Page 128 - Engineering Electromagnetics, 8th Edition

110 ENGINEERING ELECTROMAGNETICS

5.1 CURRENT AND CURRENT DENSITY
Electric charges in motion constitute a current. The unit of current is the ampere (A),
defined as a rate of movement of charge passing a given reference point (or crossing
agiven reference plane) of one coulomb per second. Current is symbolized by I, and
therefore
dQ
I = (1)
dt
Current is thus defined as the motion of positive charges, even though conduction in
metals takes place through the motion of electrons, as we will see shortly.
In field theory, we are usually interested in events occurring at a point rather
than within a large region, and we find the concept of current density, measured in
1
2
amperespersquaremeter(A/m ),moreuseful.Currentdensityisavector represented
by J.
The increment of current I crossing an incremental surface S normal to the
current density is
I = J N S

and in the case where the current density is not perpendicular to the surface,
I = J · S
Total current is obtained by integrating,

J · dS (2)
I =
S
Current density may be related to the velocity of volume charge density at a point.
Consider the element of charge Q = ρ ν ν = ρ ν S L,as shown in Figure 5.1a.
To simplify the explanation, assume that the charge element is oriented with its edges
parallel to the coordinate axes and that it has only an x component of velocity. In
the time interval t, the element of charge has moved a distance x,as indicated in
Figure 5.1b.Wehave therefore moved a charge Q = ρ ν S x through a reference
plane perpendicular to the direction of motion in a time increment t, and the resulting
current is
Q x
I = = ρ ν S
t t
As we take the limit with respect to time, we have

I = ρ ν Sv x

1 Current is not a vector, for it is easy to visualize a problem in which a total current I in a conductor of
nonuniform cross section (such as a sphere) may have a different direction at each point of a given
cross section. Current in an exceedingly fine wire, or a filamentary current,is occasionally defined as a
vector, but we usually prefer to be consistent and give the direction to the filament, or path, and not to
the current.