= Image charges for a neutral conducting sphere
{title2=$q'=-aq/d,\quad x'=a^2/d,\quad q_0=aq/d$}
A real charge $q$ at $d>a$ outside a spherical conductor of radius $a$ has the usual grounded-sphere <image charge> $q'=-aq/d$ at $x'=a^2/d$. Adding a central image $q_0=-q'$ makes the total image charge zero, representing an isolated neutral conductor. The off-centre pair has zero potential on $r=a$; the central image therefore sets the constant conductor potential to $q/(4\pi\varepsilon_0d)$. Its induced <electric dipole moment> is $p_x=q'x'=-qa^3/d^2$. Keeping the applied central field $\mathbf E_0=-q\widehat{\mathbf x}/(4\pi\varepsilon_0d^2)$ fixed gives $\mathbf p=4\pi\varepsilon_0a^3\mathbf E_0$. As $d\to\infty$, the external field becomes uniform near the sphere; its divergent constant potential can be removed without affecting the <electric field>.
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