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We have a positive charge at the origin. Far out at infinity on the x-axis we have a positive charge and at minus infinity on this minus x-axis we have a negative charge. For simplicity we will assume that the plus- and minus charge do not affect each other with their fields - maybe this assumption is what gets it wrong for me. Also we will assume that the positive charge at the origin is stationary throughout the whole time.

We define the potential energy of either charge relative to infinity. Both sitting at plus and minus infinity the sum of their potential energies is initially zero. We now move the charges simultaneously towards the origin and stop the negative one at a point -a and the positive one at the point a. Let us now assume that the energy gained by the negative charge has been used to do positive work on the positive charge during this transport such that no energy has been put into the system.

Now the sum of their potential energy must still be zero.

Now if we let go of holding the charges at point a and -a one will be shot off towards infinity and one drawn towards the origin. How does this match with the conclusion that their total energy is zero - in both cases they are gaining kinetic energy.

What am I doing wrong?