Page 70 - Engineering Electromagnetics, 8th Edition

52 ENGINEERING ELECTROMAGNETICS

3.2 GAUSS’S LAW
The results of Faraday’s experiments with the concentric spheres could be summed up
as an experimental law by stating that the electric flux passing through any imaginary
spherical surface lying between the two conducting spheres is equal to the charge
enclosed within that imaginary surface. This enclosed charge is distributed on the
surface of the inner sphere, or it might be concentrated as a point charge at the center
of the imaginary sphere. However, because one coulomb of electric flux is produced
by one coulomb of charge, the inner conductor might just as well have been a cube or a
brass door key and the total induced charge on the outer sphere would still be the same.
Certainly the flux density would change from its previous symmetrical distribution
to some unknown configuration, but +Q coulombs on any inner conductor would
produce an induced charge of −Q coulombs on the surrounding sphere. Going one
step further, we could now replace the two outer hemispheres by an empty (but
completely closed) soup can. Q coulombs on the brass door key would produce
= Q lines of electric flux and would induce −Q coulombs on the tin can. 1
These generalizations of Faraday’s experiment lead to the following statement,
which is known as Gauss’s law:

The electric flux passing through any closed surface is equal to the total charge enclosed
by that surface.

The contribution of Gauss, one of the greatest mathematicians the world has
ever produced, was actually not in stating the law as we have, but in providing a
mathematical form for this statement, which we will now obtain.
Let us imagine a distribution of charge, shown as a cloud of point charges in
Figure 3.2, surrounded by a closed surface of any shape. The closed surface may be
the surface of some real material, but more generally it is any closed surface we wish
to visualize. If the total charge is Q, then Q coulombs of electric flux will pass through
the enclosing surface. At every point on the surface the electric-flux-density vector
D will have some value D S , where the subscript S merely reminds us that D must be
evaluated at the surface, and D S will in general vary in magnitude and direction from
one point on the surface to another.
We must now consider the nature of an incremental element of the surface. An
incremental element of area S is very nearly a portion of a plane surface, and
the complete description of this surface element requires not only a statement of its
magnitude S but also of its orientation in space. In other words, the incremental
surface element is a vector quantity. The only unique direction that may be associated
with S is the direction of the normal to that plane which is tangent to the surface
at the point in question. There are, of course, two such normals, and the ambiguity
is removed by specifying the outward normal whenever the surface is closed and
“outward” has a specific meaning.

1 If it were a perfect insulator, the soup could even be left in the can without any difference in the results.