Flux density

Formal statement

Where flux is defined to be a surface integral over a vector field, the flux density is defined to be this flux per unit area.

When representing a vector field in terms of field lines, and the flux is the net number of field lines passing through a given surface, the flux density, as the flux per unit area, is a representation of the field itself, and one may find the terms field and flux density used interchangeably. However, the term is rarely used in modern physics, often not appearing at all in modern advanced texts, even though the concept of flux is still extant; the term is more often encountered in elementary physics courses, where the visualization of the fields in terms of field lines has more currency.

Mathematical statement

This illustration shows a hypothetical field, visualized with field lines, passing through a surface. The field's flux in a particular area is the total number of field lines in that area, and the flux density is the number of field lines per unit area. In the top image, the left field has a lower flux density than the right field since the left field has a lower flux (number of field lines) than the right field has for the same area. In the bottom image, both the left and right fields have the same flux density since the right field has twice the flux but in twice the area, so the flux per unit area is the same.

The definition of flux density is somewhat complicated by the different definitions of what flux is. Usually, flux means a flow (a quantity of something per unit area per unit time), and the term flux density is therefore redundant and never used; however, when dealing with fields a different definition for flux is used: properly, the field flux through a surface is the component of the field passing through that surface; in elementary physics, this quantity is proportional to the number of field lines passing through that surface. The flux density is then the "density" of this flux — that is, the rate of change of flux with respect to area; or, in elementary physics, the flux through a surface divided by the area of that surface.

Since the term flux density is only ever encountered when dealing with fields (where the second definition of flux is used), it is normally encountered when discussing electromagnetism, and one usually encounters the magnetic flux, often denoted Φ_B, more often than the electric flux, likewise often denoted Φ_E. The term may also be encountered when dealing with the gravitational field, and in general can be freely used in examining any field quantity.

For a field, $\vec F$, its properties may be described completely with the use of field lines. The strength of the field is then illustrated by its flux density, that is to say, the number of field lines passing through a unit area. In elementary physics,

$Flux\,\,Density,\,\,\phi = \frac{\Phi}{A},$

where Φ is the total number of field lines passing through the area, A. In more advanced contexts, the equation is written infinitesimally:

$Flux\,\,Density,\,\,\phi = \frac{d\Phi}{dA}.$

The field itself is then given by:

$\vec{F} = \phi\,\vec{n},$

where $\vec{n}$ is the unit vector along the surface normal of the infinitesimal area element, dA, as it appears in the equation for the flux density. Usually, this vector may be absorbed into the definition of flux density, which then becomes a vector quantity:

$\vec{\phi} = \frac{d\Phi}{dA}\vec{n},$

leaving the equation for the field as,

$\vec{F} = \vec{\phi}.$

This equation may look trivial, but it is important to remember that this equation tells us that the field (the left-hand side) can be represented in terms of field lines (the right-hand side).

The field's flux through some surface, S, (that is, the total number of field lines in S) is then the integral of this flux density:

$\Phi = \int_S \vec{F} \cdot d\vec{A},$

where $\vec{F}$ is the field and $d\vec{A}$ is the vector area element of the integral summation. See the article surface integral for the formalism of this integration.

Note carefully that the surface integral equation normally always has the field, $\vec{F}$, where one would sometimes expect to see the flux density, $\vec{\phi}$, if the flux density equation is integrated. This is because the field and its representation in terms of field lines are synonymous: the field lines are, in fact, just a particular, fairly clumsy, way of illustrating the properties of the field (its shape and its gradients). In this equation, we have generalized the right-hand side of the equation by referring to the field itself and not this particular way of illustrating it; the left-hand side (Φ, the field flux) still refers to the field lines representation, but it is to be expected that it too be replaced with a more physically relevant expression as the analysis continues (see Maxwell's electromagnetic field equations for what this is for electromagnetic fields).