ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 15

B = a x B x + a y B y + a z B z

B × A

A × B

A = a x A x + a y A y + a z A z

Figure 2-2 Graphical representation of the cross product.

2.2.4 Vector and Scalar Fields
In electromagnetic theory, a field is defined as a mathematical function of space
and time. Fields can be classified as either scalar or vector. A scalar field has a
specific value (magnitude) at every point in a region of space at each instance in
time. Figure 2-3 shows two examples of a scalar field, temperature in a block of
material and the voltage across a resistive strip. Note that each point P ( x , y , z ),
there exists a corresponding temperature T ( x , y , z ) or voltage v ( x ) at any instant
in time. Other examples of scalar fields are pressure and density. A vector field
has a variable magnitude and direction at any point in time, as illustrated with
Figure 2-4. Note that the velocity and direction of the fluid inside the pipe changes
in the vicinity of the neck-down region, so the magnitude and direction (phase)
of the vectors that describe the motion of the fluid at a given instant in time are a
function of the position in space. Other examples of vector fields are acceleration
and electric and magnetic fields.

2.2.5 Flux
A vector field, F ( x , y , z , t ), can be represented graphically by depicting a large
number of individual vectors with a specific magnitude and phase (direction);
however, this is cumbersome. A more useful method for representation of a
vector field is to use the concept of flux. Flux is a measure of how many field
vectors pass though a surface in space, as depicted in Figure 2-5a. The net flux
of vector field

F

through surface S is shown as

S F ·

ψ = d s (2-17)