Page 207 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 189

Figure 7.7 A conductor has a total current I. The line
integral of H about the closed paths a and b is equal to
I, and the integral around path c is less than I, since the
entire current is not enclosed by the path.

each micrometer of path length are quite different. Only the final answers are the
same.
We should also consider exactly what is meant by the expression “current en-
closed by the path.” Suppose we solder a circuit together after passing the conductor
once through a rubber band, which we use to represent the closed path. Some strange
and formidable paths can be constructed by twisting and knotting the rubber band, but
if neither the rubber band nor the conducting circuit is broken, the current enclosed
by the path is that carried by the conductor. Now replace the rubber band by a circular
ring of spring steel across which is stretched a rubber sheet. The steel loop forms
the closed path, and the current-carrying conductor must pierce the rubber sheet if
the current is to be enclosed by the path. Again, we may twist the steel loop, and
we may also deform the rubber sheet by pushing our fist into it or folding it in any
waywe wish. A single current-carrying conductor still pierces the sheet once, and
this is the true measure of the current enclosed by the path. If we should thread the
conductor once through the sheet from front to back and once from back to front, the
total current enclosed by the path is the algebraic sum, which is zero.
In more general language, given a closed path, we recognize this path as the
perimeter of an infinite number of surfaces (not closed surfaces). Any current-carrying
conductor enclosed by the path must pass through every one of these surfaces once.
Certainly some of the surfaces may be chosen in such a way that the conductor pierces
them twice in one direction and once in the other direction, but the algebraic total
current is still the same.
We will find that the nature of the closed path is usually extremely simple and can
be drawn on a plane. The simplest surface is, then, that portion of the plane enclosed
by the path. We need merely find the total current passing through this region of the
plane.
The application of Gauss’s law involves finding the total charge enclosed by a
closed surface; the application of Amp`ere’s circuital law involves finding the total
current enclosed by a closed path.