Not the physics I'm used to

Solving for the force of a solenoid with a plunger seems like it should be a simple equation, until I realized that the core isn't magnetized until it is induced, then there is a dipole moment, and then as it moves the core of the solenoid gradually changes from air to the core material. Of course this is a differential function. To simplify it I have come up with two different ways of thinking about it.

In terms of energy

The first and probably better way is to compare it to a vertical spring in a conservation of energy equation. Since I can calculate the magnetic energy, and differentiate it with respect to the change in y to find the force, I can equate the force of the solenoid to the force of the core mass * gravity, and plug in the

Magnetic Circuit

The second way I thought about it was as a magnetic circuit with an air gap. If the air gap is on a hinged arm, I can work out the incremental force of the field at the gap. I could also find the flux density by imagining unrolling the solenoid to represent a rectangular circuit. This would be useful for designing an enclosure. The air gap is located inside the inductor, so the flux density also changes based on the change in how much of the core is enclosed at a time.

A new appreciation for magnets

Looking into how to solve this has reignited my spark for magnets. It seems crazy that they actually work in the first place. Hopefully in my next log I will have some equations to show, and possibly some code. In later logs I plan on writing a driver based on my calculations in order to position the solenoid.