Surface electric current distributions on spheres and spheroids as sources of pure quadrupole magnetic fields
L. Medinaa, E. Ley–Koob
a Facultad de Ciencias, Universidad Nacional Autónoma de México.
b Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20–364, México, D.F., 01000, México.
]]> Recibido el 24 de enero de 2011
Abstract
Neutral atom magnetic traps and nuclear magnetic resonance imaging require internal regions with constant gradient magnetic induction fields, which are identified as pure quadrupole fields. This contribution starts from such fields in the interior of spheres and spheroids in cartesian coordinates, identifying immediately their respective scalar magnetic potentials. Next, the corresponding potentials inside and outside are constructed using spherical and spheroidal harmonic functions, respectively, except for a proportionality constant to be determined by the boundary conditions at the surface of spheres r = α, prolate ξ = ξ0 and oblate ζ = ζ0 spheroids, where the electric current sources are distributed. The negative gradients of the scalar potentials yield the respective magnetic induction fields inside (r < α, ξ < ξ0, ζ < ζ0) and outside (r > α, ξ > ξ0, ζ > ζ0). Gauss's law in its boundary condition form determines the normalization constant of the external potentials, while Ampere's law determines the electric current source distributions on the surface of the spheres and spheroids.
Keywords: Quadrupole magnetic fields and surface sources; constant gradient magnetic field; gradient coil windings; spherical and spheroidal harmonics.
Resumen
Trampas de átomos neutros e imageneología por resonancia magnética requieren de regiones internas con campos de inducción magnética de gradiente constante. Esta contribución parte de tales campos en el interior de esferas y esferoides en coordenadas cartesianas, identificando inmediatamente sus respectivos potenciales magnéticos escalares. A continuación, los potenciales interiores y exteriores correspondientes se construyen usando funciones armónicas esféricas y esferoidales, respectivamente, excepto por una constante de proporcionalidad por determinarse vía las condiciones de frontera sobre la superficie de esferas r = α, esferoides prolatos ξ = ξ0 y oblatos ζ = ζ0, donde las fuentes de corriente eléctricas se distribuyen. Los negativos de los gradientes de los potenciales escalares conducen a los campos de inducción magnética respectivos en el interior (r < α, ξ < ξ0, ζ < ζ0) y en el exterior (r > α, ξ > ξ0, ζ > ζ0). La ley de Gauss en su forma de condición de frontera determina la constante de normalización para los potenciales externos, mientras que la ley de Ampere determina las distribuciones de corriente eléctrica sobre la superficie de las esferas y esferoides.
Descriptores: Campos magnéticos y fuentes superficiales cuadrupolares; campos magnéticos de gradiente constante; embobinados de gradiente; armónicos esféricos y esferoidales.
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