Gravitational-electrostatic balance of equal charged spheres ($q^2/(4\pi {ϵ}_0)=G{M}^2$)

For two identical spheres sufficiently far apart, Coulomb's law and Newton's law of universal gravitation both give inverse-square forces. Equal-sign charges therefore balance attraction when the displayed charge-to-mass relation holds. For uniform mass density $\rho$, substituting $M=4\pi \rho a^3/3$ fixes the radius. Changing the separation does not restore balance if the charge-to-mass ratio is wrong. Spherically symmetric charge makes the monopole result exact outside each sphere; polarization can require a far-field approximation.