According to this paper, the potential of any point on the surface at a distance '$h$' from the center of some simple conducting geometric shape is given by
For a Sphere: $$V(x)=Mh^2\frac{xcos(\alpha)+hsin(\alpha)}{(x^2+h^2)^\frac{3}{2}}$$ For a Horizontal Cylinder: $$V(x)=Mh\frac{xcos(\alpha)+hsin(\alpha)}{x^2+h^2}$$ where, $M$ = Electric Dipole Moment of the Polarized body, $\alpha$ = Angle of Polarizing axis from the Horizontal, and $h$ = Distance of the center of the object from the Horizontal
Now, how to derive this? There exist multiple papers of the same kind which use the formula, but none derived it.
$My$ $Attempt$ :
I tried using the potential energy function of an Electric dipole, $p$ $(U=\frac{kpcos(\theta)}{r^2})$. Here, $cos(\theta)$ can be written as $\frac{xcos(\alpha)+hsin(\alpha)}{(x^2+h^2)^\frac{1}{2}}$ from Geometry. But then I cannot figure out how the rest of the terms, in the given formula, can possibly match this Dipole Potential Energy Function.
I even tried calculating the $V(x)$ for a Sphere by assuming charge distribution on it. Assuming that a "constant" Electric field outside the sphere was responsible in polarizing the sphere, Charge arrangement on sphere can be found. Since Electric Field inside a polarized sphere is zero, charge density of an infinitesimal ring on sphere at an angle $\theta$ from the polarizing axis must be of the form $\frac{dQ}{Rd\theta}=\lambda cos(\theta)$. But then it got quite complicated.
How to derive these formulas? Are there any other methods?
Please help me here, I've been stuck on this for way too long.