Half-immersed conducting sphere in a dielectric ($V=Q/[2\pi {ϵ}_0(1+{ϵ}_r)a]$)

An equipotential sphere centred in a flat interface between vacuum and a linear dielectric admits the same potential $C/r$ in both media. The radial field is tangential at the interface, so tangential E and normal D boundary conditions are satisfied. Displacement flux through the hemispheres gives $Q=2\pi {ϵ}_0(1+{ϵ}_r)C$, yielding the displayed sphere potential. E has equal magnitude at equal radius; D and free surface charge are larger by ${ϵ}_r$ on the dielectric side.

Integrating electrostatic pressure gives a force toward the dielectric of magnitude $Q^2({ϵ}_r-1)/[8\pi {ϵ}_0(1+{ϵ}_r{)}^2{a}^2]$. A deforming liquid interface or surface tension needs additional mechanical conditions beyond this ideal flat-interface solution.