Suppose you have two conducting spheres of different radii. Both have positive charge on them. If the spheres are far apart, of course, they repel each other in the usual Coulomb’s-Law way. You bring the two spheres closer and closer together. Does the force remain repulsive for arbitrarily small distances, or does it become attractive when the surfaces of the two spheres are sufficiently close? Does the answer depend on the values of the charges and radii?
(The idea, which should be familiar to undergraduate physics students, is that the charges move around on the surfaces of the conductors. When they’re close to each other, the positive charge on one sphere will repel the positive charge on the other, leaving negative charge nearby and potentially leading to an attractive force.)
The general solution seems to be quite difficult, but according to a Nature News article it was recently solved. The answer is that for almost all values of charges and radii, the spheres do attract each other at sufficiently close distances.
Even though the general case is quite ugly, you can turn this question into a nice puzzle, accessible to undergraduate physics students:
Show that there are some choices of charges and radii such that, at sufficiently close distances, the two spheres attract each other.
A hint, in case you need it: Consider extreme cases. That’s good advice for thinking about lots of physics problems, by the way — one of the reasons I think this is a cute puzzle to give students.