Confusion on Concept of Electric Potentials/Potential Energy

[The Head]

I am a bit confused about electric potentials and potential energy. There are three key situations that are giving me trouble:

1)
If an electron is midway between to protons, the potential energy is -ke²/r for both, and thus the total electric potential energy is twice that value. I see how to get this from the formulas, and I realize that potential energy is not a vector quantity but a scalar, but it still boggles my mind that the electron would have potential energy. What does it have the potential to do? The electric field cancels at that point and there should be no net force, right?

2) Conversely, if the electron were midway in between a proton and an electron, the sum of the two potential energies would equal zero, yet the electron would quickly move toward the proton. Why is this, because the electron will certainly move? Does this have something to do with the fact that we can choose how to define were V=0 is? It is just strange because both charges on either end of the electron in the center are trying to get it to do the same thing (move to the proton).

3) Finally, if we have a point charge that creates a potential difference in space, if you know the electric field, you can calculate using V=Ed. But say you don't, and that you define V=0 at infinity. To me that makes sense because V=kq/r, and if r tends to infinity, V tends to zero. But what happens as r approaches zero. It seems like V would become infinite, which I don't believe it does. Is this because the point charge has a finite radius in actuality, or is there something else that I am missing?

Any help or guidance would be much appreciated-- thank you!

 

[Pengwuino]

Remember, the potential energy of a point particle given by $U = {{kq_1 q_2 }\over{r}}$ is with the assumption that the potential energy is 0 at infinity. When you have a non-zero (and negative, as you can see) potential energy, even though there is no net force acting on the electron, work must be done in order to bring that electron out to infinity, where the 0 potential was initially set. If you tried to pull the electron out to infinity, you would be forced to move through points where the attractive force by the two protons was no longer 0 and work must be done to pull the electron at.

The second situation involves the fact that particles always want to move towards lower potentials. Moving towards the proton would put the electron into a lower potential. This is natural and doesn't involve an external force pulling the electron out to infinity as in the first example. In fact, in the first example, again, the electron wants to move to an area of lower potential and that point exactly between the two protons is not the lowest potential the electron can reach. Nudging it in either direction will send the electron towards the proton you nudge it to.

For the third situation, you can only find the potential using $V = E\Delta x$ for constant electric fields (which a point-charge is not as it varies as 1/r). Also, yes, as $r \to 0$, the potential goes to infinity. However, if you don't deal with the atomic scale, your charged object always has a finite radius that you can't cross. When dealing with the atomic scale, the electron is a genuine point-particle with no radius. However, the proton does have a radius. That's actually a moot point, though, because at the atomic scale, you will run into problems modeling say, a Hydrogen atom as a point electron orbiting a proton, even with a finite radius. This requires a quantum mechanical treatment.