Introductory Physics Volume Two

2.3 Conductors in Equilibrium 43
⊲ Problem 2.5
(a) Sketch the equipotentials for two like charges.
(b) Sketch the equipotentials for two opposite charges.
§ 2.3 Conductors in Equilibrium
A conductor allows it’s free charges to move through it with some
amount of ease. So, if there is a force acting on a free charge then that
charge will move. Thus if there is an electric field inside a conductor
then the free charges in the conductor will move. In some situations,
for example if the conductor is electrically insulated, the charges will
eventually reach an equilibrium configuration in which they no longer
move around. This makes sense, since if you apply a field to an isolated
conductor the charges will move around to adjust to the new force but in
the end they must settle down into the equilibrium configuration, which
is the state with the lowest energy. But this tells us something about
the electric field inside a conductor once it has reached equilibrium.
Since the charges are not moving, there can be no electric field inside
the conductor.
Theorem: Conductor in Equilibrium: Field
The electric field is zero inside a conductor at equilibrium.
This in turn tells us that there can be no net charge inside the
volume of a conductor if the conductor is in equilibrium. We can argue
this as follows. Suppose that there was a net charge in some volume
inside a conductor. By Gauss’s Law we know that there must be a
net electric flux through the surface of this volume, but the flux is the
integral of the electric field over the surface, so the electric field cannot
be zero if there is a region with a net charge. But by the previous
theorem we know that if there is an electric field inside a conductor then
the conductor is not in equilibrium. Thus there can be a net charge
inside a conductor only if the conductor is not at equilibrium.
Theorem: Conductor in Equilibrium: Charge
There is no net charge in any volume inside a conductor in equilibrium.
If a charge is placed on a conductor it will reside on the
surface of the conductor.
We can arrive at one more very important result by using the fact
that there is no field in a conductor in equilibrium. Since the field is
zero, this means that ∆V = − ∫ ⃗ E · ⃗ dr = 0 for any path inside the
conductor. This tells us that a conductor in equilibrium is all at the
same electric potential.