Magnetic Field Energy Density Confusion

[dgreenheck]

I am trying to find the approximate force imparted on a piece of iron on the axis of a finite length solenoid. One website said a good approximation was to take the difference of the magnetic field energy from when the piece of iron was directly outside the solenoid and when the piece of iron was inside the solenoid.

The formula for the magnetic field energy density is:
$\frac{1}{2}\frac{B^{2}}{μ}$

So here's the point where I am confused on: The piece of iron is attracted to the solenoid, so the energy has to go down. But I thought that putting a piece of iron in the center of a solenoid made it a stronger electromagnetic, a.k.a. the magnetic field lines are more concentrated with a high permeability core. Wouldn't this imply that the magnetic field energy density goes up, contradicting my first point? I know my fundamentals are messed up somewhere, but I can't quite figure out where.

Thanks in advance.

 

[Vailanor]

The answer is that the magnetic field energy density does indeed increase when you put a piece of iron in the center of a solenoid. This is because the higher permeability of the iron increases the strength of the magnetic field inside the solenoid, which increases the magnetic field energy density. However, the magnetic force on the iron is still attractive, because the energy of the system decreases when the iron is inserted. This is because the total energy of the system (magnetic field energy plus potential energy) decreases when the iron is inserted, and thus the net force on the iron is attractive.