a The magnetic force on a particle within the packing, but off-the vertical axis, induced by neighboring ones (in dark gray), is non-zero and along the radial direction, due to the tube finite-size, and the non-compensation of the forces from the missing particles (in white). For a vertical magnetic field, the resulting radial force is repulsive (red arrow), as confirmed by the right panel. b Numerical computation of a spatial (r,z) map of the magnetic pair-interaction radial component (averaged over the coordinate θ) for a 10 cm column, submitted to a magnetic field of Ψ = 35. Model curves for each direction of the magnetic field (c vertical, d horizontal). The light gray line shows Janssen prediction; the black squares line is the result of the computed magnetic mass loss μFr/g; the blue circles line (difference of both curves) gives the predicted apparent mass of the column.