Charged Particles: How is energy conserved? I can't seem to understand where the energy comes from when two particles accelerate.
I tried searching through the forums and it seems like the answer is always potential energy was the source of extra energy but I dont see how that applies here:
I have two charges q+ and q-.  They are sufficiently far apart such that electrostatic forces are negligible.  The initial velocity of q- is zero.  The initial velocity of q+ is some non-zero number.  This system's energy is equal to the kinetic energy of q+, correct?  As q+ approaches q-, the electrostatic force accelerates both particles toward each other.  The kinetic energy of both particles is increasing.  Prior to the particles colliding, there is quite a bit more energy in the system than we started with.  Where did this extra energy come from?  
I believe the answer is the energy was converted from potential energy to kinetic, but at time zero wasn't the potential energy just about zero?

Let's say the initial velocity of q+ is significant such that the kinetic energy is large.  If sufficiently far apart, the potential energy is approximately zero.
 A: The simple 'first year physics' answer is that the potential energy goes negative. The negative potential energy cancels out the positive kinetic energy, leaving the total energy equal to zero.
This might still feel unsatisfying, because it still looks like the kinetic energy is coming 'out of nowhere'. The real resolution is better. 
In this situation, 'electric potential energy' is just standing in for the actual energy of the electric field, which is
$$E = \frac{\epsilon_0}{2} \int \mathbf{E}^2 d\mathbf{x}.$$
This quantity is nonzero (in fact, formally infinite) when the charges are very far apart; let's call its value at this time $E_i$. After the charges have moved together, we can compute the final field energy $E_f$, and we find that $E_f < E_i$. So the kinetic energy of the particles has actually been extracted from the energy of the electric field.
A: Here is the simple high school level answer which knzhou's answer is better than.
The electric potential energy of a system of two point charges a distance $r$ apart is given by $$ E_E=\frac{kQq}{r}$$  If the charges have opposite signs than the potential energy will be a small negative value when they are far apart.  As they move closer together potential energy decreases by becoming a larger and larger value.
A: It's just that the sum of kinetic and potential energies are constant through all times. You can say that potential energy is zero when they are far away, so when they are approaching, the potential becomes more negative, as it is converted to kinetic. Or you can say that the potential energy is largely positive when they are far away, then reducing to zero when they are approaching (like gravity). 
The truth is there is no true basis for the exact potential energy at a specific location (as you might have remembered when solving for $mgh$, and choosing an arbitrary height such that the potential at that height is zero), because in reality, only the difference in potential energies from one point to another can be measured.
A: I see exactly where you are coming from. But where you are getting mixed up is assuming that potential energy is the same as the force being exerted on the particle. Yes as you move $-q$ away from $+q$, the force decreases to almost zero, but the potential energy doesn't decrease as the force does. So you could say that the origin of this energy is from the start of the universe when these two particles were first separated and the potential energy was first created.
