  We study the effect of a magnetic field on the behaviour of a conducting
elastic rod subject to a novel set of boundary conditions that, in the case of
a transversely isotropic rod, give rise to exact helical post-buckling
solutions. The equations used are the geometrically exact Kirchhoff equations
and both static (buckling) and dynamic (whirling) instability are considered.
Critical loads are obtained explicitly and are given by a surprisingly simple
formula. By solving the linearised equations about the (quasi-)stationary
solutions we also find secondary instabilities described by (Hamiltonian-)Hopf
bifurcations, the usual signature of incipient `breathing' modes. The boundary
conditions can also be used to generate and study helical solutions through
traditional non-magnetic buckling due to compression, twist or whirl.
