Fig.2 shows a type of actuator that is capable of exerting a smaller force over a larger displacement. The central cylindrical plunger is separated from the outer cylindrical casing by a uniform air gap of 0.5 mm. The coil has 500 turns. The magnetic material of the plunger and casing can be considered perfect up to 1.6 T.
a. One of the limitations of this actuator is the magnetic flux that can be supported by the plunger. If the flux density in this member is not to exceed 1.6 T, what is the maximum permissible current in the coil?
b. With the current held constant at the value found in part (a), derive an expression for the force as a function of the displacement x. Determine the maximum and minimum values of this force?
c. Suppose the coil current is increased to five times the value given in (a). This will cause the plunger to be magnetically saturated at B = 1.6 T for most of the displacement. Determine the force as a function of x for this condition. Is the force at x = 5 cm increased above the value found in (b)?

Plunger. 0.5 mm Fig. 2 x Coil -5 cm- Casing t 2 cm T