One Sided Fluxes

It is shown that a previously unknown class of magneti" zation patterns exists in planar structures which have the unique property that all the flux escapes from one surface with none leaving the other side. A simple case is a constant amplitude rotating vector magnetization where the sense of rotation dictates which surface has no flux. More complicated magnetization patterns are elucidated. The likelihood that the one"sided flux phe" nomenon occurs partially in the normal write, the contact"printing and the print"through processes of tape recording is discussed. It is concluded that significant improvements in tape recording performance would ensue if means could be found to enhance the e#ect. Introduction This paper is concerned with the symmetry of the mag" netic flux emanating from tapes, disks, and other planar magnetic structures. It is shown theoretically that a pre" viously unknown class of magnetization patterns exists which has the remarkable property that all the flux es" capes from one surface with none leaving the other side. Such one"sided fluxes are often quite surprising and in" consistent with “intuitive” concepts. In the first part of the paper the magnetic theory is de" veloped. Two proofs are given that a one"sided flux oc" curs when the magnetization is a constant amplitude rotating vector, the sense of rotation dictating which surface has no flux. The theory is then expanded to in" clude general two"dimensional magnetization patterns; it transpires that in principle an infinite set of one"sided flux magnetization patterns exists. An example of a three"dimensional magnetization pattern is introduced. Finally, the physical realizability of such one"sided fluxes is discussed. The second part of the paper deals with practical situa" tions in which the one"sided flux phenomena may occur. In recording tapes a partial e#ect may occur in the writ" ing process, and as Daniel1 recently pointed out, the contact printing and print"through processes. Little di" rect evidence for, one"sided fluxes has been reported. Until such time as deliberate e#orts are made to exploit the e#ect, it remains a magnetic curiosity. Theoretical Considerations Rotating Magnetization Vector Cas! We consider a planar structure, of thickness d, lying in the x, " plane. The upper and lower surfaces are at y = 0 and #d, respectively. Discussion is restricted to two" dimensional magnetization patterns in which the " component is either constant or zero. Suppose the mag" netization is the superposition of two sinusoids in quad" rature: mx = m0 sin kx; my = m0 coskx; mz = 0 $1% where $ is the wave number, 2&/!. We wish to solve the boundary value problem for the scalar potentials and fields above and below the sheet. The potentials within the sheet obey Poisson’s equation: !"inside = m0k coskx $2a% and Laplace’s equation above and below, !"above = 0 $2b% !"below = 0 $2c% Since the particular solution of $2a% is "$%o/$% cos kx, the general solutions are of the form: !above = Ae "ky + Be { }coskx $3a% !inside = Ce "ky + De " m0 k # $ % & ' ( coskx $3b% !below = Ee "ky + Fe { }coskx $3c% They are subject to six boundary conditions. The fields and potentials must go to zero as y becomes infinite; that is, IEEE Transactions on Magnetics, Volume 9, Number 4, December 1973, doi:10.1109/TMAG.1973.1067714' 1 !above = !below = 0 when y = ±(, respectively $4a% The tangential fields must match on the sheet upper and lower surfaces, thus !above = !inside when y = 0 $4b% !below = !inside when y = #d $4c% The normal flux density must be continuous on the up" per and lower surfaces, thus ! "#above "y = ! "#inside "y + m0 coskx; when y = 0 $4d% ! "#below "y = ! "#inside "y + m0 coskx; when y = !d $4e% The reader may verify that the solutions consonant with these restrictions are !above = 0 $5a% !inside = m0 k e "1 ( )coskx $5b% !below = m0 k 1" e ( )e coskx $5c% The remarkable fact thus emerges that the scalar poten" tial above the sheet is identically zero everywhere. Thus no flux emerges from the sheet’s upper surface. All the flux emerges from the lower surface. It may be shown that if the sense of magnetization rota" tion is reversed, so that, for example, mx = m0 sin kx $6a% my = !m0 coskx $6b% mz = 0 $6c% the potentials then become !above ' = " m0 k 1" e ( )e coskx $7a% !inside ' = " m0 k e d+y ( ) "1 ( )e coskx $7b%