= Solution
For the <gravitational-electrostatic balance of equal charged spheres>, the <shell theorem> and <Coulomb's law> give $F_g=GM^2/r^2$ and $F_e=q^2/(4\pi\epsilon_0r^2)$ for well-separated spherical particles. Their separation cancels at balance. Since $M=4\pi\rho a^3/3$,
$$
\boxed{a=\left(\frac{9q^2}{64\pi^3\epsilon_0G\rho^2}\right)^{1/6}
=\left[\frac{3|q|}{4\pi\rho\sqrt{4\pi\epsilon_0G}}\right]^{1/3}.}
$$
The two charges have the same sign, so electrostatic repulsion opposes gravitational attraction. A zero charge gives no nontrivial force balance. Spherical symmetry of the external charge distribution, or the leading far-field monopole approximation, is understood.
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