Electric potential energy in a system

According to my notes, electric potential energy of a charge at a point is the work done in bringing a charge from infinity to that point. So say, two electrons are brought from a point that's far away to a point where they are closer together with a separation of r meters.

Each electron gains energy equivalent to the work done by the external force which happens to be $$U_E= \frac{Qq}{4\pi\mathcal{E}_0r}$$. But, then why isn't the potential energy of the system equal to twice the magnitude of $$U_E$$ but of the same magnitude?

• It's not true that each electron gains that energy. Note that potential energy is always the potential energy associated with the interactions between the objects in the system. That potential energy is not the potential energy of the single electron that's brought in from infinity. It's the potential energy associated with the interactions of that electron with all the other charges in the system. – march Jul 2 at 4:46
• So if one charge is fixed and the other charge is brought from infinity towards a point where both charges are closer together, the work done on that charge goes into the system instead? – xander Jul 2 at 4:55
• Yes. Even though the work is done on only one of the particles, the system of two particles gets the potential energy. – march Jul 2 at 5:03

Electric potential energy is stored by a system consisting of at least two charges.

Let the two charges have the labels $$1$$ and $$2$$.
The force on charge $$1$$ due to charge $$2$$ is $$\vec F_{12}$$ and the force on charge $$2$$ due to charge $$1$$ be $$\vec F_{21}$$

Newton's third law tells you that $$\vec F_{12}=-\vec F_{21}$$ so the magnitude of the forces in each of the charges is the same.

The usual next step is to say that the position of one of the charges is fixed so any eternal force acting on that charge does no work.
So if it is charge $$2$$ which is fixed in position the external force acting on it $$-\vec F_{21}$$ suffers no displacement and hence does no work.

Changing the position of charge $$1$$ relative to the fixed charge $$2$$ requires an external force $$-\vec F_{12}$$ to do work as there is a displacement of this force and the amount of work done by the external force is the change in the electric potential energy of the system of two charges.

The process could have been reversed with charge $$1$$ as the fixed charge and charge $$2$$ as the charge which is moved and if the charges started and finished at the same relative positions as when charge $$1$$ was moved then the work done and hence the change in electric potential energy be the same.