The field lines of an axisymmetric magnetic field

Abstract

Willis & Young's (1987) discussion of the magnetic field lines of an axisymmetric multipole is generalized to an arbitrary axisymmetric magnetic field B outside a sphere S(a) containing all the sources of B. Explicit equations are obtained for the field lines of B in terms of the Gauss coefficients. It is shown that there are no closed field lines outside S(a). Every field line has one end which enters S(a), and the other end can enter S(a) or go to infinity; this is true even if the field line passes through and changes direction at one or more null points of B, where B=0. In any meridian plane outside S(a), B has only finitely many null points and only finitely many field lines which go to infinity. The behaviour of B and the topology of its field lines near an arbitrary null point are described. The null points on the symmetry axis are topologically and metrically different from the others, but only finitely many field lines emerge from any null point. Copyright © 1988, Wiley Blackwell. All rights reserved