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Planar Charged-particle Trajectories in Multipole Magnetic Fields : Volume 15, Issue 2 (30/11/-0001)

By Willis, D. M.

Book Id:WPLBN0003975165 Format Type:PDF Article : File Size:Pages 14 Reproduction Date:2015

Gardiner, A. R., Davda, V. N., Bone, V. J., & Willis, D. M. (-0001). Planar Charged-particle Trajectories in Multipole Magnetic Fields : Volume 15, Issue 2 (30/11/-0001). Retrieved from http://www.ebooklibrary.org/

Description
Description: Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, UK. This paper provides a complete generalization
of the classic result that the radius of curvature (ρ) of a
charged-particle trajectory confined to the equatorial plane of a magnetic
dipole is directly proportional to the cube of the particle's equatorial
distance (ϖ) from the dipole (i.e. ρ ∝ ϖ^{3}).
Comparable results are derived for the radii of curvature of all possible planar
charged-particle trajectories in an individual static magnetic multipole of
arbitrary order m and degree n. Such trajectories arise wherever
there exists a plane (or planes) such that the multipole magnetic field is
locally perpendicular to this plane (or planes), everywhere apart from possibly
at a set of magnetic neutral lines. Therefore planar trajectories exist in the
equatorial plane of an axisymmetric (m = 0), or zonal,
magnetic multipole, provided n is odd: the radius of curvature varies
directly as ϖ^{n}^{+2}. This
result reduces to the classic one in the case of a zonal magnetic dipole (n =1).
Planar trajectories exist in 2m meridional planes in the case of the
general tesseral (0 < m < n)
magnetic multipole. These meridional planes are defined by the 2m roots
of the equation cos[m(Φ – Φ_{n}^{m})]
= 0, where Φ_{n}^{m} = (1/m)
arctan (h_{n}^{m}/g_{n}^{m}); g_{n}^{m} and h_{n}^{m} denote the spherical harmonic coefficients. Equatorial planar trajectories also
exist if (n – m) is odd. The polar axis
(θ = 0,π) of a tesseral magnetic
multipole is a magnetic neutral line if m > 1. A
further 2m(n – m) neutral lines exist at
the intersections of the 2m meridional planes with the (n – m) cones defined by the (n – m) roots of the
equation P_{n}^{m}(cos θ) = 0 in the range 0 < θ < π, where P_{n}^{m}(cos θ) denotes the associated Legendre function. If (n – m) is odd, one of these cones coincides with the
equator and the magnetic field is then perpendicular to the equator everywhere
apart from the 2m equatorial neutral lines. The radius of curvature of an
equatorial trajectory is directly proportional to ϖ^{n}^{+2}
and inversely proportional to cos[m(Φ – Φ_{n}^{m})]. Since
this last expression vanishes at the 2m equatorial neutral lines, the
radius of curvature becomes infinitely large as the particle approaches any one
of these neutral lines. The radius of curvature of a meridional trajectory is
directly pro

Summary
Planar charged-particle trajectories in multipole magnetic fields