= Electrostatic attraction per unit length of a cylinder above a plane
{title2=$f=\lambda^2/(4\pi\varepsilon_0\sqrt{d^2-a^2})$}
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 $F_d=V^2(dC'/dd)/2$. Since $C'=2\pi\varepsilon_0/\operatorname{arcosh}(d/a)$ decreases with $d$, this force points toward the plane. For $d\gg a$, $\lambda\simeq2\pi\varepsilon_0V/\ln(2d/a)$ and $f\simeq\pi\varepsilon_0V^2/[d\ln^2(2d/a)]$. A total force additionally requires the cable length.
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