Sticky Magnetic Force

If particles are responsible for magnetism, then $\psi$ is responsible for its force field and naturally, the force density, $F_m$ between the magnet and a piece of magnetic material stuck on it, is,

$F_m=\cfrac{\partial\,\psi}{\partial x}|_{x=0}$

the change in $\psi$ across the contact boundaries of the two objects, at $x=0$.  This is not a good formulation because contact are made over an extended area and both objects are not point particles.

This is counter intuitive, because a strongly held material will also be strongly magnetic.  It would seem then, that the drop in $\psi$ across the material boundary should be small.  In fact, $\psi$ permeates through the material to a greater extent than the drop across the material boundary; although it provides for a greater force the drop is still comparatively small.  In this way, a strongly held nail will attracts more nails than a weakly held nail.  It would seem that this change in $\psi$ across the boundary is a fraction of (proportional to) the amount of $\psi$ that permeated through the boundary.

It is expected that $\psi$ decreases with $x$ and the force is attractive.  When $\psi$ increases in the presence of another field, the force is repulsive.  In this way, the above expression is without a negative sign.  It is the force on another body, not the force on the body exerting $\psi$.

Let's see how far can $\psi$ go...