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...
