Electrostatic induction

Electrostatic induction is redistribution of electric charge in response to an external charged object. An isolated electrical conductor can acquire positive and negative surface regions while its total electric charge stays fixed; see the contact control in electric charge.

Conductors with net charges of the same sign can attract because their internal charge distributions respond to one another. A point-charge calculation misses this electrostatic induction contribution. Exact conducting-sphere results are given in Electrostatics of Two Charged Conducting Spheres by John Lekner.^[ref]

A small conducting sphere of radius $a$ has electric polarizability $\alpha =4\pi {ϵ}_0{a}^3=a^3/k$. In the field $E\simeq k{Q}_1/R^2$ of a much more strongly charged distant source, its leading induced contribution is $-\alpha E^2/2$, in addition to $k{Q}_1{Q}_2/R$ for its fixed net charge. Taking $-dU/dR$ gives $F_R\simeq k{Q}_1{Q}_2/R^2-2k{Q}_1^2{a}^3/R^5$. The approximation requires $a/R\ll 1$ and control of the source's finite-size corrections; the threshold is not an exact finite-sphere result.