“Hidden” Momentum in a Magnetized Toroid

The toroid has uniform azimuthal magnetization M = M φ̂ in cylindrical coordinates ( , φ, z). The total magnetic moment is zero, mtotal = ∫ M dVol = 0. For uniform magnetization M, the volume density ρm = −∇ ·M of “fictitious” magnetic charges is zero. For a toroid with azimuthal magnetization, the surface density σm = M · n̂ of “fictitious” magnetic charges is also zero (where n̂ is normal to the surface). Then, there are no sources for the H-field, which is hence zero. That is, H = B− 4πM = 0 everywhere, such that B = 4πM is purely azimuthal inside the toroid and zero outside it. We take the inner and outer radii of the toroid to be a and b, respectively, and suppose the toroid has length l in the axial (z) direction large compared to b. The conductors of the cylindrical capacitor are at radii a− and b, but do not touch the toroid.