The magnetic structure of figure is a schematic view of a system designed to support a block of magnetic material (? ? ?) of mass M against the force of gravity. The system includes a permanent magnet and a winding. Under normal conditions, the force is supplied by the permanent magnet alone. The function of the winding is to counteract the field produced by the magnet so that the mass can be removed from the device. The system is designed such that the air gaps at each side of the mass remain constant at length g₀/2. Assume that the permanent magnet can be represented by a linear characteristic of the form B_m = ?_R (H_m?? H_c) and that the winding direction is such that positive winding current reduces the air-gap flux produced by the permanent magnet. Neglect the effects of magnetic fringing.

a. Assume the winding current to be zero. (i) Find the force f_fld acting on the mass in the x direction due to the permanent magnet alone as a function of x (0 x h). (ii) Find the maximum mass M _max that can be supported against gravity for 0 x h.

b. For M = M max/2, find the minimum current required to ensure that the mass will fall out of the system when the current is applied.

Depth D into the paper Winding. N turns Permanent magnet 20/2 Gravity Magnetic material mass M