= Image charges between concentric grounded spheres
{title2=$k=b/a>1$}
For radii $a<b$ and a <point charge> $q$ at $r_0\mathbf e_z$, $a<r_0<b$, set $k=b/a$. The <method of images> gives the two families $(qk^n,r_0k^{2n}\mathbf e_z)$ and $(-qa k^n/r_0,a^2k^{2n}\mathbf e_z/r_0)$ for $n\in\mathbb Z$. The first family's $n=0$ term is the physical <point charge>. All other <image charges> lie outside the gap. Sum their <Coulomb's law> <electric potentials> with factor $1/(4\pi\epsilon_0)$.
For a short proof, reflection of $(Q,z)$ in a grounded sphere of radius $R$ gives $(-RQ/z,R^2/z)$. On $r=a$ this pairs first-family index $n$ with second-family index $-n$; on $r=b$ it pairs $n$ with $1-n$. Each pair has zero boundary <electric potential>. The negative-index <electric potential> tails are $O(k^n)$ and the positive-index tails are $O(k^{-n})$, uniformly on the closed gap after excluding the physical singularity. These <geometric series> establish <absolute convergence>, boundary cancellation, and termwise differentiation away from the physical <point charge>. <Uniqueness of Poisson equation> identifies the solution. A spherical <Gaussian surface> just outside $a$ encloses the first family with $n\leq-1$ and the second with $n\leq0$, so <Gauss's law> and the <geometric series> give $Q_a=-qa(b-r_0)/(r_0(b-a))$.
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