Page 76 - Engineering Electromagnetics, 8th Edition

58 ENGINEERING ELECTROMAGNETICS

Figure 3.4 The gaussian
surface for an infinite uniform line
charge is a right circular cylinder of
length L and radius ρ. D is
constant in magnitude and
everywhere perpendicular to the
cylindrical surface; D is parallel to
the end faces.

giving

ρ L
D ρ =
2πρ
or

ρ L
E ρ =
2π 0 ρ
Comparing with Section 2.4, Eq. (16), shows that the correct result has been
obtained and with much less work. Once the appropriate surface has been chosen, the
integration usually amounts only to writing down the area of the surface at which D
is normal.
The problem of a coaxial cable is almost identical with that of the line charge and
is an example that is extremely difficult to solve from the standpoint of Coulomb’s
law. Suppose that we have two coaxial cylindrical conductors, the inner of radius a
and the outer of radius b, each infinite in extent (Figure 3.5). We will assume a charge
distribution of ρ S on the outer surface of the inner conductor.
Symmetry considerations show us that only the D ρ component is present and
that it can be a function only of ρ.A right circular cylinder of length L and radius ρ,
where a <ρ < b,is necessarily chosen as the gaussian surface, and we quickly have

Q = D S 2πρL