A single point charge at the origin induces charge onto a grounded Z-axis symmetric conductor. The induced charge is given in cylindrical coordinates as $\sigma(r, z)$ since the Z symmetry means there is no $\theta$ dependence.
Given that induced charge density from the unit point charge, what is the surface charge induced not by a point charge, but instead by a unit dipole at the origin, oriented in the +X direction?
The obvious guess at a solution is that the dipole weighting acts like a cosine, and thus the answer is a surface charge density proportional to $$\sigma_{\rm dipole}(r, z, \theta)=\sigma(r, z)\cos\theta.$$ As a check for this intuition, it does hold true for a dipole inducted charge for a conductive sphere. Spherical coordinates would be the same, but cylindrical coordinates seem more natural for this kind of problem.
The constant of proportionality is unknown, but we can integrate our (proposed) solution and find that we need to divide by the mean charge radius, $\int \sigma(r,z)\ r\ dr\ dz.$
But the hard part is proving this. I first thought to expand potential in terms of a multipole expansion of the potential in cylindrical coordinates, then seeing which terms get "shifted" with a cosine weighting of $\theta$ but this doesn't go anywhere since the multipole expansion is just 0 (for the system as a whole) or a lone unit $1/r$ term (for the conductor's contribution to the farfield expansion.)
Another interesting thought is that if the system were spherically inverted with a Kelvin transformation, we'd have the same problem but with an initial system of an axial symmetric charged capacitor and relating that surface density to the surface charge induced into the same conductor in a linear external field. Interesting, but I don't see it giving any new tool to attack the problem.
I considered a simple substitution. If we replace the central charge with a +X dipole, the system would still be in equilibrium if every bit of original induced surface charge was replaced with a proportionally scaled +X dipole as well. Those surface dipoles are not normal to the conductor. But how to transform the oriented surface dipole into an equivalent charge? The $\cos \theta$ term could come from ${p}\ \cdot\ {\hat{n}}$ but what's the justification for that?
I also tried considering the given charge density as a Green's function, but am unsure how it could be converted from the point potential Greens to the dipole version.
I'm running out of tools to attack the problem. This is for self-study, so I am eager to gain more intuition about how to think about such manipulations. Thanks for any help!
