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# Flux

Flux is the rate at which something flows through a surface. The amount of sunlight that lands on a patch of ground each second is a kind of flux. The magnitude of a river's current, that is, the amount of water that flows through a cross-section of the river each second is another kind of flux. In the most general terms, flux is a measure of passage: that is, how much stuff passes through an area in a period of time.

As a mathematical concept, flux is represented by the surface integral of a vector field,

$\Phi_f = K \int_S \mathbf{F} \cdot dA$

where K is a constant, F is a vector density, dA is the area element of the surface S, and Φf  is the resulting flux.

Pictorially (see image at right), the flux is a flow. The number of red arrows passing through a unit area is the vector 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 inner product of the vector density with the orientation vectors of the surface area elements.

We can apply this mathematic definition to many disciplines in which we see currents or forces applied through areas.

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## Meaning of flux

To better understand the concept of flux, imagine a butterfly net. The amount of air moving through the net at any given instant in time is the flux. If the wind is blowing hard, then the flux through the net is larger than before. If the net is made bigger, then the flux would be larger. 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; it will all be moving past it instead.

## Flux in chemistry

### Diffusion

Flux, or diffusion, for gaseous molecules can be related to the function:

$\Phi = 4\pi\sigma_{ab}^2\sqrt{\frac{8kT}{\pi N}}$

where N is the total number of gaseous particles, k is Boltzmann's constant, T is the relative temperature in Kelvins, and σab is the mean free path between the molecules a and b.

### Thermal systems

In thermal systems, the flux is the rate of heat flow .

## Flux in physics

### Maxwell's equations

The flux of electric and magnetic field lines is frequently discussed in electrostatics. This is because in Maxwell's equations in integral form involve integrals like above for electric and magnetic fields. For instance, Faraday's law of induction in integral form is:

$\oint_C \mathbf{E} \cdot d\mathbf{l} = -\int_{\partial C} \ {d\mathbf{B}\over dt} \cdot d\mathbf{s} = - \frac{d \Phi_D}{ d t}$

A consequence of Faraday's law is that a change in the magnetic flux through a loop of wire will create an electric potential difference in that wire. This is the basis for inductors and many electric generators.

For electromagnetic radiation, the flux of the Poynting vector through a surface is the power, or energy per unit time, passing through that surface.

### Fluid systems

In fluid systems the flux is the rate of fluid flow.

In fluid dynamics, flux is a physical rate process defined as the rate of flow or transfer of a physical quantity through an area per time. It is a key concept used in understanding fluid dynamics and related transport phenomena.

There are three basic fluxes used in the study of transport phenomena. Each type of flux has its own distinct unit of measurement along with distinct physical constants. The three basic forms of flux are defined as:

1. Momentum flux, the rate of momentum in and out of the system.
2. Heat flux, the rate of heat transfer.
3. Mass flux, the rate of mass transfer.

When dealing with one-dimensional flux, the fundamental laws that govern this process include:

• Newton's Law of Viscosity
• Fourier's Law of Convection
• Fick's Law of Diffusion.