Coaxial cylindrical capacitor

For coaxial cylinders of radii $a<b$ and length $L\gg b$, carrying charges per unit length $\pm \lambda$,The field energy is $U={\lambda }^{2}L\log (b/a)/(4\pi {ϵ}_0)=QV/2$.

An isolated conducting sphere of radius $R$ initially at potential $V_0$ has charge $Q=4\pi {ϵ}_{0}R{V}_0$. Inserting it without contact into the hollow inner conductor of a long coaxial cylindrical capacitor conserves that charge. The neutral inner conductor acquires $-Q$ on its cavity wall and $+Q$ outside. With the outer conductor grounded, the annular electric field is $Q/(2\pi {ϵ}_{0}lr)$ and the inner potential is the displayed result. The original electric potential is not conserved. End effects are neglected; exact cavity fields are geometry-dependent.

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.

A driven cylindrical conductor between two grounded coaxial conductors forms two coaxial cylindrical capacitors in parallel. For radii $a,\lambda a,2a$, the capacitance per unit length is the sum $C^{\prime }=2\pi {ϵ}_0[1/\log \lambda +1/\log (2/\lambda )]$. Writing $u=\log \lambda$ and $L=\log 2$, the function $1/u+1/(L-u)$ is strictly convex on $(0,L)$ and symmetric about $L/2$. Hence $C^{\prime }$ has its unique minimum at $\lambda =\sqrt{2}$, with value $8\pi {ϵ}_0/\log 2$. As $\lambda =1+\delta$ approaches the inner conductor, $C^{\prime }\sim 2\pi {ϵ}_0/\delta$, agreeing with the local parallel-plate capacitor approximation.