Magnetostatic interaction between two thin nanotubes

We consider here the magnetic interaction between two identical tubes, characterized by: total magnetization M, length 2L, external radius Re, internal radius Ri. Following most of the experimental realizations we consider very thin tubes, namely, (Re-Ri)$<<$Ri. We begin by considering the two nanotubes in perfect parallel alignment and we vary the separation distance D. The continuous magnetization approach is invoked using different methods to compute the interacting energy: analytic expression valid for D$<$2L, analytic integration over the interaction of elements on each tube, and numeric integration for general cases. These results are compared with two independent results: a) the tubes are far apart so they can be considered solid nanowires: b)each tube is considered as a set of elementary nanowires and a series expansion is obtained and truncated. The advantages and disadvantages of each method are discussed. The ranges of applicability of the ``handy'' approximate expressions are obtained.