Three problems are considered. In the first, a uniform magnetic field is suddenly switched on outside an infinitely long circular cylinder of incompressible con- ducting fluid. As the field diffuses into the fluid, electric currents are generated and hence a Lorentz force field. Under the assumptions of small magnetic Reynolds number and small magnetic Prandtl number, the initial flow (before it has been modified by convection or viscous diffusion of vorticity) is oalculated. In the second problem an initially uniform field is suddenly switched off outside a similar cylinder of fluid. It is shown that the switching off of the field produces a vorticity distribution identical with that produced by the switching on. The third problem considered is that of a circular cylinder of conducting fluid placed in a rapidly alternating magnetic field. It is shown that the alternating field produces vorticity at a constant rate, and a qualitative description of the resulting flow is given.