Flux (as a single scalar)

Description

Flux is two separate simple and ubiquitous concepts throughout physics and applied mathematics. Within a discipline, the term is generally used consistently, but care must be taken when comparing phenomena from different disciplines. Both concepts have mathematical rigor, enabling comparison of the underlying math when the terminology is unclear. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In electromagnetism, flux is a scalar quantity, defined as the surface integral of the component of a vector field perpendicular to the surface at each point. As will be made clear, the easiest way to relate the two concepts is that the surface integral of a flux according to the first definition is a flux according to the second definition.

Terminology

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.

One could argue, based on the work of James Clerk Maxwell, that the transport definition precedes the 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.
— James Clerk Maxwell

According to the first definition, flux may be a single vector, or flux 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 second 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 first 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 second definition. Their names in accordance with the quote (and first 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 as these fields as flows/fluxes of some sort.

Given a flux according to the second 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 first definition. Given a current such as electric current—-charge per time, current density would also be a flux according to the first 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]−1·[area]−1. The area is of the surface the property is flowing “through” or “across”. For example, the magnitude of a river’s current, i.e. the amount of water that flows through a cross-section of the river each second, or the amount of sunlight that lands on a patch of ground each second, are kinds of flux.

The formula for flux as a (single) scalar is shown here.

Related formulas

Variables

jflux (A/m2)
Icurrent (A)
Aarea (m2)