= Cylinder-plane image charges
{title2=$c=\sqrt{d^2-a^2},\quad C'=2\pi\varepsilon_0/\operatorname{arcosh}(d/a)$}
An infinite conducting cylinder of radius $a$ at centre height $d>a$ above a grounded plane has the same exterior <electric potential> as opposite <line charges> $\pm\lambda$ at heights $\pm c$. If $R_+$ and $R_-$ are distances from the positive and negative lines, $V=\lambda\ln(R_-/R_+)/(2\pi\varepsilon_0)$. The plane has $R_-=R_+$; the cylinder has $R_-/R_+=(d+c)/a$. Its voltage is therefore $\lambda\operatorname{arcosh}(d/a)/(2\pi\varepsilon_0)$, giving the displayed <capacitance per unit length>. The <Uniqueness of the Dirichlet problem> fixes the exterior field. A Gaussian contour surrounding the cylinder encloses the positive image line, so $\lambda$ is also its physical charge per unit length.
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