A closed form for the electrostatic interaction between two rod-like charged objects

We have calculated the electrostatic interaction between two rod-like charged objects with arbitrary orientations in three dimensions. We obtained a closed-form formula expressing the interaction energy in terms of the separation distance between the centers of the two rod-like objects, r, their lengths (denoted by 2l₁ and 2l₂) and their relative orientations (indicated by θ and φ). When the objects have the same length (2l₁ = 2l₂ = l), for particular values of separations, i.e. for r ≤ 0.8l, two types of minimum appear in the interaction energy with respect to θ. By employing the closed-form formula and introducing a scaled temperature t, we have also studied the thermodynamic properties of a 1D system of rod-like charged objects. For different separation distances, the dependence of the specific heat of the system to the scaled temperature has been studied. It is found that, for r < 0.8l, the specific heat has a maximum.