The maximum electric charge on a drop

Question

Imagine, we can put the amount of electric charges on a drop of water. How can we measure the maximum electric charges on that? I need a mathematical relation. I know, we can imagine the drop as an isolated spherical capacitor, but I think it is not testable or veritable. I will be thankful if someone could guide me.

Answer 1

The maximum charge on a stable liquid drop was first calculated by Rayleigh in: "On the equilibrium of liquid conducting masses charged with electricity," Philos. Mag. 14 184 (1882). The calculation assumes the drop is incompressible, and has a surface tension $\gamma$. Rayleigh writes the distortions from a spherical shape in terms of a series of essentially spherical harmonics, and works out the electrostatic energy, surface energy and volume constraint. This energy is minimized by a spherical shape for charge $$ Q < \sqrt{64 \pi^2 \gamma a_0^3 \epsilon_0}$$ in SI units, where $a_0$ is the radius of the uncharged spherical drop. For charges greater than this, the $\ell =2$ quadrupole distortions become unstable.