Helmholtz coils revisited

Abstract

Two circular coaxial current rings of radius a, separated by a distance b, are usually called Helmholtz coils if b=a. That is the well-known prescription for greatest uniformity of field in the neighborhood of the center of symmetry. Examined here is the magnetic field of a pair of current rings at a distance r from the center large compared to the ring radius a. Expanding the distant field in powers of a/r, each term may be associated with a magnetic 2k-pole source and a field falling off as (a/r)k+2. The dipole field k=1, falling as (a/r)3, has as its source the total dipole moment of the two rings. It eventually swamps the contribution of higher multipoles. The term (a/r)5 with k=3 has for its source the octupole (23-pole) moment of the pair of rings. Its coefficient vanishes if b=a. Thus the Helmholtz pair, thanks to its seemingly irrelevant specialization for uniform central field, is also endowed with zero octupole moment. The term-by-term connection between the expansion of the distant field and the expansion of the central field holds for currents entirely confined to the surface of a sphere, which is the case for the Helmholtz pair of thin current rings. The dipole field can be nullified by nesting the Helmholtz pair within a larger Helmholtz pair designed to have equal area × ampere–turns but carrying reversed current. At the cost of a minor reduction in central field and without compromising the central field’s uniformity, the total dipole moment of the four rings can thus be made zero. The current system now has zero dipole moment and zero octupole moment. As 2k -poles with k even are ruled out by symmetry, the lowest surviving multipole is the 25-pole, the ‘‘32-pole.’’ The residual distant field, of which it is now the dominant source, falls as (a/r)7. An exact calculation of the ratio of field magnitude B to central field B0 shows that B/B0 is closely proportional to (a/r)7 everywhere in the field beyond r≊5a. Already at r≊6a, B/B0 has fallen to 10−4. It is shown how equivalent results can be achieved with coils of finite cross section. This scheme can drastically reduce the ‘‘stray field’’ around a superconducting magnet. In effect, it closely confines most of the return flux. Another application involves the induction of alternating currents by precessing magnetic moments. The four coils connected in series constitute a receiving coil tightly coupled to any oscillating dipole at its center, but virtually immune to disturbance by external sources.