ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 17

Vector field

Surface (S)

Flux is a measure of how
many vector field lines
pass through a surface S

(a)

Fluid velocity flux field
(b)

Figure 2-5 (a) Definition of flux; (b) example of a flux field.

Flux is, however, useful for more than just simplifying a vector field. If a
surface S is drawn in a region of space that includes flux lines, the number of
flux lines passing through that surface is a measure of several physical quantities,
such as current or power flow. Note that if (2-17) is integrated over a closed
surface, the net flux will always be zero, assuming that no sources exist within
the volume of the closed surface. This is because the same number of flux lines
enter the volume as exit it.
To illustrate the utility of the flux concept with an example, consider current
flow in a wire. Suppose that a wire contains electric charges of density ρ (C/m 3 )
in a region and the charges have a velocity ν (m/s). The current density in the
region is calculated as

J

A / m 2 = ρν (2-18)

the instantaneous rate of charge flow per unit cross-sectional area at point P in
space. For n points in space with charge densities ρ i and velocities ν i , the current
density becomes

n

J

A / m 2

= ρ i ν i (2-19)

i = 1

Therefore, the total current flowing through a surface (e.g., the cross section of
the wire) is the sum of all the current density functions within the area of the
surface times the surface area. This calculates the total number of vectors ( J )
passing though the cross-sectional surface S of the wire, which is flux. Therefore,
the flux of the current density function is the current flowing through area S and
is calculated as

S J ·

ψ i = i = d s A (2-20)