= Half-immersed conducting sphere in a dielectric
{title2=$V=Q/[2\pi\epsilon_0(1+\epsilon_r)a]$}
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 $Q=2\pi\epsilon_0(1+\epsilon_r)C$, yielding the displayed sphere potential. E has equal magnitude at equal radius; D and free surface charge are larger by $\epsilon_r$ on the dielectric side.
Integrating <electrostatic pressure> gives a <force> toward the dielectric of magnitude $Q^2(\epsilon_r-1)/[8\pi\epsilon_0(1+\epsilon_r)^2a^2]$. A deforming liquid interface or surface tension needs additional mechanical conditions beyond this ideal flat-interface solution.
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