Page 137 - Engineering Electromagnetics, 8th Edition

CHAPTER 5 Conductors and Dielectrics 119

5.4 CONDUCTOR PROPERTIES
AND BOUNDARY CONDITIONS
Once again, we must temporarily depart from our assumed static conditions and let
time vary for a few microseconds to see what happens when the charge distribution is
suddenly unbalanced within a conducting material. Suppose, for the sake of argument,
that there suddenly appear a number of electrons in the interior of a conductor. The
electric fields set up by these electrons are not counteracted by any positive charges,
and the electrons therefore begin to accelerate away from each other. This continues
until the electrons reach the surface of the conductor or until a number of electrons
equal to the number injected have reached the surface.
Here, the outward progress of the electrons is stopped, for the material surround-
ing the conductor is an insulator not possessing a convenient conduction band. No
charge may remain within the conductor. If it did, the resulting electric field would
force the charges to the surface.
Hence the final result within a conductor is zero charge density, and a surface
charge density resides on the exterior surface. This is one of the two characteristics
of a good conductor.
The other characteristic, stated for static conditions in which no current may flow,
follows directly from Ohm’s law: the electric field intensity within the conductor is
zero. Physically, we see that if an electric field were present, the conduction electrons
would move and produce a current, thus leading to a nonstatic condition.
Summarizing for electrostatics, no charge and no electric field may exist at any
point within a conducting material. Charge may, however, appear on the surface as a
surface charge density, and our next investigation concerns the fields external to the
conductor.
Wewishtorelatetheseexternalfieldstothechargeonthesurfaceoftheconductor.
The problem is a simple one, and we may first talk our way to the solution with a
little mathematics.
If the external electric field intensity is decomposed into two components, one
tangential and one normal to the conductor surface, the tangential component is seen
to be zero. If it were not zero, a tangential force would be applied to the elements of
the surface charge, resulting in their motion and nonstatic conditions. Because static
conditions are assumed, the tangential electric field intensity and electric flux density
are zero.
Gauss’s law answers our questions concerning the normal component. The elec-
tric flux leaving a small increment of surface must be equal to the charge residing on
that incremental surface. The flux cannot penetrate into the conductor, for the total
field there is zero. It must then leave the surface normally. Quantitatively, we may
say that the electric flux density in coulombs per square meter leaving the surface
normally is equal to the surface charge density in coulombs per square meter, or
D N = ρ S .
If we use some of our previously derived results in making a more careful analysis
(andincidentallyintroducingageneralmethodwhichwemustuselater),weshouldset
up a boundary between a conductor and free space (Figure 5.4) showing tangential