PERPETUAL BATTERY

Imagine this sequence:

1 - Two different surfaces (s and S) charged with electrons (q and Q) in the way to get the same potential (v with q/s and Q/S).

2 - Resize surface "s" to get surface "S". You don't need to do work, because you're reducing charge density (q/s to q/S), and increasing potential (v to V).

3 - Now connect the separated surfaces through a load. As one surface has q/S charge density and "V" potential, and the other Q/S and "v" potential, some electrons will flow from Q/S surface to make the potential the same.

4 - As the charges flow from Q/S surface to the q/S surface, you reduce S from the Q/S surface, to keep "v" voltage. You don't have to do work to keep the voltage as the charges go out from this surface. You reduce from Q/S to q/s keeping "v" voltage in this surface. The other surface goes from q/S (V potential) to Q/S (v potential).

A graphical scheme of this method is shown below:

As you can see, there is no need of doing work against the potential field.

When increasing the surface, you're helping the electrons to go one far from each other, so no work against, only to change the surface.

When reducing the surface, electrons are going to the other surface, so the charge density and the voltage keeps the same, so no work is needed, only the work to change the surface.