There is considerable literature on this question, and closely related variations.
See: 

 - The [*Thompson problem*](https://en.wikipedia.org/wiki/Thomson_problem): Which configurations of $n$ electrons on a sphere minimize the electrostatic potential energy?
 - The [*Tammes problem*](https://en.wikipedia.org/wiki/Tammes_problem): Which configurations of $n$ points on a sphere maximizes the smallest distance between any two points? Sometimes phrased as packing $n$ congruent circles on a sphere:

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![DiskPacking][1]
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(Image from [Paul Sutcliffe](http://www.maths.dur.ac.uk/~dma0pms/my-projects/proj4_1213.html).)
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According to

> Musin, Oleg R., and Alexey S. Tarasov. "The Tammes problem for $N=14$." [arXiv:1410.2536 Abstract](http://arxiv.org/abs/1410.2536) (2014).

the Tammes problem is solved exactly for

 - For $n=3,4,6,12$ by L. Fejes Toth (1943). 
 - For $n=5,7,8,9$ by Sch&uuml;tte and van der Waerden (1951).
 - For $n=10,11$ by Danzer (1963). **Added** (*8Sep15*): Exact radius for $n=10$ by Sugimoto & Tanemura.
 - For $n=24$ by Robinson (1961).
 - For $n=13, 14$ by Musin and Tarasov (2014).
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[![N=14.][2]][2]
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Fig.1 from Musin & Tarasov: $n=14$.
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**Added** (*8Sep15*): The exact radius for $n=10$ was just found:

> Teruhisa Sugimoto, Masaharu Tanemura.
"Exact value of Tammes problem for N=10." Sep 2015.
[arXiv 1509.01768 Abstract](http://arxiv.org/abs/1509.01768).

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[![STFig1b][3]][3]
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Fig.1b from Sugimoto & Tanemura.
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  [1]: https://i.sstatic.net/fxBXg.jpg
  [2]: https://i.sstatic.net/bHCTF.png
  [3]: https://i.sstatic.net/4EtH1.png