Basis of the electric solenoid actuator, Electrical Engineering

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There is a force on the plunger required to do to move it into or out of the gap and work must be done by or against this force. The conservation of energy dictates that the change in the electrical energy stored in the magnetic field equals the work done in moving the magnetic material in or out of the core.  

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(It is usually assumed that the current will be kept constant as the plunger moves, so I is independent of x) 


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