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Normalize a point source by $\int \rho d^{3}xdz=q_1$ on the compact spatial direction. The periodic delta has the Fourier seriesWriting the potential as $\varphi (r,z)={\sum }_n{\varphi }_n(r)e^{inz/R}$ reduces the Poisson equation toAway from the source, the decaying radial solution is proportional to $e^{-|n|r/R}/r$. Its distributional normalization follows from integrating the Laplacian over a small sphere: the radial flux of $1/r$ is $-4\pi$, soThis is the contribution of the $n$th Kaluza-Klein mode, with inverse range $|n|/R$. At equal compact coordinate, the sum is a geometric series,Therefore the pair interaction energy, without the individual self-energies, isIts normalization is fixed by the stated $-4\pi R\rho$ source, rather than by importing another convention for Coulomb's law. More generally summing the phases gives the periodic electrostatic Green function with one compact dimension,An independent check uses the method of images in four spatial dimensions. Since ${\nabla }_4^2{s}^{-2}=-4{\pi }^2{\delta }^4$, its periodic image sum is $R{q}_1{\sum }_n[r^2+(z+2\pi nR{)}^2{]}^{-1}/\pi$, with the same normalization and short-distance singularity.

For $r\gg R$, only the zero Kaluza-Klein mode survives:For $r\ll R$, $\coth x=x^{-1}+x/3+O(x^3)$ givesThe leading potential thus changes from inverse distance to inverse squared distance. Differentiating gives the signed outward forceIt scales as $r^{-2}$ at large distance and as $2q_1{q}_{2}R/(\pi r^3)$ at small distance. A precision inverse-square-law test over separations approaching $R$ would reveal the extra dimension through this change of force law or its exponential corrections. The model assumes localized sources at equal compact coordinate and an electromagnetic field able to explore the circle; a field confined to ordinary spacetime would not show this effect. Finite source size and ordinary screening must be controlled when comparing a real force measurement with this point-source prediction.