= Charged conducting sphere in a slowly varying electric field
{title2=$U(R)\simeq kQ_1Q_2/R-kQ_1^2a^3/(2R^4)$}
A small conducting sphere of radius $a$ has <electric polarizability> $\alpha=4\pi\epsilon_0a^3=a^3/k$. In the field $E\simeq kQ_1/R^2$ of a much more strongly charged distant source, its leading induced contribution is $-\alpha E^2/2$, in addition to $kQ_1Q_2/R$ for its fixed net charge. Taking $-dU/dR$ gives $F_R\simeq kQ_1Q_2/R^2-2kQ_1^2a^3/R^5$. The approximation requires $a/R\ll1$ and control of the source's finite-size corrections; the threshold is not an exact finite-sphere result.
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