Page 24 - Electromagnetics

It is helpful to separate primary or “impressed” sources, which are independent of the
fields they source, from secondary sources which result from interactions between the
sourced fields and the medium in which the fields exist. Most familiar is the conduc-
tion current set up in a conducting medium by an externally applied electric field. The
impressed source concept is particularly important in circuit theory, where independent
voltage sources are modeled as providing primary voltage excitations that are indepen-
dent of applied load. In this way they differ from the secondary or “dependent” sources
that react to the effect produced by the application of primary sources.
In applied electromagnetics the primary source may be so distant that return effects
resulting from local interaction of its impressed fields can be ignored. Other examples of
primary sources include the applied voltage at the input of an antenna, the current on a
probe inserted into a waveguide, and the currents producing a power-line field in which
a biological body is immersed.

1.3.3 Surface and line source densities
Because they are spatially averaged effects, macroscopic sources and the fields they
source cannot have true spatial discontinuities. However, it is often convenient to work
with sources in one or two dimensions. Surface and line source densities are idealizations
of actual, continuous macroscopic densities.
The entity we describe as a surface charge is a continuous volume charge distributed
in a thin layer across some surface S. If the thickness of the layer is small compared to
laboratory dimensions, it is useful to assign to each point r on the surface a quantity
describing the amount of charge contained within a cylinder oriented normal to the
surface and having infinitesimal cross section dS. We call this quantity the surface
charge density ρ s (r, t), and write the volume charge density as

ρ(r,w, t) = ρ s (r, t) f (w, ),

where w is distance from S in the normal direction and in some way parameterizes the
“thickness” of the charge layer at r. The continuous density function f (x, ) satisfies

∞

f (x, ) dx = 1
−∞
and
lim f (x, ) = δ(x).
→0
For instance, we might have

2
e −x / 2
f (x, ) = √ . (1.6)
π
With this definition the total charge contained in a cylinder normal to the surface at r
and having cross-sectional area dS is

∞
dQ(t) = [ρ s (r, t) dS] f (w, ) dw = ρ s (r, t) dS,
−∞
and the total charge contained within any cylinder oriented normal to S is

Q(t) = ρ s (r, t) dS. (1.7)
S

© 2001 by CRC Press LLC