Well-distributed great circles on S^2

Let $C_1,\dots ,C_n$ denote the $1\mathrm{/}n-$neighborhood of $n$ great circles on ${\mathbb{S}}^2$. We are interested in how much these areas have to overlap and prove the sharp bounds $$\sum _{\genfrac{}{}{0px}{}{i,j=1}{i\mathrm{\ne }j}}^n\mathrm{\mid }{C}_i\cap C_j{\mathrm{\mid }}^s{\gtrsim }_s\{\begin{array}{ll}{n}^{2-2s}& if0\le s<2\\ n^{-2}\log n& ifs=2\\ n^{1-3s\mathrm{/}2}& ifs>2.\end{array}\mathrm{.}$$ For $s=1$ there are arrangements for which the sum of mutual overlap is uniformly bounded (for the analogous problem in ${\mathbb{R}}^2$ the lower bound is $\gtrsim \log n$) and there are strong connections to minimal energy configurations of $n$ charged electrons on ${\mathbb{S}}^2$ (the J. J. Thomson problem).