Introductory Physics Volume Two

1.6 Gauss’s Law 19
§ 1.6 Gauss’s Law
We have seen that the electric field due to any charge distribution
can be computed. From this relationship between the electric field and
the charge distribution, another one can be derived.
First we will define a new quantity, the electric flux through a
surface. To get an idea of what the flux represents first consider the
following story. You wish to catch some butterflies but you don’t have
time to chase them around, so you just set up your butterfly net on
a pole. The butterflies are migrating south for the winter. They are
flying by your house heading in a southerly direction, so you orient the
net so that it faces north. The number of butterflies you catch should
be proportional to both the density of butterflies in the air and the
area of the mouth of the net. The number of butterflies caught will
also depend on the orientation of the net relative to the direction the
butterflies are moving. For example, if the butterflies end up flying to
the south-west instead of directly south, you will not catch as many
since the net was not facing the optimal direction.
The electric flux is similar, it is the amount of electric field that
“passes through” a surface. There are three things that determine the
quantity of electric flux: the area of the surface, the magnitude of the
electric field and the orientation of the field relative to the surface.
To write this out clearly, we need to have a way to mathematically
represent the orientation of a surface. We will define a vector area ⃗ A
as the vector that has a magnitude equal to the area of the surface and
has a direction that is normal to the surface. A good picture to keep
in mind is a thumbtack,
A
A
the nail part of the tack is the vector area and the flat part of the
tack is the surface. This ends up being the best way to use a vector to
represent a surface. Now we can define the electric flux.