Partially filled coaxial capacitor ($C=2\pi {ϵ}_0[L+({ϵ}_r-1)x]/\ln (b/a)$)

A long coaxial cylindrical capacitor of length $L$ and radii $a,b$ has a dielectric-filled overlap of length $x$ and relative permittivity ${ϵ}_r$. Neglecting end and interface fringing, its filled and vacuum portions are in parallel because both share the same two conductors. The electric potential difference $V$ produces $E_r=V/[r\ln (b/a)]$ in either portion; the electric displacement field and electrode charge differ by the local permittivity. Adding their capacitances gives the title formula. The dielectric boundary is axial while the interior electric field is radial and tangential to that interface, consistent with the electrostatic interface boundary conditions.

At fixed applied voltage, the mechanical potential of a capacitor and its source includes $-C{V}^2/2$. Increasing the liquid overlap of a partially filled coaxial capacitor therefore pulls the liquid upward. Balancing $V^2(dC/dh)/2$ against the excess hydrostatic pressure times the annular area gives the title formula for a wide external reservoir. Its differential sensitivity is $2\Delta h/V$; it vanishes at zero bias and measures the magnitude, rather than the sign, of voltage.