e/1. The flux due to two
charges equals the sum of the
fluxes from each one. 2. When
two regions are joined together,
the flux through the new region
equals the sum of the fluxes
through the two parts.Note that although region and its surface had a definite physical
existence in our story — they are the planet Flatcat and the surface
of planet Flatcat — Gauss’ law is true for any region and surface we
choose, and in general, the Gaussian surface has no direct physical
significance. It’s simply a computational tool.
Rather than proving Gauss’ theorem and then presenting some
examples and applications, it turns out to be easier to show some ex-
amples that demonstrate its salient properties. Having understood
these properties, the proof becomes quite simple.
self-check K
Suppose we have a negative point charge, whose field points inward,
and we pick a Gaussian surface which is a sphere centered on that
charge. How does Gauss’ theorem apply here? .Answer, p. 106010.6.2 Additivity of flux
Figure e shows two two different ways in which flux is additive.
Figure e/1, additivity by charge, shows that we can break down a
charge distribution into two or more parts, and the flux equals the
sum of the fluxes due to the individual charges. This follows directly
from the fact that the flux is defined in terms of a dot product,E·A,
and the dot product has the additive property (a+b)·c=a·c+b·c.
To understand additivity of flux by region, e/2, we have to con-
sider the parts of the two surfaces that were eliminated when they
were joined together, like knocking out a wall to make two small
apartments into one big one. Although the two regions shared this
wall before it was removed, the area vectors were opposite: the di-
rection that is outward from one region is inward with respect to
the other. Thus if the field on the wall contributes positive flux to
one region, it contributes an equal amount of negative flux to the
other region, and we can therefore eliminate the wall to join the two
regions, without changing the total flux.
10.6.3 Zero flux from outside charges
A third important property of Gauss’ theorem is that it only
refers to the chargeinsidethe region we choose to discuss. In other
words, it asserts that any charge outside the region contributes zero
to the flux. This makes at least some sense, because a charge outside
the region will have field vectors pointing into the surface on one
side, and out of the surface on the other. Certainly there should
be at least partial cancellation between the negative (inward) flux
on one side and the positive (outward) flux on the other. But why
should this cancellation be exact?
To see the reason for this perfect cancellation, we can imagine
space as being built out of tiny cubes, and we can think of any charge
distribution as being composed of point charges. The additivity-by-
charge property tells us that any charge distribution can be handled
Section 10.6 Fields by Gauss’ law 643