  We analyze the stability of two charged conducting spheres orbiting each
other. Due to charge polarization, the electrostatic force between the two
spheres deviates significantly from $1/r^2$ as they come close to each other.
As a consequence, there exists a critical angular momentum, $L_c$, with a
corresponding critical radius $r_c$. For $L > L_c$ two circular orbits are
possible: one at $r > r_c$ that is stable and the other at $r < r_c$ that is
unstable. This critical behavior is analyzed as a function of the charge and
the size ratios of the two spheres.
