Electrostatic attraction per unit length of a cylinder above a plane ($f={\lambda }^2/(4\pi {\epsilon }_0\sqrt{d^2-a^2})$)

For the cylinder-plane image charges geometry, the attractive force is the displayed quantity per unit length. One derivation uses the effective energy including the voltage source: at fixed cylinder voltage $V$, the force toward increasing centre height is $F_d=V^2(d{C}^{\prime }/dd)/2$. Since $C^{\prime }=2\pi {\epsilon }_0/\mathrm{arcosh}(d/a)$ decreases with $d$, this force points toward the plane. For $d\gg a$, $\lambda \simeq 2\pi {\epsilon }_{0}V/\ln (2d/a)$ and $f\simeq \pi {\epsilon }_0{V}^2/[d{\ln }^2(2d/a)]$. A total force additionally requires the cable length.