In the various subfields of
physics, there
exist two common usages of the term
flux, both
with rigorous mathematical frameworks.
- In the study of transport
phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the
amount that flows through a unit area per unit time Flux
in this definition is a vector.
- In the field of electromagnetism and mathematics, flux is usually the integral of a vector quantity over a finite surface. It
is an integral operator and acts on a
vector field as do the gradient, divergence and curl found in vector analysis. The result of this
integration is a scalar quantity.
The magnetic flux is thus the integral
of the magnetic vector field B over a surface, and the electric flux is defined similarly. Using this
definition, the flux of the Poynting
vector over a specified surface is the rate at which
electromagnetic energy flows through that surface. Confusingly, the
Poynting vector is sometimes called the power flux, which
is an example of the first usage of flux, above. It has units of
watts per square
metre (W/m2)
In this context, flux has a primary mathematical definition in
terms of a surface integral which
uses the vectors that represent the force which is causing the flux
being studied:
\mbox{Flux}=\iint_{S} \vec{F}\cdot\hat{n} dS,
where \vec{F} is the vector field,
\hat{n} is the normal unit vector which
is perpendicular to the surface S, and
dS is the differential surface element.
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 mathematical definitions, there are
many more loose usages found in fields such as biology.
Transport phenomena
Origin of the term
The word
flux comes from
Latin:
fluxus means "flow", and
fluere is "to flow". As
fluxion, this term was introduced
into
differential calculus by
Isaac Newton.
Flux definition and theorems
Flux is surface bombardment rate. There are many fluxes used in the
study of transport phenomena. Each type of flux has its own
distinct unit of measurement along with distinct physical
constants. Six of the most common forms of flux from the transport
literature are defined as:
- Momentum flux, the rate of
transfer of momentum across a unit area
(N·s·m−2·s−1). (Newton's
law of viscosity,)
- Heat flux, the rate of heat 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.)
- Chemical flux, the rate of
movement of molecules across a unit area
(mol·m−2·s−1). (Fick's law of diffusion)
- Volumetric flux, the rate of
volume flow across a unit area
(m3·m−2·s−1). (Darcy's law of groundwater flow)
- Mass flux, the rate of mass flow across a unit area
(kg·m−2·s−1). (Either an alternate form of
Fick's law that includes the molecular mass, or an alternate form
of Darcy's law that includes the density)
- Radiative flux, the amount of
energy moving in the form of photons at a
certain distance from the source per steradian per second
(J·m−2·s−1). Used in astronomy to determine
the magnitude and spectral class of a star. Also acts as a
generalization of heat flux, which is equal to the radiative flux
when restricted to the infrared spectrum.
- Energy flux, the rate of transfer of
energy through a unit area
(J·m−2·s−1). The radiative flux and heat flux
are specific cases of energy flux.
These fluxes are vectors at each point in space, and have a
definite magnitude and direction. Also, one can take the
divergence of any of these fluxes to determine
the accumulation rate of the quantity in a control volume around a
given point in space. For
incompressible flow, the divergence of
the volume flux is zero.
Chemical diffusion
Chemical molar flux of a component A in an
isothermal,
isobaric
system is defined in above-mentioned
Fick's first law as:
- \overrightarrow{J_A} = -D_{AB} \nabla c_A
where:
- *D_{AB} is the diffusion coefficient (m2/s)
of component A diffusing through component B,
- *c_A is the concentration (mol/m3) of species A.
This flux has units of mol·m
−2·s
−1, and fits
Maxwell's original definition of flux.
Note: \nabla ("
nabla") denotes the
del operator.
For dilute gases, kinetic molecular theory relates the diffusion
coefficient
D to the particle density
n =
N/
V, the molecular mass
m, the collision
cross section \sigma, and
the
absolute temperature
T by
- D = \frac{1}{3} \frac{1}{\sqrt 2 n\sigma}\sqrt{\frac{8kT}{\pi
m}}
where the second factor is the
mean free
path and the square root (with
Boltzmann's constant k) is the
mean
velocity of the particles.
In turbulent flows, the transport by eddy motion can be expressed
as a grossly increased diffusion coefficient.
Quantum mechanics
In
quantum mechanics, particles of
mass m in the state \psi(r,t) have a probability density defined as
- \rho = \psi^* \psi = |\psi|^2. \,
So the probability of finding a particle in a unit of volume, say
d^3x, is
- |\psi|^2 d^3x. \,
Then the number of particles passing through a perpendicular unit
of area per unit time is
- \mathbf{J} = -i \frac{\hbar}{2m} \left(\psi^* \nabla \psi -
\psi \nabla \psi^* \right). \,
This is sometimes referred to as the "flux density".
Electromagnetism
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.
The flux visualized.
The rings show the surface boundaries. The red arrows stand for the
flow of charges, fluid particles, subatomic particles, photons,
etc. The number of arrows that pass through each ring is the
flux.
As a mathematical concept, flux is represented by the
surface
integral of a vector field,
- \Phi_f = \int_S \mathbf{E} \cdot \mathbf{dA}
where:
- *E is a vector field of
Electric Force,
- *dA is the vector area of
the surface S, directed as the surface normal,
- *\Phi_f 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
influx is counted positive;
the opposite is the
outflux.
The
divergence theorem states
that the net outflux through a closed surface, in other words the
net outflux from a 3D region, is found by adding the local net
outflow from each point in the region (which is expressed by the
divergence).
If the surface is not closed, it has an oriented curve as boundary.
Stokes' theorem states that the flux
of the
curl of a vector field is
the
line integral of the vector field
over this boundary. This path integral is also called
circulation, especially in
fluid dynamics. Thus the curl is the circulation density.
We can apply the flux and these theorems to many disciplines in
which we see currents, forces, etc., applied through areas.
Maxwell's equations
The flux of
electric and
magnetic field lines is frequently discussed
in
electrostatics. This is because
Maxwell's equations in integral
form involve integrals like above for electric and magnetic
fields.
For instance,
Gauss's law states that
the flux of the electric field out of a closed surface is
proportional to the
electric charge
enclosed in the surface (regardless of how that charge is
distributed). The constant of proportionality is the reciprocal of
the
permittivity of free space.
Its integral form is:
- \oint_A \epsilon_0 \mathbf{E} \cdot d\mathbf{A} = Q_A
where:
- * \mathbf{E} is the electric field,
- *d\mathbf{A} is the area of a differential square on
the surface A with an outward facing surface normal defining its direction,
- * Q_A \ is the charge enclosed by the surface,
- * \epsilon_0 \ is the permittivity of free space
- *\oint_A is the integral over the surface
A.
Either \oint_A \epsilon_0 \mathbf{E} \cdot d\mathbf{A} or \oint_A
\mathbf{E} \cdot d\mathbf{A} is called the
electric
flux.
If one considers the flux of the electric field vector,
E, for a tube near a point charge in the field the
charge but not containing it with sides formed by lines tangent to
the field, the flux for the sides is zero and there is an equal and
opposite flux at both ends of the tube. This is a consequence of
Gauss's Law applied to an inverse square field. The flux for any
cross-sectional surface of the tube will be the same. The total
flux for any surface surrounding a charge q is
q/ε
0.
In free space the electric displacement vector
D =
ε
0 E so for any bounding surface the
flux of
D = q, the charge within it. Here the
expression "flux of" indicates a mathematical operation and, as can
be seen, the result is not necessarily a "flow".
Faraday's law of
induction in integral form is:
- \oint_C \mathbf{E} \cdot d\mathbf{l} = -\int_{\partial C} \
{d\mathbf{B}\over dt} \cdot d\mathbf{s} = - \frac{d \Phi_D}{ d
t}
where:
- *\mathrm{d}\mathbf{l} is an infinitesimal element (differential) of the closed curve
C (i.e. a vector with
magnitude equal to the length of the
infinitesimal line element, and
direction given by the tangent to the
curve C, with the sign determined by the integration
direction).
The
magnetic field is denoted by
\mathbf{B} . Its flux is called the
magnetic flux. The time-rate of change of the
magnetic flux through a loop of wire is minus the
electromotive force created in that
wire. The direction is such that if current is allowed to pass
through the wire, the electromotive force will cause a current
which "opposes" the change in magnetic field by itself producing a
magnetic field opposite to the change. This is the basis for
inductors and many
electric generators.
Poynting vector
The flux of the
Poynting vector
through a surface is the electromagnetic
power, or
energy per
unit
time, passing through that surface. This
is commonly used in analysis of
electromagnetic radiation, but has
application to other electromagnetic systems as well.
Biology
In general, 'flux' in
biology relates to
movement of a substance between compartments. There are several
cases where the concept of 'flux' is important.
- The movement of molecules across a membrane: in this case, flux
is defined by the rate of diffusion or
transport of a substance across a permeable membrane. Except in the case of active
transport, net flux is directly proportional to the concentration difference across the membrane,
the surface area of the membrane, and
the membrane permeability
constant.
- In ecology, flux is often considered at
the ecosystem level - for instance,
accurate determination of carbon fluxes
using techniques like eddy
covariance (at a regional and global level) is essential for
modeling the causes and consequences of global warming.
- Metabolic flux refers to the rate
of flow of metabolites along a metabolic pathway, or even through a
single enzyme. A calculation may also be made
of carbon (or other elements, e.g. nitrogen) flux. It is dependent
on a number of factors, including: enzyme concentration; the
concentration of precursor, product, and intermediate metabolites;
post-translational
modification of enzymes; and the presence of metabolic
activators or repressors. Metabolic control analysis and
flux balance analysis provide
frameworks for understanding metabolic fluxes and their
constraints.
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