 Planar quadrupole potential

This electric potential is harmonic and independent of $z$. Its electric field is linear in transverse position, with a saddle at the origin. For $A>0$, field lines approach along the $x$ axis and leave along the $y$ axis; off-axis field lines satisfy $xy=\mathrm{constant}$. Equipotentials satisfy $x^2-y^2=\mathrm{constant}$. This is a local planar electric quadrupole geometry, not the decaying far field of an isolated three-dimensional quadrupole.