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

By Willis, D. M.

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

Title: Planar Charged-particle Trajectories in Multipole Magnetic Fields : Volume 15, Issue 2 (30/11/-0001)  
Author: Willis, D. M.
Volume: Vol. 15, Issue 2
Language: English
Subject: Science, Annales, Geophysicae
Collections: Periodicals: Journal and Magazine Collection (Contemporary), Copernicus GmbH
Publication Date:
Publisher: Copernicus Gmbh, Göttingen, Germany
Member Page: Copernicus Publications


APA MLA Chicago

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

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 ∝ ϖ3). 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 ϖn+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(ΦΦnm)] = 0, where Φnm = (1/m) arctan (hnm/gnm); gnm and hnm denote the spherical harmonic coefficients. Equatorial planar trajectories also exist if (nm) is odd. The polar axis (θ = 0,π) of a tesseral magnetic multipole is a magnetic neutral line if m > 1. A further 2m(nm) neutral lines exist at the intersections of the 2m meridional planes with the (nm) cones defined by the (nm) roots of the equation Pnm(cos θ) = 0 in the range 0 < θ < π, where Pnm(cos θ) denotes the associated Legendre function. If (nm) 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 ϖn+2 and inversely proportional to cos[m(ΦΦnm)]. 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

Planar charged-particle trajectories in multipole magnetic fields


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