Let's consider a 2D electromagnetic field defined in a square domain $[0,\Lambda]^2$, with periodic boundary conditions, with a constant charge distribution, uniform all over the aforementioned domain:
$$ \mathscr{L}=-\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + j_\mu A^\mu$$ $$ j_\mu=(0,k,k) \hspace{26pt} k \in \mathbb{R} $$
I have to calculate the contribution to the energy of the system given by the charge distribution, in the limit of small $k$.
I can't figure out how to write down the energy for the charge distribution in terms of $k$. I always end up with formulas involving the stress-energy tensor, which in turns depends upon $F_{\mu \nu}$ which in turn depends upon the actual dynamics of the system. My intuition, along with analogies with some similar problems tells me that a simple (maybe approximate) elegant answer should be given in terms of $k^2$.
I have tried writing down the partition function for the system in the path integral formalism, performing the gaussian integral and evaluating the term $j_\mu (K^{-1})^{\mu \nu} j_\nu$, but that does not seem to help.
For more fun one can also consider the case where the photon has mass: $\mathscr{L}'=\mathscr{L}+m A_\mu A^\mu$ in the limit of $m \approx \Lambda$, so that Yukawa potential of the interaction for the gapful photon is still quite similar to the Coulomb one.
