Offset uniformly charged spherical shells ($r=\sqrt{x^2+a^2/4}$)

Place a fixed shell of radius $a$ and charge $-q$ at $(0,a/2)$ and a fixed shell of radius $2a$ and charge $+q$ at $(0,-a/2)$. On the positive $x$ axis their centre distances coincide. The electrostatic shell theorem makes the electric field vanish until $x=\sqrt{3}a/2$, gives magnitude $k_{e}q/(x^2+a^2/4)$ until $x=\sqrt{15}a/2$, and gives $k_{e}qa/(x^2+a^2/4{)}^{3/2}$ thereafter. At each ideal charged surface the field has distinct one-sided limits. The net charge is zero but the electric dipole moment is $-qa\hat{\mathbf{y}}$.