Page 25 - Electromagnetics Handbook

We may describe a line charge as a thin “tube” of volume charge distributed along
some contour . The amount of charge contained between two planes normal to the
contour and separated by a distance dl is described by the line charge density ρ l (r, t).
The volume charge density associated with the contour is then
ρ(r,ρ, t) = ρ l (r, t) f s (ρ, ),
where ρ is the radial distance from the contour in the plane normal to and f s (ρ, ) is
a density function with the properties

∞
f s (ρ, )2πρ dρ = 1
0
and
δ(ρ)
lim f s (ρ, ) = .
→0 2πρ
For example, we might have
2
e −ρ / 2
f s (ρ, ) = . (1.8)
π 2
Then the total charge contained between planes separated by a distance dl is
∞

dQ(t) = [ρ l (r, t) dl] f s (ρ, )2πρ dρ = ρ l (r, t) dl
0
and the total charge contained between planes placed at the ends of a contour is

Q(t) = ρ l (r, t) dl. (1.9)

We may define surface and line currents similarly. A surface current is merely a
volume current confined to the vicinity of a surface S. The volume current density may
be represented using a surface current density function J s (r, t), defined at each point r
on the surface so that

J(r,w, t) = J s (r, t) f (w, ).
Here f (w, ) is some appropriate density function such as (1.6), and the surface current
vector obeys ˆ n · J s = 0 where ˆ n is normal to S. The total current flowing through a strip
of width dl arranged perpendicular to S at r is

∞
dI (t) = [J s (r, t) · ˆ n l (r) dl] f (w, ) dw = J s (r, t) · ˆ n l (r) dl
−∞
where ˆ n l is normal to the strip at r (and thus also tangential to S at r). The total current
passing through a strip intersecting with S along a contour is thus

I (t) = J s (r, t) · ˆ n l (r) dl.

We may describe a line current as a thin “tube” of volume current distributed about
some contour and flowing parallel to it. The amount of current passing through a
plane normal to the contour is described by the line current density J l (r, t). The volume
current density associated with the contour may be written as

J(r,ρ, t) = ˆ u(r)J l (r, t) f s (ρ, ),

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