Dielectric-rise voltmeter ($\Delta h={ϵ}_0({ϵ}_r-1)V^2/[\rho g(b^2-a^2)\ln (b/a)]$)

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.