Image charges for a neutral conducting sphere ($q^{\prime }=-aq/d,x^{\prime }=a^2/d,q_0=aq/d$)

A real charge $q$ at $d>a$ outside a spherical conductor of radius $a$ has the usual grounded-sphere image charge $q^{\prime }=-aq/d$ at $x^{\prime }=a^2/d$. Adding a central image $q_0=-q^{\prime }$ 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 $q/(4\pi {\epsilon }_{0}d)$. Its induced electric dipole moment is $p_x=q^{\prime }{x}^{\prime }=-q{a}^3/d^2$. Keeping the applied central field ${\mathbf{E}}_0=-q\hat{\mathbf{x}}/(4\pi {\epsilon }_0{d}^2)$ fixed gives $\mathbf{p}=4\pi {\epsilon }_0{a}^3{\mathbf{E}}_0$. As $d\to \infty$, the external field becomes uniform near the sphere; its divergent constant potential can be removed without affecting the electric field.