Minimum radius of N congruent circles on a sphere, placed optimally, such that the sphere is covered by the circles? What is the minimum (radius/radii/range of such) of N congruent circles that are placed (optimally) on a sphere in such a way that they cover the entire surface of the sphere?

For 2 circles, a easier visualization would be having two points, growing them into circles and to keep growing them until they touch each other... Then keep growing them until the... 'tunnel-thing' of area covered by the circles is as... wide as the circumference of the sphere that you want, or something...
Tarnai and Gaspar (1991) state that


  How must a sphere be covered by $n$ equal circles so that the angular radius of the circles will be as small as possible?  In this paper, conjectured solutions of this problem for $n=15$ to $20$ are given and some sporadic results for $n&gt;20$ $(n=22,26,38,42,50)$ are presented.


and


  Solutions have been established for $n=2$ to $7$ and for $n=10,12,14$; conjectured solutions exist for $n=8,9,11,13,16,20,32,72,122,132$


and they provide the references.


Reference:

T. Tarnai and Zs. Gáspár, Covering a sphere by equal circles, and the rigidity of its graph, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 110, Issue 01, July 1991, pp. 71-89.
Start with in-circles of faces of the only possible 5 Platonic solids. What you say "Circles" are spherical caps. Unless the regular polygons grow at their corners the sphere cannot be fully covered.