The nonaxisymmetric Tayler instability of toroidal magnetic fields due to axial electric currents is studied for conducting incompressible fluids between two coaxial cylinders without endplates .
The inner cylinder is considered as so thin that even the limit of R _ { in } \to 0 can be computed .
The magnetic Prandtl number is varied over many orders of magnitudes but the azimuthal mode number of the perturbations is fixed to m = 1 .
In the linear approximation the critical magnetic field amplitudes and the growth rates of the instability are determined for both resting and rotating cylinders .
Without rotation the critical Hartmann numbers do not depend on the magnetic Prandtl number but this is not true for the growth rates .
For given product of viscosity and magnetic diffusivity the growth rates for small and large magnetic Prandtl number are much smaller than those for Pm = 1 .
For gallium under the influence of a magnetic field at the outer cylinder of 1 kG the resulting growth time is 5 s. The minimum electric current through a container of 10 cm diameter to excite the kink-type instability is 3.20 kA .
For a rotating container both the critical magnetic field and the related growth times are larger than for the resting column .