Expression for the force exerted on the iron plunger, Electrical Engineering

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Solenoids find application in a variety of electrically controlled valves. The magnetic structure shown in Figure is a simplified representation of a solenoid in which the flux in the air gap activates the motion of the iron plunger.

(a) Develop a general expression for the force exerted on the iron plunger and comment on its dependence on position x.

(b) Determine the current through the coil of N = 100 turns to pull the plunger to x = a, given that a = 1 cm, lg = 1 mm, and the spring constant is k = 1 N/m. Assume the permeability of the magnetic structure to be infinite and neglect fringing.

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