Magnetostatic modes of gyrotropic cylinder and bar

Abstract

We present a theory of magnetostatic modes which are wavelike in the direction parallel to the axis of an infinitely long, ferromagnetic bar of square cross-section and are localized at the faces and edges of the bar. We write H = -▽θ{symbol}, where φ is a magnetic scalar potential, and obtain as the equations determining θ{symbol}, Σ μαβ∂2σ ∂Xα∂Xβ=0 for the interior of the bar, and ▽2θ{symbol} = 0 for the exterior, where μαβ(ω) is the magnetic permeability tensor of the bar. We transform to cylindrical coordinates and solve the resulting second order partial differential equations by assuming a solution for θ{symbol} of the form θ{symbol}(r, θ, z) = eiqxf(r, θ). The use of cylindrical coordinates allows us to project out of our expressions for θ{symbol} the parts which belong to each of the irreducible representations of C4, the group of proper rotations which leave the bar invariant. This has the consequence that the boundary conditions need to be applied along only one side of the bar, and are then automatically satisfied along the remaining three sides as well. The boundary conditions that θ{symbol} and the normal component of B be continuous across the bar surfaces are satisfied at a discrete set of points along one side of the bar, and two coupled eigenvalue equations are obtained which are solved simultaneously for the frequencies of the corresponding magnetostatic modes. The convergence of this method is found to be quite good. We also present the dispersion curves for the magnetostatic modes of a gyrotropic right circular cylinder which are localized at the surface of the cylinder. © 1977.