in Totești Commun ...
sup>−1. 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
A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
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:
where:
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
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
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.
Finally, flux as a vector field:
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, ), 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. trictly speaking, this is an abuse of notation because the "arg max" cannot directly compare vectors; we take the vector with the biggest norm instead.]
Properties
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
A wind vane, weather vane, or weathercock is an instrument used for showing the direction of the wind. It is typically used as an architectural ornament to the highest point of a building. The word ''vane'' comes from the Old English word , m ...
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 , then
where · is the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of the unit vectors. That 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
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
of j over a surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
''S'', gives the proper flowing per unit of time through the surface.
A (and its infinitesimal) is the vector area In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an ''oriented area'' in three dimensions.
Every bounded surface in three dimensions can be associated with a ...
, combination of the magnitude of the area through which the property passes, ''A'', and a unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
normal to the area, . The relation is .
Unlike in the second set of equations, the surface here need not be flat.
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):
Transport fluxes
Eight of the most common forms of flux from the transport phenomena literature are defined as follows:
# Momentum flux
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechan ...
, the rate of transfer of momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
across a unit area (N·s·m−2·s−1). ( Newton's law of viscosity)
# Heat flux
Heat flux or thermal flux, sometimes also referred to as ''heat flux density'', heat-flow density or ''heat flow rate intensity'' is a flow of energy per unit area per unit time. In SI its units are watts per square metre (W/m2). It has both a ...
, the rate of heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
flow across a unit area (J·m−2·s−1). ( Fourier's law of conduction) (This definition of heat flux fits Maxwell's original definition.)[
# ]Diffusion flux
Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...
, the rate of movement of molecules across a unit area (mol·m−2·s−1). (Fick's law of diffusion
Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion eq ...
)
# Volumetric flux
In fluid dynamics, the volumetric flux is the rate of volume flow across a unit area (m3·s−1·m−2).
Volumetric flux has dimensions of volume/(time*area). The density of a particular property in a fluid's volume, multiplied with the volumetri ...
, the rate of volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
flow across a unit area (m3·m−2·s−1). ( Darcy's law of groundwater flow)
# Mass flux
In physics and engineering, mass flux is the rate of mass flow. Its SI units are kg m−2 s−1. The common symbols are ''j'', ''J'', ''q'', ''Q'', ''φ'', or Φ (Greek lower or capital Phi), sometimes with subscript ''m'' to indicate mass is the ...
, the rate of mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
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.)
# Radiative flux
Radiative flux, also known as radiative flux density or radiation flux (or sometimes power flux density), is the amount of Power (physics), power radiated through a given area, in the form of photons or other elementary particles, typically measure ...
, the amount of energy transferred in the form of photons
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alway ...
at a certain distance from the source per unit area per second (J·m−2·s−1). Used in astronomy to determine the magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
and spectral class
In astronomy, stellar classification is the classification of stars based on their spectral characteristics. Electromagnetic radiation from the star is analyzed by splitting it with a prism or diffraction grating into a spectrum exhibiting the ...
of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the electromagnetic spectrum.
# Energy flux Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:
# Total rate of energy transfer (not per unit area); SI units: W = J⋅s−1.
# Specific rate of energy transfe ...
, the rate of transfer of energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
through a unit area (J·m−2·s−1). The radiative flux and heat flux are specific cases of energy flux.
# Particle flux
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from su ...
, the rate of transfer of particles through a unit area (umber of particles
Umber is a natural brown earth pigment that contains iron oxide and manganese oxide. In its natural form, it is called raw umber. When calcined, the color becomes warmer and it becomes known as burnt umber.
Its name derives from ''terra d'ombr ...
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
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
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
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...
, the divergence of the volume flux is zero.
Chemical diffusion
As mentioned above, chemical molar flux of a component A in an isothermal
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and a ...
, isobaric system
In thermodynamics, an isobaric process is a type of thermodynamic process in which the pressure of the system stays constant: Δ''P'' = 0. The heat transferred to the system does work, but also changes the internal energy (''U'') of ...
is defined in Fick's law of diffusion
Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion eq ...
as:
where the nabla symbol
The nabla symbol
The nabla is a triangular symbol resembling an inverted Greek delta:Indeed, it is called ( ανάδελτα) in Modern Greek. \nabla or ∇. The name comes, by reason of the symbol's shape, from the Hellenistic Greek word f ...
∇ denotes the gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
operator, ''DAB'' is the diffusion coefficient (m2·s−1) of component A diffusing through component B, ''cA'' is the concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', ''number concentration'', an ...
( mol/m3) of component A.
This flux has units of mol·m−2·s−1, and fits Maxwell's original definition of flux.
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
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
, and the absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
''T'' by
where the second factor is the mean free path
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
and the square root (with the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas 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
In quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, particles of mass ''m'' in the quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
''ψ''(r, ''t'') have a probability density
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
defined as
So the probability of finding a particle in a differential volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV = \ ...
d3r is
Then the number of particles passing perpendicularly through unit area of a cross-section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Ab ...
per unit time is the probability flux;
This is sometimes referred to as the probability current or current density, or probability flux density.
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,
:
:
where F is a vector field, and d''A'' is the vector area In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an ''oriented area'' in three dimensions.
Every bounded surface in three dimensions can be associated with a ...
of the surface ''A'', directed as the surface normal
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
. For the second, n is the outward pointed unit normal vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction ve ...
to the surface.
The surface has to be orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
, 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
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of th ...
.
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
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
(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
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
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
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
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
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
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
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
).
If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
states that the flux of the curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
of a vector field is the line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
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
Electricity is the set of physics, physical Phenomenon, phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagne ...
(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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
component of the electric field over a given area. Hence, units of electric flux are, in the MKS system
The MKS system of units is a physical system of measurement that uses the metre, kilogram, and second (MKS) as base units. It forms the base of the International System of Units (SI), though SI has since been redefined by different fundamental ...
, newtons
The newton (symbol: N) is the unit of force in the International System of Units (SI). It is defined as 1 kg⋅m/s, the force which gives a mass of 1 kilogram an acceleration of 1 metre per second per second. It is named after Isaac Newton in r ...
per coulomb
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
times meters squared, or N m2/C. (Electric flux density is the electric flux per unit area, and is a measure of strength of the normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
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
In electromagnetism, electric flux is the measure of the electric field through a given surface, although an electric field in itself cannot flow.
The electric field E can exert a force on an electric charge at any point in space. The electric fi ...
are used, one for the E-field:
:
and one for the D-field (called the electric displacement
In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
):
:
This quantity arises in Gauss's law
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
– which states that the flux of the electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
E out of a closed surface
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...
is proportional to the electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
''QA'' enclosed in the surface (independent of how that charge is distributed), the integral form is:
:
where ''ε''0 is the permittivity of free space
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
.
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.
In free space the electric displacement
In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
is given by the constitutive relation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and appro ...
D = ''ε''0 E, so for any bounding surface the D-field flux equals the charge ''QA'' 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
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
) having the unit Wb/m2 ( Tesla) is denoted by B, and magnetic flux
In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the weber ( ...
is defined analogously:
:
with the same notation above. The quantity arises in Faraday's law of induction
Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic inducti ...
, where the magnetic flux is time-dependent either because the boundary is time-dependent or magnetic field is time-dependent. In integral form:
:
where ''d'' is an infinitesimal vector line element
In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc l ...
of the closed curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
, with magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
equal to the length of the infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
line element, and direction given by the tangent to the curve , 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
In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal or ) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical ''transd ...
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 inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s and many electric generator
In electricity generation, a generator is a device that converts motive power (mechanical energy) or fuel-based power (chemical energy) into electric power for use in an external circuit. Sources of mechanical energy include steam turbines, gas ...
s.
Poynting flux
Using this definition, the flux of the Poynting vector
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...
S over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before:
:
The flux of the Poynting vector
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...
through a surface is the electromagnetic power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
, or energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
per unit time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
, passing through that surface. This is commonly used in analysis of electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
, 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.[ p.357] It has units of watt
The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James Wa ...
s per square metre
The square metre ( international spelling as used by the International Bureau of Weights and Measures) or square meter (American spelling) is the unit of area in the International System of Units (SI) with symbol m2. It is the area of a square w ...
(W/m2).
SI radiometry units
See also
* AB magnitude The AB magnitude system is an astronomical magnitude system. Unlike many other magnitude systems, it is based on flux measurements that are calibrated in absolute units, namely spectral flux densities.
Definition
The ''monochromatic'' AB magnit ...
* Explosively pumped flux compression generator
An explosively pumped flux compression generator (EPFCG) is a device used to generate a high-power electromagnetic pulse by compressing magnetic flux using high explosive.
An EPFCG only ever generates a single pulse as the device is physically d ...
* Eddy covariance
The eddy covariance (also known as eddy correlation and eddy flux) is a key atmospheric measurement technique to measure and calculate vertical turbulent fluxes within atmospheric boundary layers. The method analyses high-frequency wind and scala ...
flux (aka, eddy correlation, eddy flux)
* Fast Flux Test Facility
The Fast Flux Test Facility (FFTF) is a 400 MW thermal, liquid sodium cooled, nuclear test reactor owned by the U.S. Department of Energy.
It does not generate electricity. It is situated in the ''400 Area'' of the Hanford Site, which is located ...
* Fluence
In radiometry, radiant exposure or fluence is the radiant energy ''received'' by a ''surface'' per unit area, or equivalently the irradiance of a ''surface,'' integrated over time of irradiation, and spectral exposure is the radiant exposure per un ...
(flux of the first sort for particle beams)
* Fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
* Flux footprint Flux footprint (also known as atmospheric flux footprint or footprint) is an upwind area where the atmospheric flux measured by an instrument is generated. Specifically, the term flux footprint describes an upwind area "seen" by the instruments meas ...
* Flux pinning
Flux pinning is a phenomenon that occurs when flux vortices in a type-II superconductor are prevented from moving within the bulk of the superconductor, so that the magnetic field lines are "pinned" to those locations. The superconductor must be a ...
* Flux quantization
The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconduct ...
* Gauss's law
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
* Inverse-square law
In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understo ...
* Jansky
The jansky (symbol Jy, plural ''janskys'') is a non- SI unit of spectral flux density, or spectral irradiance, used especially in radio astronomy. It is equivalent to 10−26 watts per square metre per hertz.
The ''flux density'' or ''mono ...
(non SI unit of spectral flux density)
* Latent heat flux
Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process — usually a first-order phase transition.
Latent heat can be understo ...
* Luminous flux
In photometry, luminous flux or luminous power is the measure of the perceived power of light. It differs from radiant flux, the measure of the total power of electromagnetic radiation (including infrared, ultraviolet, and visible light), in th ...
* Magnetic flux
In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the weber ( ...
* Magnetic flux quantum
The magnetic flux, represented by the symbol , threading some contour or loop is defined as the magnetic field multiplied by the loop area , i.e. . Both and can be arbitrary, meaning can be as well. However, if one deals with the superconducti ...
* Neutron flux
The neutron flux, φ, is a scalar quantity used in nuclear physics and nuclear reactor physics. It is the total length travelled by all free neutrons per unit time and volume. Equivalently, it can be defined as the number of neutrons travelling ...
* Poynting flux
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...
* Poynting theorem
In electrodynamics, Poynting's theorem is a statement of conservation of energy for electromagnetic fields developed by British physicist John Henry Poynting. It states that in a given volume, the stored energy changes at a rate given by the work ...
* Radiant flux
In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the Spec ...
* Rapid single flux quantum
In electronics, rapid single flux quantum (RSFQ) is a digital electronic device that uses superconducting devices, namely Josephson junctions, to process digital signals. In RSFQ logic, information is stored in the form of magnetic flux quanta ...
* Sound energy flux
Sound power or acoustic power is the rate at which sound energy is emitted, reflected, transmitted or received, per unit time. It is defined as "through a surface, the product of the sound pressure, and the component of the particle velocity, at ...
* Volumetric flux
In fluid dynamics, the volumetric flux is the rate of volume flow across a unit area (m3·s−1·m−2).
Volumetric flux has dimensions of volume/(time*area). The density of a particular property in a fluid's volume, multiplied with the volumetri ...
(flux of the first sort for fluids)
* Volumetric flow rate
In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes ). I ...
(flux of the second sort for fluids)
Notes
*
*
Further reading
*
External links
* {{Wiktionary-inline
Physical quantities
Vector calculus
Rates