= Sphere-plane capacitance
{title2=$C(R,h)$}
For a conducting sphere of radius $R$ at center height $h>R$ above a grounded infinite plane,
$$
C(R,h)=4\pi\epsilon R\sinh\alpha\sum_{n=1}^{\infty}\frac1{\sinh(n\alpha)},
\qquad\cosh\alpha=\frac hR.
$$
The distance $h$ is measured to the center, so the surface gap is $h-R$. Relative to the <capacitance of an isolated conducting sphere>, the enhancement is $c(R/h)=C(R,h)/(4\pi\epsilon R)$.
The <method of images> gives a constructive proof. At fixed sphere potential $V_s$, start with $q_0=4\pi\epsilon RV_s$ at height $z_0=h$ and place its opposite image below the plane. Cancel that image's nonconstant sphere potential by the successive charges
$$
q_{n+1}=\frac R{h+z_n}q_n,
\qquad z_{n+1}=h-\frac{R^2}{h+z_n},
$$
with opposite mirror charges below the plane at every step. The resulting sphere potential is $V_s$ and the plane potential is zero. Since $q_n=4\pi\epsilon RV_s\sinh\alpha/\sinh((n+1)\alpha)$, summing the sphere charges gives the displayed <capacitance>. In particular $c(y)=1+y/2+O(y^2)$ for small $y$.
The exact sphere-plane boundary-value problem is also treated in https://arxiv.org/abs/1206.6034[Behunin and colleagues' analysis of sphere-plane electrostatics].
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