This problem was invented by the physics professor at the community college near me which he used on a past final exam:
A pendulum consists of a massless rod of length L and a conducting ball of mass m at the bottom of the rod. A charge of +q is placed on this ball and the pendulum is held at a position displaced theta degrees to the left of the vertical. A second pendulum identical to the first (except that a charge of -q is placed on the ball) is mounted from the same pivot point and held at a position displaced theta degrees to the right of vertical. Then the two balls are released from rest at the same time. The two balls collide without deformation (i.e. elastic collision) and the two charges are neutralized. The question is, what angle are the pendulums relative to the vertical when they achieve their maximum height after the collision?
The professor posted this solution (which I don't agree with):
Let $y_i$ and $y_f$ be the heights of the pendulum balls above the ground at their initial and final positions (final being the max height achieved) and let r be the initial distance between the two pendulums. The work required to assemble the two charges from infinity is $k(+q)(-q)/r = -k q^2/r$. So the total initial energy is $2mgy_i – kq^2/r$ and the total final energy is $2mgy_f$ (since there no longer is any charge). Since the initial and final energies must be equal we see that $y_f < y_i$ a result that the professor admits is not intuitively obvious. He then goes on to compute the final angle (a computation I won't bore you with).
I have no training in physics other than the freshman physics course I took in college 45 years ago, but still this result did not sit well with me. As the balls are falling the balls are accelerated faster than they would be with gravity alone and this energy has to go somewhere. So I conclude that the final position will be higher than the initial position.
I attempted to explain in an email to the professor why his computation was not correct. I suggested that it was improper to treat the E field as electrostatic as he did because the rapid charge neutralization violates the electrostatic assumption of stationary or slowly moving charges. In electrostatics we have path independence meaning that the work required to move a collection of particles from position A to position B is the negative of the work required to move them back to position A. If we consider A to be the initial pendulum position and B to be any lower position, there is an electric force from A to B, but in the reverse direction there is no electric force because the charges have been neutralized. So we don't have path independence and we can't pick an arbitrary zero reference (such as infinity in this case).
I also suggested that we can't even compute the final height (or angle) with the information given because the size of the balls were not specified. Smaller balls would mean that the two charges get closer together before neutralizing and thus more energy would be gained on the way down.
Apparently the professor does not agree with my objections and is sticking with his belief that his original answer is correct. I'm hoping that I get some replies that will either solidify my position or point out the error of my ways.
Also I had some other concerns about the question that I didn't even mention to the professor. I seem to recall that an accelerating charge radiates energy, so some energy will be lost as the two charges are accelerated downwards towards the collision. I'm not so worried about that, because the balls will still be gaining energy overall due to the electric force. Perhaps the acceleration will be small enough that this energy loss could be neglected in a practical situation. However what about the charge neutralization? An elastic collision is an idealization which implies that the balls only touch for an infinitesimal time which means that the charges have to neutralize in essentially zero time. Would charges moving that fast produce an infinite B field? Is an elastic collision a practical idealization in this situation? In a real world example of such a construction would the energy loss due to the neutralization be small enough to ignore?