The efficiency and the demagnetization field of a general Halbach cylinder

The maximum magnetic efficiency of a general multipole Halbach cylinder of order p is found as function of p. The efficiency is shown to decrease for increasing absolute value of p. The optimal ratio between the inner and outer radius, i.e. the ratio resulting in the most efficient design, is also found as function of p and is shown to tend towards smaller and smaller magnet sizes. Finally, the demagnetizing field in a general p-Halbach cylinder is calculated, and it is shown that demagnetization is largest either at cos2pφ=1 or cos2pφ=-1. For the common case of a p=1 Halbach cylinder the maximum values of the demagnetizing field are either at φ=0,π at the outer radius, where the field is always equal to the remanence, or at φ=±π/2 at the inner radius, where it is the magnitude of the field in the bore. Thus to avoid demagnetization the coercivity of the magnets must be larger than these values.