We study the effect of a magnetic field on the behaviour of a slender conducting elastic structure subject to end forces. Both statical (buckling) and dynamical (whirling) instability are considered and we also compute post-buckling configurations. The theory used is the geometrically exact Cosserat rod theory. We consider two types of boundary conditions: the traditional welded boundary conditions (for tether-like applications) and a novel set of boundary conditions that give rise to exact helical post-buckling solutions. Our results are relevant for current designs of electrodynamic space tethers and potentially for future applications in nano- and molecular wires.

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