Introductory Physics Volume Two

20 Electric Field 1.6
Definition: Electric Flux
The electric flux (φ e ) through a surface is
φ e = ⃗ E · ⃗A
if the field is uniform over the surface. If the field is not uniform
then one must integrate over the surface,
∫
φ e = ⃗E · dA ⃗
We see that the dot product represents the dependence of the flux
on the relative orientation of the surface and the field. When thinking
of the flux integral over a surface it can be helpful to imagine gluing
tacks to the surface with the nail part sticking out, so that you end up
with a spiky covering of the surface. Each tack represents one small
surface element dA ⃗ and the integral is the sum of the flux’s through all
of the small surface element.
Now we can state the theorem that relates the electric field to the
charge density.
Theorem: Gauss’s Law
The electric flux out of any closed surface is proportional to the
total charge enclosed within the surface.
∮
⃗E · dA ⃗ = Q in
ɛ 0
Note that the integral of the electric field over the surface does
not depend on the charge density outside the surface in any way. For
example if there is no charge inside the surface then the integral must
be zero.
Example
We can use Gauss’s law in order to find the electric field strength.
Here is an example of how this can be done. The result of this example
is also generally useful.