I would like an electric potential with specific qualities, namely, quadratic in two dimensions. On the one hand, I could try and do a little thought experiment, as to which boundary gives the desired properties. Realistically though on the length scale I need it, the odds of producing the boundary are zilch. Much easier in terms of the multipole expansion. With this in mind, and knowing that copper wire along a cylinder makes for an easy setup, and that at least four terms are required for a quadratic potential, the following should work: V(x=0,z=a)=k, V(x=0,z=-a)=k, V(x=-a,z=0)=-k, V(x=a,z=0)=-k, with each wire along y. Assuming long wires, the potential for all four in the x/z plane is:
To confirm this has the desired behavior near the origin (and hopefully over some reasonable distance), I turn to the Taylor multi-variable expansion. Example partial derivatives:
Substituting into the expansion:
does indeed reveal that V ~ (2/{a*a})[z*z-x*x], as desired:
