paramagnetism & affect of polarity in multi-magnet arrays

[Widdekind]

Employing Mathematica to visualize the cumulative magnetic field, of multiple magnetic dipoles, apparently the region of high-field-strength, near the magnets, extends slightly farther away, when the magnetics are arranged, with alternating polarities. Apparently, alternating polarities "re-captures field lines", keeping more field lines near the array of magnets, so strengthening the field slightly.

 

Paramagnets respond to external magnetic fields, with parallel-pointing magnetization, linearly proportional to the applied external field. So, the strength of the force, acting on the paramagnet, is [math]\propto \left( B \circ \nabla \right) B[/math]. Employing vector calculus identities, and the Maxwell equations, that winds up being [math]\propto \nabla \left( B \circ B \right)[/math], i.e. paramagnets respond to the applied magnetic field energy density.