Interaction Term in EM Lagrangian - Explained

[Anypodetos]

The (classical, relativistic) Lagrangian for electrodynamics contains the field energy density -F_μνF^μν/4 and the interaction term -A_μj^μ. I understand the maths of that - for one thing, the equations of motion turn out right if you plug this into the Euler Lagrange equantion.
Now I recall having learned that you can explain the forces between charged particles solely with the field energy: pushing 2 electrons together increases field energy because it goes with the square of the field strength, and pushing an electron and a positron together decreases field energy. If this is true, why do we need the interaction term at all? What am I missing?

 

[Orodruin]

If you do not have an interaction term, the charges do not generate a non-zero electromagnetic field.

Furthermore, it is worth noting that you are considering the static limit when you are doing that type of calculation.

 

[Anypodetos]

Perfect answer, thanks! Makes me feel a bit stupid, though.