Electrostatic potential created by two finite-width infinite-length strips at opposite potential $\pm V_0$ I'm tackling with this problem but it seems I'm going in the wrong direction.

Consider a 3D space where the potential is specified over the plane $z=0$ , assuming it has value $V_0$ and $-V_0$ on adjacent infinite stripes $a/2$ wide. Assume this stripes are oriented with their "long" edge parallel to the $y$ axis. Find the potential $V(r)$ at great distance from the plane.

I assumed the boundary condition to the infinite is that the potential must go to zero.
I've tried to solve this this way:
using some heuristic arguments and the method of charge images by considering what the system will look like as seen at great distance from the plane. I imagined it could look like a series of charged wires interspaced by $a/2$ from each other located far behind the plane, their linear density of charge being assigned so that the potential on the plane is the one assigned. This way the equipotential surface of each wire should look like a plane and since they're interspaced I thought this could lead to the specified potential on the plane. However this setup doesn't work because doing this the potential at great distance from the wires would be infinite.
Maybe the problem can be solved easily with the method of Green functions but I was just trying to solve it by reasoning than by calculating.
Hope someone will be interested by this problem, like I am.
 A: 
Find the potential $V(r)$ at great distance from the plane.

This is basically code for "find the leading-order contribution to the multipolar expansion of the field at large distances".
The equivalent for 3D multipoles would be two charged finite wires placed between the origin and $(\pm a,0,0)$, where (at large enough distances) it is appropriate to approximate each wire as a point charge, which then combine to create a 3D dipole field of the form $V\sim\mathbf d\cdot \mathbf r/r^3$.
For your case, the 3D spherical multipoles must be replaced with cylindrical multipoles: instead of point charges, you now approximate each stip of charge as a single line charge (whose potential then goes as $\ln(\rho)$), add up the two opposing contributions, and approximate at large distances similarly to the spherical-dipole case.
(As a hint: you must get a final result proportional to $\frac{\partial}{\partial y}\ln(\rho)$, and it pays to spend the time to understand why that connection works.)
The tricky part with your problem is connecting this solution (and, specifically, the global constant that determines how strong it must be) to the known data about your source, specifically $V_0$. What you really need is the dipole-moment density of your source (i.e. how much dipole moment along $x$ exists in a strip of unit length $\Delta$ along $y$). If you were given strips of constant surface charge, this would be a trivial calculation, but the setup instead gives you conducting planes at constant potential. To be honest with you, I don't know what they're expecting you to do with that one, and I'm not even convinced that it produces a well-behaved answer, due to the discontinuous nature of the potential at the boundary condition.
But hopefully this is enough to get you pointed in useful directions.
