The vector potential and stored energy of thin cosine (nθ) helical wiggler magnet

Expressions for pure multipole field components that are present in helical devices have been derived from a current distribution on the surface of an infinitely thin cylinder of radius R. The strength of such magnetic fields varies purely as a Fourier sinusoidal series of the longitudinal coordinate Z in proportion to cos(nθ- ωmz), where ωm = (2m-1)π/L, L denotes the half-period and m = 1, 2, 3 etc. As an alternative to describing such field components as given by the negative gradient of a scalar potential function (Appendix A), one of course can derive these same fields as the curle of a vector potential function rvec A--specifically one for which ∇ x ∇ x rvec A = 0 and ∇· rvec A = 0. It is noted that we seek a divergence-free vector that exhibits continuity in any of its components across the interface r = R, a feature that is free of possible concern when applying Stokes' theorem in connection with this form of vector potential. Alternative simpler forms of vector potential, that individually are divergence-free in their respective regions (r < R and r > R), do not exhibit full continuity on r = R and whose curl evaluations provide in these respective regions the correct components of magnetic field are not considered here. Such alternative forms must differ merely by the gradient of scalar functions that with the divergence-free property are required to be ''harmonic'' (∇2Ψ = 0)