Electricity and magnetism/Electrostatics/The potential energy of a system of electric charges

How to calculate the electric potential energy of a system of charges? A charge placed in an electric potential has an electric potential energy . If we add up the electric potential energies of all the charges with the formula , we do not find the right result. Why ? To calculate the potential energy of a system, we must calculate the energy spent or received when it is assembled. When we assemble electric charges, the electric field they must pass through is not the same at the beginning and at the end of the assembly. The potential that we calculate when the system is assembled is not the potential of the field that the charges encountered when they were assembled, it is only the potential of the field that a new charge would encounter if brought near the system. The first of the assembled charges does not encounter any electric force field to be put into place, because no charge is present before it. The second charge meets the field produced by the first. The third charge meets the field produced by the first two, and so on until the last charge, which encounters the field produced by all the previous ones. Example: the potential energy of a charged capacitor To charge a capacitor, electrons must be moved from one of its plates to the other. The first charge moved encounters no field, because the capacitor is not charged. The work that must be done to move it is therefore . If the capacitor is already charged by a charge , a new charge encounters an electric force field, therefore a potential difference between its starting point and its arrival point . So is therefore the electric potential energy of a capacitor charged by a charge , where is its capacitance and the potential difference between its plates.