Page 26 - Electromagnetics Handbook

where ˆ u is a unit vector along , ρ is the radial distance from the contour in the plane
normal to , and f s (ρ, ) is a density function such as (1.8). The total current passing
through any plane normal to at r is
∞

I (t) = [J l (r, t)ˆ u(r) · ˆ u(r)] f s (ρ, )2πρ dρ = J l (r, t).
0
It is often convenient to employ singular models for continuous source densities. For
instance, it is mathematically simpler to regard a surface charge as residing only in the
surface S than to regard it as being distributed about the surface. Of course, the source
is then discontinuous since it is zero everywhere outside the surface. We may obtain a
representation of such a charge distribution by letting the thickness parameter in the
density functions recede to zero, thus concentrating the source into a plane or a line. We
describe the limit of the density function in terms of the δ-function. For instance, the
volume charge distribution for a surface charge located about the xy-plane is

ρ(x, y, z, t) = ρ s (x, y, t) f (z, ).

As → 0 we have

ρ(x, y, z, t) = ρ s (x, y, t) lim f (z, ) = ρ s (x, y, t)δ(z).
→0
It is a simple matter to represent singular source densities in this way as long as the
surface or line is easily parameterized in terms of constant values of coordinate variables.
However, care must be taken to represent the δ-function properly. For instance, the
density of charge on the surface of a cone at θ = θ 0 may be described using the distance
normal to this surface, which is given by rθ − rθ 0 :

ρ(r,θ,φ, t) = ρ s (r,φ, t)δ (r[θ − θ 0 ]) .

Using the property δ(ax) = δ(x)/a, we can also write this as

δ(θ − θ 0 )
ρ(r,θ,φ, t) = ρ s (r,φ, t) .
r
1.3.4 Charge conservation

There are four fundamental conservation laws in physics:conservation of energy, mo-
mentum, angular momentum, and charge. These laws are said to be absolute; they have
never been observed to fail. In that sense they are true empirical laws of physics.
However, in modern physics the fundamental conservation laws have come to represent
more than just observed facts. Each law is now associated with a fundamental symme-
try of the universe; conversely, each known symmetry is associated with a conservation
principle. For example, energy conservation can be shown to arise from the observation
that the universe is symmetric with respect to time; the laws of physics do not depend
on choice of time origin t = 0. Similarly, momentum conservation arises from the obser-
vation that the laws of physics are invariant under translation, while angular momentum
conservation arises from invariance under rotation.
The law of conservation of charge also arises from a symmetry principle. But instead
of being spatial or temporal in character, it is related to the invariance of electrostatic
potential. Experiments show that there is no absolute potential, only potential difference.
The laws of nature are invariant with respect to what we choose as the “reference”

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