In class on Wednesday (1/27/99), a question arose about whether potential
differences are different for positive and negative charges placed between
two charged plates. The answer is an unequivocal "NO."
The first way to see this is by going back to what I
said when I first introduced electric potential. The reason that we have
this concept to begin with is that it does not depend on what kind of
charges we're moving through a potential difference--unlike potential
energy. It is a property of the region we're interested in and it arises
from the charge distributions that create the electric field, which are
separate from the positive and negative charges that we're talking about
now. In our case, the charges on the plates (below) create the electric field
and the potential difference. Both of these quantities (field and potential)
exist even before we put charges (positive or negative) in the region.
Recall from class that for a uniform electric field (as exists in the region
between the charged plates shown above):
if we go from the negative plate to the positive (opposite the electric field) and
if we go from the positive plate to the negative (in the same direction as the field). We say that the positive plate is at higher potential than the negative plate. This is true no matter what kind of charge we put in the middle.
What is different for the two kinds of charge in the middle is the potential
energy (and also the force). Think about gravity again. When a rock is farther
from where it wants to go, it has greater potential energy. If we put a positive
charge near the positive plate, it's far from where it wants to go. Thus it has
more potential energy than when it's near the negative plate. Because of our
positive test charge convention, positive charges have higher potential energy
where the potential is higher. Now, if we put an electron near the positive
plate, that's like putting the rock near the ground. It's already near its
goal, and thus has lower potential energy. So for negative charges, higher
potential means lower potential energy. But they both have the same potential
because V = U/q.
Perhaps the best way to visualize this paradox is to return to the example
of point charges. Recall that
Also recall that for the point charge we defined the potential energy to
be zero at when the charge q is infinitely far away from Q. In that case,
when both q and Q are positive, the potential energy is positive. It's zero
at infinity and it gets larger as you move the two charges closer together.
Remember, the more a charge (or a mass) is separated from where it wants to
be, the greater is its potential energy. For two positive charges, they want
to be as far away as possible.
When we have -q and +Q, the situation is different. The potential energy is
still zero at infinity, but now as -q moves toward +Q, it's moving
toward where it wants to go and thus is losing potential energy.
The only way that can happen is if the potential energy is negative for
attractive forces. The plot below shows the potential energy as a function
of r for both cases.
This plot clearly shows that the potential energy moves in opposite directions
for positive and negative charges. In fact, if you understand this graph,
you should have a good grasp of why electric potential is the same even though
what happens to the positive charge is different than what happens to the
negative charge. The key lies in that electric potential difference is defined
as
This means that the potential experienced by both positive and negative q is the same as they move in the vicinity of +Q. For example, as a positive charge goes from a to b, its potential energy decreases and (since it's positive), we see that it also experiences a decreasing electric potential. For a negative charge going from a to b, the potential energy decreases, but it also experiences decreasing potential because its charge is negative. The plot of V(r) for a positive point charge is shown below. As indicated, positive charges move to regions of lower potential and negative charges move to regions of higher potential.
I hope that this helps to clear up this topic. It is a tricky concept to
comprehend, so please let me know if it's still not clear.