This is very classical (also known as "Thomson problem" usually on the sphere but the same asymptotic results hold for other sets as well - Hausdorff dimension is important).
Whenever
$\inf_{\mu}\int_{A\times A}\frac{1}{|x-y|^{s}} d\mu(x) d\mu(y)<\infty,$
where infimum is taken with respect to all probability measures supported on, say a compact set $A \subset \mathbb{R}^{k}$, (this usually happens when $0\leq s <d_{H}(A),$ here $d_{H}(A)$ is the Hausdorff dimension of the set $A$) then one has $\frac{1}{N^{2}} \inf_{x_{1}, \ldots, x_{N} \in A} \sum_{1\leq i \neq j \leq N}\frac{1}{|x_{i}-x_{j}|^{s}} \to \inf_{\mu}\int_{A\times A}\frac{1}{|x-y|^{s}} d\mu(x) d\mu(y)$ as $N$ goes to infinity. In other words $s$-Riesz energy $E_{s}(A):=\inf_{x_{1}, \ldots, x_{N} \in A} \sum_{1\leq i \neq j \leq N}\frac{1}{|x_{i}-x_{j}|^{s}} \sim C N^{2}$.
The standard references is potential theory
Landkof, N. S., Foundations of modern potential theory. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften. Band 180. Berlin-Heidelberg-New York: Springer-Verlag. X, 424 p. Cloth DM 88.00; $ 27.90 (1972). ZBL0253.31001., Springer-Verlag
Mattila, Pertti, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Studies in Advanced Mathematics. 44. Cambridge: Univ. Press. xii, 343 p. (1995). ZBL0819.28004.
Another interesting scenario is when $s=d_{H}(A)$ (in this case $E_{s}(A)\sim C n^{2} \ln n$); and when $s>d_{H}(A)$ (in this case $E_{s}(A) \sim C N^{1+\frac{s}{d}}$). You may look at
Hardin, D. P.; Saff, E. B., Discretizing manifolds via minimum energy points, Notices Am. Math. Soc. 51, No. 10, 1186-1194 (2004). ZBL1095.49031.
and references therein.