Magnetic Forces

The forces between a coil and some ferromagnetic material is highly depending on the distance from the magnet. An approximation of the stray field (for large r)

B = Bo * r ^ (-3)

and the field gradient is

dB/dr = - 3 Bo * r ^ (-4)

The force is proportional to the product of the induced field in the ferromagnet and the field gradient. The induced field is proportional to the stray field, for low fields, but gets constant when the ferromagnet saturates. Therefore the force between the coil and a ferromagnetic piece is proportional to the power of -4 of the distance in high fields

F ~ r ^ (-4)

and proportional to the power of -7 of the distance in low fields.

When the distance is kept constant, the relation between the force and the central field is quadratic for low fields, and gets linear for higher fields.