General Expressions for the Magnetic Flux Density Produced by Axially Magnetized Toroidal Permanent Magnets

This paper presents analytical-integral expressions which evaluate the magnetic flux density radial and axial components' on- and off-axis of axially magnetized toroidal permanent magnets. The pieces, when used in magnetic electron beam focusing structures called periodic-permanent magnets (PPMs), can be employed in microwave vacuum electronics devices. These expressions can be adapted to computational algorithms, especially those developed based on mathematical commercial codes, aiding the investigation of the effect of magnetic fields on electron beam dynamics. Comparisons between analytical-integral expressions and theoretical and experimental data are also presented and discussed.