Definition:Flux/Informal Definition

Definition

Let $\mathbf V$ be a vector field which acts on a region of space $R$.

Let $S$ be a surface embedded in $R$.

The concept of flux is most easily pictured in the context of the flow of fluids.

Let there be a fluid whose flow is defined by the vector $\mathbf v$.

Imagine a small loop of wire enclosing an area element $\delta S$ placed in the fluid, perpendicular to the direction of $\mathbf v$.

The flux $F$ of fluid through $\delta S$ is the rate of flow of fluid through the loop.

In this case this is the product of speed and area:

$F = \size {\mathbf v} \delta S$

Now suppose $\delta S$ is rotated so that the normal of $\delta S$ is now at some angle $\psi$ to the direction of $\mathbf v$.

Looking along the direction of $\mathbf v$, the area through which the fluid can flow is now $\delta S \cos \phi$

$F = \size {\mathbf v} \delta S \cos \phi$

Hence by definition of dot product:

$F = \mathbf v \cdot \delta \mathbf S$

where $\delta \mathbf S$ is now considered as a vector area.

Illustration

$\qquad$

As can be seen, more flux passes through $\delta S$ when it is positioned perpendicular to the direction of $\mathbf v$, as in the diagram on the left, than when not, as in the diagram on the right.

Sources

• 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.4$ Gauss's Law: $1.4.1$ The flux of a vector field