= Image charge for a grounded conducting sphere
{title2=$q'=-aq/R,\quad\mathbf p'=a^2\mathbf p/R^2$}
For a charge $q$ at $\mathbf p$, $R=|\mathbf p|>a$, outside a grounded sphere of radius $a$, the displayed image gives zero potential on the sphere and decay at infinity. On $|\mathbf x|=a$, one has $|\mathbf x-\mathbf p'|=(a/R)|\mathbf x-\mathbf p|$, proving boundary cancellation. The induced force is inward, of magnitude $q^2aR/[4\pi\varepsilon_0(R^2-a^2)^2]$, from <Coulomb's law> applied to the image field. The self-field is excluded.
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