[Physics] A point between two charges has an electric potential of zero, but a charge placed at this point will gain kinetic energy. Why

Question

If I have a positive charge of +q and a negative charge of -q that are set a distance of r apart from each other, their midpoint will have an electric potential of zero. However, if I put a test charge into this midpoint and release it, it will gain kinetic energy, and I am under the impression that it comes from some sort of potential energy. But according to my book, that midpoint that it was placed at has no potential? Where does the kinetic energy come from?

Answer 1

The actual value of the potential is not meaningful, what is meaningful is the gradient: the field on a particle is given by $\vec{E} = -\vec{\nabla} V$. Equivalently, what is meaningful is only the potential difference between 2 points.

Notice that one can define a new potential differing from the old by an additive constant: $V' = V + c$ and the force will still be the same. So we have a redundancy in the system when describing potential energy, this is a gauge degree of freedom. But nevermind the jargon for now.

In your case, the test charge (positive)'s derivative of its potential energy will give rise to a force towards the negative charge, and so it moves towards the negative charge. Conservation of energy holds, and the gain in KE must be compensated by a loss (i.e. difference) of PE.

The potential being $0$ at the midpoint is a consequence of choosing $c$ such that the potential at the point $r\to \infty$ has $V \to 0 $, but one can equally well define the potential at $\infty$ to be $1$ GeV, for example, and nothing physical would change.