This is very classical (also known as "Thomson problem"  usually on the sphere but the same 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
 

<cite authors="Landkof, N. S.">_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](https://zbmath.org/?q=an:0253.31001).</cite>, Springer-Verlag

<cite authors="Mattila, Pertti">_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](https://zbmath.org/?q=an:0819.28004).</cite>


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 

<cite authors="Hardin, D. P.; Saff, E. B.">_Hardin, D. P.; Saff, E. B._, [**Discretizing manifolds via minimum energy points**](http://www.ams.org/notices/200410/200410-toc.html), Notices Am. Math. Soc. 51, No. 10, 1186-1194 (2004). [ZBL1095.49031](https://zbmath.org/?q=an:1095.49031).</cite> 

and references therein.