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The circuit consists of a single magnetic material piece (like soft iron) and a solenoid half immersed on it, as in the image below:

Magnetic circuit

I'll feed DC current to the solenoid, and I'm trying to imagine the figure made by the magnetic field lines in the piece and more specifically: if they are going to "cross" the solenoid, producing a force on it.

Could you help me figuring it out?

Thanks!

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Even before drawing magnetic field lines, it is to be expected that solenoid and iron will attract each and hence the solenoid will be pulled down into the iron.

To have a concrete situation I assume the current flows in the direction as given by the orange arrows. The magnetic field lines of solenoid and iron will look like this:

enter image description here

You see, that in the upper half of the solenoid the magnetic field vector $\vec{B}$ has not only a component vertically upwards, but also a component radially outwards. And especially across the gap between the iron pieces there is a strong magnetic field pointing radially outwards. Determining the Lorentz force $$\Delta\vec{F}= I\Delta\vec{\ell}\times\vec{B}$$ acting on a piece of wire $\Delta\vec{\ell}$ there, you find that the force $\Delta\vec{F}$ has a component downwards. Hence the solenoid is pulled down.

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