The following discussion involves Gauss' Law for
gravitation, but can of course be used for the
electric field too.
Consider a uniform volume mass density in the shape of
a sphere or an infinitely long cylinder. Using Gauss'
Law, with appropriate Gaussian surfaces, one can
easily find the form of the gravitational fields
inside these configurations.
Now, my understanding of Gauss' Law is that it only
tells me what the gravitational field is on the
Gaussian surface due to the mass inside the Gaussian
surface. It does not tell me anything about the
gravitational field on the Gaussian surface due to
mass outside the surface. So, what to do about the
mass outside for a spherical or cylindrical mass
distribution. It's not clear to me, using symmetrical
arguments, how one can establish that the mass outside
the Gaussian surface in these situations contributes
zero gravitational field on the Gaussian surface.
Perhaps I'm missing something.
One can of course start with a spherical or infinite
cylindrical shell of mass and integrate using Newton's
Law of Gravitation as is done in Tipler for a
spherical shell. Then having the knowledge that the
field is zero inside these shell distributions, one
can conclude that the mass outside the Gaussian
surface for uniform volume mass densities for
spherical or cylindrical distributions contributes
nothing on the Gaussian surface and only the mass
inside is important.
So, it appears to me, that one must first show that
the gravitational field is zero inside a spherical
shell or infinitely long cylindrical shell of mass
without using Gauss' Law, before we can use Gauss' Law
to find the field inside uniform volume mass density
spherical or infinitely long cylindrical mass
configurations.
Again, Tipler has done this with analytical calculus
using Newton's Law of gravitation and in the past, I
suggested students read this section in Tipler if they
wanted to see this is true. I, however, was looking
for an easier way for them to understand this. So, I
have written two documents, one for the spherical
shell and one for the infinite cylindrical shell,
which use numerical sums to show the fields inside are
zero and drop off accordingly outside the mass
distributions. The documents are at: